Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00082485/00001
## Material Information- Title:
- Synthesis and analysis of real single-sideband signals for communication systems
- Creator:
- Couch, Leon W. (
*Dissertant*) George, T. S. (*Thesis advisor*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1968
- Copyright Date:
- 1968
- Language:
- English
- Physical Description:
- xiv, 125 leaves. : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Amplitude modulation ( jstor )
Analytics ( jstor ) Autocorrelation ( jstor ) Bandwidth ( jstor ) Electric potential ( jstor ) Entire functions ( jstor ) Modulated signal processing ( jstor ) Noise spectra ( jstor ) Signals ( jstor ) Sine function ( jstor ) Communication research ( lcsh ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Signal theory (Telecommunication) ( lcsh ) System analysis ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis--University of Florida, 1968.
- Bibliography:
- Bibliography: leaves 121-123.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 030419387 ( ALEPH )
16961426 ( OCLC ) AER8332 ( NOTIS )
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SYNTHESIS AND ANALYSIS OF REAL SINGLE-SIDEBAND SIGNALS FOR COMMUNICATION SYSTEMS By Leon Worthington Couch, II A Dissertation Presented to the Graduate Council of The University of Florida in Partial Fulfillment of the Reauirements for the Degree of Doctor of Philosophy UNIVERSITY OF FLORIDA 1968 Copyright by Leon Worthington Couch, II 1968 DEDICATION The author proudly dedicates this dissertation to his parents, Mrs, Leon Couch and the late Rev. Leon Couch, and to his wife, Margaret Wheland Couch, ACKNOWLEDGMENTS The author wishes to express sincere thanks to some of the many people who have contributed to his Ph.D program. In particular, acknowledgment is made to his chairman, Professor T. S. George, for his stimulating courses, sincere discussions, and his professional example. The author also appreciates the help of the other members of his super- visory committee. Thanks are expressed to Professor R. C. Johnson and the other members of the staff of the Electronics Research Section, Department of Electrical Engineering for their comments and suggestions. The author is also grateful for the help of Miss Betty Jane Morgan who typed the final draft and the final manuscript. Special thanks are given to his wife, Margaret, for her inspi- ration and encouragement. The author is indebted to the Department of Electrical Engi- neering for the teaching assistantship which enabled him to carry out this study and also to Harry Diamond Laboratories which supported this work in part under Contract DAAG39-67-C-0077, U. S. Army Materiel Com- mand. TABLE OF CONTENTS ACKNOWLEDGMENTS . . LIST OF FIGURES . KEY TO SYMBOLS . ABSTRACT . . CHAPTER I I. INTRODUCTION . . .1, MATHEMATICAL PRELIMINARIES . I. SYNTHESIS OF SINGLE-SIDEBAND SIGNALS * V. EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN 4.1. Example 1: Single-Sideband AM with Suppressed-Carrier ....... * 4,2 Example 2: Single-Sideband PM * 4.3. Example 3: Single-Sideband FM * 4,4. Example 4: Single-Sideband a * V. ANALYSIS OF SINGLE-SIDEBAND SIGNALS * 5. 1 Three Additional Equivalent Realiza 5.2, Suppressed-Carrier Signals * 5.3. Autocorrelation Functions *. 5.4. Bandwidth Considerations .. *. 5.4-1. Mean-type bandwidth * 5.4-2. RMS-type bandwidth . 5.4-3, Equivalent-noise bandwidth * 5 5, Efficiency . . 5.6, Peak-to-Average Power Ratio .* * v Page * iv * viii * x * xiii * 1 4 * 9 * 18 tions * * . 18 19 28 * 30 . 3 42 * .* 43 *. 44 * 45 * .* 45 . 46 * * * * Page VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS .. .48 6o1. Example 1: Single-Sideband AM With Suppressed Carrier. ................ 48 62. Example 2: Single-Sideband PM . 51 6.3. Example 3: Single-Sideband FM * 68 6.4. Example 4: Single-Sideband a . 71 VII, COMPARISON OF SOME SYSTEMS . .. 75 7.1. Output Signal-to-Noise Ratios . 76 7 1-1 AM system . . 76 7.1-2. SSB-AM-SC system . 77 7.1-3. SSB-FM system . .. 78 7.1-4. FM system . . .. 84 7.1-5. Comparison of signal-to-noise ratios. 85 7.2, Energy-Per-Bit of Information .......... *89 7.2-1. AM system . . .. 93 7.2-2. SSB-AM-SC system . . 93 7.2-3. SSB-FM system . .. 94 7.2-4. FM system . . 94 7.2-5. Comparison of energy-per-bit for various systems *. ..............** 95 7.3. System Efficiencies ................. 97 7,3-1. AM system . . .. 98 7.3-2c SSB-AM-SC system . . 98 7.3-3. SSB-FM system . .. 98 7.3-4. FM system . . .. 99 7.3-5. Comparison of system efficiencies 100 Page VIII. SUMMARY . . .. 102 APPENDIX I. PROOFS OF SEVERAL THEOREMS . .. .105 II. EVALUATION OF e j(x + jy) . . 119 REFERENCES . . . .. 121 BIOGRAPHICAL SKETCH . . . 124 LIST OF FIGURES Figure 1. Voltage Spectrum of a Typical m(t) Waveform *. 2. Voltage Spectrum of the Analytic Signal Z(t) .* 3. Voltage Spectrum of an Entire Function of an Analytic Signal ....................... 4. Voltage Spectrum of the Positive Frequency-Shifted Entire Function of the Analytic Signal *. 5. Voltage Spectrum of the Synthesized Upper Single- Sideband Signal . . . 6. Voltage Spectrum of the Negative Frequency-Shifted Entire Function of an Analytic Signal . 7. Voltage Spectrum of the Synthesized Lower Single- Sideband Signal . . 8. Phasing Method for Generating USSB-AM-SC Signals - 9. USSB-PM Signal Exciter--Method I . . 10. USSB-FM Signal Exciter . . 11. Envelope-Detectable USSB Signal Exciter ...... . 12. Square-Law Detectable USSB Signal Exciter ...... 13. USSB-PM Signal Exciter--Method 11 . . 14. USSB-PM Signal Exciter--Method III . . 15. Power Spectrum of a(t) * 16. AM Coherent Receiver . . 17. SSB-AM-SC Receiver . . . 18. SSB-FM Receiver . . . 19. Output to Input Signal-to-Noise Power Ratios for Several Systems .* . Page 9 11 * 16 * 20 . 22 . 24 * 26 . 27 * 53 . 54 * 67 * 76 * 78 * 78 * 86 vi11 * * * * Figure Page 20 Output Signal-to-Noise to Input Carrier-to-Noise Ratio for Several Systems .* ..................... 87 21. Output Signal-to-Noise to input Signal-to-Normalized- Noise Power Ratio for Various Systems .......... 90 22, Output Signal-to-Noise to Input Carrier-to-Normalized- Noise Power Ratio for Various Systems . 91 23. Comparison of Energy-per-Bit for Various Systems .. .96 24. Efficiencies of Various Systems . 101 25. Contour of Integration . . .. 107 26. Contour of Integration .................. 115 KEY TO SYMBOLS A0 = Amplitude Constant AM = Amplitude-Modulation b = Baseband Bandwidth (rad/s) B = RF Signal Bandwidth (rad/s) Cb = Baseband Channel Capacity CB = RF Channel Capacity Ci = Input Carrier Power (C/N)i = Input Carrier-to-Noise Ratio (C/N)I = Input Carrier-to-Normalized-Noise Ratio D = Modulator Transducer Constant FM = Frequency-Modulation F(w) = Voltage Spectrum F(-) = The Fourier Transform of (*) g(W) = U(W) + jV(W) = An Entire Function GN = Gaussian Noise LSSB = Lower Single-Sideband m(t) = Modulating Signal or a Real Function of the Modulating Signal (see e(t) below) M = Either Multiplex or Figure of Merit Ni = Input Noise Power NI = Normalized Input Noise Power P()) = Power Spectral Density PM = Phase-Modulation R(,) ; Autocorrelation Function Re(-) = Real Part of (*) RF = Radio Freouency Si = Input Signal Power So: = Output Signal Power (S/N)i = Input Signal-to-Noise Ratio (S/N)I = Input Signal-to-Normalized-Noise Ratio (S/N)o = Output Signal-to-Noise Ratio SC = Suppressed-Carrier USSB = Upper Single-Sideband -(W) = The "Suppressed-Carrier" Function of U(W) V(W) = The "Suppressed-Carrier" Function of V(W) X(t) = A Real Modulated Signal XL = Lower Single-Sideband Modulated Signal XU = Upper Single-Sideband Modulated Signal Z(t) = m(t) + jm(t) = The Analytic Signal of m(t) a = Modulation (as defined in the text) S= System Efficiency 6 = Modulation Index n = Efficiency e(t) = Modulating Signal (when m(t) is not the Modulating Signal) o0 = Variance om Average Power of m(t) W = Angular Frequency wrms = RMS-Type Bandwidth AO = Eouivalent-Noise Bandwidth S = Mean-Type Bandwidth * = The Convolution Operator (.)* = The Conjugate of (*) (*) = The Hilbert Transform of (*) (*) = The Averaging Operator Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SYNTHESIS AND ANALYSIS OF REAL SINGLE-SIDEBAND SIGNALS FOR COMMUNICATION SYSTEMS By Leon Worthington Couch, II June, 1968 Chairman: Professor T. S. George Major Department: Electrical Engineering A new approach to single-sideband (SSB) signal design and ana- lysis for communications systems is developed. It is shown that SSB signals may be synthesized by use of the conjugate functions of any entire function where the arguments are the real modulating signal and its Hilbert transform. Entire functions are displayed which give the SSB amplitude-modulated (SSB-AM), SSB frequency-modulated (SSB-FM), SSB envelope-detectable, and SSB square-law detectable signals. Both upper and lower SSB signals are obtained by a simple sign change. This entire generating function concept, along with analytic signal theory, is used to obtain generalized formulae for the properties of SSB signals, Formulae are obtained for (1) equivalent realizations for a given SSB signal, (2) the condition for a suppressed-carrier SSB signal, (3) autocorrelation function, (4) bandwidth (using various-de- finitions), (5) efficiency of the SSB signal, and (6) peak-to-average power ratio. The amplitude of the discrete carrier term is found to be xiii equal to the absolute value of the entire generating function evaluated at the origin provided the modulating signal is AC coupled. Examples of the use of these formulae are displayed where these properties are evaluated for stochastic modulation. The usefulness of a SSB signal depends not only on the pro- perties of the signal but on the properties of the overall system as well. Consequently, a comparison of AM, SSB-AM, SSB-FM, and FM systems is made from the overall viewpoint of generation, transmission with additive Gaussian noise, and detection. Three figures of merit are used in these comparisons: (1) Output signal-to-noise ratios, (2) Energy-per-bit of information, and (3) System efficiency. In summary, the entire generating function concept is a new tool for synthesis and analysis of single-sideband signals. xiv CHAPTER I INTRODUCTION In recent years the use of single-sideband modulation has become more and more popular in communication systems. This is due to certain advantages such as conservation of the frequency spectrum and larger post- detection signal-to-noise ratios in suppressed carrier single-sideband systems when comparison is made in terms of total transmitted power. A single-sideband communication system is a system which generates a real signal waveform from a real modulating signal such that the Fourier transform, or voltage spectrum, of the generated signal is one-sided about the carrier frequency of the transmitter. In conventional amplitude-modu- lated systems the relationship between the real modulating waveform and the real transmitted signal is given by the well-known formula: XAM(t) = Ao [1 + m(t)] cos Wot m(t)|j 1 (1.1) where Ao is the amplitude constant of the transmitter m(t) is the modulating (real) waveform Wu is the carrier frequency of the transmitter. Likewise, frequency-modulated systems generate the transmitted waveform: t XFM(t) = Ao cos [Wot + D f m(t')dt'] (12) where A0 is the amplitude constant of the transmitter m(t) is the modulating (real) waveform w0 is the frequency of the transmitter D is the transducer constant of the modulator. Now, what is the corresponding relationship for a single-sideband system? Oswald, and Kuo and Freeny have given the relationship: XSSB-AM(t) = Ao [m(t) cos "ot m(t) sin owt] (1.3) where A0 is the amplitude constant of the transmitter m(t) is the modulating signal m(t) is the Hilbert transform of the modulating signal (o is the frequency of the transmitter [1, 2]. This equation represents the conventional upper single-sideband suppressed- carrier signal, which is now known as a single-sideband amplitude-modulated suppressed-carrier signal (SSB-AM-SC). It will be shown here that this is only one of an infinitely denumerable set of single-sideband signals. In- deed, it will be shown that any member of the set can be represented by XSSB(t) = Ao [U(m(t), m(t))cos wot T V(m(t), m(t)) sin wot] (1.4) where Ao is the amplitude constant of the transmitter U(x,y) and V(x,y) are the conjugate functions of any entire function m(t) is the modulating (real) waveform m(t) is the Hilbert transform of m(t) Wo is the transmitter frequency. Various properties of these single-sideband signals will be analyzed in 3 general for the whole set, and some outstanding members of the set will be chosen for examples to be examined in detail. It should be noted that Bedrosian has classified various types of modulation in a similar manner; however, he does not give a general repre- sentation for single-sideband signals [3]. CHAPTER II MATHEMATICAL PRELIMINARIES Some properties of the Hilbert transform and the corresponding analytic signal will be examined in this chapter. None of the material presented in this chapter is new; in fact, it is essentially the same as that given by Papoulis except for some changes in notation [4]. How- ever, this background material will be very helpful in derivations pre- sented in Chapter III and Chapter V The Hilbert transform of m(t) is given by 1. m(x)dx 1 m(t) =- P m(t) -- (2,1) St-x Ttt where (*) is read "the Hilbert transforms of (*)" P denotes the Cauchy principal value indicates the convolution operation. The inverse Hilbert transform is also defined by Ea (2.1) except that a minus sign is placed in front of the right-hand side of the equation. It is noted that these definitions differ from those used by the mathema- ticians by a trivial minus sign. It can be shown, for example, that the Hilbert transform of cos wot is sin mot when 0o > 0 and that the Hilbert transform of a constant is zero. A list of Hilbert transforms has been compiled and published under work done at the California Institute of Tech- nology on the Bateman Manuscript Project [5]. The Fourier transform of m(t) is given by Fm(w) = [-j sgn (w)] Fm(a) (2.2) where + 1 W > 0 sgn (w) = 0 0 = (2.3) 1 W < 0 and Fm(w) is the Fourier transform of m(t). In other words, the Hilbert transform operation is identical to that performed by a -900 all-pass linear (ideally non-realizable) network. From Eq. (2.2), it follows that F^(w) = [-j sgn (w)]2 Fm(w) = -Fm(W) (2.4) or M(t) = -m(t). (2.5) The (complex) analytic signal associated with the real signal m(t) is defined by Z(t) = m(t) + jm(t). (2.6) The Fourier transform of Z(t) follows by the use of Ea. (2.2), and it is FZ(w) = Fm(w) + j[-j sgn (w)] Fm(w) or 2Fm(w) > FZ(w)= Fm(w) W = 0 (2.7) L 0 W < j Now suppose that m(t) is a stationary random process with auto- correlation Rmm(t) and power spectrum Pmm(w)o Then the power spectrum of m(t) is Pim(w) = Pmm(w) 1-J sgn () = Pmm(w). (2.8) This is readily seen by use of the transfer function of the Hilbert trans- form operator given by Eq. (2.2). Then, by taking the inverse Fourier transform of Eq. (2.8), it follows that Rmf(-) = Rmm(r). (2.9) The cross-correlation function is obtained as follows: Rim(T) = m(t + T)m(t) S-m(t + T A)m(t)dx where () is the averaging operator. Thus where (.) is the averaging operator. Thus Rmm(i) = Rmm(). (2.10) It follows that the spectrum of the cross-correlation function is given by fmmif) [-j sgn (w)] Pmalnw) (2,11) it. is i..'fd that Pmm(() is a purely imaginary function since Pmm(w) is a real function. Then 1 Rmm(L ) = I [-j sgn (,)] Pmm(w) ejwT which, for Pmm(w) a real even function, reduces to " '0 Pinfri L ) sin t d. Thus the cross-correlation function is an odd function of L: Rnm(i) = -Rmm(-) = -m(t [)m(t) = -m(t + 1)mr(t) or Rmm(' ) :. -Rm(-') = -Rmm(i)o (2.12) (2.13) (2.14) The autocui elation for the analytic signal is found as follows: RZZ(i) = Z(tt+)Z*(t) = [m(t+) + jm(t+r)] [m(t) jm(t)] m(tr,)m(t) + m(t+T)m(t) + jm(t+. )m(t) jm(t+r)m(t) = Rmm(') + RWm(T) + jRim(T) jRmm(i). Using Eqs. (2.9), (2.10) and (2.14) we obtain RZZ(T) = 2[Rmm(i) + jRm(t)] = 2[Rmmn() + 3Rmm(s)]. (2.15) Thus (1/2)Rzz(T) is an analytic signal associated with Rmm(v). By use of Eq. (2.7) it follows that "4Pmm() W 0 PZZ(W) = 2Pmm(w) a 0. (2 16) L 0 < O_ CHAPTER I ; SYNTHESIS OF SINGLE: -SiUDEAIND GN'LS Eq. (1.4), which specifies thc set of single-sideband signals that can be generated from a given modulating waveform or process, will be derived in this chapter. The equation must be a real function of a real input waveform, m(t), since it represents the generating function of a physically realizable system--the single-1sideband transmitter--and, in general, it is non-linear. Analytic signal techniques will be used in the derivation. It will be shown that if we have a complex function k(z) where k(z) is analytic for z = x + jy in the upper half-plane (UHP), then the voltage spectrum of k(x,O) k(t) is zero for < 0. In order to synthesize real SSB signals from a real modulating waveform, an UHP analytic generating function of the complex time real modulating process must be found regardless of the particular (physically realizable) wave- form that the process assumes. Let m(t) be either the real baseband modulating signal or a reaL function of the baseband modulating signal e(t), Then the amplitude of the voltage spectrum of m(t) is double sided about the origin, for ex- ample, as shown by Figure 1. Fm( i) Figure 1. Voltage Spectrum of a Typical m(t) Waveform 10 Since m(t) is generated by a physically realizable process, it con- tains finite power for a finite time interval. This, of course, is equiva- lent to saying that m(t) is a finite energy signal or, in mathematical terms, it is a L2 function. By Theorem 91 of Titchmarsh, m(t) is also a member of the L2 class of functions almost everywhere [6]. Now the complex signal Z(t) is formed by Z(t) = m(t) + jm(t). (3,1) It is recalled that Z(t) is commonly called an analytic signal in the literature. By Theorem 95 of Titchmarsh there exists an analytic func- tion (regular for y > 0), Z (z), such that as y 0 Z1(x + jy) Z(t) = m(t) + jm(t) x t for almost all t and, furthermore, Z(t) is a Ll (-, function [6] It follows by Theorem 48 of Titchmarsh that the Fourier transform of Z(t) exists [6]. Theorem I: If k(z) is analytic in the UHP then the spectrum of k(t,O), denoted by Fk(w), is zero for all w < 0, assuming that k(t,0) is Fourier trans- formable, For a proof of this theorem the reader is referred to Appendix I. Thus the voltage spectrum of Z(t) is zero for w < 0 by Theorem I since Z(t) takes on values of the analytic function Z7(z) almost every- where along the x axis. Furthermore, since Z(t) is an analytic signal-- that is, it is defined by Eq. (3.1)--its voltage spectrum is given by Eq. (2.7), which is Fz(,) F j where Fm(w) is the voltage spectrum of the signal m(t) Figure 2 for our example used in Figure 1. ,> j F ) This is shown in Figure 2 Voltage Spectrum of the Analytic Signal Z(t) Figure 2. Voltage Spectrum of the Analytic Signal Z(t) Now let a function g(W) be given such that g(W) = U(ReW,ImW) + jV(ReW,ImW) (3.3) where g(W) is an entire function of the complex variable W. Theorem. II: If Z(z) is an analytic function of z in the UHP and if g(W) is an entire function of W, then g[Z(z)] is an analytic function of z in the UH z-plane, A proof of this theorem may be found in Appendix I. Thus g[Z1(z)] is an analytic function of z in the UH z-plane, and by Theorem I g[Z(t)] has a single-sided spectrum: the spectrum being zero for w < 0. This is illustrated by Figure 3, where Fg(w) = F[g(Z(t))]. r1 (3 2) SFga () Figure 3. Voltage Spectrum of an Entire Function of an Analytic Signal Now multiply the complex baseband signal g[Z(t)] by eJ"ut to translate the signal up to the transmitting frequency, o,. It is noted that g[Z,(z)] and ejWoz for mo > 0 are both analytic functions in the UH z-plane, By the Lemma to Theorem I in Appendix I, g[Z,(z)]eJawz is ana- lytic in the UH z-plane. Therefore, by Theorem I the voltage spectrum of g[Z(t)]e JOt is one sided about the origin. Furthermore, F[g(Z(t))eJoWt 1 TeF[g(Z(t))] F[eJ t] = Fg(w) 6(w-Lo) or F[g(Z(t))ej ot] Fg(w-wo) 'o 0 (3.4) This spectrum is illustrated in Figure 4. Figure 4. Voltage Spectrum of the Positive Frequency- Shifted Entire Function of the Analytic Signal The real upper single-sideband signal can now be obtained from the complex single-sideband signal, g[Z(t)]eJot, by taking the real part, This is seen from Theorem III, Theorem III. If h(z) is analytic for all z in the UHP and F[h(x,O)] = Fh(w), then for wc > 0, Fh(w-w)o a wo F{Re[h(x,0)eJwox]} = 0 i W < mo -Fi(46-w.) w s-w This theorem is proved in Appendix I. Thus the upper single-sideband signal for a given entire function is XUSSB(t) = Re{g[Z(t)l]ejot} = Re{[U(ReZ(t),ImZ(t)) + jV(ReZ(t),ImZ(t))]ej 't} = Re{[U(m(t),m(t)) + jV(m(t),m(t))]ejot} )ejwut i i I XUSSB(t) = U(m(t),m(t)) cos t- V(m(t),m(t)) sin wmt I where U(ReW,ImW) is the real part of the entire function g(W) V(ReW,ImW) is the imaginary part of g(W) m(t) is either the modulating signal or a real function of the modulating signal e(t) m(t) is the Hilbert transform of m(t), Using Theorem III the voltage spectrum of XUSSB(t) is FUX(w) = F[XUSSB(t)] = Fg(- ) 0 F*(-w-0) g , O -< coo , ii < O 0 < -- (3.6) This spectrum is illustrated by Figure 5. IFUX( )i -WO 0 Figure 5. Voltage Spectrum of the Synthesized Upper Single-Sideband Signal The lower single-sideband signal can be synthesized in a similar manner from the complex baseband signal. Now we need to translate the complex baseband signal down to the transmitting frequency instead of up, (3.5) 15 as in the upper single-sideband synrthc:.. Then the Fourier transform of the down-shifted complex baseband signal is F[g(Z(t))e-Lot] [g ] 2;r F[e-Jut] Fg(v) I(ulJo) F[g(Z(t))e-Jmot] Fg(w+u4) , WU (3,7) This spectrum is illustrated in Figure 6. IF[g(Z(t))e-jot] Figure 6. Shifted W0 W4 Voltage Spectrum of the Negative Frequency- Entire Function of an Analytic Signal Theorem IV: If h(z) is analytic for all z in UHP and F[h(x,O)] = Fh(w) where Fh(Q) = 0 for all > wo, then for wo > 0 F{Re[h(x,O)e-JmWx] 0 Fh(w+wo) , O < WO , i > Wo ,0 > W ~-W This theorem is proved in Appendix I. Thus the real lower single-sideband signal for a given entire function is XLSSB(t) = Reg(Z(t))e-j 't = Re{[U(m(t),m(t)) t jV(m(t),fi(t))]e-j o t XLSSB(t) = U(m(t),m(t)) cos wot + V(m(t),m(t)) sin wt. Using Theorem IV the voltage spectrum of XLSSB(t) is (3.8) FLX(M) = F[XLSSB(t)] = F*(-wtwo) 0 Fg(W+Wo) , 0 < < WO , Ki > Wo , O m u -W It is noted that the requirements that Fg(w) be zero for w > wo is to prevent spectral overlap at the origin. This requirement is satisfied (for all practical purposes) for wo at radio frequencies. The spectrum of FLX(M) is illustrated by Figure 7. FLX(w) (3.9) -WO I O w Figure 7. Voltage Spectrum of the Synthesized Lower Single-Sideband Signal To summarize, it has been shown that once an entire function 9gW) is chosen, then an upper or lower single-sideband signal can be obtained from the signal m(t). The signal m(t) is either the modulating signal or a real function of the modulating signal b(t). The generalized ex- pressions, which represent SSB signals, are given by Eq, (3 5) for the USSB signal and by Eq. (3.8) for the LSSB signal. These expressions are obviously the transfer functions that are implemented by the upper and lower single-sideband transmitters respectively. Since there are dn in- finitely denumerable number of entire functions, there are an inrinuiely denumerable number of upper and lower single-sideband signals that can be generated from any one modulation process, In Chapter IV some specific entire functions will be chosen to illustrate some well-known single- sideband signals. CHAPTER IV EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN Specific examples of upper single-sideband signal design will now be presented. Entire functions will be chosen to give signals which have various distinct properties. In Chapter VI these properties will be ex- amined in detail. Only upper sideband examples are presented here since the corresponding lower sideband signals are given by the same equation except for a sign change (Eq. (3.5) and Eqo (3.8)). 4.1. Example 1: Single-Sideband AM With Suppressed-Carrier This is the conventional type of single-sideband signal that is now widely used by the military, telephone companies, and amateur radio operators. It will be denoted here by SSB-AM-SC. Let the entire function be g!(W) = W (4.1) and let m(t) be the modulating signal. Then substituting the corresponding analytic signal for W g (Z(t)) = m(t) + jm(t) or UM(m(t),6(t)) = m(t) and Vj(m(t),m(t)) = m(t). (4.2 a,b) Substituting Eqs. (4.2a) and (4o2b) into Eq, (3.5) we obtain the upper single-sideband signal: XUSSB-AM-SC(t) = m(t) cos wot m(t) sin wot (4.3) where m(t) is the modulating audio or video signal and m(t) is the Hil- bert transform of m(t). It is assumed that m(t) is AC coupled so that it will have a zero mean. The upper single-sideband transmitter corresponding to the gene- rating function given by Eq. (4.3) is illustrated by the block diagram in Figure 8. It is recalled that this is the well-known phasing method for generating SSB-AM-SC signals [7, 8],, 4.2. Example 2: Single-Sideband PM Single-sideband phase-modulation was described by Bedrosian in 1962 [3]. To synthesize this type of signal, denoted by SSB-PM, use the entire function: g,(W) = eJ (44) Let m(t) be the modulating audio or video signal. Then g2(Z(t)) = e(m(t) + j(t)) = e-(t) em(t) or U2(m(t),m(t)) = e-(t) cos m(t) (4.5a) m(t) cos Radio Frequency Oscillator, wo XUSSB-AM-SC(t) +"^ -Modulated RF Output Phasing Method for Generating USSB-AM-SC Signals m(t) Input Hilbert Transform {-90 Phase Shift over Spectrum of m(t)} Figure 8. and V2(m(t),m(t)) = e-m(t) sin m(t). (4.5b) Substituting Eqs. (4.5a) and (4.5b) into Eq. (3.5) we obtain the upper single-sideband signal: XUSSB-PM(t) = e-(t) cos m(t) cos wot e-m(t) sin m(t) sin wot or XUSSB-PM(t) = e-m(t) cos (cot + m(t)). (4.6) It is again assumed that the modulation m(t) is AC coupled so that its mean value is zero. The single-sideband exciter described by Eq. (4.6) is shown in Figure 9. 4.3. Example 3: Single-Sideband FM Single-sideband frequency-modulation is very similar to SSB-PM in that they are both angle modulated single-sideband signals. In fact the equations for SSB-FM are identical to those given in Section 4.2 ex- cept that t m(t) = D f e(t)dt (47) -00 where e(t) is now the modulating signal (instead of m(t)) and D is the transducer constant. Experiments with SSB-FM signals have been conducted by a number of persons and are reported in the literature [9, 10]. m(t) Modulating Input XUSSB-PM( Modulated RF Output cos (mot+m( Figure 9. USSB-PM Signal Exciter--Method I Phase Modulator at Radio Frequency mo Hilbert Transform {-90 Phase Shift over Spectrum of m(t)} The SSB-FM exciter as described by Eas. (4.6) and (4.7) is given in Figure 10. 4.4. Example 4: Single-Sideband a The term single-sideband a (SSB-a) will be used to denote a sub- class of single-sideband signals which may be generated from a particular entire function with a real parameter a. This notation was first used by Bedrosian [3]. Let the entire function be g3(W) = e"W (4.8) where a is a real parameter, and let m(t) = ln[l + e(t)] (4.9) where e(t) is the video or audio modulation signal which is amplitude limited such that le(t)| < 1. It is assumed that m(t) is AC coupled (that is, it has a zero mean). Note that these assumptions are usually met by communications systems since they are identical to the restrictions in conventional AM modulations systems. Then g3(Z(t)) = e[1m(t)+jm(t)]= e m(t) emja(t) or U3(m(t),6(t) = eam(t) cos (am(t)) (4.10a) Figure 10. USSB-FM Signal Exciter V3 (m(t),m(t)) = em(.t) sin (am(t)). (410b) Substituting Eqs. (4.10a) and (4.10b) into Eq. (3.5), the SSB-a signal is XUSSB-a(t) = em"(t) cos (am(t)) cos wot eam(t) sin (am(t)) sin wot or XUSSB-a(t) = eam(t) cos (mot + am(t)). (4.11) In terms of the input audio waveform, Eq. (4.11) becomes XUSSB(t) = ealn[l+e(t)] cos (wot + aln[l+e(t)]) or XUSSB_-(t) = [1+e(t)] cos (Wot + aln[l+e(t)]). (4.12) For a = 1 we have an envelope-detectable SSB signal, as is readily seen from Ea. (4.12). Voelcker has recently published a paper demon- strating the merits of the envelope-detectable SSB signal [11]. The real- ization of Eq. (4.12) is shown in Figure 11. For a = 1/2 we have a square-law detectable SSB signal. This type of signal has been studied in detail by Powers [121. Figure 12 gives the block-diagram realization for the square-law detectable SSB exciter. e(t) Modulating Input DC Level of +1 XUSSB-o = 1(t) [1+e(t)] Phase Modulator at Radio Frequency w0 Envelope-Detectable USSB Signal Exciter Hilbert Transform {-90 Phase Shift over Spectrum of In[1+e(t)]} cos (w~t+i'[l +e(t) I Figure 11. e(t) Modulating Input Positive Square I [l+e(t)]1 Root Circuit Square-Law Detectable USSB Signal Exciter Figure 12. CHAPTER V ANALYSIS OF SINGLE-SIDEBAND SIGNALS The generalized SSB signal, that was developed in Chapter III, will now be analyzed to determine such properties as equivalent gener- alized SSB signals, presence or absence of a discrete carrier term, autocorrelation functions, bandwidths, efficiency, and peak-to-averaige power ratio. Some of these properties will depend only on the entire function associated with the SSB signal, but most of the properties will be a function of the statistics of the modulating signal as well. 5.1. Three Additional Equivalent Realizations Three equivalent ways (in general) for generating an upper SSB signal will now be found in addition to the realization given by Eq. (3.5). Similar expressions will also be given for lower SSB signals which are equivalent to Eq. (3.8). It is very desirable to know as many equivalent realizations as possible since any ore of them might be the most econom- ical to implement for particular SSB signal. Theorem V: If h(x,y) = U(x,y) + jV(x,y) is analytic in the UHP (including UH-) then h(t,O) = U(t,O) + j[O(t,O)+k1] (5.1) or h(t,O) = [-(t,O)+k2] + jV(t,0) (5.2) or h(t,O) = [-V(t,O)+k2] + j[O(t,O)+k1] (5.3) where T- k, = lim f V(R cos e,R sin e)de a real constant (5.4) R** 0 k2 = lim U(R cos e,R sin e)de a real constant (5.5) R-- o A proof of this theorem is given in Appendix I. Theorem V may be applied to ,the generalized SSB signal by letting h(z) = g(Z1(z)) where g(.) is an entire function of (.), Z1(z) is analytic in the UHP, and lim Z (z) = lim Z (t + jy) = m(t) + j6(t). Thus Theorem V yO y+O gives three additional equivalent expressions for g(Z(t)) in addition to g(z(t)) = U(m(t),m(t)) + jV(m(t),M(t)) (5.6) which was used in the derivation in Chapter III. Therefore, following the same procedure as in Chapter III, equivalent upper SSB signals may be found. Using Eq. (5.1) we have for the first equivalent representation of Eq. (3.5): XUSSB(t) = Re{g(Z(t))ejwOt} = Re{g(m(t),m(t))eJwot} = Re{[U(m(t),m(t) + jU(m(t),m(t)) + jkl]eJWOt} or XUSSB(t) = U(m(t),m(t)) cos wot [U(m(t),m(t))+ k,] sin o0t. (5.7) Using Eq. (5.2) the second equivalent representation is XUSSB(t) = [-V(m(t),m(t))+k2 cos mot V(m(t),m(t)) sin mot. (5.8) Using Eq. (5.3) the third equivalent representation is XUSSB(t) = [-V(m(t),m(t))+k2] cos mot [U(m(t),m(t))+k1],sin wot. (5.9) Likewise the three lower SSB signals, which are equivalent to Eq. (3.8), are XLSSB(t) = U(m(t),m(t)) cos awt + [U(m(t),m(t))+k1] sin mot (5.10) XLSSB(t) = [-V(m(t),m(t))+k2] cos mot + V(m(t),6(t)) sin wot (5.11) and XLSSB(t) = [-9(m(t),m(t))+k2] cos w0t + [U(m(t),i(t))+kj] sin wot.(5.12) It should be noted, however, that if for a given entire function k, and k2 are both zero, then all four representations for the USSB or the LSSB signals are identical since by Theorem V, U = -V and V = U under these conditions. 5.2. Suppressed-Carrier Signals The presence of a discrete carrier term appears as impulses in the (two-sided) spectrum of transmitted signal at frequencies wo and -woo The impulses may have real, purely imaginary, or complex-valued weights depending on whether the carrier term is cos wot, sin mot, or a com- bination of the two. Thus the composite voltage spectrum of the modulated signal consists of a continuous part due to the modulation plus impulse functions at w0 and -mo if there is a discrete carrier term. As defined here, the "continuous" part may contain impulse functions for some types of modulation, but not at the carrier frequency. Taking the inverse Fourier transform of the composite voltage spectrum it is seen that if there is a discrete carrier term, the time waveform must be expressible in the form: X(t) = [f1(t)+c1] cos Wot [f2(t)+c2] sin mot (5.13) where cI and c2 are due to the discrete carrier f1(t) and f2(t) are due to the continuous part of the spectrum and have zero mean values. Thus Eq. (5.13) gives the condition that c2 and cl are not both zero if there is a discrete carrier term. To determine the condition for a discrete carrier in an upper SSB signal, Eq. (5.13) will be identified with Eq. (5o9), which represents the whole class of upper SSB signals. It is now argued that both U and V have a zero mean value if the modulating process is stationary. This is seen as follows: U(m(t),m(t))= P U(m(t'),m(t')) dt' . -00 But U[m(t'),m(t')] = c, a constant, since m(t) is wide-sense stationary. Thus U(m(t),m(t)) = IP c--dt' = 0. -00 Likewise V has a zero mean value. Then, identifying Eq. (5.13) with Eq. (5.9), it is seen that fi(t) 4 -V(m(t),m(t)) (5.14a) f2(t) = U(m(t),m(t)) (5.14b) c, = kg and c2 k (5.14c,d) Similarily, for lower SSB signals Eq. (5.13) can be identified with Eq. (5.12). Thus the SSB signal has a discrete carrier provided that k1 and k2 are not both zero. As an aside, it is noted that the criterion for a discrete car- rier, given by Eq. (5.13), is not limited to SSB signals; it holds for alt modulated signals. For AM Eq. (5.13) can be identified with Eq. (1.1). Here f (t) Aom(t) (5.15a) f2(t) = 0 (5.15b) c, = Ao and c2 = 0 (5.15c,d) because m(t) has a zero mean due to AC coupling in the modulator of the transmitter. Thus for AM it is seen that there is a discrete carrier term of amplitude c, = A which does not depend on the modulation. For FM Eq. (1.2) can be expanded. Then, assuming sinusoidal modulation of fre- quency wa, we obtain XFM(t) = [Ao cos (- cos wat)] cos ot wa [A0 sin (-- cos wat)] sin wot. (5.16) To identify Eq. (5.16) with Eq. (5.13) we have to find the DC terms of f (t) + c A cos (- cos mat) 1 0 a and f (t) + c2 Ao sin (-cos wat). These are C, = A0 COS (L COS Wat) wa SAo T cos ( cos at)dt = A ( D) (5.17a) wa and c2 = A sin ( D cos wat) Sa T = Ao0 sin (-k- cos wat)dt T 0 wa = 0 (5.17b) Then for sinusoidal frequency-modulation it is seen that the discrete carrier term has an amplitude of AoJo(D/ta) which may or may not be zero depending on the modulation index D/wa. Consequently, for FM it is seen that the discrete carrier term may or may not exist depending on the modulation. Prof. T. S. George has given the discrete carrier condition for the case of FM Gaussian noise [13]. Continuing with our SSB signals, it will now be shown that k, and k2 depend only on the entire function associated with the SSB signal and not on the modulation. From Theorem IV we have k = I lim R-*oo k2 = lim SRT f V[ml(R cos e,R sin e) m (R cos e,R sin e)]de 0 U[m1(R coS e,R sin e) m(R cos e,R sin e)]de 0 where U and V are the real and imaginary parts of the entire function Z1(z) = m1(z) + jm1(z) is the analytic function associated with the analytic signal Z(t) of m(t). It is seen that if (5.18a) lim m (R cos e,R sin e) = 0 0 e s R+o1 lim m,(R cos e,R sin e) = 0 R-+o (5.18b) , 0 O then k and k2 depend only on U and V of the entire function and not on m. Thus we need to show that Ea. (5.18a) and (5.18b) are valid. By the theory of Chapter III there exists a function Z1(z) = m (z) + j0(z) which is analytic in the UHP such that (almost everywhere) lim Z1(t + jy) y* = Z(t) = m(t) + jm(t) where the Fourier transform of Z(t), F(o), is L.2(-o, o). Then we have F(w)ejzwdw. It follows that lim IZ (Rej')I2 R+.- = lim (-)2 R -*O " [F(w)][e-(R sin O)wej(R cos o)]dJ2 By use of Schwarz's inequality this becomes lim 1Z1(Rej3) |2 R-o 12 f IF(w)12d } {lim R-xo e-2(R sin O)adw,} But F(M) e L ("', -) so that f 0 IF(w) 12dw K. Vim R ~ e-(2R sin e)wdw = 0 e di= , 0 < 0 < T. Therefore we have lim jZ(ReJ )I (-)2 K 0 = 0 Also Z, (z) = , 0 < e < n. For e = 0 or e = 2(+) e = 0 lim Z I(ReJ)I = 0 R-o since Z(t) e L2(-, 2). Then lim jZi(ReJe)I = lim IZ (R cos e,R sin O)| = 0 0 s e s R-co R- which shows that Eqs. (5.18a) and (5.18b) are valid statements. Thus, the presence (kI and k2 not both zero) or the absence (k, = k = 0) of a discrete carrier depends only on the entire function associated with the SSB signal and not on the modulation. Furthermore, it is seen that the amplitude of the discrete carrier is given by the magnitude of the entire function evaluated at the origin (of the W plane), and the power in the discrete carrier is one-half the square of the magnitude. For every generalized USSB signal represented by Eq. (3.5), there exists a corresponding suppressed-carrier USSB signal: XUSSB-SC(t) = '-(m(t),(t)) cos Wot *(m(t),m(t)) sin Wot (5.19) where the notation SC and denote the suppressed-carrier functions. But what are these functions U and W? The condition for a suppressed carrier is that ki = k2 = 0. By comparing Eq. (5.19) with Eq. (5.9) it follows that th -V and V Furthermore by Theorem V of Section 5.1, U = -V + kg and V = U + ki, Thus U -V = U k2 (5,20) and V- U = V ki. (5,21) It is also noted that 4 and + are a unique Hilbert transform pair. That is, V- is the Hilbert transform of i, and U- is the inverse Hilbert trans- form of -. This is readily shown by taking the Hilbert transform of Eq. (5.20), comparing the result with Eq. (5.21), and by taking the in- verse Hilbert transform of Eq. (5.21), comparing the result with Eq. (5.20). Thus Eq. (5,19) may be re-written as XUSSB-SC(t) = tt(m(t),'(t)) cos wot -(m(t),m(t)) sin o0t (5.22) or XUSSB-SC(t) = -V(m(t),m(t)) cos wot V(m(t),m(t)) sin wot (5.23) where U and V-are given by Eq. (5.20) and Eq. (5.21). It is interesting to note that the form of the USSB signal given above checks with the expression given by Haber [14]. He indicates that if a process n(t) has spectral components only for I)w > wo then n(t) can be represented by n(t) = s(t) cos wot 9(t) sin mot. (5.24) Thus Eq. (5.22) checks with Eq. (5.24) where U = s(t), and Eq. (5.23) checks also where -V E s(t). The corresponding representations for LSSB suppressed-carrier signals are given by XLSSB-SC(t) = tJ(m(t),m(t)) cos wot + 4(m(t),m{t)) sin mot (5.25) and XLSSB-SC(t) = --(m(t),m(t)) cos wot + *(m(t),m(t)) sin mot (5.26) where 4- and V are given by Eq. (5.20) and Eq. (5.21). This representation also checks with that given by Haber for pro- cesses with spectral components only for Iwl < wo which is n(t) = s(t) cos wot + s(t) sin mot. (5o27) 5.3. Autocorrelation Functions The autocorrelation function for the generalized SSB signal and the corresponding suppressed-carrier SSB signal will now be derived. Using the result of Chapter III, it is known that the generalized upper SSB signal can be represented by XUSSB(t) = Re{g(m(t),m(t))ej(ot+)} (5.28) where a uniformly distributed phase angle > has been included to account for the random start-up phase of the RF oscillator in the SSB exciter. Then, using Middleton's result [15], the autocorrelation of the USSB sig- nal is RXU(t) = XUSSB(t+T)XUSSB(t) = Re{eJwoTRg(-)} (5.29) where Rg(T) = g(M(t+T)mi(t+T))g*(m(t),rm(t)) (5.30) and g(m(t),m(t)) = U(m(t),m(t)) + jV(m(t),m(t)). (5.31) The subscript XU indicates the USSB signal. For the generalized LSSB signal the corresponding formulae are XLSSB(t) = Re{g(m(t),rn(t))eJj(bOt+f) } and IRXL(T) = Re{e-JWOTRg(T)}. These equations can be simplified if we consider the autocorre- lation for the continuous part of the spectrum of the SSB signal. The suppressed DC carrier version of g, denoted by gSC, will first be found in terms of g, and then the corresponding autocorrelation function Rg-sc() (5.32) (5.33) will be determined in terms of Rg(T). By examining Eqo (5o19) and comparing this equation to Eq. (35), with the aid of Eq. (3.3) it is seen that the suppressed DC carrier version of g is given by gsc(m(t),m(t)) = f(m(t),m(t)) + jW(m(t),m(t)) (5.34) where 4 and V are the suppressed-carrier functions defined by Eq. (5.20) and Eq. (5.21). Then it follows that g(m(t),m(t)) = gsc(m(t),m(t)) + [k2+jk,]o (5.35) It is noted that the mean value of gSC is zero. This is readily seen via Eqs. (5.34), (5.20), and (5.21) since it is recalled that the mean value of U and V was shown to be zero in Section 5,2. Then, using Eq. (5.35), the autocorrelation of g is obtained in terms of the autocorrelation of 9SC: Rg(T) = Rg-SC(T) + (k 2+k ). (5.36) Therefore the autocorrelation functions for the USSB signal, Eq. (5.29), and the LSSB signal, Eq. (5.33), become RXU(T) = Re{eJwo( 22+k22) + Rg-SC(r)]} (5o37) and RXL(') = Re{&WoT[ k2+k22) + Rg-SC(T)]} (5.38) It may be easier to calculate the autocorrelation for the USSB or LSSB signal using this representation rather than that of Eq. (5.29) and Eq. (5,33) since RgSC(T) may be easier to calculate than Rg(r). This is shown below. A simplified expression for Rg-SC(T) will now be derived. First, it is recalled from Section 5.2 that and V- are a unique Hilbert trans- form pair. Thus gSC, given by Eq. (5.34), can be expressed in terms of two analytic signals: gsc(m(t),m(t)) = t(m(t),m(t)) + j4(m(t),m(t)) (5.39) and gSC(m(t),m(t)) = -V(m(t),m(t)) + j-(m(t),m(t)) (5.40) where Eq. (5.39) is the analytic signal associated with and Eq. (5.40) is the analytic signal associated with -V-. Using Eq. (5.39) and Eq. (2.15), the autocorrelation of gSC is given by Rg-SC(T) = 2[R (T) + JR4.(T)] (5.41) or by using Eqs. (5.40), (2.15),and (2.9) Rg-SC(T) = 2[R.(T) + jR (T)]o (5.42) Thus Rg-SC(T) may be easier to calculate than R (T) since only Ri,(T) or R .(T) is needed. This, of course, is assuming that the Hilbert trans- form is relatively easy to obtain. On the other hand Rg(T) may be calcu- lated directly from g(m(t),m(t)) or indirectly by use of RUU.(T), Rvv(), RUV(T), and RVU(T). The autocorrelation functions for the generalized USSB and LSSB signals having a suppressed-carrier are readily given by Eq. (5.37) and Eq. (5.38) with k, = k2 = 0: RXU-SC(i) = Re{eJmwTRg-SC(T)} = CR. o) cos W"o Rtt() Sin wfo = RV(7T) cos wo' RW.(T) sin moT RXL-SC(t) = -Re{ eeoRg-SC()} = R 1.(,) COs WOT + RiJ.(T) sin wor = R ,(T) cos W0T + Rv.(T) sin wor. (5.43a) (5.43b) (5.43c) (5.44a) (5.44b) (5.44c) It follows that the power spectral density of any of these SSB signals may be obtained by taking the Fourier transform of the appro- priate autocorrelation function presented above. 5.4. Bandwidth Considerations The suppressed-carrier autocorrelation formulae developed above will now be used to calculate bandwidths of SSB signals. It is noted that the suppressed-carrier formulae are needed instead of the "total sig- nal" formulae since, from the engineering point of view, the presence or absence of a discrete carrier should not change the bandwidth of the sig- nal. Various definitions of bandwidth will be used [16, 17]1 5o4-1. Mean-type bandwidth Since the spectrum of a SSB signal is one-sided about the carrier frequency, the average frequency as measured from the carrier frequency is a measure of the bandwidth of the signal: f WPg.SC(w)d Rg-Sc(O) 0 = ------- (5.45) f Pg-SC(w)dw Rg-SC(O) where Pg_SC(w) is the power spectral density of gSC(m(t),m(t))and the prime indicates the derivative with respect to r. The relationship is valid whenever R'_SC (0) and Rgsc(O) exist. Substituting Eg. (5.41) into Eq. (5.45) we have 2[R (O) + jR (O)] I 2[R*(0) + jRi(0O)] But it recalled that Rju.(T) is an even function of T and, from Chapter II, RUS.(T) is an odd function of T. Then R%,(0) = R(O) = 0 and it follows that R44u(O) R(0) P"Ry(0) Rft(O) (5,46) I It is noted that this formula is applicable whenever Ru.(0) and R. (O) or R.(0O) and R,,(0) exist. That is, R,44(0), R 4(0), R (O), and R^_(0) may or may not exist since Rg-SC(T) is analytic almost everywhere (Theorem 103 of Titchmarsh [6]). 5o4-2, RMS-type bandwidth The rms bandwidth, wrms, may also be obtained. CO 2 2 PgSC()d -Rg-SC(O) (wrms)2 2 -0o f Pg-SC(w)dw Rg-SC(O) -Substituting Eq (5.41) once again, we have00 Substituting Eq. (5.41) once again, we have (rms)2 (5.47) -2[R~(0) + jR (0)] 2[R i (0) + JRi (O)] Since RUU(T) is an odd function of T, R (O) = u(0) = 0, and we have 2 -R (0) R-BB(0) -RW(0) R4.(0) (5.48) It is noted that this formula is applicable whenever R 1(0) and Ry,.(0) or R,4(0) and RW (0) exist. 5,4-3. Equivalent-noise bandwidth The equivalent-noise bandwidth, Aw, for the continuous part of the power spectrum is defined by (2Aw) Pg-SC(O)] 2= Pg-SC(w)d = Rg-SC(O) -00 (5.49) But PgSC(O) = f Rg-SC(t)dT COO Thus (AoW) = 1 Rg-SC(O) f Rg-SC(r)dT -00 Substituting for Rg-SC(T) by using Eq. (5.41) or Eq. (5.42) we obtain (noting once again that R (T) is even and R (r) is odd) TT i7 (Aw) (5.50) 1 f- R,,(T)dT R1 f Rv(T)dr R(0) Rw() ) 5.5. Efficiency A commonly.Used definition of efficiency for modulated signals is [18] n = Sideband Power/Total Power. I - (5o51) This definition will be used to obtain a formula expressing the efficiency for the generalized SSB signal. Using Eq. (5.43) and Eq. (5.44) the side- band power in either the USSB or LSSB signal is RXU-SC(O) = RXL-SC(O) = RWo(O) = RW(O) (552) It is also noted that Rg-SC(O) is not equal to the total power in the real-signal sidebands since gSC is a complex (analytic) baseband signal; instead, (l/2)Re[Rg_sc(O)] : Rwu(O) = Rwv(O) is the total real-signal power. This is readily seen from Eq. (5.43a) and Eq. (5.44a). Similarily the total power in either the USSB or LSSB signal is obtained from Eq. (5.37) or Eq. (5.38): Rxu(O) = RXL(O) = [kl2 + k2 + 2R4(0)] = 1[k 2 + k22 + 2R (O)] (5.53) Thus the efficiency of a SSB signal is 2R 4U(0) 2RV(0) S= (5.54) k 2 + k2 + 2R (0) k2 + k2 + 2R4.(0) 1 2 +12R2 5.6. Peak-to-Average Power Ratio The ratio of the peak-average (over one cycle of the carrier- frequency) to the average power for the SSB signal may also be obtained. The expression for the peak-average power over one carrier- frequency cycle of a SSB signal is easily obtained with the aid of Eq. (3.5) and Eq. (3.8). Recalling that U and V are relatively slow time-varying functions compared to cos mot and sin mot, we have for the peak-average power: P pv {[U(m(t),m(t))] + [V(m(t),m(t))]l} pt i tpeak (5.55) where tpeak is the value of t which gives the maximum value for Eq. (5.55). Using Eq. (5.20) and Eq. (5.21), Pp-Av can also be written as Pp-Av -= {[ + k 2 + [+ k i]2 t = tpeak 2 2 = {[U-+ k2] + [*-+ kj] t tpeak peak = {[-V-+ k21 + [2 + kI t peak . The average power of the SSB signal was given previously by Eq, (5.53). Thus the expression for peak-to-average power ratio for the generalized SSB signal is Pp-Av {[U(m(t),m(t))] + [V(m(t),m(t))] It = t (556a) -- peak PAv k 2 + k 2 + 2R (0) 2 o6 (2 {[U(m(t)m(t))] + [V(m(t),m(t))] t = tpeak (556b) k2 + k2 + 2Rv (0) {[U(m(t),m(t))+k2]2 + [4(m(t),m(t))+k,]2} it tpeak (5 56c) k 2 + k22 + 2R.4(0) {[-V-(m(t),m(t))+k2]2 + [V-(m(t),m(t))+k]21} = ____________________t tpeak. (5.56d) k, 2 + k22 + 2R .(0) Several equivalent representations have been given for peak-to-average power since one representation may be easier to use than another for a particular SS1B signal. CHAPTER VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS The examples of SSB signals that were presented in Chapter IV will now be analyzed using the techniques which were developed in Chapter V. 6.1. Example 1: Single-Sideband AM With Suppressed Carrier The constants kI and k2 will first be determined to show that indeed we have a suppressed carrier SSB signal. By substituting Eq. (4.2b) into Eq. (5.4) we have Tr k = lim m(R cos e,R sin e)de 0 But from Eq. (5.18b) it follows that lim m1(R cos e,R sin e) = 0 0 < e < . R+-> Thus k, = 0 (6.1) Similarily substituting Eq. (4.2a) into Eq. (5.5) we have k = lim P m(R cos e,R sin e)de = 0 (6.2) 2 R-wc J since lim m(R cos e,R sin e) = 0 for 0 < e < x from Eq. (5.18a). Further- R-x more, since both k and k are zero, the equivalent realizations for the SSB signals, as given by the equations in Section 5.1, reduce identically to the phasing method of generating SSB-AM-SC signals (which was given previously in Figure 8). The autocorrelation for the SSB-AM-SC signal is readily given by use of Eq. (4.2a) and Eq. (5.20). Thus U-(m(t),m(t)) = m(t). (6.3) Then the autocorrelation of the suppressed-carrier USSB-AM signal is given via Eq. (5.43b), and it is RXU-SC-SSB-AM() = Rmm () cos Wmo Rmm(T) sin wmo. (6.4) Likewise, by use of Eq. (5o44b) the autocorrelation for the suppressed- carrier LSSB-AM signal is RL-SC-SSB-AM() = Rmm(T) cosw + mm(t) sin wor. (6.5) From Eq. (6.4) it follows that the spectrum of the USSB-AM-SC signal is just the positive-frequency spectrum of the modulation shifted up to a0 and the negative-frequency spectrum of the modulation shifted down to -wo. That is, there is a one-to-one correspondence between the spectrum of this SSB signal and that of the modulation. This is due to the fact that the corresponding entire function for the signal, g(W) = W, is a linear function of W. Consequently, the bandwidths for this SSB signal are identical to those for the modulation. This is readily shown below. The mean-type bandwidth is given by use of Eq. (6.3) in MSSB-AM where 'm = Rmm(0), the power in the rms bandwidth is (when the numerator and denominator exist) Eq. (5.46): Rmm(O) Rmm(O) (6.6) Rmm(O) m the modulating signal. By using Eq. (5.48) M = (0) (6.7) whenever Rmm(O) and 'm exist. By using Eq. (5.50) the equivalent-noise bandwidth is (Am)SSBAM = (6.8) f- Rmm(r)dT Thus the bandwidths of the SSB-AM-SC signal are identical to those of the modulating process m(t). The efficiency of the SSB-AM-SC signal is obtained by using Eq. (5.54): 2Rmm(0) nSC-SSB-AM - 2Rmm(0) (6.9) The peak-to-average power ratio for the SSB-AM-SC signal follows from Eq. (5.56c), and it is Pp-Av {[m(t)]2 + [m(t)]2}t = tpeak PAV /SC-SSB-AM 2*m (6.10) 6.2. Example 2: Single-Sideband PM The SSB-PM signal has a discrete carrier term. This is shown by calculating the constants k, and k2. Substituting Eq. (4.5b) into Eq. (5.4) we have k I lim e-m,(R cos e,R sin e)sin [m,(R cos e,R sin e)]de. 0 But from Eqs. (5.18a) and (5.18b) lim m1(R cos e,R sin e) = 0 for R-o 0 _s e < T and lim m (R cos e, R sin e) = 0 for 0 e rr. Thus R--o ki = 0. (6.11) Likewise, substituting Eq. (4,5a) into Eq. (5.5) we have k = e-0 cos 0 de = 1. (6.12) IT J 0 Thus the SSB-PM signal has a discrete carrier term since k2 = 1 f 0. There are equivalent representations for the SSB-PM signal since k and k2 are not both zero. For example, for the upper sideband signal, equivalent representations are given by Egs. (5.7) and (5.8). It is noted that Eq. (5.8) is identical to Eq. (5.9) for the USSB-PM signal since k, = 0. Thus the two equivalent representations are: XUSSB-PM(t) e= e(t)cos m(t)] cos Wot e acos m(t)] sin wot (6.13) and XUSSB-PM(t) =-(e-(t)sin m(t))+1]cos wot [e-m(t)sin m(t)]sin ot. (6.14) The USSB-PM exciters corresponding to these equations are shown in Figure 13 and Figure 14. They may be compared to the first realization method given in Figure 9. The autocorrelation function for the SSB-PM signal will now be examined. In Section 6.1 the autocorrelation for the SSB-AM-SC signal was obtained in terms of the autocorrelation function of the modulation. This was easy to obtain since 4 = m(t). However, for the SSB-PM case 4 and Vare non-linear functions of the modulation m(t). Consequently, the density function for the modulation process will be needed in order to obtain the autocorrelation of the SSB-FM signal in terms of Rmm(T). To calculate the autocorrelation function for the SSB-PM signal, first RV_(r) will be obtained in term of Rmm(T). Using kL = 0, Eq.(5.21), and Eq. (4.5b) we have V(m(t),m(t)) = V(m(t),m(t)) = e-m(t) sin m(t). (6Jo5) Then Rm jm(t) -jm(t) -Jm(t-)- jm(t-r) R (1) = e ep-m(t-T) e-2j W e 2, e-2/_ m(t) Modulating Input jcos -.t XUSSB-PM(t) OutDut USSB-PM Signal Exciter--Method II e4(t) cos m(t) Figure 13. m(t) Modulating Input e-m(t) sin m(t) (e sin m(t)) Balanced [-(e tsin m(t})+1]cos wot Modulator + - DC Level of +1 RF Oscillator at mro i sin wot -90 Phase _ ~ shift at wO XUSSB-PM(t) Modulated RF Output [e-&(t) sin m(t)]sin Figure 14. USSB-PM Signal Exciter--Method III or R(r) ej[xi(tr)+jy(t)+jy )] ej[X2(tT)+jy(tT)] eJ[x3(t,.)+jy(t,T)] + ej[x4(t,T)+jy(t,T)] (6.16) where x1(t,r) = m(t) m(t-T.) x2(t,T) m(t) + m(t-r) x3(t,t) -m(t) m(t-T) = -x2(t,T) x4(t,T) E -m(t) + m(t-t) = -x (t,T) y(t,r) H i(t) + i(t-T). Now Zet the modulation m(t) be a stationary Gaussian process with zero mean. Then x1(t,T), X2(t,T), X3(t,T), x11(t,r), and y(t,T) are Gaussian processes since they are obtained by linear operations on m(t). They are also stat- ionary and have a zero mean value. It follows that x (t,T), y(t,r); x2(t,i), y(t,7); x3(t,T), y(t,T); and x4(t,r), y(t,r) are jointly Gaussian since the probability density of the input and output of a linear system is jointly Gaussian when the input is Gaussian [15]. For example, to show that x,(t,T) and y(t,r) are jointly Gaussian, a linear system with inputs m(t) and m(t-r) can readily be found such that the output is y(t,r). Now the averaging operation in Eq. (6.16) can be carried out by using the fol- lowing formula which is derived in Appendix II: eJ{x(t)+jy(t)} = e-{x +j2Pxy-ay2} (6.17) where x(t) and y(t) are jointly Gaussian processes with zero mean, x2 = x2(t) y2 = 2(t) uxy = x(t)y(t) . Thus 2 2---- -7 O' [m(t)-m(t-t)] = 2[am -Rmm(T)] ax r[m(t)+m(t-r)]2 2[om2+Rmm(T)] 2 OX3 [-m(t)-m(t-T)]= 2[ om2+Rmm( )] x 2 = [-m(t)+m(t-t)]2= 2[am2-Rmm()] and y 2 [m(t)+m(t-1i)] = 2[cm2+Rmm([)] . From Chapter II -Rmm() so that it is recalled that Rm(0) = 0 and Rm(i) = -Rlm() = the p averages are = [m(t)-m(t-r)][m(t)+m(rt-T)] = -2Rmm() [m(t-(tT( = Sx3Y and X y = -[m(t)-m(t-T)][m(t)+m(t-r)] 2Rmm(T) . Ix y I Ix2 = Therefore, using Eq. (6.17), Eq. (6.16) becomes R (T) = e-{2[am -Rmm(T)] + j2[-2Rmm(T)] 2[om2+Rmm(T)]} e-2{2[m2 +Rmm(i)] + j2.0 2[am2+Rmm(T)]} .- e-{2[m2+Rmm(,)] + j2 2[a m2+Rmm(T)]} + 1 e-{2[am2 -Rmm(T)] + j2[2Rmm(l)] 2[Om2+Rmm(T)]} which reduces to RVW-SSB-PM-GN(r) = {e2Rmm(T) cos (2Rmm(r)) 1} (6.18) where W -SSB-PM-GN denotes the autocorrelation of the imaginary part of the entire function which is associated with the suppressed-carrier SSB- PM signal with Gaussian noise modulation. It is noted that Eq. (6.18) is an even function of i, as it should be, since it is the autocorrelation of the real function V(m(t),6(t)). Furthermore R (0) is zero when Rmm(O) = 0, as it should be, since the power in any suppressed-carrier signal should be zero when the modulating power is zero. The autocorrelation of the USSB-PM signal is now readily obtained for the case of Gaussian noise modulation by substituting Eq. (6.18) into Eq. (5.42) and using Eq. (5.37): RXU-SSB-PM-GN(T) = Re eJ0 T{[e2Rmm() cos (2Rmm(())] + j[e Rmm T)cos (2Rmm(T)]}] (6.19) Likewise, the autocorrelation of the LSSB-PM signal may be obtained by using Eq. (5.38). The autocorrelation of the suppressed-carrier USSB-PM signal with Gaussian modulation is given by using Eq. (5.43a): RXU-SC-SSB-PM-GN(t) = Re [eJ m([e2Rmm(t) cos (2Rmm(r)) 1] + j[e2Rmm( cos (2Rmm(r)]} (6.20) Similarly, the autocorrelation of the suppressed-carrier LSSB-FM signal may be obtained by using Eq. (5,44a). The mean-type bandwidth will now be evaluated for the SSB-PM signal assuming Gaussian noise modulation. From Eq. (6.18) we obtain 001 e2Rmm( ) cos (2Rmm(x)dA P(tA)2 Then 1 e2Rmm() cos [2Rmm(x)]dx R V_(O) P 2 (6o21) and from Eq. (6.18) R *(O) = [e2%m 1] (6.22) where m = m2 is the average power of m(t). Substituting Eqs. (6.21) and (6.22) into Eq,, (5.46) we have the mean-type bandwidth for the Gaussian noise modulated SSB-PM signal: 17 P e2Rmm(A) cos[2Rmm(A)]dA (W)SSB-PM-GN (6-23) e29m -1 where m is the noise power of m(t), It is seen that Eq. (6.23) may or may not exist depending on the autocorrelation of m(t). The mis-type bandwidth can be obtained with the help of the second derivative of Eq. (6.18): R () {-e2Rmm() + {-e2Rmm(r) + {-e2Rmm () + { e2Rmm ( ) + {-e2Rmm(,) + { e2Rmm(t) sin [2Rmm(T)]} 2[Rmm(T)]2 cos [2Rmm(T)]J 2[Rmm(T)l sin [2Rmm(T)]} Rmm () cos [2Rmm(I)]} 2[Rmm(i)]2 sin [2Rmm(T)]} cos [2Rmm(T)]} Imm Rmm (T) RW(O) = e2m {Rmm(0) 2[Rmm(0)]2} Substituting Eq. (6.24) and Eq. (6.22) into Eq. (5.48) we have for the rms-type bandwidth of the SSB-PM signal with Gaussian noise modulation: /2(2[Rnm(O)12 Rm(O)} 6(; ?q) v'rmsJssB-PM-GN 1 e-2m This expression for the rms bandwidth may or may not exist depending on the autocorrelation of m(t). It is interesting to note that Mazo and Salz have obtained a formula for the rms bandwidth in terms of different para- meters [19]. However both of these formulae give the same numerical re- sults, as we shall demonstrate by Eqo (6.29). Thus (6.24) 2RMM(T)MM(T) t \1 A ll ? The equivalent-noise bandwidth is obtained by substituting Eq. (6.18) into Eq. (5.50): (AW) = 1 e2Rmm() cos [2Rmm(T)] 1} dT [e22m-l]. or 'T(e2m 1) (Am)SSB-PM-GN = (6.26) S{e2Rmm(T) cos [2Rmm(T)] l}dT It is noted that the equivalent-noise bandwidth may exist when the formu- lae for the other types of bandwidth are not valid because of the non- existence of derivatives of Rmm(t) at T = 0. It is obvious that the actual numerical values for the bandwidths depend on the specific autocorrelation function of the Gaussian noise. For example, the rms bandwidth of the SSB-PM signal will now be calculated for the particular case of Gaussian modulation which also has a Gaussian spectrum. Let 2 -w) Pm(w) = e 232 where Pm(w) is the spectrum of m(t) no = m is the total noise power in m(t) 2a is the "variance" of the spectrum. The autocorrelation corresponding to this spectrum is Rmm(1) = e 0, (6.27) The Hilbert transform of Rmm(T) is also needed and is obtained by the frequency domain approach. It is recalled from Chapter II that P () = mm -j Pmm() 0 j Pmm S> 0 Then SRmm(T) ~ = T) P .1 2i 00 f Pim(w) ejwt dw 0 -,2 0 -W S e2 ej de J e2-' e2a ejTd] f f -0 0 which reduces to w 2 integral is evaluated by using the of the Bateman Manuscript Project, 1 [5]: S 2 - 2V" o sin wT dm . formula obtained from page 73, Tables of Integral Transforms, Erf Re a > 0 \2v/a / Erf (x) e't dt. O This #18, vol. where a < 0 Thus Rmm J) (- 0 e e-22T2) Erf (-4 2) Rmm(T) = -j Rmm() Erf or . \/j2 / (6.28) From Eao. (6.27) it follows that Rmm(O) = -oo2 and from Ea. (6.28) we have Rim(O) = 2 /27 Substituting these two equations into Ea. (6.25) we get (Grms) 2, -e-2o-- Thus if m(t) has a Gaussian spectrum and if the modulation has a Gaussian density function, the SSB-FM signal has the rms bandwidth: 20o2 [4( oo/n) + ] ("rms)SSB-PM-GN = l -2' -- (6.29) where *o is the total noise power in m(t) a'is the "variance" in the spectrum of m(t). This has the same numerical value as that obtained from the result given by Mazo and Salz [19]. The result may also be compared to that given by Kahn and Thomas for the SSB-PM signal with sinusoidal modulation [20]. From Eq. (19) of their work (wrms)ssB-PM-S = wa6 (6.30) where ma is the frequency of the sinusoidal modulation and 6 is the modu- lation index. For comparison purposes, equal power will be used for m(t) in Eq. (6.30) as was used for m(t) in Eq. (6.29). Then Eq. (6,30) becomes (wrms)SSB-PM-S = /2 wa 'u (6.31) Thus it is seen that for Gaussian modulation the rms bandwidth is propor- tional to the power in m(t) when the power is large (%o > > T/4), and for sinusoidal modulation the rms bandwidth is proportional to the square root of the power m(t). The efficiency for the SSB-PM signal with Gaussian modulation will now be obtained. Substituting En. (6.22) into Eq. (5.54) we have e2m-_l tSSB-PM-GN = + (e2m-1l) or SSB-PM-N e2m (6.32) where 1m is the noise power of m(t). The peak-average to average power ratio for Gaussian m(t) is given by use of Eas. (4.5a), (4.5b), and (6.22) in Ea. (5.56b): Pv VAv {[e-m(t) cos m(t)] [el(t) sin m(t)]2 t tpeak 1 + (e-2mm-l) (6,33) Note that m(t) may take on large negative values because it has a Gaussian density function (since it was assumed at the outset that the modulation was Gaussian), However, it is reasoned that for all practical purposes, m(t) takes on maximum and minimum values of +3am and -3am volts where am is the standard deviation of m(t). This approximation is useful only for small values of am since e+2(3am) approximates the peak power only when the exponential function does not increase too rapidly for larger values of am. Thus the peak-to-average power ratio for the SSB-PM signal with Gaussian noise modulation is -Av ( v )SSB-PM-GN Se6am e6/im-2 m e - when 'm is small. It is noted that the efficiency and the peak-to-average power ratio depend on the total power in the Gaussian modulation process and not on the shape of the modulation spectrum. On the other hand the autocorre- lation function and bandwidth for the SSB signal depend on the spectral (6.34) shape of the modulation as well. The dependence of bandwidth on the spectrum of the Gaussian noise modulation will be illustrated by another example. Consider the narrow- band modulation process: m(t) = a(t) cos (wat + 4) (6.35) where a(t) is the (double-sideband) suppressed-subcarrier amplitude modulation ma is the frequency of the subcarrier ( is a uniformly distributed independent random phase due to the subcarrier oscillator. That is, we are considering a SSB signal which is phase modulated by the m(t) given above. Then Rmm(T) = Raa(f) cos wa{ (6.36) where Raa(T) is the autocorrelation of the subcarrier modulation a(t). Rmm(T) can be obtained from Eq. (6.36) by use of the product theorem [21]. Thus, assuming that the highest frequency in the power spectrum of a(t) is less than wa, Rmm(T) = Raa() sin aT (6.37) Furthermore let a(t) be a Gaussian process; then m(t) is a narrow-band Gaussian process. This is readily seen since Eq. (6.35) may be expanded as follows: m(t) = [a(t) cos (wat+q) a(t) sin (wat+f)] + [a(t) cos (a t+p) + a(t) sin (wat+4)] (6.38) The terms in the brackets are the USSB and LSSB parts of the suppressed- subcarrier signal m(t). But these USSB and LSSB parts are recognized as the well-known representation for a narrow-band Gaussian process. Thus m(t) is a narrow-band Gaussian process. Now the previous expressions for bandwidth, which assume that m(t) is Gaussian, may be used. The mean-type bandwidth for the multi- plexed SSB-PM signal is then readily given via Eq. (6.23), and it is 00 eRaa(X) cos wax cos[Raa(x) sin wax]dA -________ ______________ (6.39) (u)M-SSB-PM-GN ~ e8a 1 where ag is the average power of the Gaussian distributed subcarrier modulation a(t). Obtained in a similar manner, the rms bandwidth is Wa2 ( a+1) Raa(0) (wrms)M-SSB-PM-GN = e*a (6.40) and the equivalent-noise bandwidth is f[e2a-1] (Aw)MSSB-PMN = 0 (6.41) f eRaa(T) cos waT cos[Raa(r) sin waT] -00 Thus, it is seen once again that the bandwidth depends on the spectrum of the modulation, actually the subcarrier modulation a(t). To obtain a numerical value for the rms bandwidth of the multi- plexed SSB-PM signal assume that the spectrum of a(t) is flat over a Wo < w"a 67 Pa()M 0 W W-+ Figure 15. Power Spectrum of a(t) From Figure 15 we have -Wo or NoWo sin Wo. Raa(r) (6.42) and 0a 0 (6.43) IT Then Raa(O) (6.44) 31 Substituting the last two equations into Ea. (6.40) we obtain the rms bandwidth for the SSB-PM multiplexed signal: 2 NOWO( N0W0 N oWo_ a -- -- +1 + - (mrms)M-SSB-PM-GN = NoWo/f (6.45) 1 l -e-W/ where ma is the subcarrier frequency No is the amplitude of the spectrum of the subcarrier Gaussian noise modulation Wo is the bandwidth of the subcarrier noise modulation. Thus the rms bandwidth is proportional to the power in the subcarrier modulation as No becomes large. 6.3. Example 3: Single-Sideband FM As was indicated in Section 4.3. the representation for the SSB-FM signal is very similar to that for the SSB-PM signal. In fact it will be shown below that all the formulae for the properties of the SSB-PM signal (which were obtained in the previous section) are directly applicable to the SSB-FM signal. The SSB-FM signal has a discrete carrier term since the entire function for generating the SSB-FM signal is identical to that for the SSB-PM signal, which has a discrete carrier term. The other properties of the SSB-FM signal follow directly from those of the SSB-PM signal if the autocorrelation of m(t) can be obtained in terms of the spectrum for the frequency modulating signal e(t). It is recalled from Eq. (4.7) that t m(t) = D ef (t')dt', (6.46) -00 First, the question arises: Is m(t) stationary if e(t) is stationary? The answer to this question has been given by Rowe; however, it is not very satisfactory since he says that m(t) may or may not be stationary [22]. However, it will be shown that m(t), as given by Eq. (6.46), is stationary in the strict sense if e(t) is stationary in the strict sense; and, furthermore, m(t) is wide-sense stationary if e(t) is wide-sense stationary. It is recalled that if y(t) = L[x(t)] where L is a linear time-invariant operator, then y(t) is strict-sense stationary if x(t) is strict-sense stationary and that y(t) is wide-sense stationary if x(t) is wide-sense stationary [4]. Since the integral is a linear operator, we need to show only that it is time-invariant, that is to show that y(t+e) = L[x(t+E] or I (tl)dt f e(t2+e)dt2 e(tl)dt1 This is readily seen to be true by making a change in the variable, letting tI = t2 + e. Thus, if 8(t) is stationary, then m(t) is stationary. Moreover, in the same way it is seen that if m(t) had been defined by t m,(t) = D e,(t')dt' (6.47) to then m (t) is not necessarily stationary for e,(t) stationary since the system is time-varying (i.e. it was turned on at to). But this should not worry us because, as Middleton points out, aZZll physically realizable systems have non-stationary outputs since no physical process could have started out at t = -m and continued without some time variation in the parameters 05]. However, after the "time-invariant" physical systems have reached steady-state we may consider them to be stationary processes-- provided there is a steady state. Thus by letting t -m we are con- sidering the steady-state process m(t) which we have shown to be stationary. Now the autocorrelation of m(t) can be obtained by using power-spectrum techniques since m(t) has been shown to be stationary. From Eq. (6.46) we have e(t) dm(t) (6.48) Then in terms of power-spectrum densities P ee() = W2Pmm(a) (6.49) As Rowe points out, Pmm(w) must eventually fall off faster than k/w2, where k is a constant, if e(t) is to contain finite power; and if Pmm(w) = k/w2, Pee(w) will be flat and, consequently, white noise. Thus we have a condition for the physical realizability of m(t): Pmm(w) falls off faster than -6 db/octave at the high end. This condition is satisfied by physi- cal systems since they do not have infinite frequency response. From Eq. (6.49) we have 1 Pmm(W) PeO() (6650) Immediately we see that if POe() takes on a constant value as IwI + 0 and at w = 0, m(t) will contain a large amount of power with spectral components concentrated about the origin. In other words, m(t) has a large block of power, located infinitely close to the origin which is infinitely large. Thus m(t) contains a slowly varying "DC" term with a period T and m2(t) -- -. By examining Eqo (6.46) we obtain the same result from the time domain. That is, for Pee(w) equal to a constant, e(t) contains a finite amount of power located infinitely close to the origin which appears as a slowly varying finite "DC" term in e(t) such that T -. Then by Eq. (6.46), m(t) has a infinite amplitude and, consequently, infinite power, In other words, the system does not have a steady-state output condition if the input has a power around w = 0. Thus, this system is actually conditionally stable, the output being bounded only if the input power spectrum has a slope greater than or equal to +6 db/octave near the origin (and, consequently, zero at the origin) as seen from Eq. (6.50). It is interesting to note that for the case of FM, eJm(t) is stationary regardless of the shape of the spectral density Pee(w). This is due to the fact that ejm(t) is bounded regardless of whether m(t) is bounded or not. From Eq. (6.50) we can readily obtain Rmm(r) for any input process e(t) which has a bounded output process m(t). Thus 00 Rmm()) : 2 eJWT dw (6.51) -00 Furthermore, R"m(0), Rmm(M), and Rmm(O) may be obtained in terms of Pee(w). By substituting for these quantities in the equations of Section 6.2, the properties of a SSB-FM signal can be obtained in terms of the spectrum of the modulating process. 6.4. Example 4: Single-Sideband a The SSB-a signal has a discrete carrier term. This is readily shown by calculating the constants k, and k2. Substituting Eq. (4.10b) into Eq. (5.4) we have iT ki = 'I lim em1(R cos eR sin e) sin am,(R cos e,R sin e)de . SR_00 f 0 But lim m,(R cos e, R sin e) = 0, for 0 s e s 7 and lim m (R cos e, R--o R4- R sin e) = 0 for 0 s e To, Thus k, = 0. (6.52) Likewise, substituting Eq. (4.10a) into Eq. (5.5), we have k2 = 1. (6.53) Thus the SSB-a signal has a discrete carrier term. It follows that equivalent representations for the SSB-a signal are possible since k2 0. This is analogous to the discussion on equiva- lent representations for SSB-PM signals (Section 6.2) so this subject will not be pursued further. The autocorrelation function for the SSB-a signal will now be ob- tained in terms of Rmm(T)o Using Eq. (5.21) and Eq. (4.10b) we have RW(T) = [eem(t) sin am(t)][eam(t-[) sin am(t-T)] or R (,) = k{ea[m(t)+m(t-T)]} eja[m(t)-m(t-T)] -eja[m(t)+m(t-T)] + {eam(t)+m(t-)]} {eJ[-m(t)-m(t-)] + ej[-m(t)+(t-T)}. (6.54) The density function of m(t) has to be specified in order to carry out this average. It is recalled that m(t) is related to the modulating signal e(t) by the equation: m(t) = In [l+e(t)] o Now assume that the density function of the modulation is chosen such that m(t) is a Gaussian random process of all orders. Eq. (6.54) can then be evaluated by the procedure that was used to evaluate Eq. (6.16). Assuming a Gaussian m(t), Eq. (6.54) becomes RW(T)SSB-a-GN = {e2a2Rmm(T) cos [2a2mm(T)] 1} o (6,55) But this is identical to Eq. (6.18) except for the scale factor a2. Thus for envelope detectable SSB (a = 1) with m(t) Gaussian, the auto- correlation and spectral density functions are identical to those for the SSB-PM signal with Gaussian m(t). Moreover, the properties are identical for SSB-a and SSB-PM signals having Gaussian m(t) processes such that (*m)SSB-PM = a2(m)SSB-a_ It is also seen that if |e(t)l < < 1 most of the time then m(t) : e(t). Thus, when e(t) is Gaussian with a small variance, m(t) is approximately Gaussian most of the time. Then Eq. (6.55) becomes RW(t)SSB-a-GN {e22Ree () cos [2a2Ree(f)] 1} (6.56) when le(t)l < < 1 most of the time. Consequently, formulae for the auto- correlation functions analogous to Eqs. (6.19) and (6.20), may be further simplified to a function of Ree(i) instead of Rmm(t). Then the auto- correlation functions for USSB-a and LSSB-a signals, assuming Gaussian modulation e(t) with a small variance, are RXU-SSB-a-GN(r) Re ejo'T {[e2a2Re(T) cos (2otee(T))] + j RLe2a2Ree() cos (2a2Reo())]} (6)57) and RXL-SSB-a-GN(,) Re e-jwo([e2a2Ree(T) cos (2c2(ee(T))] + j [e22Ree) cos (2W2 ee(r))]} (6.58) The efficiency is readily obtained by substituting Eq. (6.56) into Eq. (5.54): nSSB-a-GN = 1 e"22m (659) where Pm is the power in the Gaussian m(t) and le(t)| < < 1. This result may be compared for a = 1 to that given by Voelcker for envelope-detectable SSB(a = 1) [11]. That is, if e(t) has a small variance, then m(t) = e(t); and Eq. (6.59) becomes nSSB-a-GN 1. 1 e-2e2 z 20e2, (6o60) This agrees with Voelcker's result (his Eq. (38)) when the variance of the modulation is small--the condition for Eqo (6o60) to be valid. The expressions for the other properties of the SSB-a signal, such as bandwidths and peak-to-average power ratio, will not be examined further here since it was shown above that these properties are the same as those obtained for the SSB-PM signal when (Wm)SSB-PM '= 2(m)SSB-a as long as m(t) is Gaussian. CHAPTER VII COMPARISON OF SOME SYSTEMS In the two preceding chapters properties of single-sideband sig- nals have been studied. However, the choice of a particular modulation scheme also depends on the properties of the receiver. For example, the entire function g(W) = W2 can be used to generate a SSB signal, but there is no easy way to detect this type of signal. In this chapter a comparison of various types of modulated sig- nals will be undertaken from the overall system viewpoint (i.e. generation, transmission and detection). Systems will be compared in terms of the degradation of the modulating signal which appears at the receiver out- put when the modulated RF signal plus Gaussian noise is present at the input. This degradation will be measured in terms of three figures of merit: 1. The signal-to-noise ratio at the receiver output 2o The signal energy required at the receiver input for a bit of information at the receiver output when com- arison is made with the ideal system (Here the ideal system is defined as a system which requires a minimum amount of energy to transmit a bit of information as predicted by Shannon's formula.) 3. The efficiency of the system as defined by the ratio of the RF power required by an ideal system to the RF power required by an actual system.(Here the ideal sys- 75 tem is taken to be a system which has optimum trade-off between predetection signal bandwidth and postdetection signal-to-noise ratio.) Comparison of AM, FM, SSB-AM-SC, and SSB-FM systems will be made using these three figures of merit. It is clear that these comparisons are known to be valid only for the conditions specified; that is, for the given modulation density function, and detection schemes which are used in these comparisons. 7.1. Output Signal-to-Noise Ratios 7.1-1. AM system Consider the coherent receiver as shown in Figure 16 where the input AM signal plus narrow-band Gaussian noise is given by X(t) + ni(t) = {Ao[l+6 sin wmt] cos wot} + {xc(t) cos )ot xs(t) sin wmot (7.1) where X(t) is the input signal, ni(t) is the input noise with a flat spec- trum over the bandwidth 2wm, and 6 is the modulation index. AC Couple X(t)+ni(t) Low Pass ___ Output 2k cos cot Figure 16. AM Coherent Receiver Then the output signal-to-noise power ratio, where Aok6 sin wmt is the output signal, is given by (S/N)o (S/N)i (7.2) 1 + _26 or (S/N)o 62(C/N)i, (7.3) where (S/N)i = The input signal-to-noise power ratio (C/N)i = The input carrier-to-noise power ratio and the spectrum of the noise is taken to be flat over the IF bandpass which is 2wm(rad/s). 7.1-2, SSB-AM-SC system Consider the coherent receiver (Figure 16) once again, where the input is a SSB-AM-SC signal plus narrow-band Gaussian noise. Then the input signal plus noise is X(t) + ni(t) = {Ao[m(t) cos wot m(t) sin wot]} + [xc(t) cos wot Xs(t) sin mot] (7.4) where m(t) = 6 sin mt and xs(t) = xc(t) if the IF passes only upper sideband components. The input noise is assumed to have a flat spectrum over the bandwidth m.o Then the output signal-to-noise power ratio, where Ak6 sin wmt is the output signal, is given by [23] (S/N)o = (S/N)i (7.5) where the spectrum of the noise is taken to be flat over the IF bandpass which is wm(rad/s)o It is interesting to note that the same result is obtained from a more complicated receiver as given in Figure 17. However, in some practi- cal applications the receiver in Figure 17 may give much better perform- ance due to better lower sideband noise rejection. That is, in Figure 17 the lower sideband noise is eliminated as the result of the approximate Hilbert transform filter realized about w = 0; whereas, in Figure 16 the lower sideband noise is rejected by the IF filter realized about w = mo. Thus, in order to obtain equal lower sideband noise rejection in both receivers, the IF bandpass for the receiver in Figure 16 would have to have a very steep db/octave roll-off characteristic at w = wo. Low Pass Filter X(t)+ni(t) + Output 2k cos wot 2k sin mot + Low Pass Hilbert Filter Filter Figure 17. SSB-AM-SC Receiver 7.1-3. SSB-FM system Now consider a FM receiver which is used to detect a SSB-FM sig- nal plus narrow-band Gaussian noise as shown in Figure 18. X(t)+ni(t) FM Output Receiver Figure 18. SSB-FM Receiver The input signal plus noise is given by X(t) + ni(t) = Aoe-(t) cos [wot + m(t)] + ni(t) (7.6) where Ao = The amplitude of carrier o =s The radian frequency of the carrier m(t) = D _t v(t) dt m(t) = m(t) = The Hilbert transform of m(t) ni(t) = Narrow-band Gaussian noise with power spectral density Fo over the (one-sided spectral) IF band and v(t) is the modulation on the upper SSB-FM signal. The independent narrow-band Gaussian noise process may be represented by ni(t) = R(t) cos [mot + (t)j = xc(t) cos wot xs(t) sin wot where xs(t) = xc(t) since the IF passes only the frequencies on the upper sideband of the carrier frequency. Then the phase of the detector output is obtained from Eq. (7.6) and is p(t) = k tan-1 (7.7) which reduces to i(t) = km(t) + k tan- K R(t) sin [p(t) m(t)] (7,8) where k is a constant due to the detector. The detector output voltage is given by dt)- o Eq. (7.8) is identical to the phase output when the input is conventional FM plus noise except for the factor e -(t) For large input signal-to-noise ratios (i.e. Aoe-m(t) > > R(t) most of the time), Eq. (7.8) becomes kR(t) p(t) = km(t) + ---- sin [p(t) m(t)] (7.9) Aoe-m(t) dno(t) Then the noise output voltage is -d--- where kem(t) n (t) = R(t) sin [<(t) m(t)]. (7.10) Ao Now the phase p(t) is uniformly distributed over 0 to 2r since the input noise is a narrow-band Gaussian process. Then for m(t) deterministic, [<(t) m(t)] is distributed uniformly also. Furthermore, R(t) has a Rayleigh density function. Then it follows that R(t) sin [q(t) m(t)] is Gaussian (at least to the first order density) and, using Rice's formulation [24, 25], R(t) sin [p(t) m(t)] = xs(t) = l 2F(n) a sin [(Un- )t + en] n=l 2i-T where F(w) = F0 is the input noise spectrum and {en} are independent random variables uniformly distributed over 0 to 2i. Actually it is known that the presence of modulation produces some clicks in the out- put [26], but this effect is not considered here. Eq. (7.10) then be- comes kem(t) _____ ke r no(t) = 2F(wn) -w sin [(Pn-wo)t + On] Ao n=l 2' or dno(t) kem --- dt Ao nt 1 2F(tn) (wn-wo) cos [(n-mo)t+ On] dt A0 n f n 2n ke(t) dm(t) i F(n) Am n] + t-- 1 Fn) sin [(n-)t + n Ao Ldt-] n=1 2Tn Noting that {on} are independent as well as uniformly distributed and that the noise spectrum is zero below the carrier frequency, the output noise power is dno(t) 2 Wm Wm k k e2m(t) F + 2 e2m(t) 2dO(t)] id ^-dT2 Ao2 t F 2dw + e 2- Fd k 2 (t F m3 k2 2m(t) dm(t) F0 2 [ -- 2 m (7.11) Ao2 2 3 A0 dt 2n where ,-) is the averaging operator and wm is the baseband bandwidth (rad/s) o Now let v(t) = -Am cos Wmt then, averaging over t, we have 2e / 2 S2m(tj- wm e26 cos Wmt dt = Io(26) (7.12) and e2m(t) ^l(t)2 1 (m6)2 [io(26) 12(26)] = m 26 I (26) (7.13) 2 m 1 where 6 = DAm/wm, the modulation index Eq. (7.13) into Eq, (7,11) we obtain for the output noise power k2Fo 0m3 N 0 0 1 (2 1a' (7.14) Referring to Eq. (7.9), the output signal power is k2 2 = -T (DAm) . 2 dkm(t) 0 dt Then the output signal-to-noise ratio is k2 (DAm)2 S Ak2L 2TWn A 02 3 AO2 2 Fo 2-- 21T m I-o(26) 1 + 61 2 1 + - 2 i(26) (7.16) 611(26) Referring to Eq. (7.6), the signal power into the detector is Si = Ao2 e2(t) cos2 [.ot + m(t)] = 1 A 2 e2(t) =A2 e~~~(2 1 2 = Ao 2 Io(26). Kahn and Thomas have given the ratio of the rms bandwidths (taken about (7.15) (S/N), = (S/N)o = (7.17) Substituting Eq. (7.12) and 83 the mean of the one-sided spectrum) for a SSB-FM signal to a conventional FM signal [20], and it is BSSB-FM BFM (7.18) I 1 2(26) 102(26) It is known that the bandwidth (in rad/s) of a FM signal is approxi- mately BFM = 2(6+1)wm. (7,19) Thus, to the first approximation, the SSB-FM bandwidth is BSSB-FM 2 I 2(2 ) 2 1 2= (7.20) (6+1)w . m Then, taking the IF bandwidth to be that of the SSB-FM signal, the input noise power is V 0 N B SSB-FM' (7 21) Consequently, the input signal-to-noise ratio is Io(26) (S/N)i = (7,22) mm (6+1) /2 I 2(26) 1- 2) Io (26) Fo 4 - 2-u Using Eq. (7.16) and Eq. (7.22), we have (7.23) for the case of SSB-FM plus Gaussian noise into a FM detector. The signal-to-noise output can also be obtained in terms of the unmodulated-signal-to-noise ratio (i.e. the carrier-to-noise power at the input). From Ea, (7.6) we obtain (S/N)i lo(26) (C/N)i and Eq. (7.23) becomes iO2(26) 6 62(6+1) /2 1 ---- (S/N)o = 102(26) (C/N)i Io(26) + 61,(26) (7.24) (7.25) where (C/N)i is the carrier-to-noise power ratio. 7.1-4. FM system The signal-to-noise ratio at the output of a FM receiver for a FM signal plus narrow-band Gaussian noise at the input can be obtained by the same procedure as used above for SSB-FM. The factor e (t)of Eq. (7.6) is replaced by unity, and the bandwidth of the input noise is given by Eqo (7.19). Then the output signal-to-noise ratio becomes (S/N)o = 3 62(6+1) (S/N)i 6 62(6+1) /2 21 12(6) (S/N)o = 2 (S/N)i 102(26) + T 61o(26)11(26) (7.26) when the input signal-to-noise ratio is large. It is also noted that (S/N)i = (C/N)i. (7.27) 7.1-5. Comparison of signal-to-noise ratios A comparison of the various modulation systems is now given by plotting (S/N)o/(S/N)i as a function of the modulation index by use of Eqs. (7.2), (7.5), (7.23), and (7.26). This plot is shown in Figure 19. Likewise (S/N)o/(C/N)i as a function of the modulation index is shown in Figure 20, where Eqs. (7.3), (7.25), (7.26), and (7.27) are used. It is noted that in both of these figures the noise power band- width was determined by the signal bandwidth. When systems are compared in terms of signal-to-noise ratios, a useful criterion is the output signal-to-noise ratio from the system for a given RF signal power in the channel--that is, (S/N)o/Si. This result can be obtained from (S/N)o/(S/N)i, which was obtained previously for each system, if the input noise, Ni, is normalized to some convenient constant. This is done, for example, by taking only the noise power in the band 2mm (rad/s) for measurement purposes. (The actual input noise power of each system is not changed, just the measurement of it.) Then the normalized input noise power for all the systems is F0 N = 2wm 2n where the subscript I denotes the normalized power. Then the ratio (S/N)o/(S/N), is identical to Ni[(S/N)o/Si] where NJ is the constant de- fined above. Thus, to within the multiplicative constant NJ, comparison of (S/N)o/(S/N)I for the various systems is a comparison of the output 3.5 ---- I I 3.0 - I FM (S/N)o f (S/N)i I / SSB-FM--FM Detection 1.5 5 -- SSB-AM-SC 1 ____ _ 0.5 / / AM--Coherent / Detection 0 0.5 1.0 1.5 2.0 2.5 Modulation Index (6) Figure 19. Output to Input Signal-to-Noise Power Ratios for Several Systems |

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4; Pa() 1 Wf 0)** Figure 15. Power Spectrum of a(t) From Figure 15 we have Raa(T) W 1 o 'aa^; 77 J -o' ~WA eJTdw or Raa(T) NqW0 /sin W t * \ W0t (6.42) and ^ = N W oo (6.43) Then ii , Raa(o) -N0W0' (6.44) Substituting the last two equations into Eq. (6.40) we obtain the rms bandwidth for the SSB-PM multiplexed signal: rTTMoyw. i ^ + nw ^rms^M-SSB-PM-SN- 3tt 1 e-Nowo/lT (6.45) where u>a is the subcarrier frequency N0 is the amplitude of the spectrum of the subcarrier Gaussian noise modulation WQ is the bandwidth of the subcarrier noise modulation. 106 and that these partial derivatives are continuous. W = wrw2 = (Ux + j)(U2 + jV2) = (U^- v1v2) + j(v1u2 + V2U1) = U + jV. Then and aU = u, 3Uz + u, Mi. V, av2 - v, Ml ax 1 ax 2 ax 1 ax 2 ax av Vi 3U9 u2 aVi v2 aUi + u1 a V2 _ - + + 3y sy ay ay ay By substituting Eqs. (I-1 a) and (I-lb) into Eq. (1-4), 3V1 aV aU2 all = Un - + U0 i- + V, ay ax ax av2 ax + v. ax (1-3) (1-4) (1-5) But Eq. (1-5) is identical to Eq. (1-3) and the partial derivatives are continuous. Thus, the condition of Eq. (I-2a) is satisfied. Also, and 3V 3U2. + y Ml _l u Ml x 11 Ml = V, + V, + u, ax ax ax ax ax Then aU 3y = u, au2 ay + u. alii ay - v, av2 ay - Vc av1 ay ay \ ax ax ax ax (1-6) (1-7) and all the partial derivatives are continuous. By comparing Eq. (1-6) with Eq. (1-7) it is seen that the condition of Eq. (I-2b) is satisfied. Therefore W(z) is analytic in the UHP. 37 where the notation SC and denote the suppressed-carrier functions. But what are these functions tt and Â¥? The condition for a suppressed carrier is that kx = k2 = 0. By comparing Eq. (5.19) with Eq. (5.9) it follows that it = -V and if- = 0. Furthermore by Theorem V of Section 5.1, U = -V + k2 and V = + kx. Thus it = -V = U k2 (5.20) and Â¥ = U = V kr (5.21) It is also noted that it and Â¥ are a unique Hilbert transform pair. That is, Â¥ is the Hilbert transform of it, and it is the inverse Hilbert trans form of 3t. This is readily shown by taking the Hilbert transform of Eq. (5.20), comparing the result with Eq. (5.21), and by taking the in verse Hilbert transform of Eq. (5.21), comparing the result with Eq. (5.20). Thus Eq. (5.19) may be re-written as XUSSB-SC^ = ,m(t)) cos w0t it(m(t),m(t)) sin wot (5.22) or XUSSB-SC^ = "^(t) ,m(t)) cos w0t V where it and Â¥ are given by Eq. (5.20) and Eq. (5.21). It is interesting to note that the form of the USSB signal given _ * above checks with the expression given by Haber [14]. He indicates that if a process n(t) has spectral components only for |w| > wq then n(t) can be represented by Copyright by Leon Worthington Couch, II 1968 116 or for h(z) = U(x,y) + jV(x,y) 0 = P U(x,0) + jV(x,0) x-t dx Jtt[U(t,0) + jV(t,0)] + lim R-+00 [U(R cos e,R sin 0) + jV(R cos 0,R sin R(cos 0 + j sin 0) t Aside: calculate the term: t f R[U+jV][-sin 0 + j cos 0] I im j -- R-x J (R cos e-t) + jR sin 0 d0 1 im Â¡ R-* ~ R[U + jV][t sin 0 + j(R-t cos 0)] (R cos 0-t)2 + R2 sin2 0 d0 lim R400 {[Ut sin 0 + V(t cos 0-R)] + j[Vt sin 0 + U(R-t R 2t cos 0 + t2/R For finite t, lim (t cos 0 R) = -R, lim (R-t cos 0) = R, and R-Kjo R--K lim [R 2t cos 0 + t2/R] = R, Thus Eq. (1-30) becomes R-x 1 im R->oo {[Ut sin 0 VR] + j[Vt sin 0 + UR]} R d0 1 im * R-* 'Ut sin 0 R - V + j Vt sin 0 R + U de 0)]RjeJed0 (1-29) COS 0)]} de. (1-30) (1-31) Since U and V are real and imaginary parts of a function which is analytic f//7aj Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Couch, Leon TITLE: Synthesis and Analysis of Real Single Side Band... PUBLICATION 1968 DATE: I, L^>ov\ CoucM as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees-of the-University f-Elafida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant ol permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be llmittPtb those specifically allowed by "Fair Use" as prescribed by the teams,nf United States copyright legislatiop-fcf. Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate image- andTxf:;basfi"'versions'as appropriate and to provide and enhance access using search-software. cU-1 This grant of permissions prohibits use of the digitized versions for commercial use or profit. Signature of Copyright Holder Loon Gwc^i Printed or Typed Name of Copyright Holder/Licensee Personal Information Blurred Date of Signature APPENDIX II EVALUATION OF eo(x + J'y) Assume: x and y are joint Gaussian random variables, bbth having zero mean values. To show: e^x ^ = e"^x +J2uxy~^y2l when x and y are joint Gaussian random variables. The joint density function is 1 p(x,y) 2lTaXCTy(l "P T- e 2ax2-ay2^p2) [ay2x2-2axavpXy+0x2y2] Then J[x+jy] _2TTaxay(l-p2)^ 00 00 r ,j(x+jy)e 2ax2ay2(1-p2) [ay2x2*2ax0ypxy+ax2y2] dxdy oo oo r - 2ax2(l-p2) x2-2(^~ py+jax2(l-p2)j x+ CTy2 u -oo co y2+y2a/(l-p2)) dxdy oo oo r' J 00 00 2ax2(l-p2) [x-k(y| [2 - 2ax2(l-p2) L | y k^y)+^y2+y2ax2(l-p2l| V dxdy where k(y) = ~ py + jox2(l-p2) ay 119 Page VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS 48 6.1c Example 1: Single-Sideband AM With Suppressed Carrier* ...... 48 6.2. Example 2: Single-Sideband PM 51 6.3. Example 3: Single-Sideband FM 68 6.4. Example 4: Single-Sideband a 71 VII. COMPARISON OF SOME SYSTEMS 75 7.1. Output Signal-to-Noise Ratios .... 76 7.1-1 AM system ....... ...... 76 7.1-2, SSB-AM-SC system ..... 77 7.1-3. SSB-FM system 78 7.1-4. FM system 84 7.1-5. Comparison of signal-to-noise ratios* 85 7.2. Energy-Per-Bit of Information 89 7.2-1. AM system 93 7.2-2. SSB-AM-SC system 93 7.2-3. SSB-FM system 94 7.2-4, FM system 94 7.2-5. Comparison of energy-per-bit for various systems 95 7.3. System Efficiencies 97 7.3-1. AM system ..... ... 98 7.3-2. SSB-AM-SC system ...... 93 7.3-3. SSB-FM system ....... 93 7.3-4. FM system 99 7.3-5. Comparison of system efficiencies 100 vi ACKNOWLEDGMENTS The author wishes to express sincere thanks to some of the many people who have contributed to his Ph.D program. In particular, acknowledgment is made to his chairman, Professor T. S. George, for his stimulating courses, sincere discussions, and his professional example. The author also appreciates the help of the other members of his super visory committee. Thanks are expressed to Professor R. C. Johnson and the other members of the staff of the Electronics Research Section, Department of Electrical Engineering for their comments and suggestions. The author is also grateful for the help of Miss Betty Jane Morgan who typed the final draft and the final manuscript. Special thanks are given to his wife, Margaret, for her inspi ration and encouragement. The author is indebted to the Department of Electrical Engi neering for the teaching assistantship which enabled him to carry out this study and also to Harry Diamond Laboratories which supported this work in part under Contract DAAG39-67-C-0077, U. S. Army Materiel Com mand. Figure 10. USSB-FM Signal Exciter 65 shape of the modulation as well. The dependence of bandwidth on the spectrum of the Gaussian noise modulation will be illustrated by another example. Consider the narrow- band modulation process: m(t) = a(t) cos (ojat + (f>) (6.35) where a(t) is the (double-sideband) suppressed-subcarrier amplitude modulation a is the frequency of the subcarrier <Â¡> is a uniformly distributed independent random phase due to the subcarrier oscillator. That is, we are considering a SSB signal which is phase modulated by the m(t) given above. Then Rmm(T) = *2 Raa(t) oos ^a1 (6.36) where Raa(T) is the autocorrelation of the subcarrier modulation a(t). Rmm(T) can obtained from Eq. (6.36) by use of the product theorem [21]. Thus, assuming that the highest frequency in the power spectrum of a(t) is less than o>a, Rmmi'O ~ Raa(x) sin a1 > (6.37) Furthermore let a(t) be a Gaussian process; then m(t) is a narrow-band Gaussian process. This is readily seen since Eq. (6.35) may be expanded as follows: m(t) = %[a(t) cos (wat+<|>) a(t) sin (cjat+cÂ¡>)] + %[a(t) cos (cgt+tj)) + a(t) sin (ojat+<)>)] (6.38) 78 more complicated receiver as given in Figure 17 However, in some practi cal applications the receiver in Figure 17 may give much better perform ance due to better lower sideband noise rejection. That is, in Figure 17 the lower sideband noise is eliminated as the result of the approximate Hilbert transform filter realized about oj = 0; whereas, in Figure 16 the lower sideband noise is rejected by the IF filter realized about u = to0. Thus, in order to obtain equal lower sideband noise rejection in both receivers, the IF bandpass for the receiver in Figure 16 would have to have a very steep db/octave roll-off characteristic at 7.1-3. SSB-FM system Now consider a FM receiver which is used to detect a SSB-FM sig nal plus narrow-band Gaussian noise as shown in Figure 18. X{t)+n-j (t) FM Receiver Output Figure 18. SSB-FM Receiver 55 or (6.16) where XjU.r) = m(t) m(t-r) x2(t,x) = m(t) + m(t-r) x3(t,x) = -m(t) m(t-x) -x2(t,r) X4(t,r) = -m(t) + m(t-r) = -x j(t jt) y(t,r) = m(t) + m(t-x) Now let the modulation m(t) be a stationary Gaussian process with zero mean. Then x^t,/), x2(t,i), x3(t,T), x4(t,x), and y(t,x) are Gaussian processes since they are obtained by linear operations on m(t). They are also stat ionary and have a zero mean value. It follows that x (t,x), y(t,t); x?_(t jt) j y(t,x); x 3 (t j i), y (t, T); and x4(t,T)s y(t,r) are jointly Gaussian since the probability density of the input and output of a linear system is jointly Gaussian when the input is Gaussian [15]. For example, to show that Xj(t,x) and y(t,r) are jointly Gaussian, a linear system with inputs m(t) and m(t-r) can readily be found such that the output is y(t,x). Now the averaging operation in Eq. (6.16) can be carried out by using the fol lowing formula which is derived in Appendix II: ej{x(t)+jy(t)} = e-Js{ox2+j2yXy-ay2} (6.17) where x(t) and y(t) are jointly Gaussian processes with zero mean, 2 = = X2(t) oy2 = y2(t) time-varying functions compared to cos ta0t and sin wot, we have for the peak-average power: 47 Pp_Av = %{[U(m(t),m(t))]2 + [V(m(t),m(t))]/}I P 't t, (5.55) where tpea(< is the value of t which gives the maximum value for Eq. (5.55). Using Eq. (5.20) and Eq. (5.21), Pp_Av can also be written as P Av [4M- k2]2 [* + k,]2) | p peak 2 2 = ^(Du- + k9] + [ti- + k.] } L . r Vak = %{[4+ k]2 + |> + kL]2}11 t 1 xpeak . The average power of the SSB signal was given previously by Eq. (5.53). Thus the expression for peak-to-average power ratio for the generalized SSB signal is (5.56a) (5.56b) (5.56c) (5.56d) Several equivalent representations have been given for peak-to-average power since one representation may be easier to use than another for a particular SS>B signal. Pn-Av f[U(m(t).ii(t))]2 + [V(m(t),m(t))]2) |t t = _____ Lpeak PAv k;i2 + k2 + 2R(JU(0) {[U(m(t),m(t))]2 + [V(m(t),m(t))]2} L , k,2 + k22 + 2RW(0) {[Wm(t).m(t))+k,]2 + [ = ^ 1 ~ Lpeak kj2 + k22 + 2Rw(0) {[-Â¥(m(t),m(t))+k2]2 + [Â¥-(m(t),m(t))+k1]2} | = t ''peak. k;i2 + k22 + 2RW(0) 68 Thus the niis bandwidth is proportional to the power in the subcarrier modulation as N0 becomes large. 6.3/ Example 3: Single-Sideband FM As was indicated in Section 4.3. the representation for the SSB-FM signal is very similar to that for the SSB-PM signal. In fact it will be shown below that all the formulae for the properties of the SSB-PM signal (which were obtained in the previous section) are directly applicable to the SSB-FM signal. The SSB-FM signal has a discrete carrier term since the entire function for generating the SSB-FM signal is identical to that for the SSB-PM signal, which has a discrete carrier term. The other properties of the SSB-FM signal follow directly from those of the SSB-PM signal if the autocorrelation of m(t) can be obtained in terms of the spectrum for the frequency modulating signal e(t). It is recalled from Eq. (4.7) that t (6.46) First, the question arises: Is m(t) stationary if e(t) is stationary? The answer to this question has been given by Rowe; however, it is not very satisfactory since he says that m(t) may or may not be stationary [22] However, it will be shown that m(t), as given by Eq. (6.46), is stationary in the strict sense if e(t) is stationary in the strict sense; and, furthermore, m(t) is wide-sense stationary if e(t) is wide-sense stationary It is recalled that if y(t) L[x(t)l 109 in the UH z-plane and that they are continuous in the UHP also. Now, 3V2 3V2 3Ux aV2 3V (1-12) 3y 3U1 3y 3 V1 3y Substituting Eqs. (1-9) and (1-10) into Eq. (1-12), Eq. (1-12) becomes (1-13) Thus the condition given by Eq. (I-11 a) is satisfied in the region where these derivatives exist and are continuous. Similarly, 3V2 3V2 311} 3V2 3Vi (1-14) 3X 3 U x 3X 3 V i 3X Substituting Eqs. (1-9) and (I-10) into Eq. (1-14), Eq. (1-14) becomes Thus the condition given by Eq. (I-llb) is satisfied in the region where these derivatives exist and are continuous. It is now argued that the derivatives exist and are continuous for z in the UHP. This is true because for any z in the UHP, including UH , z(z) may take on any value in the finite W plane. Also, the derivatives of Ux and V1 with respect to x and y exist and are continuous for z in the UHP, and the derivatives of U2 and V2 with respect to U} and \1 exist and are continuous anywhere in the finite W plane. Thus the con ditions given by Eqs. (I-11 a) and (I-llb) are satisfied, and g[Z(z)] is analytic in the UH z-plane. or 6 Fz(ui)' 2Fm(w) oj > 0 Fm(w) ai = 0 0 oj < Q (2.7) Now suppose that m(t) is a stationary random process with auto correlation Rmm([) anc* power spectrum Pmm(u))o Then the power spectrum of m(t) is Fmm(w) = pmm(w) l~J s9n (w) I Pmmi^)- (2.8) This is readily seen by use of the transfer function of the Hilbert trans form operator given by Eq. (2.2). Then, by taking the inverse Fourier transform of Eq. (2.8), it follows that Rmm(T) ~ Rmm(1) (2.9) The cross-correlation function is obtained as follows: Rmm(T) = (t + T)m(t) m(t + i A)m(t)dA TT A Rmm(t-A)dA where () is the averaging operator. Thus Rmm(i~ Rmm(T) (2.10) 73 Assuming a Gaussian m(t), Eg. (6.54) becomes COS [2a2R|7im(x)] 1} (6.55) But this is identical to Eg. (6.18) except for the scale factor a2. Thus for envelope detectable SSB (a = 1) with m(t) Gaussian, the auto correlation and spectral density functions are identical to those for the SSB-PM signal with Gaussian m(t). Moreover, the properties are identical for SSB-a and SSB-PM signals having Gaussian m(t) processes such that (ipm)SSB-PM 0(2 ^m^sSB-a* It is also seen that if |e(t)| < < 1 most of the time then m(t) = e(t). Thus, when e(t) is Gaussian with a small variance, m(t) is approximately Gaussian most of the time. Then Eg. (6.55) becomes 2a2Ree(T) cos [2a2Ree(x)] 1} (6.56) RW(l)SSB-a-GN ~ when |e(t)| < < 1 most of the time. Conseguently, formulae for the auto correlation functions analogous to Egs. (6.19) and (6.20), may be further simplified to a function of Ree(x) instead of Rrnm(t). Then the auto correlation functions for USSB-cx and LSSB-a signals, assuming Gaussian modulation e(t) with a small variance, are RXU-SSB-a-GN (t) = H Re eJoT{te22Ree(T) cos (2<,2l?ee(t))] j [e2"R86(T) cos (2c.^e(0)]> + (6,57) 88 signal-to-noise ratios for the systems for a given RF signal power. This procedure is commonly used for system comparisons [23]. Likewise, a comparison of output signal-to-noise ratios for vari ous systems for a given aarrier power can be carried out by comparing (S/N)0/(C/N)j, where the subscript I denotes the normalized input noise power once again. The AM, SSB-AM-SC, SSB-FM, and FM systems will now be compared by using this procedure. For the AM system (S/N)i = (S/N)j so that Eq. (7.2) becomes (S/N)0 1 + (S/N)! (7.28) and, likewise, Eq. (7.3) becomes (S/N)0 = 2 (C/N)]. For the SSB-AM system Eq. (7.5) becomes (S/N)0 = 2(S/N)i. (7.29) (7.30) For the SSB-FM system, Eq. (7.23) becomes (7.31) and Eq. (7.25) becomes (S/N)0 662 10(2<$) + | 61,(26) (C/N)j (7.32) 41 It may be easier to calculate the autocorrelation for the USSB or LSSB signal using this representation rather than that of Eq (5.29) and Eq. (5.33) since Rg_$c(x) may be easier to calculate than Rg(x). This is shown below. A simplified expression for Rg_$c(T) will now be derived. First, it is recalled from Section 5.2 that if and -V- are a unique Hilbert trans form pair. Thus g$c given by Eq. (5.34), can be expressed in terms of two analytic signals: gsc(m(t),m(t)) = tf(m(t) ,m(t)) + jit(m(t) ,m(t)) (5.39) and gsc(m(t),m(t)) = -Â¥-(m(t) ,m(t)) + jÂ¥-(m(t) ,m(t)) (5.40) where Eq. (5.39) is the a^lytic signal associated with -H-and Eq. (5.40) is the analytic signl associated with -Â¥. Using Eq. (5.39) and Eq. (2.15), the autocorrelation of g5g is given by Rg-SC(T) 2[:Rw(t) + jRw(r)] (5.41) or by using Eqs. (5.40), (2.15), and (2.9) Rg-SC(T) = 2[rw(t) + jRw(t)]. (5.42) Thus Rg_$c(x) may be easier to calculate than Rg(x) since only Rw(x) or RyyM is needed. This, of course, is assuming that the Hilbert trans- by use of Eqs. (4.5a), (4.5b), and (6.22) in Eq. (5.56b): 64 i[e m (t) cos m(t)] + [e rn(t) sin m(t)]2} 1 + (e 1) tpeak or e-2rn(t) e t = tpeak Si'm (6.33) Note that m(t) may take on large negative values because it has a Gaussian density function (since it was assumed at the outset that the modulation was Gaussian). However, it is reasoned that for all practical purposes, m(t) takes on maximum and minimum values of +3am and 40^ volts where cm is the standard deviation of m(t). This approximation is useful only for small values of crm since e+^m) approximates the peak power only when the exponential function does not increase too rapidly for larger values of am. Thus the peak-to-average power ratio for the SSB-PM signal with Gaussian noise modulation is (6.34) when is small. It is noted that the efficiency and the peak-to-average power ratio depend on the total power in the Gaussian modulation process and not on the shape of the modulation spectrum. On the other hand the autocorre lation function and bandwidth for the SSB signal depend on the spectral 85 when the input signal-to-noise ratio is large. It is also noted that (S/N)i = (C/N)j. (7.27) 7.1-5. Comparison of signal-to-noise ratios A comparison of the various modulation systems is now given by plotting (S/N)0/(S/N)^ as a function of the modulation index by use of Eqs. (7.2), (7.5), (7.23), and (7.26). This plot is shown in Figure 19. Likewise (S/N)0/(C/N)j as a function of the modulation index is shown in Figure 20, where Eqs. (7.3), (7.25), (7.26), and (7.27) are used. It is noted that in both of these figures the noise power band width was determined by the signal bandwidth. When systems are compared in terms of signal-to-noise ratios, a useful criterion is the output signal-to-noise ratio from the system for a given RF signal power in the channel--that is, (S/N)0/S-Â¡, This result can be obtained from (S/N)0/(S/N)-Â¡, which was obtained previously for each system, if the input noise, N-Â¡, is normalized to some convenient constant. This is done, for example, by taking only the noise power in the band 2oim (rad/s) for measurement purposes. (The actual input noise power of each system is not changed, just the measurement of it.) Then the normalized input noise power for all the systems is F0 [\| _ 2u)pi 2ir where the subscript I denotes the normalized power. Then the ratio (S/N)0/(S/N)j is identical to Nj[(S/N)0/Si] where Nj is the constant de fined above. Thus, to within the multiplicative constant Nj, comparison of (S/N)0/(S/N)j for the various systems is a comparison of the output Using Eg. (2.2) and the difinition for the Fourier transform, we F^(w) = FReh(a)) + j[-j sgn (w)] FReh(w) + j^irk^U) or FRehM *sFh(w) a) > 0 Ff-i(a))-j2-rrki6(ai) w s 0 Also, it is recalled that FReh(-) = FReh() This is seen from Re[h(x Thus from Eqs. (1-18) and (1-19) we obtain ,0)]eJtXdx Re[h(x,0)]e"J)Xdx FReh() JgF^(a)) a) > 0 [F^uO-J^irk^w)] = [F^i-wJ+jZTrk^i-w)] w = 0 %Ffi(-o)) a) < 0 Proof of Theorem III: By aid of Eg. (1-23) we have 111 obtain (1-18) (1-19) h(x,0)ejox = {U(x,0)+j[U(x,0)+k1]}ejx . 94 7.2-3. SSB-FM system To obtain M(6) for the SSB-FM system with (S/N)Q = 27.5 db, it follows from Eq. (7.25) that (C/N)-f = 23.3 db for 6 = 1. Also, for the SSB-FM system Eq. (7.38) becomes M(6) _ / I12(26)1 -I 2(6+1) o)m /2 /l I (26)(C/N)i y I02(26) 0.693 com log 6 62 (6+1) /2 /l - 1 + T7(W In2(2.) -. (7.43) 10(26) + | 61^26) (C/N)i For (C/N)^ = 23.3 db, Eq. (7.43) reduces to 7.2-4. FM system To obtain M() for the FM system, for (S/N)0 = 27.5 db, it follows from Eqs. (7.26) and (7.27) that (C/N)-,* = 12 db (which is just above the threshold) for 6=2. Also, for the FM system, Eq. (7.38) becomes [2 (6+1) oom] (C/N) M(6) = : . 0.693 aw log2 [1+3 62(6+1)(C/N)i] (7.45) 112 Then ReCh(x90)e^a>x3> = f (U(x,0) cos to0x [Uix.Oj+k^sin wox}eja)Xdx U(x.O) >s(ej,oX + e1"x) + J[0{x,O)+k1]yejx-e-;ilX)>e-jxdx 00 CO = h J U(x,0)e"J^"w^xdx + % f U(x,0)e"J'^a)+)^xdx 00 CO 00 00 + 3h C U(x,O)e'J^"w0^xdx jig f U(x,0)e'^w+w^xdx v/ J 00 oo 00 00 + Jhk1 f ej("w+)o)xdx jj5k1 f ej(_)"a)o)xdx and by using Eq. (2.2) and the Fourier transform of U(x,0), '{Re[h(x,0)e,;)oX]}= FRe^()-)0) + % FR(ahU+o>0) Reh' + 3h [-J sgn (o-jq)3 FReh(c-)0) j% [-j sgn (w+a)0)]FReh(cD+iD0) + 3h 2Trk1(-to+Jo) 3h 2-rrk16(-cj-u)o). (1-20) Using Eq. (1-17) from the Lemma to Theorem III to evaluate FReh(*) in Eq. (1-20), Eq. (1-20)becomes %F^(t->o) w > wq 0 |o)| < uo %F^(-o)-w0) w -w0 F{Re[h(x,0)eja)x]} = Figure Page 20 Output Signal-to-Noise to Input Carrier-to-Noise Ratio for Several Systems 87 21. Output Signal-to-Noise to Input Signal-to-Normalized- Noise Power Ratio for Various Systems 90 22. Output Signal-to-Noise to Input Carrier-to-Normalized- Noise Power Ratio for Various Systems 91 23 Comparison of Energy-per-Bit for Various Systems ....... 96 24,. Efficiencies of Various Systems 101 25. Contour of Integration 107 26o Contour of Integration 115 40 will be determined in terms of Rg(-r). By examining Eq. (5.19) and comparing this equation to Eq. (3.5), with the aid of Eq. (3.3) it is seen that the suppressed DC carrier version of g is given by gsc(m(t),m(t)) = if(m(t),m(t)) + jÂ¥-(m(t) ,m(t)) (5.34) where -0-and M- are the suppressed-carrier functions defined by Eq. (5.20) and Eq. (5.21). Then it follows that g(m(t),m(t)) = gsc(m(t) ,m(t)) + [k2+jk1]0 (5.35) It is noted that the mean value of ggQ is zero. This is readily seen via Eqs. (5.34), (5.20), and (5.21) since it is recalled that the mean value of 0 and V was shown to be zero in Section 5.2. Then, using Eq. (5.35), the autocorrelation of g is obtained in terms of the autocorrelation of 9sc; Rg(0 = Rg-sc(x) + (kiZ+k22) (5.36) Therefore the autocorrelation functions for the USSB signal, Eq. (5.29), and the LSSB signal, Eq. (5.33), become Rxu(T) = ^(^[(k^2) + Rg-SC (T 1 and Rxl(i) = %Re{eJwoT[(kl2+k22) + Rg_sc(x)]}o (5.37) (5.38) i 14 or XuSSB^t) = U(m(t) ,rn(t)) cos w0t V(m(t) ,m(t)) sin oj0t (3.5) where U(ReW,ImW) is the real part of the entire function g(W) V(ReW,ImW) is the imaginary part of g(W) m(t) is either the modulating signal or a real function of the modulating signal e(t) m(t) is the Hilbert transform of m(t). Using Theorem III the voltage spectrum of Xu$$g(t) is FUXU) = ^[XusSBt)] %Fg(w-tQ) to > , j co j < to. lgFg(-w-aJ0) a) < -u0 (3.6) This spectrum is illustrated by Figure 5. The lower single-sideband signal can be synthesized in a similar manner from the complex baseband signal Now we need to translate the complex baseband signal down to the transmitting frequency instead of up, KEY TO SYMBOLS AM b B Cb Cb Ci (C/N)i (C/N )i D FM F(a>) F(-) g(w) GN LSSB m(t) M Ni P(u) = Amplitude Constant = Amplitude-Modulation = Baseband Bandwidth (rad/s) = RF Signal Bandwidth (rad/s) = Baseband Channel Capacity = RF Channel Capacity = Input Carrier Power = Input Carrier-to-Noise Ratio = Input Carrier-to-Normalized-Noise Ratio = Modulator Transducer Constant = Frequency-Modulation = Voltage Spectrum = The Fourier Transform of (*) = U(W) + jV(W) = An Entire Function = Gaussian Noise = Lower Single-Sideband = Modulating Signal or a Real Function of the Modulating Signal (see e(t) below) = Either Multiplex or Figure of Merit - Input Noise Power = Normalized Input Noise Power = Power Spectral Density x CHAPTER I INTRODUCTION In recent years the use of single-sideband modulation has become more and more popular in communication systems. This is due to certain advantages such as conservation of the frequency spectrum and larger post detection signal-to-noise ratios in suppressed carrier single-sideband systems when comparison is made in terms of total transmitted power, A single-sideband communication system is a system which generates a real signal waveform from a real modulating signal such that the Fourier transform, or voltage spectrum, of the generated signal is one-sided about the carrier frequency of the transmitter. In conventional amplitude-modu lated systems the relationship between the real modulating waveform and the real transmitted signal is given by the well-known formula: XAM(t) = A0 H + m(t)] cos ojQt |m(t)| <1 (11) where A() is the amplitude constant of the transmitter m(t) is the modulating (real) waveform oj0 is the carrier frequency of the transmitter. Likewise, frequency-modulated systems generate the transmitted waveform: Xp^(t) = A0 cos [w0t + D / tm(t")dt'] (1.2) 1 101 g(db) Figure 24. Efficiencies of Various Systems 50 signal are identical to those for the modulation. This is readily shown below. The mean-type bandwidth (when the numerator and denominator exist) is given by use of Eq. (6.3) in Eq. (5.46): (6.6) where = Rmm(0), the power in the modulating signal. By using Eq. (5.48) the rms bandwidth is (rms^SSB-AM -tim(O) I'm (6.7) whenever R^m(0) and exist. By using Eq. (5.50) the equivalent-noise bandwidth is ir (A^SSB-AM = oo 4^ S Rmrn(T)dT (6.8) Thus the bandwidths of the SSB-AM-SC signal are identical to those of the modulating process m(t). The efficiency of the SSB-AM-SC signal is obtained by using Eq. (5.54): iSC-SSB-AM 2Rmm(0) = 1 . 2Rmm(0) (6.9) CHAPTER II MATHEMATICAL PRELIMINARIES Some properties of the Hilbert transform and the corresponding analytic signal will be examined in this chapter. None of the material presented in this chapter is new; in fact, it is essentially the same as that given by Papon1 is except for some changes in notation [4]. How ever, this background material will be very helpful in derivations pre sented in Chapter III and Chapter' V The Hilbert transform of m(t) is given by co (2.1) where () is read !lthe Hilbert transforms of ()" P denotes the Cauchy principal value * indicates the convolution operation. The inverse Hilbert transform is also defined by Eo (2.1) except that a minus sign is placed in front of the right-hand side of the equation. It is noted that these definitions differ from those used by the mathema ticians by a trivial minus sign. It can be shown, for example, that the Hilbert transform of cos oo0t is sin to0t when w0 > 0 and that the Hilbert transform of a constant is zero, A list of Hilbert transforms has been compiled and published under work done at the California Institute of Tech nology on the Bateman Manuscript Project [5], 4 31 signal consists of a continuous part due to the modulation plus impulse functions at u0 and -o>0 if there is a discrete carrier term. As defined here, the "continuous" part may contain impulse functions for some types of modulation, but not at the carrier frequency. Taking the inverse Fourier transform of the composite voltage spectrum it is seen that if there is a discrete carrier term, the time waveform must be expressible in the form: X(t) = [f^tj+cj cos ai0t [f2(t)+c2] sin iOgt (5.13) where c1 and c are due to the discrete carrier f^t) and f2(t) are due to the continuous part of the spectrum and have zero mean values. Thus Eq. (5.13) gives the condition that c2 and Cj are not both zero if there is a discrete carrier term. To determine the condition for a discrete carrier in an upper SSB signal, Eq. (5.13) will be identified with Eq. (5.9), which represents /V A the whole class of upper SSB signals. It is now argued that both U and V have a zero mean value if the modulating process is stationary. This is seen as follows: U(m(t),Â£(t))-lp f df . But U[m(t'),m(t')] = c, a constant, since m(t) is wide-sense stationary. Thus 00 U(m(t) ,m(t)) = ^ P J' 00 dt' = 0. Figure 8 Phasing Method for Generating USSB-AM-SC Signals we have 70 (6.48) Then in terms of power-spectrum densities (6.49) As Rowe points out, Pmm(oo) must eventually fall off faster than k/ where k is a constant, if e(t) is to contain finite power; and if Pmm(to) = k/to2, Pqq(w) will be flat and, consequently, white noise. Thus we have a condition for the physical realizability of m(t): Pmm(u)) falls off faster than -6 db/octave at the high end. This condition is satisfied by physi cal systems since they do not have infinite frequency response. From Eq. (6.49) we have (6.50) Pmm() ~ C Immediately we see that if P9g(to) takes on a constant value as |w| ->- 0 and at o> 0, m(t) will contain a large amount of power with spectral components concentrated about the origin. In other words, m(t) has a large block of power, located infinitely close to the origin which is infinitely large. Thus m(t) contains a slowly varying "DC" term with a period T and m2(t) - . By examining Eq. (6.46) we obtain the same result from the time domain. That is, for Pe0(w) equal to a constant, e(t) contains a finite amount of power located infinitely close to the origin which appears as a slowly varying finite "DC" term in e(t) such that T * Then by Eq. (6.46), m(t) has a infinite amplitude and, 7 It follows that the spectrum of the cross-correlation function is given by Pmm() ~ l-"j sgn (w)3 PfTirii(tlJ) (2.11) it is noted that Pmm(w) is a purely imaginary function since Pmm(w) is a real function. Then 00 Rmm(c) ~ T7 f (w)] Pmm(uduJ l~ '* *00 which5 for Prnm(u)) a real even function, reduces to Rmm( i) = J Pmm(^) s,n wt du) (2/12) 11 0 Thus the cross-correlation function is an odd function of r: fynm(T) ~ Rmm(_T) = -rh(t T)m(t) -m(t + i)m(t) (2/13) or wo = -W-O = -WO. (2.14) The autocorrelation for the analytic signal is found as follows: RZZ( ) = Z(tH)Z*(iy = [m(t+-r) + jm(t+x)] [m(t) jm(tj] = m(t+i)rn(t) + m(t+r)m(t) + jm(t+: )m(t) jm(t+r)m(t) - Rmm() f Rmm(T) + JRmm(1) jRmm(TK Figure 9. USSB-PM Signal Exciter--Method I 17 To summarize, it has been shown that once an entire function g(W) is chosen, then an upper or lower single-sideband signal can be obtained from the signal m(t) The signal m(t) is either the modulating signal or a real function of the modulating signal e(t). The generalized ex pressions, which represent SSB signals, are given by Eq (35) for the USSB signal and by Eq, (3,8) for the LSSB signal. These expressions are obviously the transfer functions that are implemented by the upper and lower single-sideband transmitters respectively. Since there are an in finitely denumerable number of entire functions, there are an infinitely denumerable number of upper and lower single-sideband signals that can be generated from any one modulation process, In Chapter IV some specific entire functions will be chosen to illustrate some well-known single sideband signals. LIST OF FIGURES Figure Page 1. Voltage Spectrum of a Typical m(t) Waveform 9 2. Voltage Spectrum of the Analytic Signal Z(t) 11 3. Voltage Spectrum of an Entire Function of an Analytic Signal ]2 4. Voltage Spectrum of the Positive Frequency-Shifted Entire Function of the Analytic Signal ..... 13 5. Voltage Spectrum of the Synthesized Upper Single- Sideband Signal 14 6. Voltage Spectrum of the Negative Frequency-Shifted Entire Function of an Analytic Signal 15 7. Voltage Spectrum of the Synthesized Lower Single- Sideband Signal 16 8. Phasing Method for Generating USSB-AM-SC Signals 20 9. USSB-PM Signal Exciter--Method I .... 22 10. USSB-FM Signal Exciter 24 11. Envelope-Detectable USSB Signal Exciter 26 12. Square-Law Detectable USSB Signal Exciter 27 13. USSB-PM Signal Exciter--Method 11 ...... 53 14. USSB-PM Signal Exciter--Method III 54 15. Power Spectrum of a(t) 67 16. AM Coherent Receiver ....... 76 17. SSB-AM-SC Receiver 78 18. SSB-FM Receiver 78 19. Output to input Signal-to-Noise Power Ratios for Several Systems ....... 86 vi i i It follows that 35 CO T-i (z) = '2~ ^ F())e^Zwd). '00 lim |Z1(Rej0)|2 R-H oo = lim (-)2| ^ [FU)][e~(R sln 0)uej(R cos e)]du, R -Xx> A By use of Schwarz's inequality this becomes lim |Z (ReJ0) |2 R-* (f)2c 'll 00 CO r |F(a)) |2doj} {lim f e2(R sin e)Wh 0 R- 0 But F(to) e L2 (-5 ) so that / | F(to) 12dto < K. Also lim R 7-00 e-(2R sin e)du) 0 < 0 < TT . Therefore we have lim 1Z, (ReJ6) | < {-f K 0 = 0 , R 0 < 0 < 1T . 23 The SSB-FM exciter as described by Eqs. (4.6) and (4.7) is given in Figure 10. 4.4. Example 4; Singlet-Sideband a The term single-sideband a (SSB-a) will be used to denote a sub class of single-sideband signals which may be generated from a particular entire function with a real parameter a. This notation was first used by Bedrosian [3], Let the entire function be g3W eaW (4.8) where a is a real parameter, and let m(t) = 1n[1 + e(t)] (4.9) where e(t) is the video or audio rnodulation signal which is amplitude limited such that |e(t)| <1. It is assumed that m(t) is AC coupled (that is, it has a zero mean). Note that these assumptions are usually met by communications systems since they are identical to the restrictions in conventional AM modulations systems. Then g3(Z(t)) e' 1a[m(t)+jm(t)]= eam(t) ejam(t) or am(t) cos (am(t)) U3(m(t),m(t) = e (4.10a) SYNTHESIS AND ANALYSIS OF REAL SINGLE-SIDEBAND SIGNALS FOR COMMUNICATION SYSTEMS By Leon Worthington Couch, II A Dissertation Presented to the Graduate Council of The University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy UNIVERSITY OF FLORIDA 1968 83 the mean of the one-sided spectrum) for a SSB-FM signal to a conventional FM signal [20], and it is BSSB-FM _____ a fz bfm I1z(26) UHzT) (7.18) It is known that the bandwidth (in rad/s) of a FM signal is approxi mately Bfm = 2(+l (7.19) Thus, to the first approximation, the SSB-FM bandwidth is _ / 112(26) ' bssb-fm2 /2 y1 'S+1>V <7-20> Then, taking the IF bandwidth to be that of the SSB-FM signal, the input noise power is i SSB-FM n (7.21) Consequently, the input signal-to-noise ratio is Ag2 Iq(26) (S/N), - Fo 4 a)m (6+1) /2 Ii2(26) I02(26) (7.22) ] 1 Eq (2.7), which is Fz(uj) 2FmU) 03 > F rn (0) 9 U3 = 0 5 0) < (3 2) where Fm(a)) is the voltage spectrum of the signal m(t), This is shown in Figure 2 for our example used in Figure 1 Now let a function g(W) be given such that g(W) i U(ReW,ImW) + jV(ReW,ImW) (3.3) where g(W) is an entire function of the complex variable W. Theorem II: If Z(z) is an analytic function of z in the UHP and if g(W) is an entire function of W, then g[Z(z)] is an analytic function of z in the UH z-plane, A proof of this theorem may be found in Appendix I. Thus g[Z.(z)] is an analytic function of z in the UH z-plane, and by Theorem I g[Z(t)] has a single-sided spectrum: the spectrum being zero for w < 0. This is illustrated by Figure 3, where Fg(w) = F[g(Z(t))]. 21 and V2(m(t) ,m(t)) = e"rn(t) sin m(t). (4.5b) Substituting Eqs. (4.5a) and (4.5b) into Eq. (3.5) we obtain the upper single-sideband signal: or XyssB-PM^) = e~m^ cos K1 + (4.6) It is again assumed that the modulation m(t) is AC coupled so that its mean value is zero. The single-sideband exciter described by Eq. (4.6) is shown in Figure 9. 4.3. Example 3: Single-Sideband FM Single-sideband frequency-modulation is very similar to SSB-PM in that they are both angle modulated single-sideband signals. In fact the equations for SSB-FM are identical to those given in Section 4.2 ex cept that t (4.7) where e(t) is now the modulating signal (instead of m(t)) and D is the transducer constant. Experiments with SSB-FM signals have been conducted by a number of persons and are reported in the literature [9, 10]. 2 where A0 is the amplitude constant of the transmitter m(t) is the modulating (real) waveform u)0 is the frequency of the transmitter D is the transducer constant of the modulator. Now, what is the corresponding relationship for a single-sideband system? Oswald, and Kuo and Freeny have given the relationship: XsSB-AM^) = Ao Cm(t) cos <*>0t m(t) sin co01] (1.3) where A0 is the amplitude constant of the transmitter m(t) is the modulating signal m(t) is the Hilbert transform of the modulating signal co0 is the frequency of the transmitter [1, 2]. This equation represents the conventional upper single-sideband suppressed- carrier signal, which is now known as a single-sideband amplitude-modulated suppressed-carrier signal (SSB-AM-SC). It will be shown here that this is only one of an infinitely denumerable set of single-sideband signals. In deed, it will be shown that any member of the set can be represented by xSSB(t) = Ao [u(mU)> m(t)) cos a)0t + V(m(t), m(t)) sin co0t] (1.4) where A0 is the amplitude constant of the transmitter U(x,y) and V(x,y) are the conjugate functions of any entire function m(t) is the modulating (real) waveform m(t) is the Hilbert transform of m(t) to0 is the transmitter frequency. Various properties of these single-sideband signals will be analyzed in 44 It is noted that this formula is applicable whenever Ryy(O) and Ryy(O) or Rw(0) and R^(0) exist. That is, RyyJO), RW(Q), Ry.(0), and R^_(0) may or may not exist since ^g_sc^T) 1S ana^ytic almost everywhere (Theorem 103 of Titchmarsh [6]). 5.4-2. RMS-type bandwidth The rms bandwidth, wrms, may also be obtained. oo f pg-SC()d Rg-SC^0^ oo (5.47) Substituting Eq. (5.41) once again, we have (rms) -2[Ryy_(0) + jRyy.(0) ] 2[RW(0) + j%(0)] Since Ryy(x) is an odd function of x, Ryy(O) = Ryy(O) =0, and we have 2 (rms) -R^(0) o J 1 Rw(0) c o w) j -R^.(0) rms J c o i c o (5.48) It is noted that this formula is applicable whenever R^y.(0) and Rw(0) or R^O) and R^(0) exist. PM RO Re O RF Si S0 (S/N)i (S/N)j (S/N)0 SC USSB (w) -v-(w) X(t) XL XU Z(t) a 5 6 n e(t) o2 0) wrms - Phase-Modulation = Autocorrelation Function = Real Part of (0 = Radio Freauency -- Input Signal Power * Output Signal Power = Input Signal-to-Noise Ratio = Input Signal-to-Normalized-Noise Ratio = Output Signal-to-Noise Ratio = Suppressed-Carrier = Upper Single-Sideband = The "Suppressed-Carrier" Function of U(W) = The "Suppressed-Carrier" Function of V(W) = A Real Modulated Signal = Lower Single-Sideband Modulated Signal = Upper Single-Sideband Modulated Signal = m(t) + jm(t) = The Analytic Signal of m(t) = a Modulation (as defined in the text) = System Efficiency = Modulation Index = Efficiency = Modulating Signal (when m(t) is not the Modulating Signal) - Variance - Average Power of m(t) = Angular Frequency = RMS-Type Bandwidth xi CHAPTER VII COMPARISON OF SOME SYSTEMS In the two preceding chapters properties of single-sideband sig nals have been studied. However, the choice of a particular modulation scheme also depends on the properties of the receiver. For example, the entire function g(W) W can be used to generate a SSB signal, but there is no easy way to detect this type of signal. In this chapter a comparison of various types of modulated sig nals will be undertaken from the overall system viewpoint {i.e* generation, transmission and detection). Systems will be compared in terms of the degradation of the modulating signal which appears at the receiver out put when the modulated RF signal plus Gaussian noise is present at the input. This degradation will be measured in terms of three figures of merit: 1. The signal-to-noise ratio at the receiver output 2. The signal energy required at the receiver input for a bit of information at the receiver output when com an son is made with the ideal system (Here the ideal system is defined as a system which requires a minimum amount of energy to transmit a bit of information as predicted by Shannon's formula.) 3. The efficiency of the system as defined by the ratio of the RF power required by an ideal system to the RF power required by an actual system,(Here the ideal sys- 75 29 or h(t,0) = [-V(t,0)+k2] + J[U(t#0)+k1] (5.3) where . TV k. = lim i V(R cos e,R sin e)de a real constant (5.4) v R-* 0 TV k2 = lim | U(R cos e,R sin e)de a real constant (5.5) " R-X JQ A proof of this theorem is given in Appendix I. Theorem V may be applied to the generalized SSB signal by letting h(z) = g(Zx(z)) where g(*) is an entire function of () ^(z) is analytic in the UHP, and lim Zx(z) = lim Z (t + jy) = m(t) + jm(t). Thus Theorem V y+0 y ">0 gives three additional equivalent expressions for g(Z(t)) in addition to g(z(t)) = U(m(t),m(t)) + jV(m(t),m(t)) (5.6) which was used in the derivation in Chapter III. Therefore, following the same procedure as in Chapter III, equivalent upper SSB signals may be found. Using Eq. (5.1) we have for the first equivalent representation of Eq. (3.5): xUS$B(t) = Re{g(Z(t))ej,Ot} = Re{g(m(t),m(t))e^o^} = Re [U(m(t) ,m(t) + jU(m(t) ,m(t)) + jk1]eJ'ot} XyssB^) U(m(t),m(t)) cos w0t [U(m(t) ,m(t)) + kj sin taQt. (5.7) or 46 This definition will be used to obtain a formula expressing the efficiency for the generalized SSB signal. Using Eq. (5.43) and Eq. (5.44) the side band power in either the USSB or LSSB signal is Rxu_sc(o) rxl-sc(0) = W) = y) (5.52) It is also noted that Rg-SC(O) is not equal to the total power in the real-signal sidebands since ggQ is a complex (analytic) baseband signal; instead, (1 /2)Re[Rg_^(-.(0)3 Ryy(O) = Ryy(O) is the total real-signal power. This is readily seen from Eq. (5.43a) and Eq. (5.44a). Similarily the total power in either the USSB or LSSB signal is obtained from Eq. (5.37) or Eq. (5.38): Rxu(0) = rxl(0) = %[ki2 + k22 + 2W0)] = Hlk* + k22 + 2RW(0)] (5.53) Thus the efficiency of a SSB signal is 2Rnn(0) 2RW(0) TTCT VTv" n = ----- = 0 (5.54) kx2 + k2 + 2Rm(0) k;|2 + k22 + 2RW(0) 5.6. Peak-to-Average Power Ratio The ratio of the peak-average (over one cycle of the carrier- frequency) to the average power for the SSB signal may also be obtained. The expression for the peak-average power over one carrier- frequency cycle of a SSB signal is easily obtained with the aid of Eq. (3.5) and Eq. (3.8). Recalling that U and V are relatively slow CHAPTER VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS The examples of SSB signals that were presented in Chapter IV will now be analyzed using the techniques which were developed in Chapter V. 6.1. Example 1: Single-Sideband AM With Suppressed Carrier The constants k1 and k2 will first be determined to show that indeed we have a suppressed carrier SSB signal. By substituting Eq. (4.2b) into Eq. (5.4) we have 7T 0 But from Eq. (5.18b) it follows that lim m^R cos e,R sin e) = 0 0 < e < tt .. R-x Thus k = 0 . (6.1) Similarily substituting Eq. (4.2a) into Eq. (5.5) we have IT (6.2) o 48 16 Thus the real lower single-sideband signal for a given entire function is XLSSB(t) = Re{g(Z(t))eJVc} = Re [U(m(t) ,m(t)) + jV(m(t) ,rii(tj )]e"Ju3ot) or XLSSB^^ = U(m(t) ,m(t)) cos co0t + V(m(t) ,m(t)) sin u>0t. (3.8) Using Theorem IV the voltage spectrum of XL<^g(t) is Flx(o>) i[XLSSB(t)] JgFg (-(J+-a)0 ) 5 0 < 0) < COq 0 | o Â¡ > Wq hFq ()+)q ) 0 > a) > to 0 (3.9) It is noted that the requirements that Fg(oj) be zero for m > w0 is to prevent spectral overlap at the origin. This requirement is satisfied (for all practical purposes) for ojq at radio frequencies. The spectrum of FBx(^) is illustrated by Figure 7. Figure 7. Voltage Spectrum of the Synthesized Lower Single-Sideband Signal 8 Using Eqs. (2.9), (2.10) and (2.14) we obtain Rzz(t) = 2[Rmm(t) + J Rmm(T)J = 2[Rmm(T) + jRmmU)]. (2.15) Thus (1/2)R2(t) is an analytic signal associated with Rmm(-t). By use of Eq. (2.7) it follows that 4Pmm(a>) w > 0 Pzz^) 2Pmm(a)) > w = 0 V 3 1 (2 16) 87 Figure 20. Output Signal-to-No1se to Input Carrler- to-No1se Ratio for Several Systems 117 in the open UHP, U and V are finite as R for 0 < 0 < rr. Thus 1 im R-x Ut sin 6 R 0 and 1 im R-* Vt sin e R = 0 for 0 < e < tt, and Eq. (1-31) becomes 11m {-V + jU} de R-" 0 (1-32) Substituting the right side of Eq. (1-32) for the right-hand term on the right side of Eq. (1-29), Eq. (1-29) becomes O P / dx + j P V(x,0) d.. x-t ax - jirU(t.O) + irV(t,0) - 1im / Vde + j Tim I Ude R-x J R-^> J (1-33) Setting the real and imaginary parts of Eq. (1-33) equal to zero we get 0 = P U(x,0) x-t dx + ttV (t ,0) 1 im R-* Vde (I-34a) and 0 = P V(x,0) d.. x-t ax TT TrU(t,0) + lim f Ude R-* vi (I-34b) Thus and V(t,0) = U(t,0) + lim \ Vde 11 R-*= x U(t,0) = -V(t,0) + lim f Ude 17 R-x J (I-35a) (I-35b) 0 115 and k2 = 11m [ U(R cos 0,R sin e)de a real constant. (1-27) ^ R-x > 0 Proof of Theorem V; By Cauchy's Theorem h(x) dz = 0 (1-28) Jc z-t for c as shown in Figure 26 since h(z) is analytic in the UHP, where t is real and finite. Figure 26. Contour of Integration Thus for e > 0 0 = lim Â£-0 t-e h(x,0) x-t dx + h(t+ee^e)ee'-l'ej Je eec d9 + I dx t+e 11. I h(Re^)RjeJe de R-x ReJ0-t or 0 = P J rh(x-P>. dx jirh(t.O) + lim f MReJ' ^R-J-e~- de X-t R^oo J DoJ0. ReJ -t 89 For the FM system, Eo. (7.26) becomes (S/N)0 = 3 62(S/N)j and (S/N)Q = 3 62(C/N)j. (7.33) (7.34) A comparison of the output signal-to-noise ratios for the vari ous modulation systems can be made now for a given input signal or car rier power by using these equations. (S/N)0/(S/N)j as a function of modulation index is plotted for various systems in Figure 21. Likewise (S/N)0/(C/N)j is shown in Figure 22. From these figures, it is concluded that FM gives the greatest signal-to-noise ratio at the detector output for high index, followed by SSB-AM. For low index (6 < 1), SSB-AM is best, followed by SSB-FM and FM which have about the same (S/N)0, and AM gives the lowest (S/N)0. 7.2. Energy-Per-Bit of Information The concept of RF energy required per bit of received information is used by Raisbeck for comparing SSB-AM and FM systems [27]. This will be extended to AM and SSB-FM systems in this section. The (received) capacity of the system is given by [28] Cb > (b/2ir) log2 [1 + (S/N)Q] (7.35) where b is the baseband bandwidth (rad/s) (S/N)0 is the output signal-to-noise power ratio. Eq. (7.35) becomes an equality when the output noise is Gaussian. 123 30. W.G. Tuller, "Theoretical Limits on the Rate of Information," Proa, IREj vol. 37, 1949. 31. R.E.A.C. Paley and N. Wiener, "Fourier Transforms in the Complex Domain," Am. Math, Soa. Colloq. Publ. vol. 10, 1934. SYNTHESIS AND ANALYSIS OF REAL SINGLE-SIDEBAND SIGNALS FOR COMMUNICATION SYSTEMS By Leon Worthington Couch, II A Dissertation Presented to the Graduate Council of The University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy UNIVERSITY OF FLORIDA 1968 Copyright by Leon Worthington Couch, II 1968 DEDICATION The author proudly dedicates this dissertation to his parents, Mrs,. Leon Couch and the late Rev, Leon Couch, and to his wife, Margaret Wheland Couch, ACKNOWLEDGMENTS The author wishes to express sincere thanks to some of the many people who have contributed to his Ph.D program. In particular, acknowledgment is made to his chairman, Professor T. S. George, for his stimulating courses, sincere discussions, and his professional example. The author also appreciates the help of the other members of his super visory committee. Thanks are expressed to Professor R. C. Johnson and the other members of the staff of the Electronics Research Section, Department of Electrical Engineering for their comments and suggestions. The author is also grateful for the help of Miss Betty Jane Morgan who typed the final draft and the final manuscript. Special thanks are given to his wife, Margaret, for her inspi ration and encouragement. The author is indebted to the Department of Electrical Engi neering for the teaching assistantship which enabled him to carry out this study and also to Harry Diamond Laboratories which supported this work in part under Contract DAAG39-67-C-0077, U. S. Army Materiel Com mand. table of contents Page ACKNOWLEDGMENTS iv LIST OF FIGURES viii KEY TO SYMBOLS x ABSTRACT xiii CHAPTER I. INTRODUCTION 1 II. MATHEMATICAL PRELIMINARIES 4 III. SYNTHESIS OF SINGLE-SIDEBAND SIGNALS 9 IV, EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN 18 4.1. Example 1: Single-Sideband AM with Suppressed-Carrier 18 4,2 Example 2: Single-Sideband PM . 19 4.3. Example 3: Single-Sideband FM 21 4.4. Example 4: Single-Sideband a 23 V.ANALYSIS OF SINGLE-SIDEBAND SIGNALS .... 28 51. Three Additional Equivalent Realizations 28 5.2. Suppressed-Carrier Signals 30 5.3. Autocorrelation Functions 38 5.4. Bandwidth Considerations 42 5.4-1. Mean-type bandwidth 43 5.4-2, RMS-type bandwidth 44 5.,4-3. Equivalent-noise bandwidth 45 5 5 Efficiency 45 5,6 Peak-to-Average Power Ratio 46 v Page VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS 48 6.1c Example 1: Single-Sideband AM With Suppressed Carrier* ...... 48 6.2. Example 2: Single-Sideband PM 51 6.3. Example 3: Single-Sideband FM 68 6.4. Example 4: Single-Sideband a 71 VII. COMPARISON OF SOME SYSTEMS 75 7.1. Output Signal-to-Noise Ratios .... 76 7.1-1 AM system ....... ...... 76 7.1-2, SSB-AM-SC system ..... 77 7.1-3. SSB-FM system 78 7.1-4. FM system 84 7.1-5. Comparison of signal-to-noise ratios* 85 7.2. Energy-Per-Bit of Information 89 7.2-1. AM system 93 7.2-2. SSB-AM-SC system 93 7.2-3. SSB-FM system 94 7.2-4, FM system 94 7.2-5. Comparison of energy-per-bit for various systems 95 7.3. System Efficiencies 97 7.3-1. AM system ..... ... 98 7.3-2. SSB-AM-SC system ...... 93 7.3-3. SSB-FM system ....... 93 7.3-4. FM system 99 7.3-5. Comparison of system efficiencies 100 vi Page VIII. SUMMARY 102 APPENDIX L PROOFS OF SEVERAL THEOREMS 105 II. EVALUATION OF e j(x + Jy^ 119 REFERENCES 121 BIOGRAPHICAL SKETCH 124 vi i LIST OF FIGURES Figure Page 1. Voltage Spectrum of a Typical m(t) Waveform 9 2. Voltage Spectrum of the Analytic Signal Z(t) 11 3. Voltage Spectrum of an Entire Function of an Analytic Signal ]2 4. Voltage Spectrum of the Positive Frequency-Shifted Entire Function of the Analytic Signal ..... 13 5. Voltage Spectrum of the Synthesized Upper Single- Sideband Signal 14 6. Voltage Spectrum of the Negative Frequency-Shifted Entire Function of an Analytic Signal 15 7. Voltage Spectrum of the Synthesized Lower Single- Sideband Signal 16 8. Phasing Method for Generating USSB-AM-SC Signals 20 9. USSB-PM Signal Exciter--Method I .... 22 10. USSB-FM Signal Exciter 24 11. Envelope-Detectable USSB Signal Exciter 26 12. Square-Law Detectable USSB Signal Exciter 27 13. USSB-PM Signal Exciter--Method 11 ...... 53 14. USSB-PM Signal Exciter--Method III 54 15. Power Spectrum of a(t) 67 16. AM Coherent Receiver ....... 76 17. SSB-AM-SC Receiver 78 18. SSB-FM Receiver 78 19. Output to input Signal-to-Noise Power Ratios for Several Systems ....... 86 vi i i Figure Page 20 Output Signal-to-Noise to Input Carrier-to-Noise Ratio for Several Systems 87 21. Output Signal-to-Noise to Input Signal-to-Normalized- Noise Power Ratio for Various Systems 90 22. Output Signal-to-Noise to Input Carrier-to-Normalized- Noise Power Ratio for Various Systems 91 23 Comparison of Energy-per-Bit for Various Systems ....... 96 24,. Efficiencies of Various Systems 101 25. Contour of Integration 107 26o Contour of Integration 115 KEY TO SYMBOLS AM b B Cb Cb Ci (C/N)i (C/N )i D FM F(a>) F(-) g(w) GN LSSB m(t) M Ni P(u) = Amplitude Constant = Amplitude-Modulation = Baseband Bandwidth (rad/s) = RF Signal Bandwidth (rad/s) = Baseband Channel Capacity = RF Channel Capacity = Input Carrier Power = Input Carrier-to-Noise Ratio = Input Carrier-to-Normalized-Noise Ratio = Modulator Transducer Constant = Frequency-Modulation = Voltage Spectrum = The Fourier Transform of (*) = U(W) + jV(W) = An Entire Function = Gaussian Noise = Lower Single-Sideband = Modulating Signal or a Real Function of the Modulating Signal (see e(t) below) = Either Multiplex or Figure of Merit - Input Noise Power = Normalized Input Noise Power = Power Spectral Density x PM RO Re O RF Si S0 (S/N)i (S/N)j (S/N)0 SC USSB (w) -v-(w) X(t) XL XU Z(t) a 5 6 n e(t) o2 0) wrms - Phase-Modulation = Autocorrelation Function = Real Part of (0 = Radio Freauency -- Input Signal Power * Output Signal Power = Input Signal-to-Noise Ratio = Input Signal-to-Normalized-Noise Ratio = Output Signal-to-Noise Ratio = Suppressed-Carrier = Upper Single-Sideband = The "Suppressed-Carrier" Function of U(W) = The "Suppressed-Carrier" Function of V(W) = A Real Modulated Signal = Lower Single-Sideband Modulated Signal = Upper Single-Sideband Modulated Signal = m(t) + jm(t) = The Analytic Signal of m(t) = a Modulation (as defined in the text) = System Efficiency = Modulation Index = Efficiency = Modulating Signal (when m(t) is not the Modulating Signal) - Variance - Average Power of m(t) = Angular Frequency = RMS-Type Bandwidth xi Alo = Equivalent-Noise Bandwidth nr = Mean-Type Bandwidth * = The Convolution Operator ()* = The Conjugate of () () = The Hilbert Transform of () () = The Averaging Operator XI 1 Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SYNTHESIS AND ANALYSIS OF REAL SINGLE-SIDEBAND SIGNALS FOR COMMUNICATION SYSTEMS By Leon Worthington Couch, II June, 1968 Chairman: Professor T. S. George Major Department: Electrical Engineering A new approach to single-sideband (SSB) signal design and ana lysis for communications systems is developed. It is shown that SSB signals may be synthesized by use of the conjugate functions of any entire function where the arguments are the real modulating signal and its Hilbert transform. Entire functions are displayed which give the SSB amplitude-modulated (SSB-AM), SSB frequency-modulated (SSB-FM), SSB envelope-detectable, and SSB square-law detectable signals. Both upper and lower SSB signals are obtained by a simple sign change. This entire generating function concept, along with analytic signal theory, is used to obtain generalized formulae for the properties of SSB signals Formulae are obtained for (1) equivalent realizations for a given SSB signal, (2) the condition for a suppressed-carrier SSB signal, (3) autocorrelation function, (4) bandwidth (using various-de finitions), (5) efficiency of the SSB signal, and (6) peak-to-average power ratio. The amplitude of the discrete carrier term is found to be xi i i equal to the absolute value of the entire generating function evaluated at the origin provided the modulating signal is AC coupled. Examples of the use of these formulae are displayed where these properties are evaluated for stochastic modulation. The usefulness of a SSB signal depends not only on the pro perties of the signal but on the properties of the overall system as well. Consequently, a comparison of AM, SSB-AM, SSB-FM, and FM systems is made from the overall viewpoint of generation, transmission with additive Gaussian noise, and detection. Three figures of merit are used in these comparisons: (1) Output signal-to-noise ratios, (2) Energy-per-bit of information, and (3) System efficiency. In summary, the entire generating function concept is a new tool for synthesis and analysis of single-sideband signals. xiv CHAPTER I INTRODUCTION In recent years the use of single-sideband modulation has become more and more popular in communication systems. This is due to certain advantages such as conservation of the frequency spectrum and larger post detection signal-to-noise ratios in suppressed carrier single-sideband systems when comparison is made in terms of total transmitted power, A single-sideband communication system is a system which generates a real signal waveform from a real modulating signal such that the Fourier transform, or voltage spectrum, of the generated signal is one-sided about the carrier frequency of the transmitter. In conventional amplitude-modu lated systems the relationship between the real modulating waveform and the real transmitted signal is given by the well-known formula: XAM(t) = A0 H + m(t)] cos ojQt |m(t)| <1 (11) where A() is the amplitude constant of the transmitter m(t) is the modulating (real) waveform oj0 is the carrier frequency of the transmitter. Likewise, frequency-modulated systems generate the transmitted waveform: Xp^(t) = A0 cos [w0t + D / tm(t")dt'] (1.2) 1 2 where A0 is the amplitude constant of the transmitter m(t) is the modulating (real) waveform u)0 is the frequency of the transmitter D is the transducer constant of the modulator. Now, what is the corresponding relationship for a single-sideband system? Oswald, and Kuo and Freeny have given the relationship: XsSB-AM^) = Ao Cm(t) cos <*>0t m(t) sin co01] (1.3) where A0 is the amplitude constant of the transmitter m(t) is the modulating signal m(t) is the Hilbert transform of the modulating signal co0 is the frequency of the transmitter [1, 2]. This equation represents the conventional upper single-sideband suppressed- carrier signal, which is now known as a single-sideband amplitude-modulated suppressed-carrier signal (SSB-AM-SC). It will be shown here that this is only one of an infinitely denumerable set of single-sideband signals. In deed, it will be shown that any member of the set can be represented by xSSB(t) = Ao [u(mU)> m(t)) cos a)0t + V(m(t), m(t)) sin co0t] (1.4) where A0 is the amplitude constant of the transmitter U(x,y) and V(x,y) are the conjugate functions of any entire function m(t) is the modulating (real) waveform m(t) is the Hilbert transform of m(t) to0 is the transmitter frequency. Various properties of these single-sideband signals will be analyzed in 3 general for the whole set, and some outstanding members of the set will be chosen for examples to be examined in detail. It should be noted that Bedrosian has classified various types of modulation in a similar manner; however, he does not give a general repre sentation for single-sideband signals [3], CHAPTER II MATHEMATICAL PRELIMINARIES Some properties of the Hilbert transform and the corresponding analytic signal will be examined in this chapter. None of the material presented in this chapter is new; in fact, it is essentially the same as that given by Papon1 is except for some changes in notation [4]. How ever, this background material will be very helpful in derivations pre sented in Chapter III and Chapter' V The Hilbert transform of m(t) is given by co (2.1) where () is read !lthe Hilbert transforms of ()" P denotes the Cauchy principal value * indicates the convolution operation. The inverse Hilbert transform is also defined by Eo (2.1) except that a minus sign is placed in front of the right-hand side of the equation. It is noted that these definitions differ from those used by the mathema ticians by a trivial minus sign. It can be shown, for example, that the Hilbert transform of cos oo0t is sin to0t when w0 > 0 and that the Hilbert transform of a constant is zero, A list of Hilbert transforms has been compiled and published under work done at the California Institute of Tech nology on the Bateman Manuscript Project [5], 4 5 The Fourier transform of m(t) is given by FfiU) = [-j sgn (o)] Fm(to) (2.2) where sgn (u) + 1 u) > 0 0 a) = 0 - 1 to < 0_ (2.3) and Fm(j) is the Fourier transform of m(t). In other words, the Hilbert transform operation is identical to that performed by a -90 all-pass linear (ideally non-realizable) network. From Eq. (2.2), it follows that F*(u>) = [-j sgn (w)]2 Fm(u>) -Fm(co) (2.4) or m(t) = -m(t). (2.5) The (complex) analytic signal associated with the real signal m(t) is defined by Z(t) = m(t) + jm(t). (2.6) The Fourier transform of Z(t) follows by the use of Eq. (2.2), and it is Fz(<>) = Fm(o>) + j[-j sgn (w)] Fm(w) or 6 Fz(ui)' 2Fm(w) oj > 0 Fm(w) ai = 0 0 oj < Q (2.7) Now suppose that m(t) is a stationary random process with auto correlation Rmm([) anc* power spectrum Pmm(u))o Then the power spectrum of m(t) is Fmm(w) = pmm(w) l~J s9n (w) I Pmmi^)- (2.8) This is readily seen by use of the transfer function of the Hilbert trans form operator given by Eq. (2.2). Then, by taking the inverse Fourier transform of Eq. (2.8), it follows that Rmm(T) ~ Rmm(1) (2.9) The cross-correlation function is obtained as follows: Rmm(T) = (t + T)m(t) m(t + i A)m(t)dA TT A Rmm(t-A)dA where () is the averaging operator. Thus Rmm(i~ Rmm(T) (2.10) 7 It follows that the spectrum of the cross-correlation function is given by Pmm() ~ l-"j sgn (w)3 PfTirii(tlJ) (2.11) it is noted that Pmm(w) is a purely imaginary function since Pmm(w) is a real function. Then 00 Rmm(c) ~ T7 f (w)] Pmm(uduJ l~ '* *00 which5 for Prnm(u)) a real even function, reduces to Rmm( i) = J Pmm(^) s,n wt du) (2/12) 11 0 Thus the cross-correlation function is an odd function of r: fynm(T) ~ Rmm(_T) = -rh(t T)m(t) -m(t + i)m(t) (2/13) or wo = -W-O = -WO. (2.14) The autocorrelation for the analytic signal is found as follows: RZZ( ) = Z(tH)Z*(iy = [m(t+-r) + jm(t+x)] [m(t) jm(tj] = m(t+i)rn(t) + m(t+r)m(t) + jm(t+: )m(t) jm(t+r)m(t) - Rmm() f Rmm(T) + JRmm(1) jRmm(TK 8 Using Eqs. (2.9), (2.10) and (2.14) we obtain Rzz(t) = 2[Rmm(t) + J Rmm(T)J = 2[Rmm(T) + jRmmU)]. (2.15) Thus (1/2)R2(t) is an analytic signal associated with Rmm(-t). By use of Eq. (2.7) it follows that 4Pmm(a>) w > 0 Pzz^) 2Pmm(a)) > w = 0 V 3 1 (2 16) CHAPTER III SYNTHESIS OF SINGLE-SIDEBAND SIGNALS Eq. (1.4), which specifies the set of single-sideband signals that can be generated from a given modulating waveform or process, will be derived in this chapter. The equation must be a real function of a real input waveform, m(t), since it represents the generating function of a physically realizable system--the single-sideband transmitter--and, in general, it is non-linear. Analytic signal techniques will be used in the derivation. It will be shown that if we have a complex function k(z) where k(z) is analytic for z = x + jy in the upper half-plane (UHP), then the voltage spectrum of k(x,0) k(t) is zero for w < 0. In order to synthesize real SSB signals from a real modulating waveform, an UHP analytic generating function of the complex time veal modulating process must be found regardless of the particular (physically realizable) wave form that the process assumes. Let m(t) be either the real baseband modulating signal or a veal function of the baseband modulating signal e(t), Then the amplitude of the voltage spectrum of m(t) is double sided about the origin, for ex ample, as shown by Figure 1. 9 i o Since m(t) is generated by a physically realizable process, it con tains finite power for a finite time interval This, of course, is equiva lent to saying that m(t) is a finite energy signal or, in mathematical terms, it is a L2 function. By Theorem 91 of Titchmarsh, m(t) is also a member of the L2 class of functions almost everywhere [6]. Now the complex signal Z(t) is formed by Z(t) = m(t) + jm(t). (3,1) It is recalled that Z(t) is commonly called an analytic signal in the literature. By Theorem 95 of Titchmarsh there exists an analytic func tion (regular for y > 0), Z^z), such that as y -* 0 Zj(x + jy) Z(t) = m(t) + jm(t) x = t for almost all t and, furthermore, Z(t) is a L2 (-*>, function [6] It follows by Theorem 48 of Titchmarsh that the Fourier transform of Z(t) exists [6], Theorem I: If k(z) is analytic in the UHP then the spectrum of k(t,0), denoted by F^oj), is zero for all to < 0, assuming that k(t,0) is Fourier trans formable. For a proof of this theorem the reader is referred to Appendix I. Thus the voltage spectrum of Z(t) is zero for ui < 0 by Theorem I since Z(t) takes on values of the analytic function Z,(z) almost every where along the x axis. Furthermore, since Z(t) is an analytic signal-- that is, it is defined by Eq. (3=1)its voltage spectrum is given by ] 1 Eq (2.7), which is Fz(uj) 2FmU) 03 > F rn (0) 9 U3 = 0 5 0) < (3 2) where Fm(a)) is the voltage spectrum of the signal m(t), This is shown in Figure 2 for our example used in Figure 1 Now let a function g(W) be given such that g(W) i U(ReW,ImW) + jV(ReW,ImW) (3.3) where g(W) is an entire function of the complex variable W. Theorem II: If Z(z) is an analytic function of z in the UHP and if g(W) is an entire function of W, then g[Z(z)] is an analytic function of z in the UH z-plane, A proof of this theorem may be found in Appendix I. Thus g[Z.(z)] is an analytic function of z in the UH z-plane, and by Theorem I g[Z(t)] has a single-sided spectrum: the spectrum being zero for w < 0. This is illustrated by Figure 3, where Fg(w) = F[g(Z(t))]. 12 Figure 3. Voltage Spectrum of an Entire Function of an Analytic Signal Now multiply the complex baseband signal g[Z(t)] by eJuJot to translate the signal up to the transmitting frequency, oi0. It is noted that g[Zj(z)] and eju)z for ojq > 0 are both analytic functions in the UH z-plane By the Lemma to Theorem I in Appendix I, g[Z; (z)]ev,l'Uo2 is ana lytic in the UH z-plane. Therefore, by Theorem I the voltage spectrum of g[Z(t)]eJU)ot is one sided about the origin. Furthermore, FCgUltiJeK4] = -LF[g(z(t))] * (- 7T = Fg (oo) 6(w-o)0) or F[gU(t))e^ot] = Fg(ld-u)q) o>0 > 0 (3.4) This spectrum is illustrated in Figure 4. 13 \F[q(l(t))e^^} i Figure 4 Voltage Spectrum of the Positive Frequency- Shifted Entire Function of the Analytic Signal The real upper single-sideband signal can now be obtained from the complex single-sideband signal, g[Z(t)]e'-*a,ot, by taking the real part. This is seen from Theorem III Theorem III, If h(z) is analytic for all z in the UHP and F[h(x,0)] e Fh(w), then for > 0, F{Re[h(x,0)eJwox]} ^sFp-! (oCQ ) O) > 0)g o 0) < 0),. J O),. This theorem is proved in Appendix I. Thus the upper single-sideband signal for a given entire function is XUSSB(t) = ReigCZUne^} = Re{[U(ReZ(t) ,lmZ(t)) + jV^ReZ(t),lmZ(t))]eJwot} = Re{[U(m(t) ,m(t)) + jV(mU) *m(t J)]eJa)^t} 14 or XuSSB^t) = U(m(t) ,rn(t)) cos w0t V(m(t) ,m(t)) sin oj0t (3.5) where U(ReW,ImW) is the real part of the entire function g(W) V(ReW,ImW) is the imaginary part of g(W) m(t) is either the modulating signal or a real function of the modulating signal e(t) m(t) is the Hilbert transform of m(t). Using Theorem III the voltage spectrum of Xu$$g(t) is FUXU) = ^[XusSBt)] %Fg(w-tQ) to > , j co j < to. lgFg(-w-aJ0) a) < -u0 (3.6) This spectrum is illustrated by Figure 5. The lower single-sideband signal can be synthesized in a similar manner from the complex baseband signal Now we need to translate the complex baseband signal down to the transmitting frequency instead of up, 15 as in the upper single-sideband synthesis. Then the Fourier transform of the down-shifted complex baseband signal is F[g(St))e-ot] F[g(Zt))] * - Fn {W ) (u>+\on) or [g(Z(t))ej)ot] = Fg(u+u0) a), > 0. (3,7) This spectrum is illustrated in Figure 6. Figure 6. Voltage Spectrum of the Negative Frequency- Shifted Entire Function of an Analytic Signal Theorem IV: If h(z) is analytic for all z in UHP and F[h(x,0)] e F^(oj) where Fh(fi) = 0 for all n > go0, then for oj0 > 0 y{Re[h(x,0)e'JUJx] ^(-W+uJg) 0 0 < j < )q M > uo JgFh (tD+u)0} 0 > oo > - 16 Thus the real lower single-sideband signal for a given entire function is XLSSB(t) = Re{g(Z(t))eJVc} = Re [U(m(t) ,m(t)) + jV(m(t) ,rii(tj )]e"Ju3ot) or XLSSB^^ = U(m(t) ,m(t)) cos co0t + V(m(t) ,m(t)) sin u>0t. (3.8) Using Theorem IV the voltage spectrum of XL<^g(t) is Flx(o>) i[XLSSB(t)] JgFg (-(J+-a)0 ) 5 0 < 0) < COq 0 | o Â¡ > Wq hFq ()+)q ) 0 > a) > to 0 (3.9) It is noted that the requirements that Fg(oj) be zero for m > w0 is to prevent spectral overlap at the origin. This requirement is satisfied (for all practical purposes) for ojq at radio frequencies. The spectrum of FBx(^) is illustrated by Figure 7. Figure 7. Voltage Spectrum of the Synthesized Lower Single-Sideband Signal 17 To summarize, it has been shown that once an entire function g(W) is chosen, then an upper or lower single-sideband signal can be obtained from the signal m(t) The signal m(t) is either the modulating signal or a real function of the modulating signal e(t). The generalized ex pressions, which represent SSB signals, are given by Eq (35) for the USSB signal and by Eq, (3,8) for the LSSB signal. These expressions are obviously the transfer functions that are implemented by the upper and lower single-sideband transmitters respectively. Since there are an in finitely denumerable number of entire functions, there are an infinitely denumerable number of upper and lower single-sideband signals that can be generated from any one modulation process, In Chapter IV some specific entire functions will be chosen to illustrate some well-known single sideband signals. CHAPTER IV EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN Specific examples of upper single-sideband signal design will now be presented* Entire functions will be chosen to give signals which have various distinct properties. In Chapter VI these properties will be ex amined in detail. Only upper sideband examples are presented here since the corresponding lower sideband signals are given by the same equation except for a sign change (Eq. (3.5) and Eq. (3.8)). 4.1. Example 1: Single-Sideband AM With Suppressed-Carrier This is the conventional type of single-sideband signal that is now widely used by the military, telephone companies, and amateur radio operators. It will be denoted here by SSB-AM-SC, Let the entire function be 9l(W)=W (4.1) and let m(t) be the modulating signal. Then substituting the corresponding analytic signal for W gx(Z(t)) = m(t) + jm(t) or (m(t),iii(t)) = m(t) and Vx(m(t),m(t)) = m(t). (4.2 a,b) 18 19 Substituting Eqs. (4.2a) and (4.2b) into Eq. (3.5) we obtain the upper single-sideband signal: XUSSB-AM-SC^ cos o1 m(t) sin Jot (4.3) where m(t) is the modulating audio or video signal and m(t) is the Hil bert transform of m(t). It is assumed that m(t) is AC coupled so that it will have a zero mean. The upper single-sideband transmitter corresponding to the gene rating function given by Eq, (4.3) is illustrated by the block diagram in Figure 8. It is recalled that this is the well-known phasing method for generating SSB-AM-SC signals [7, 8], 4.2. Example 2: Single-Sideband PM Single-sideband phase-modulation was described by Bedrosian in 1962 [3]. To synthesize this type of signal, denoted by SSB-PM, use the entire function: (4.4) Let m(t) be the modulating audio or video signal. Then g2(Z(t)) = eJ(m^ + ^(t)) = or U2(m(t),iii(t)) = e-"i(t) cos m(t) (4.5a) Figure 8 Phasing Method for Generating USSB-AM-SC Signals 21 and V2(m(t) ,m(t)) = e"rn(t) sin m(t). (4.5b) Substituting Eqs. (4.5a) and (4.5b) into Eq. (3.5) we obtain the upper single-sideband signal: or XyssB-PM^) = e~m^ cos K1 + (4.6) It is again assumed that the modulation m(t) is AC coupled so that its mean value is zero. The single-sideband exciter described by Eq. (4.6) is shown in Figure 9. 4.3. Example 3: Single-Sideband FM Single-sideband frequency-modulation is very similar to SSB-PM in that they are both angle modulated single-sideband signals. In fact the equations for SSB-FM are identical to those given in Section 4.2 ex cept that t (4.7) where e(t) is now the modulating signal (instead of m(t)) and D is the transducer constant. Experiments with SSB-FM signals have been conducted by a number of persons and are reported in the literature [9, 10]. Figure 9. USSB-PM Signal Exciter--Method I 23 The SSB-FM exciter as described by Eqs. (4.6) and (4.7) is given in Figure 10. 4.4. Example 4; Singlet-Sideband a The term single-sideband a (SSB-a) will be used to denote a sub class of single-sideband signals which may be generated from a particular entire function with a real parameter a. This notation was first used by Bedrosian [3], Let the entire function be g3W eaW (4.8) where a is a real parameter, and let m(t) = 1n[1 + e(t)] (4.9) where e(t) is the video or audio rnodulation signal which is amplitude limited such that |e(t)| <1. It is assumed that m(t) is AC coupled (that is, it has a zero mean). Note that these assumptions are usually met by communications systems since they are identical to the restrictions in conventional AM modulations systems. Then g3(Z(t)) e' 1a[m(t)+jm(t)]= eam(t) ejam(t) or am(t) cos (am(t)) U3(m(t),m(t) = e (4.10a) Figure 10. USSB-FM Signal Exciter 25 and V3(m(t)9m(t)) = em^ sin (am(t)). (4,10b) Substituting Eqs. (4.10a) and (4.10b) into Eq. (3.5), the SSB-a signal is XUSSB-a^ eam^ cos (om(t)) cos w0t - eam^ sin .(am(t)) sin ajQt or XUSSB-a^ ~ cos (0t + (4.11) In terms of the input audio waveform, Eq. (4.11) becomes XUSSB-a(t) = ealn[1+e(t)] cos ()Qt + an[l+e(t)]) or XUSSB-a^ = [l+e(t)]a cos u0t + a1n[1+e(t)]). (4.12) For a = 1 we have an envelope-detectable SSB signal, as is readily seen from Eq. (4.12). Voelcker has recently published a paper demon strating the merits of the envelope-detectable SSB signal [11]. The real ization of Eq. (4.12) is shown in Figure 11. For a = 1/2 we have a square-law detectable SSB signal. This type of signal has been studied in detail by Powers [12]. Figure 12 gives the block-diagram realization for the square-law detectable SSB exciter. Figure 11. Envelope-Detectable USSB Signal Exciter e(t) Figure 12. Square-Law Detectable USSB Signal Exciter CHAPTER V ANALYSIS OF SINGLE-SIDEBAND SIGNALS The generalized SSB signal, that was developed in Chapter III, will now be analyzed to determine such properties as equivalent gener alized SSB signals, presence or absence of a discrete carrier term, autocorrelation functions, bandwidths, efficiency, and peak-to-averajge power ratio. Some of these properties will depend only on the entire function associated with the SSB signal, but most of the properties will be a function of the statistics of the modulating signal as well. 5.1. Three Additional Equivalent Realizations Three equivalent ways (in general) for generating an upper SSB signal will now be found in addition to the realization given by Eq. (3.5). Similar expressions will also be given for lower SSB signals which are equivalent to Eq. (3.8). It is very desirable to know as many equivalent realizations as possible since ally orle of them might be the most econom ical to implement for particular SSB signal. Theorem V: If h(x,y) = U(x,y) + jV(x,y) is analytic in the HP (including UH) then h (t ,0) = U(t,0) + jiOU^+kj] (5.1) or h(t,0) = [4(t,0)+k2] + jV(t,0) (5.2) 28 29 or h(t,0) = [-V(t,0)+k2] + J[U(t#0)+k1] (5.3) where . TV k. = lim i V(R cos e,R sin e)de a real constant (5.4) v R-* 0 TV k2 = lim | U(R cos e,R sin e)de a real constant (5.5) " R-X JQ A proof of this theorem is given in Appendix I. Theorem V may be applied to the generalized SSB signal by letting h(z) = g(Zx(z)) where g(*) is an entire function of () ^(z) is analytic in the UHP, and lim Zx(z) = lim Z (t + jy) = m(t) + jm(t). Thus Theorem V y+0 y ">0 gives three additional equivalent expressions for g(Z(t)) in addition to g(z(t)) = U(m(t),m(t)) + jV(m(t),m(t)) (5.6) which was used in the derivation in Chapter III. Therefore, following the same procedure as in Chapter III, equivalent upper SSB signals may be found. Using Eq. (5.1) we have for the first equivalent representation of Eq. (3.5): xUS$B(t) = Re{g(Z(t))ej,Ot} = Re{g(m(t),m(t))e^o^} = Re [U(m(t) ,m(t) + jU(m(t) ,m(t)) + jk1]eJ'ot} XyssB^) U(m(t),m(t)) cos w0t [U(m(t) ,m(t)) + kj sin taQt. (5.7) or 30 Using Eq. (5.2) the second equivalent representation is XUSSB^ = t-V(m(t),m(t))+k2] cos w0t V(m(t),m(t)) sin w0t. (5.8) Using Eq. (5.3) the third equivalent representation is XUSSB^ = iv(m(t;) iti(t))+k23 cos (o0t [U(m(t) ,m(t))+k1], sin u)Qt (5.9) Likewise the three lower SSB signals, which are equivalent to Eq. (3.8), are It should be noted, however, that if for a given entire function k: and k2 are both zero, then all four representations for the USSB or the LSSB signals are identical since by Theorem V, U = -V and V = 0 under these conditions. 5.2. Suppressed-Carrier Signals The presence of a discrete carrier term appears as impulses in the (two-sided) spectrum of transmitted signal at frequencies w0 and - depending on whether the carrier term is cos a)0t, sin u)0t, or a com bination of the two. Thus the composite voltage spectrum of the modulated 31 signal consists of a continuous part due to the modulation plus impulse functions at u0 and -o>0 if there is a discrete carrier term. As defined here, the "continuous" part may contain impulse functions for some types of modulation, but not at the carrier frequency. Taking the inverse Fourier transform of the composite voltage spectrum it is seen that if there is a discrete carrier term, the time waveform must be expressible in the form: X(t) = [f^tj+cj cos ai0t [f2(t)+c2] sin iOgt (5.13) where c1 and c are due to the discrete carrier f^t) and f2(t) are due to the continuous part of the spectrum and have zero mean values. Thus Eq. (5.13) gives the condition that c2 and Cj are not both zero if there is a discrete carrier term. To determine the condition for a discrete carrier in an upper SSB signal, Eq. (5.13) will be identified with Eq. (5.9), which represents /V A the whole class of upper SSB signals. It is now argued that both U and V have a zero mean value if the modulating process is stationary. This is seen as follows: U(m(t),Â£(t))-lp f df . But U[m(t'),m(t')] = c, a constant, since m(t) is wide-sense stationary. Thus 00 U(m(t) ,m(t)) = ^ P J' 00 dt' = 0. 32 Likewise V has a zero mean value. Then, identifying Eg. (5.13) with Eq. (5.9), it is seen that fx(t) S -V(m(t),m(t)) (5.14a) f2(t) = U(m(t),m(t)) (5.14b) 1 = k2 and c2 = k . (5.14c ,d) Similarily, for lower SSB signals Eq. (5.13) can be identified with Eq. (5.12). Thus the SSB signal has a discrete carrier provided that kx and k2 are not both zero. As an aside, it is noted that the criterion for a discrete car rier, given by Eq. (5.13), is not limited to SSB signals; it holds for all modulated signals. For AM Eq. (5.13) can be identified with Eq. (1.1) Here fj(t) = A0m(t) (5.15a) f2(t) 5 0 (5.15b) 5 Ao and c2 = 0 (5.15c,d) because m(t) has a zero mean due to AC coupling in the modulator of the transmitter. Thus for AM it is seen that there is a discrete carrier term of amplitude c1 = AQ which does not depend on the modulation. For FM Eq. (1.2) can be expanded. Then, assuming sinusoidal modulation of fre quency wa, we obtain 33 XPM(t) = tAn cos (zr cos wat)] COS u)nt '0 'Wa a*'-1 -0' - [An sin (cos to t)] sin wnt. (5.16) To identify Eq. (5.16) with Eq (5.13) we have to find the DC terms of and f. (t) + c. e An cos (cos coat) i i o wa a and f0(t) + c2 = A0 sin (- cos u)at). wa These are c, = An cos ( cos w,t) i u M- a Ao C D / cos ( cos wat)dt v/ 0) a Vo<Â£> (5.17a) c. = A sin ( cos wat) 2 o>a _ A0 T o 0 J sin (^~ cos aiat)dt (5.17b) Then for sinusoidal frequency-modulation it is seen that the discrete carrier term has an amplitude of AQJ0(D/ul^) which may or may not be zero depending on the modulation index D/wa. Consequently, for FM it is seen that the discrete carrier term may or may not exist depending on the modulation. Prof. T. S. George has given the discrete carrier condition 34 for the case of FM Gaussian noise [13]. Continuing with our SSB signals, it will now be shown that kx and k2 depend only on the entire function associated with the SSB signal and not on the n]odulation. From Theorem IV we have IT k = lim V[m,(R cos e,R sin e) m. (R cos e,R sin e)]de 1 IT D J 1 i R-* o and IT k, = lim / U[m.,(R co$ e,R sin e) m(R cos e,R sin )]de 17 R^ o where U and V are the real and imaginary parts of the entire function l1{z) = m^z) + jm^z) is the analytic function associated with the analytic signal Z(t) of m(t). It is seen that if lim m (R cos e,R sin e) = 0 0 < e < v (5.18a) R-x and lim mJR cos e,R sin e) = 0 0 < e < tt (5.18b) R-* then kx and k2 depend only on U and V of the entire function and not on m. Thus we need to show that Eq. (5.18a) and (5.18b) are valid. By the theory of Chapter III there exists a function Z^z) = m^z) + jm^z) which is analytic in the UHP such that (almost everywhere) 11^ Zx(t + jy) y = Z(t) = m(t) + jm(t) where the Fourier transform of Z(t), F(ai), is l2(-, ). Then we have It follows that 35 CO T-i (z) = '2~ ^ F())e^Zwd). '00 lim |Z1(Rej0)|2 R-H oo = lim (-)2| ^ [FU)][e~(R sln 0)uej(R cos e)]du, R -Xx> A By use of Schwarz's inequality this becomes lim |Z (ReJ0) |2 R-* (f)2c 'll 00 CO r |F(a)) |2doj} {lim f e2(R sin e)Wh 0 R- 0 But F(to) e L2 (-5 ) so that / | F(to) 12dto < K. Also lim R 7-00 e-(2R sin e)du) 0 < 0 < TT . Therefore we have lim 1Z, (ReJ6) | < {-f K 0 = 0 , R 0 < 0 < 1T . 36 For e = 0 or 0 = tt Z (+>) ,0 = 0 Tim Â¡Z (Reje) R->~ Z(-oo) = 0 since Z(t) e L (-*>, ). Then lim |Z1(ReJ0)| = 1 im |Z?L(R cos 0, R sin 0) | = 0 0 < 0 < rr R-x Rx which shows that Eqs. (5.18a) and (5.18b) are valid statements. Thus, the presence (kx and k2 not both zero) or the absence (kx = k2 = 0) of a discrete carrier depends only on the entire funtion associated with the SSB signal and not on the modulation. Furthermore, it is seen that the amplitude of the discrete carrier is given by the magnitude of the entire function evaluated at the origin (of the W plane), and the power in the discrete carrier is one-half the square of the magnitude. For every generalized USSB signal represented by Eq. (3.5), there exists a corresponding sppressed-carrier USSB signal: XUSSB-SC^ = ,^Km('t) ,m(t)) cos w0t Â¥-(m(t) ,m(t)) sih wot (5.19) 37 where the notation SC and denote the suppressed-carrier functions. But what are these functions tt and Â¥? The condition for a suppressed carrier is that kx = k2 = 0. By comparing Eq. (5.19) with Eq. (5.9) it follows that it = -V and if- = 0. Furthermore by Theorem V of Section 5.1, U = -V + k2 and V = + kx. Thus it = -V = U k2 (5.20) and Â¥ = U = V kr (5.21) It is also noted that it and Â¥ are a unique Hilbert transform pair. That is, Â¥ is the Hilbert transform of it, and it is the inverse Hilbert trans form of 3t. This is readily shown by taking the Hilbert transform of Eq. (5.20), comparing the result with Eq. (5.21), and by taking the in verse Hilbert transform of Eq. (5.21), comparing the result with Eq. (5.20). Thus Eq. (5.19) may be re-written as XUSSB-SC^ = ,m(t)) cos w0t it(m(t),m(t)) sin wot (5.22) or XUSSB-SC^ = "^(t) ,m(t)) cos w0t V where it and Â¥ are given by Eq. (5.20) and Eq. (5.21). It is interesting to note that the form of the USSB signal given _ * above checks with the expression given by Haber [14]. He indicates that if a process n(t) has spectral components only for |w| > wq then n(t) can be represented by 38 n(t) = s(t) cos wot s(t) sin wot. (5.24) Thus Eq. (5.22) checks with Eq, (5.24) where it = s(t), and Eq. (5.23) checks also where -V e s(t). The corresponding representations for LSSB suppressed-carrier signals are given by XLSSB-SC^ = -y-(m(t),rn(t)) cos w0t + tt(m(t),m{t)) sin w0t (5.25) and XLSSB-SC^ = "'^'(m(t) ,m(t)) cos w0t + Â¥(m(t) ,m(t)) sin w0t (5.26) where it and-V-are given by Eq. (5.20) and Eq. (5.21). This representation also checks with that given by Haber for pro cesses with spectral components only for |w| < wo which is n(t) = s(t) cos wot + Â§(t) sin w0t. (5.27) 5.3. Autocorrelation Functions The autocorrelation function for the generalized SSB signal and the corresponding suppressed-carrier SSB signal will now be derived. Using the result of Chapter III, it is known that the generalized upper SSB signal can be represented by XUSSB j(w0t+ (t) = Re{g(m(t),m(t))e' (5.28) 39 where a uniformly distributed phase angle 4> has been included to account for the random start-up phase of the RF oscillator in the SSB exciter. Then, using Middleton's result [15], the autocorrelation of the USSB sig nal is RXU^ XUSSB^t+T^XUSSB^t^ %Re{eJa)TRg(T)} (5.29) where Rg(t) = g(m(t+x),m(t+T)) g*(m(t),m(t)) (5.30) and g(m(t),m(t)) = U(m(t),m(t)) + jV(m(t) ,m(t)). (5.31) The subscript XU indicates the USSB signal. For the generalized LSSB signal the corresponding formulae are XL$SB(t) = Re{g(m(t),m(t))e"J^t+
(5.32)and Rxl(t) = %Re{e"J"wTRg(T)}. (5.33) These equations can be simplified if we consider the autocorre lation for the continuous part of the spectrum of the SSB signal. The suppressed DC carrier version of g, denoted by g$c> will first be found in terms of g, and then the corresponding autocorrelation function Rg_sc(T) 40 will be determined in terms of Rg(-r). By examining Eq. (5.19) and comparing this equation to Eq. (3.5), with the aid of Eq. (3.3) it is seen that the suppressed DC carrier version of g is given by gsc(m(t),m(t)) = if(m(t),m(t)) + jÂ¥-(m(t) ,m(t)) (5.34) where -0-and M- are the suppressed-carrier functions defined by Eq. (5.20) and Eq. (5.21). Then it follows that g(m(t),m(t)) = gsc(m(t) ,m(t)) + [k2+jk1]0 (5.35) It is noted that the mean value of ggQ is zero. This is readily seen via Eqs. (5.34), (5.20), and (5.21) since it is recalled that the mean value of 0 and V was shown to be zero in Section 5.2. Then, using Eq. (5.35), the autocorrelation of g is obtained in terms of the autocorrelation of 9sc; Rg(0 = Rg-sc(x) + (kiZ+k22) (5.36) Therefore the autocorrelation functions for the USSB signal, Eq. (5.29), and the LSSB signal, Eq. (5.33), become Rxu(T) = ^(^[(k^2) + Rg-SC (T 1 and Rxl(i) = %Re{eJwoT[(kl2+k22) + Rg_sc(x)]}o (5.37) (5.38) i 41 It may be easier to calculate the autocorrelation for the USSB or LSSB signal using this representation rather than that of Eq (5.29) and Eq. (5.33) since Rg_$c(x) may be easier to calculate than Rg(x). This is shown below. A simplified expression for Rg_$c(T) will now be derived. First, it is recalled from Section 5.2 that if and -V- are a unique Hilbert trans form pair. Thus g$c given by Eq. (5.34), can be expressed in terms of two analytic signals: gsc(m(t),m(t)) = tf(m(t) ,m(t)) + jit(m(t) ,m(t)) (5.39) and gsc(m(t),m(t)) = -Â¥-(m(t) ,m(t)) + jÂ¥-(m(t) ,m(t)) (5.40) where Eq. (5.39) is the a^lytic signal associated with -H-and Eq. (5.40) is the analytic signl associated with -Â¥. Using Eq. (5.39) and Eq. (2.15), the autocorrelation of g5g is given by Rg-SC(T) 2[:Rw(t) + jRw(r)] (5.41) or by using Eqs. (5.40), (2.15), and (2.9) Rg-SC(T) = 2[rw(t) + jRw(t)]. (5.42) Thus Rg_$c(x) may be easier to calculate than Rg(x) since only Rw(x) or RyyM is needed. This, of course, is assuming that the Hilbert trans- 42 form is relatively easy to obtain On the other hand Rg(x) may be calcu lated directly from g(m(t),m(t)) or indirectly by use of Ry^t), Rvv(x), Ruv(t), and Rvu(t). The autocorrelation functions for the generalized USSB and LSSB signals having a suppressed-carrier are readily given by Eq. (5.37) and Eg. (5.38) with kj = k, 0: (5.43a) (5.43b) (5.43c) and (5.44a) (5.44b) (5.44c) It follows that the power spectral density of any of these SSB signals may be obtained by taking the Fourier transform of the appro priate autocorrelation function presented above. 5.4. Bandwidth Considerations The suppressed-carrier autocorrelation formulae developed above will now be used to calculate bandwidths of SSB signals. It is noted RXL-SC Rw(t) cos O)0x + R^t) sin 000 W Rw(t) cos UQT + Rw(x) sin CjOq ' RXU-SC^ %Re{eJTRg_sc(T)} - Ryy.(t) cos o>qT Ryy( :) sin coot w t) cos wot Ryy(x) sin u)0t 43 that the suppressed-carrier formulae are needed' instead of the "total sig nal" formulae since, from the engineering point of view, the presence or absence of a discrete carrier should not change the bandwidth of the sig nal Various definitions of bandwidth will be used [16, 17], 5.4-1. Mean-type bandwidth Since the spectrum of a SSB signal is one-sided about the carrier frequency, the average frequency as measured from the carrier frequency is a measure of the bandwidth of the signal: f wPg_ScU)du j Rg_sc() oo - = *' (5 45) CO v ' f Pg-SC^^ Rg-SC(O) 00 where Pg_^c(w) is the power spectral density of g$c(m(t) ,m(t))and the prime indicates the derivative with respect to t. The relationship is valid whenever Rg_sg(0) and Rg_$c^ exist. Substituting Eg. (5.41) into Eq. (5.45) we have f[R' (0) + JR' (0)] j trtr trtr " a'1 o 2[RW(0) + jRyyjO)] But it recalled that Ryy(x) is an even function of t and, from Chapter II, RyU(t) is an odd function of t. Then Ryy(O) = Ryy(O) = 0 and it follows that Ryy.(0) Rw(o) Ryy-(o) Rw(o) (5.46) 44 It is noted that this formula is applicable whenever Ryy(O) and Ryy(O) or Rw(0) and R^(0) exist. That is, RyyJO), RW(Q), Ry.(0), and R^_(0) may or may not exist since ^g_sc^T) 1S ana^ytic almost everywhere (Theorem 103 of Titchmarsh [6]). 5.4-2. RMS-type bandwidth The rms bandwidth, wrms, may also be obtained. oo f pg-SC()d Rg-SC^0^ oo (5.47) Substituting Eq. (5.41) once again, we have (rms) -2[Ryy_(0) + jRyy.(0) ] 2[RW(0) + j%(0)] Since Ryy(x) is an odd function of x, Ryy(O) = Ryy(O) =0, and we have 2 (rms) -R^(0) o J 1 Rw(0) c o w) j -R^.(0) rms J c o i c o (5.48) It is noted that this formula is applicable whenever R^y.(0) and Rw(0) or R^O) and R^(0) exist. 45 5*4-3. Equivalent-noise bandwidth the equivalent-noise bandwidth, Aw* for the continuous part of the power spectrum is defined by (2aw) 27 pg-scC0) 2tt Pg-SC(w)du = Rg_sc() (5.49) But Thus 00 Pg-Sc(O) = f Rg-Sc(T>dT on 00 (Aco) = g-sc (0) Rg-SC(-r)di Substituting for Rg_sc(T) by using Eq. (5.41) or Eq. (5.42) we obtain (noting once again that R^Ct) is even and Ryy(-r) is odd) (5.50) 5.5. Efficiency A commonly.Used definition of efficiency for modulated signals is [18] n = Sideband Power/Total Power. 5.51 46 This definition will be used to obtain a formula expressing the efficiency for the generalized SSB signal. Using Eq. (5.43) and Eq. (5.44) the side band power in either the USSB or LSSB signal is Rxu_sc(o) rxl-sc(0) = W) = y) (5.52) It is also noted that Rg-SC(O) is not equal to the total power in the real-signal sidebands since ggQ is a complex (analytic) baseband signal; instead, (1 /2)Re[Rg_^(-.(0)3 Ryy(O) = Ryy(O) is the total real-signal power. This is readily seen from Eq. (5.43a) and Eq. (5.44a). Similarily the total power in either the USSB or LSSB signal is obtained from Eq. (5.37) or Eq. (5.38): Rxu(0) = rxl(0) = %[ki2 + k22 + 2W0)] = Hlk* + k22 + 2RW(0)] (5.53) Thus the efficiency of a SSB signal is 2Rnn(0) 2RW(0) TTCT VTv" n = ----- = 0 (5.54) kx2 + k2 + 2Rm(0) k;|2 + k22 + 2RW(0) 5.6. Peak-to-Average Power Ratio The ratio of the peak-average (over one cycle of the carrier- frequency) to the average power for the SSB signal may also be obtained. The expression for the peak-average power over one carrier- frequency cycle of a SSB signal is easily obtained with the aid of Eq. (3.5) and Eq. (3.8). Recalling that U and V are relatively slow time-varying functions compared to cos ta0t and sin wot, we have for the peak-average power: 47 Pp_Av = %{[U(m(t),m(t))]2 + [V(m(t),m(t))]/}I P 't t, (5.55) where tpea(< is the value of t which gives the maximum value for Eq. (5.55). Using Eq. (5.20) and Eq. (5.21), Pp_Av can also be written as P Av [4M- k2]2 [* + k,]2) | p peak 2 2 = ^(Du- + k9] + [ti- + k.] } L . r Vak = %{[4+ k]2 + |> + kL]2}11 t 1 xpeak . The average power of the SSB signal was given previously by Eq. (5.53). Thus the expression for peak-to-average power ratio for the generalized SSB signal is (5.56a) (5.56b) (5.56c) (5.56d) Several equivalent representations have been given for peak-to-average power since one representation may be easier to use than another for a particular SS>B signal. Pn-Av f[U(m(t).ii(t))]2 + [V(m(t),m(t))]2) |t t = _____ Lpeak PAv k;i2 + k2 + 2R(JU(0) {[U(m(t),m(t))]2 + [V(m(t),m(t))]2} L , k,2 + k22 + 2RW(0) {[Wm(t).m(t))+k,]2 + [ = ^ 1 ~ Lpeak kj2 + k22 + 2Rw(0) {[-Â¥(m(t),m(t))+k2]2 + [Â¥-(m(t),m(t))+k1]2} | = t ''peak. k;i2 + k22 + 2RW(0) CHAPTER VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS The examples of SSB signals that were presented in Chapter IV will now be analyzed using the techniques which were developed in Chapter V. 6.1. Example 1: Single-Sideband AM With Suppressed Carrier The constants k1 and k2 will first be determined to show that indeed we have a suppressed carrier SSB signal. By substituting Eq. (4.2b) into Eq. (5.4) we have 7T 0 But from Eq. (5.18b) it follows that lim m^R cos e,R sin e) = 0 0 < e < tt .. R-x Thus k = 0 . (6.1) Similarily substituting Eq. (4.2a) into Eq. (5.5) we have IT (6.2) o 48 49 since 1 im m(R cos e,R sin e) = 0 for 0 < e < ir from Eq (5.18a). Further- R-* more, since both k and k2 are zero, the equivalent realizations for the SSB signals, as given by the equations in Section 5.1, reduce identically to the phasing method of generating SSB-AM-SC signals (which was given previously in Figure 8). The autocorrelation for the SSB-AM-SC signal is readily given by use of Eq. (4.2a) and Eq (5.20). Thus y-(m(t) ,m(t)) = m(t). (6.3) Then the autocorrelation of the suppressed-carrier USSB-AM signal is given via Eq. (5.43b), and it is RXU-SC-SSB-AM^ Rmm^ cos WT Rmm^x^ sin oT* (6.4) Likewise, by use of Eq. (5.44b) the autocorrelation for the suppressed- carrier LSSB-AM signal is RXL-SC-SSB-AM^ WT) cos oT r(t) mm sin From Eq. (6.4) it follows that the spectrum of the USSB-AM-SC signal is just the positive-frequency spectrum of the modulation shifted up to on and the negative-frequency spectrum of the modulation shifted down to oj0. That is, there is a one-to-one correspondence between the spectrum of this SSB signal and that of the modulation. This is due to the fact that the corresponding entire function for the signal, g(W) = W, is a linear function of W. Consequently, the bandwidths for this SSB 50 signal are identical to those for the modulation. This is readily shown below. The mean-type bandwidth (when the numerator and denominator exist) is given by use of Eq. (6.3) in Eq. (5.46): (6.6) where = Rmm(0), the power in the modulating signal. By using Eq. (5.48) the rms bandwidth is (rms^SSB-AM -tim(O) I'm (6.7) whenever R^m(0) and exist. By using Eq. (5.50) the equivalent-noise bandwidth is ir (A^SSB-AM = oo 4^ S Rmrn(T)dT (6.8) Thus the bandwidths of the SSB-AM-SC signal are identical to those of the modulating process m(t). The efficiency of the SSB-AM-SC signal is obtained by using Eq. (5.54): iSC-SSB-AM 2Rmm(0) = 1 . 2Rmm(0) (6.9) 51 The peak-to-average power ratio for the SSB-AM-SC signal follows from Eq. (5.56c), and it is f[m(t)]2 + [ii(t)]2} tpeak SC-SSB-AM 2 (6.10) m 6.2. Example 2: Single-Sideband PM The SSB-PM signal has a discrete carrier term. This is shown by calculating the constants k1 and k2. Substituting Eq. (4.5b) into Eq. (5.4) we have K .-Inin f cos e-R sin e)s1n [m,(R cos e.R sir, e)]de. 1 7r R-K. J 1 But from Eqs. (5.18a) and (5.18b) lim m^R cos e,R sin e) = 0 for R-X 0 < e < Ti and lim m.(R cos e, R sin e) = 0 for 0 < e < tt. Thus R-**> kl = 0. (6.11) Likewise, substituting Eq. (4.5a) into Eq. (5.5) we have IT k = C e" cos 0 de = 1. (6.12) 2 TT J 0 Thus the SSB-PM signal has a discrete carrier term since k2 = 1 f 0. There are equivalent representations for the SSB-PM signal since k and k are not both zero. For example, for the upper sideband signal, 1 2 equivalent representations are given by Egs. (5.7) and (5.8). It is noted that Eq. (5.8) is identical to Eq. (5.9) for the USSB-PM signal 52 since k1 = 0. Thus the two equivalent representations are: X USSB PM^ = [e"m^cos m(t)] cos a)0t [_e '"'"'cos mQt;J sin w0t (6.13) and XUssB_p|v|(t) = [-^;sin m(t))+l]cos oi0t [e_r"^sin m(t)]sin w0t. (6.14) The USSB-PM exciters corresponding to these equations are shown in Figure 13 and Figure 14. They may be compared to the first realization method given in Figure 9. The autocorrelation function for the SSB-PM signal will now be examined. In Section 6.1 the autocorrelation for the SSB-AM-SC signal was obtained in terms of the autocorrelation function of the modulation. This was easy to obtain since 44 = m(t). However, for the SSB-PM case 44 and -V-are non-linear functions of the modulation m(t). Consequently, the density function for the modulation process will be needed in order to obtain the autocorrelation of the SSB-FM signal in terms of Rrnm(T) To calculate the autocorrelation function for the SSB-PM signal, first Ryy(t) will be obtained in term of R^fx). Using k;L = 0, Eq. (5.21), and Eq. (4.5b) we have V-(m(t) ,m(t)) a V(m(t),m(t)) = e_m(t) sin m(t). (6.15) Then Figure 13* USSB-PM Signal ExciterMethod II 55 or (6.16) where XjU.r) = m(t) m(t-r) x2(t,x) = m(t) + m(t-r) x3(t,x) = -m(t) m(t-x) -x2(t,r) X4(t,r) = -m(t) + m(t-r) = -x j(t jt) y(t,r) = m(t) + m(t-x) Now let the modulation m(t) be a stationary Gaussian process with zero mean. Then x^t,/), x2(t,i), x3(t,T), x4(t,x), and y(t,x) are Gaussian processes since they are obtained by linear operations on m(t). They are also stat ionary and have a zero mean value. It follows that x (t,x), y(t,t); x?_(t jt) j y(t,x); x 3 (t j i), y (t, T); and x4(t,T)s y(t,r) are jointly Gaussian since the probability density of the input and output of a linear system is jointly Gaussian when the input is Gaussian [15]. For example, to show that Xj(t,x) and y(t,r) are jointly Gaussian, a linear system with inputs m(t) and m(t-r) can readily be found such that the output is y(t,x). Now the averaging operation in Eq. (6.16) can be carried out by using the fol lowing formula which is derived in Appendix II: ej{x(t)+jy(t)} = e-Js{ox2+j2yXy-ay2} (6.17) where x(t) and y(t) are jointly Gaussian processes with zero mean, 2 = = X2(t) oy2 = y2(t) 56 and yxy 25 x(t)y(t) . Thus oxj ~ [m(t)-m(t-t )F = 2[am -R^ir)] cx2 [m(t)+m(t-i)]T 2[orn2+Rmm( r)] ox? [-m(t)-m(t-x )]2 = 2[am2+Rmm(x)] axl C-nri(t)+m(t-r)]2- 2[am2-RtTim(x)] and a/ [l(t)+l(t-,)]2 2[am2+Rnlm(r)] . From Chapter II it is recalled that Rm^(0) = 0 and R^r) = -R-m(x) - -Rmm(x) so that the y averages are yx v = [m(t)-m(t- r )][m(t)+m(t-i)] -2Rmm(t) iy yx v = [m(t)-m(t-x)][m(tT+m(t-x)] = 0 2J Mv = -[m(t)+m(t-r)][m(t)+m(tX)] = 0 and %y = -[m(t)-m(t-x)][m(t)+m(t-x)] = 2Rmrtl(x) . 57 Therefore, using Eq. (6.17), Eq, (6.16) becomes -%{2[am -Rmm(r)] + j2[-2Rmm(-t)] 2[am2+Rmm(T)]} u e-^i2[om +Rmm(t)] + j20 2[am2+Rmm(T)]} _ ^ e-%{2[am2+Rmt11(T)] + j20 2[am2+Rrnm(t)]} + ^ eJs{2[o|T| -Rmm(T)3 + j2[2Rmm(x)] 2[am2+Rmm(i:)]} which reduces to (6.18) where W -SSB-PM-GN denotes the autocorrelation of the imaginary part of the entire function which is associated with the suppressed-carrier SSB- PM signal with Gaussian noise modulation. It is noted that Eq. (6.18) is an even function of t, as it should be, since it is the autocorrelation of the real function -V-(m(t) ,m(t)) Furthermore Ryy(O) is zero when Rmii)(0) 0, as it should be, since the power in any suppressed-carrier signal should be zero when the modulating power is zero. The autocorrelation of the USSB-PM signal is now readily obtained for the case of Gaussian noise modulation by substituting Eq. (6.18) into Eq. (5.42) and using Eq. (537): R XU-SSB-PM-GN (t) = % Re ejT{[e2R>(x) Cos (2Rmm(t))] (6.19) 58 Likewise, the autocorrelation of the LSSB-PM signal may be obtained by using Eq. (5.38). The autocorrelation of the suppressed-carrier USSB-PM signal with Gaussian modulation is given by using Eq. (543a): RXU-SC-SSB-PM-GN^ = ** Re e^oT{[e2Rmm(0 cos (2Rmm(T)) 1] + j[e2lWT) cos (2Rmm(T)]: (6.20) Similarly, the autocorrelation of the suppressed-carrier LSSB-FM signal may be obtained by using Eq. (5.44a). The mean-type bandwidth will now be evaluated for the SSB-PM signal assuming Gaussian noise modulation. From Eq. (6.18) we obtain R^(t) = 1 r e2R"(x) cos (2Rmm(i)di 2J (t-x)2 Rvv(0) 2 it ,2Rmm(x) cos [2Rmm(x)]d/ (6.21) and from Eq. (6.18) Rw(0) = %[e2^ l] (6.22) where = om2 is the average power of m(t). Substituting Eqs. (6.21) and (6.22) into Eq. (5.46) we have the mean-type bandwidth for the Gaussian noise modulated SSB-PM signal: oo P J~ ~T e2FWA) cos[2Rmm(x)]dx A () SSB-PM-GN 32^m (6.23) 59 where is the noise power of m(t) It is seen that Eq, (6,23) may or may not exist depending on the autocorrelation of m(t). The rms-type bandwidth can be obtained with the help of the second derivative pf Eq. (6.18): C(i) i-e2R(t) sin [2Rmm(T)]> 2[Rm(,)]2 *VV- + (-e2R(r) cos [2Rmm(T)]) 2[R^(t)]2 + t-e2R(l) sin [2Rmm(T)]> R^(t) + t e2R(T> cos [2Rmnl(-r)]> 2[R(Of + <-e2R"(T) sin [2R(t)]} 2R^(T)t(,) + { e2R" Rw() = e 2^m {RÂ¡>) 2C4(0)]2> (6.24) Substituting Eq. (6.24) and Eq. (6.22) into Eq. (5.48) we have for the rms-type bandwidth of the SSB-PM signal with Gaussian noise modulation: (wrms) /2{2[RmlO)f R"(0)} SSB-PM-GN 1 e -2iPm (6.25) This expression for the rms bandwidth may or may not exist depending on the autocorrelation of m(t). It is interesting to note that Mazo and Salz have obtained a formula for the rms bandwidth in terms of different para meters [19] However both of these formulae give the same numerical re sults, as we shall demonstrate by Eq. (6.29). 60 The equivalent-noise bandwidth is obtained by substituting Eq. (6.18) into Eq. (5.50): (Ato) %{e2Rmm(r) cos [26mrT1(T)] 1} dt %[e^m-l] ^ or (Aw) ;(e2^i 1) SSB-PM-GN {e2Rmm(T) co$ [2Rrnm(T)] 1 }di (6.26) It is noted that the equivalent-noise bandwidth may exist when the formu lae for the other types of bandwidth are not valid because of the non existence of derivatives of Rmm(i:) at t = 0. It is obvious that the actual numerical values for the bandwidths depend on the specific autocorrelation function of the Gaussian noise. For example, the rms bandwidth of the SSB-PM signal will now be calculated for the particular case of Gaussian modulation which also has a Gaussian spectrum. Let 2 -0) 0 where Pm(u) is the spectrum of m(t) 4>0 = is the total noise power in m(t) o2 is the "variance" of the spectrum. The autocorrelation corresponding to this spectrum is Rmm(r) ~ Vr 1 2 2 -Ho t (6.27) 61 The Hilbert transform of Rmm(T) is also needed and is obtained by the frequency domain approach. It is recalled from Chapter II that P*(o)) = mm j PmmU) > w > 0 , cu 0 j Pmn ) < 0 Then Wt) = ^im 2tr -U)2 t -00 iaiS ei#* ejwTdu> f e2^ eJ<*Tdw which reduces to fynm^) /2rrcr f St ~coz ... sin oox dw . This integral 1s evaluated by using the formula obtained from page 73, #18, of the 8ateman Manuscript Project, Tab1es of In teg ral fra ns forms, vol. 1 [5]: 00 / 1 ..2 eaX sin xy dx e ^ Erf(~4=. y) Z/a \2/a / Re a > 0 where Erf (x) * Jr f -t e z dt. 62 Thus Rmm(T) = "J i^o e 2 T j 0T or Rmm(T) ~ " Rmm(T) a n . From Ea. (6.27) it follows that Rmm() = -Vo2 and from Ea. (6.28) we have Rmm(O) s ~pzr /2tt Substituting these two equations into Ea. (6.25) we get ( ta rms 2 1 e~Z>po Thus if m(t) has a Gaussian spectrum and if the modulation has density function, the SSB-FM signal has the rms bandwidth: (wrms^sSB-PM-GN 2i|>0az[4(ip0/;ir) + 1] 1 e where t|>0 is the total noise power in m(t) a4is the "variance" in the spectrum of m(t). (6.28) Gaussian (6.29) 63 This has the same numerical value as that obtained from the result given by Mazo and Salz [19]. The result may also be compared to that given by Kahn and Thomas for the SSB-PM signal with sinusoidal modulation [20]. From Eg. (19) of their work ^rms)$SB-,PM-S = ua^ (6.30) where wa is the frequency of the sinusoidal modulation and 6 is the modu lation index. For comparison purposes, equal power will be used for m(t) in Eq. (6.30) as was used for m(t) in Eq. (6.29). Then Eq. (6.30) becomes (k>rms)ssB-PM-S = ^ 3 *^o (6.31) Thus it is seen that for Gaussian modulation the rms bandwidth is propor tional to the power in m(t) when the power is large (ip0 > > tt/4), and for sinusoidal modulation the rms bandwidth is proportional to the square root of the power m(t). The efficiency for the SSB-PM signal with Gaussian modulation will now be obtained. Substituting Eo. (6.22) into Eq. (5.54) we have e2+ra-i "SSB-PM-GN + or - i p"2^m nSSB-PM-GN (6.32) where is the noise power of m(t). The peak-average to average power ratio for Gaussian m(t) is given by use of Eqs. (4.5a), (4.5b), and (6.22) in Eq. (5.56b): 64 i[e m (t) cos m(t)] + [e rn(t) sin m(t)]2} 1 + (e 1) tpeak or e-2rn(t) e t = tpeak Si'm (6.33) Note that m(t) may take on large negative values because it has a Gaussian density function (since it was assumed at the outset that the modulation was Gaussian). However, it is reasoned that for all practical purposes, m(t) takes on maximum and minimum values of +3am and 40^ volts where cm is the standard deviation of m(t). This approximation is useful only for small values of crm since e+^m) approximates the peak power only when the exponential function does not increase too rapidly for larger values of am. Thus the peak-to-average power ratio for the SSB-PM signal with Gaussian noise modulation is (6.34) when is small. It is noted that the efficiency and the peak-to-average power ratio depend on the total power in the Gaussian modulation process and not on the shape of the modulation spectrum. On the other hand the autocorre lation function and bandwidth for the SSB signal depend on the spectral 65 shape of the modulation as well. The dependence of bandwidth on the spectrum of the Gaussian noise modulation will be illustrated by another example. Consider the narrow- band modulation process: m(t) = a(t) cos (ojat + (f>) (6.35) where a(t) is the (double-sideband) suppressed-subcarrier amplitude modulation a is the frequency of the subcarrier <Â¡> is a uniformly distributed independent random phase due to the subcarrier oscillator. That is, we are considering a SSB signal which is phase modulated by the m(t) given above. Then Rmm(T) = *2 Raa(t) oos ^a1 (6.36) where Raa(T) is the autocorrelation of the subcarrier modulation a(t). Rmm(T) can obtained from Eq. (6.36) by use of the product theorem [21]. Thus, assuming that the highest frequency in the power spectrum of a(t) is less than o>a, Rmmi'O ~ Raa(x) sin a1 > (6.37) Furthermore let a(t) be a Gaussian process; then m(t) is a narrow-band Gaussian process. This is readily seen since Eq. (6.35) may be expanded as follows: m(t) = %[a(t) cos (wat+<|>) a(t) sin (cjat+cÂ¡>)] + %[a(t) cos (cgt+tj)) + a(t) sin (ojat+<)>)] (6.38) 66 The terms in the brackets are the USSB and LSSB parts of the suppressed- subcarrier signal m(t). But these USSB and LSSB parts are recognized as the well-known representation for a narrow-band Gaussian process. Thus m(t) is a narrow-band Gaussian process. Now the previous expressions for bandwidth, which assume that m(t) is Gaussian, may be used. The mean-type bandwidth for the multi plexed SSB-PM signal is then readily given via Eq. (6.23), and it is (<*>) M-SSB-PM-GN oo f eRaa(x) cos waA cos[RaaU) sin coaA]dA 00 e^a 1 (6.39) where tpa is the average power of the Gaussian distributed subcarrier modulation a(t). Obtained in a similar manner, the rms bandwidth is (rms^.ssB.pM. GN a W1) Raa(0) 1 e^a (6.40) and the equivalent-noise bandwidth is (aw) r[e2*a-l] M-SSB-PM-GN J' eRaa^T^ cos aT Cos[Raa(x) sin toax] (6.41) Thus, it is seen once again that the bandwidth depends on the spectrum of the modulation, actually the subcarrier modulation a(t). To obtain a numerical value for the rms bandwidth of the multi plexed SSB-PM signal assume that the spectrum of a(t) is flat over |o)| < w0 < )a. 67 4; Pa() 1 Wf 0)** Figure 15. Power Spectrum of a(t) From Figure 15 we have Raa(T) W 1 o 'aa^; 77 J -o' ~WA eJTdw or Raa(T) NqW0 /sin W t * \ W0t (6.42) and ^ = N W oo (6.43) Then ii , Raa(o) -N0W0' (6.44) Substituting the last two equations into Eq. (6.40) we obtain the rms bandwidth for the SSB-PM multiplexed signal: rTTMoyw. i ^ + nw ^rms^M-SSB-PM-SN- 3tt 1 e-Nowo/lT (6.45) where u>a is the subcarrier frequency N0 is the amplitude of the spectrum of the subcarrier Gaussian noise modulation WQ is the bandwidth of the subcarrier noise modulation. 68 Thus the niis bandwidth is proportional to the power in the subcarrier modulation as N0 becomes large. 6.3/ Example 3: Single-Sideband FM As was indicated in Section 4.3. the representation for the SSB-FM signal is very similar to that for the SSB-PM signal. In fact it will be shown below that all the formulae for the properties of the SSB-PM signal (which were obtained in the previous section) are directly applicable to the SSB-FM signal. The SSB-FM signal has a discrete carrier term since the entire function for generating the SSB-FM signal is identical to that for the SSB-PM signal, which has a discrete carrier term. The other properties of the SSB-FM signal follow directly from those of the SSB-PM signal if the autocorrelation of m(t) can be obtained in terms of the spectrum for the frequency modulating signal e(t). It is recalled from Eq. (4.7) that t (6.46) First, the question arises: Is m(t) stationary if e(t) is stationary? The answer to this question has been given by Rowe; however, it is not very satisfactory since he says that m(t) may or may not be stationary [22] However, it will be shown that m(t), as given by Eq. (6.46), is stationary in the strict sense if e(t) is stationary in the strict sense; and, furthermore, m(t) is wide-sense stationary if e(t) is wide-sense stationary It is recalled that if y(t) L[x(t)l 69 where L is a linear time-invariant operator, then y(t) is strict-sense stationary if x(t) is strict-sense stationary and that y(t) is wide-sense stationary if x(t) is wide-sense stationary [4]. Since the integral is a linear operator, we need to show only that it is time-invariant, that is to show that y(t+e) = L Jx(t+E)j or t+e e(t1)dt1 = j e(t2+e)dtr This is readily seen to be true by making a change in the variable, letting t = t2 + e. Thus, if e(t) is stationary, then m(t) is stationary. Moreover, in the same way it is seen that if m(t) had been defined by t mj (t) = D 1 (t')dt' (6.47) tQ then m (tj is not necessarily stationary for e^t) stationary since the system is time-varying (i.e. it was turned on at tQ). But this should not worry us because, as Middleton points out, all physically realizable systems have non-stationary outputs since no physical process could have started out at t = - and continued without some time variation in the parameters D5]. However, after the "time-invariant" physical systems have reached steady-state we may consider them to be stationary processes provided there is a steady state. Thus by letting tQ -> - we are con sidering the steady-state process m(t) which we have shown to be stationary Now the autocorrelation of m(t) can be obtained by using power-spectrum techniques since m(t) has been shown to be stationary. From Eq. (6.46) we have 70 (6.48) Then in terms of power-spectrum densities (6.49) As Rowe points out, Pmm(oo) must eventually fall off faster than k/ where k is a constant, if e(t) is to contain finite power; and if Pmm(to) = k/to2, Pqq(w) will be flat and, consequently, white noise. Thus we have a condition for the physical realizability of m(t): Pmm(u)) falls off faster than -6 db/octave at the high end. This condition is satisfied by physi cal systems since they do not have infinite frequency response. From Eq. (6.49) we have (6.50) Pmm() ~ C Immediately we see that if P9g(to) takes on a constant value as |w| ->- 0 and at o> 0, m(t) will contain a large amount of power with spectral components concentrated about the origin. In other words, m(t) has a large block of power, located infinitely close to the origin which is infinitely large. Thus m(t) contains a slowly varying "DC" term with a period T and m2(t) - . By examining Eq. (6.46) we obtain the same result from the time domain. That is, for Pe0(w) equal to a constant, e(t) contains a finite amount of power located infinitely close to the origin which appears as a slowly varying finite "DC" term in e(t) such that T * Then by Eq. (6.46), m(t) has a infinite amplitude and, 71 consequently, infinite pwer. In other words, the system does not have a steady-state output condition if the input has a power around oi =0. Thus, this system ia actually conditionally stable, the output being bounded only if the input power spectrum has a slope greater than or equal to +6 db/octave near the origin (and, consequently, zero at the origin) as seen from Eq. (6,50). It is interesting to note that for the case of FM, ejm^ is stationary regardless of the shape of the spectral density Pqq(ua). This is due to the fact that ejm^ is bounded regardless of whether m(t) is bounded or not. From Eq. (6.50) we can readily obtain Rmm(ir) for any input process e(t) which has a bounded output process m(t). Thus oo WO = J- f ^-ej3T du (6.51) IT J tl) 00 Furthermore, R^m(0), Rmm(T), and Rmm(0) may be obtained in terms of Pee(w). By substituting for these quantities in the equations of Section 6.2, the properties of a SSB-FM signal can be obtained in terms of the spectrum of the modulating process. 6.4. Example 4: Single-Sideband g The SSB-a signal has a discrete carrier term. This is readily shown by calculating the constants kT and k2. Substituting Eq. (4.10b) into Eq. (5.4) we have IC 1 11m f e"MR cos e*R s1n 6> sin am, (R cos 9.R sin 8)d8 . 1 n R-Mo J 1 .0 But lim m^R cos e, R sin e) = 0, for 0 < e s: tt and lim m^R cos e, R-x R-x R sin e) = 0 for 0 Â£ e $ it. Thus kx =0. (6.52) 72 Likewise, substituting Eq. (4.10a) into Eq. (5.5), we have k2 =1. (6.53) Thus the SSB-a signal has a discrete carrier term. It follows that equivalent representations for the SSB-a signal are possible since k2 f 0. This is analogous to the discussion on equiva lent representations for SSB-PM signals (Section 6.2) so this subject will not be pursued further. The autocorrelation function for the SSB-a signal will now be ob tained in terms of Rmm(T) Using Eq. (5.21) and Eq. (4.10b) we have Rw(t) = [eam(t) sin am(t)][eam(t"T) sin am(t-t)] or r (t) = 53{ea[m(t)+m(t-T)]} {eja[m(t)-m(t-T)] _eja[m(t)+m(t-T)]} + %{ea^(t)+m(t-r)]} {_eja[-m(t)-m(t-T)] + eja[-m(t)+m(t-r)]}. (6.54) The density function of m(t) has to be specified in order to carry out this average. It is recalled that m(t) is related to the modulating signal e(t) by the equation: m(t) In [1+(t)H . Now assume that the density function of the modulation is chosen such that m(t) is a Gaussian random process of all orders. Eq. (6.54) can then be evaluated by the procedure that was used to evaluate Eo. (6.16). 73 Assuming a Gaussian m(t), Eg. (6.54) becomes COS [2a2R|7im(x)] 1} (6.55) But this is identical to Eg. (6.18) except for the scale factor a2. Thus for envelope detectable SSB (a = 1) with m(t) Gaussian, the auto correlation and spectral density functions are identical to those for the SSB-PM signal with Gaussian m(t). Moreover, the properties are identical for SSB-a and SSB-PM signals having Gaussian m(t) processes such that (ipm)SSB-PM 0(2 ^m^sSB-a* It is also seen that if |e(t)| < < 1 most of the time then m(t) = e(t). Thus, when e(t) is Gaussian with a small variance, m(t) is approximately Gaussian most of the time. Then Eg. (6.55) becomes 2a2Ree(T) cos [2a2Ree(x)] 1} (6.56) RW(l)SSB-a-GN ~ when |e(t)| < < 1 most of the time. Conseguently, formulae for the auto correlation functions analogous to Egs. (6.19) and (6.20), may be further simplified to a function of Ree(x) instead of Rrnm(t). Then the auto correlation functions for USSB-cx and LSSB-a signals, assuming Gaussian modulation e(t) with a small variance, are RXU-SSB-a-GN (t) = H Re eJoT{te22Ree(T) cos (2<,2l?ee(t))] j [e2"R86(T) cos (2c.^e(0)]> + (6,57) 74 and RXL-SSB-a-GN^ ~ Re e"JuoT{[e2a2Ree(T) cos (2a2fr0e(r))] + J [e 2a2Ree(t) cos (2a2tee(r))]} (6.58) The efficiency is readily obtained by substituting Eq. (6.56) into Eq. (5.54): 'SSB-a-GN = 1 5-2a2(J>m (6.59) where is the power in the Gaussian m(t) and |e(t)| < < 1. This result may be compared for a = 1 to that given by Voelcker for envelope-detectable SSB(a = 1) [11]. That is, if e(t) has a small variance, then m(t) = e(t); and Eq. (6.59) becomes nSSB-oi-GN : 1 e 00 ~ 2o0/. (6.60) This agrees with Voelcker"s result (his Eq. (38)) when the variance of the modulation is small-~the condition for Eq. (6.60) to be valid. The expressions for the other properties of the SSB-ct signal, such as bandwidths and peak-to-average power ratio, will not be examined further here since it was shown above that these properties are the same as those obtained for the SSB-PM signal when Um)sSB-PM ~ a2(iJm^sSB-a m(t) is Gaussian. as long as CHAPTER VII COMPARISON OF SOME SYSTEMS In the two preceding chapters properties of single-sideband sig nals have been studied. However, the choice of a particular modulation scheme also depends on the properties of the receiver. For example, the entire function g(W) W can be used to generate a SSB signal, but there is no easy way to detect this type of signal. In this chapter a comparison of various types of modulated sig nals will be undertaken from the overall system viewpoint {i.e* generation, transmission and detection). Systems will be compared in terms of the degradation of the modulating signal which appears at the receiver out put when the modulated RF signal plus Gaussian noise is present at the input. This degradation will be measured in terms of three figures of merit: 1. The signal-to-noise ratio at the receiver output 2. The signal energy required at the receiver input for a bit of information at the receiver output when com an son is made with the ideal system (Here the ideal system is defined as a system which requires a minimum amount of energy to transmit a bit of information as predicted by Shannon's formula.) 3. The efficiency of the system as defined by the ratio of the RF power required by an ideal system to the RF power required by an actual system,(Here the ideal sys- 75 76 tern is taken to be a system which has optimum trade-off between predetection signal bandwidth and postdetection signal-to-noise ratioo) Comparison of AM, FM, SSB-AM-SC, and SSB-FM systems will be made using these three figures of merit. It is clear that these comparisons are known to be valid only for the conditions specified; that is, for the given modulation density function, and detection schemes which are used in these comparisons. 7.1, Output Signal-to-Noise Ratios 7.1-1. AM system Consider the coherent receiver as shown in Figure 16 where the input AM signal plus narrow-band Gaussian noise is given by X(t) + n.Â¡(t) = {A0[l + 6 sin tmt] cos co0t} + ixc(t) cos )Qt xs(t) sin cdQt} (7.1) where X(t) is the input signal, n-Â¡(t) is the input noise with a flat spec trum over the bandwidth 2wm> and 6 is the modulation index. X(t)+nj(t) Low Pass Fi 1 ter AC Couple 2k cos wqL Figure 16. AM Coherent Receiver Output Then the output signal-to-noise power ratio, where A0k6 sin ojmt is the output signal, is given by (S/N) 0 = 6 2 1 + %62 (S/N)1 (7.2) 77 or (7.3) where (S/N)-j The input signal-to-noise power ratio (C/N)i = The input carrier-to-noise power ratio and the spectrum of the noise is taken to be flat over the IF bandpass which is 2a)m(rad/s). 7.1-2. SSB-AM-SC system Consider the coherent receiver (Figure 16) once again, where the input is a SSB-AM-SC signal plus narrow-band Gaussian noise. Then the input signal plus noise is X(t) + ni(t) = iA0[m(t) cos w0t m(t) sin o>0t]} + [xc(t) cos (O0t xs(t) sin oj0t] (7.4) where m(t) = 6 sin wt m and xs(t) = xc(t) if the IF passes only upper sideband components. The input noise is assumed to have a flat spectrum over the bandwidth )m. Then the output signal-to-noise power ratio, where AQk6 sin wmt is the output signal, is given by [23] (7.5) (S/N)Q = (S/N)i where the spectrum of the noise is taken to be flat over the IF bandpass which is cjjm(rad/s). It is interesting to note that the same result is obtained from a 78 more complicated receiver as given in Figure 17 However, in some practi cal applications the receiver in Figure 17 may give much better perform ance due to better lower sideband noise rejection. That is, in Figure 17 the lower sideband noise is eliminated as the result of the approximate Hilbert transform filter realized about oj = 0; whereas, in Figure 16 the lower sideband noise is rejected by the IF filter realized about u = to0. Thus, in order to obtain equal lower sideband noise rejection in both receivers, the IF bandpass for the receiver in Figure 16 would have to have a very steep db/octave roll-off characteristic at 7.1-3. SSB-FM system Now consider a FM receiver which is used to detect a SSB-FM sig nal plus narrow-band Gaussian noise as shown in Figure 18. X{t)+n-j (t) FM Receiver Output Figure 18. SSB-FM Receiver 79 The input signal plus noise is given by X(t) + n^(t) A0e"^^^ cos [o>0t + m(t)] + n-j(t) (7.6) where A0 ~ The amplitude of carrier u>o The radian frequency of the carrier m(t) = D /t v(t) dt m(t) = m(t) ~= The Hilbert transform of m(t) nj(t) Narrow-band Gaussian noise with power spectral density F0 over the (one-sided spectral) IF band and v(t) is the modulation on the upper SSB-FM signal. The independent narrow-band Gaussian noise process may be represented by n^(t) = R(t) cos [w0t + 4>(t)J = xc(t) cos w0t x$(t) sin w0t where xs(t) = xc(t) since the IF passes only the frequencies on the upper sideband of the carrier frequency. Then the phase of the detector output is obtained from Eq. (7.6) and is ;p(t) = k tan which reduces to A0e~^ sin m(t) + R(t) sin A0e^ cos m(t) + R(t) cos (7.7) xp(t) km(t) + k tan^ R(t) sin [ A0e-^ + R(t) cos [m(t) (7.8) where k is a constant due to the detector. The detector output voltage is given by Eq. (7.8) is identical to the phase output when the input is conventional FM plus noise except for the factor e_n1' . 80 For large input signal-to-noise ratios {i.e. A0e-m^ > > R(t) most of the time), Eq (7o8) becomes kR(t) ip(t) km(t) + sin [ (7.9) dn0(t) Then the noise output voltage is where n (t) = R(t) sin [ (7.10) Now the phase (t) is uniformly distributed over 0 to 2n since the input noise is a narrow-band Gaussian process. Then for m(t) deterministic, U(t) m(t)J is distributed uniformly also. Furthermore, R(t) has a Rayleigh density function. Then it follows that R(t) sin [ is Gaussian (at least to the first order density) and, using Rice's formulation [24, 25], where F(u>) = F0 is the input noise spectrum and {n} are independent random variables uniformly distributed over 0 to 2-rr. Actually it is known that the presence of modulation produces some clicks in the out put [26], but this effect is not considered here. Eq. (7.10) then be comes 81 or dn0(t) dt kem(t) % r___ ~Yo J~2F(con) K-w0) COS [(wn-w0)t + 0n] + ke(t) Ao CO z n=l ^2F{n) 2. sin C(n-u.0)t + 8]. Noting that {en} are independent as well as uniformly distributed and that the noise spectrum is zero below the carrier frequency, the output noise power is N o dn0(t) dt m F^d. + -- e2S dm (t) dt 2-it ^m F0dw o k2 e2m(t)' ^ m3 + k2 e2m(t) dm(t) Ao2 J 2 rr 3 /\ 2 no dt - _ (7.11) where () is the averaging operator and wm is the baseband bandwidth (rad/s) Now let v(t) = -Am cos tomt then, averaging over t, we have 2m(t) w, 2u/c| m 2tt m e26 cos dt = In(26) and 2m(t) dm(t) dt I (m6)2 [10(26) Iz(2)] 1^6 1,(26) (7.12) (7.13) 82 where 6 DAm/tom, the modulation index. Substituting Eq, (7,12) and Eq (7,13) into Eq, (7,11) we obtain for the output noise power k2F0i%3 2ttA02 ll0(2i) +isl1(26) Referring to Eq. (7.9), the output signal power is 2 = y Then the output signal-to-noise ratio is dkm(t) dt (7,14) (7.15) (S/N)0 1 1 - 1.(26) + 61,(26) 3 0 2 1 (S/N)n - A0262 2 <%, 1 Io(26) + I 01,(26)1 2ir 3 2 (7.16) Referring to Eq. (7.6), the signal power into the detector is Si = A02 e2"1^ cos2 [o)0t + m(t)] = A02 e2^^^ jA02 I0(26) (7.17) Kahn and Thomas have given the ratio of the rms bandwidths (taken about 83 the mean of the one-sided spectrum) for a SSB-FM signal to a conventional FM signal [20], and it is BSSB-FM _____ a fz bfm I1z(26) UHzT) (7.18) It is known that the bandwidth (in rad/s) of a FM signal is approxi mately Bfm = 2(+l (7.19) Thus, to the first approximation, the SSB-FM bandwidth is _ / 112(26) ' bssb-fm2 /2 y1 'S+1>V <7-20> Then, taking the IF bandwidth to be that of the SSB-FM signal, the input noise power is i SSB-FM n (7.21) Consequently, the input signal-to-noise ratio is Ag2 Iq(26) (S/N), - Fo 4 a)m (6+1) /2 Ii2(26) I02(26) (7.22) 84 Using Eo. (7.16) and Eq. (7.22), we have 6 6 2(6+1) /2 / 1 (S/N)0 = Ij (26) TTT In2(2)+| 610(26)1,(26) (S/N) -f (7.23) for the case of SSB-FM plus Gaussian noise into a FM detector. The signal-to-noise output can also be obtained in terms of the unmodulated-signal-to-noise ratio [i.e. the carrier-to-noise power at the input). From Eq. (7.6) we obtain (S/N)-f = 10(26) (C/N)i (7.24) and Eq. (7.23) becomes (S/N)o = Il2(26) 6 62(6+1) /2 1 _ 10 2 (^ 6) Iq(26) + | 6I1(26) (C/N)i where (C/N)^ is the carrier-to-noise power ratio. (7.25) 7.1-4. FM system The signal-to-noise ratio at the output of a FM receiver for a FM signal plus narrow-band Gaussian noise at the input can be obtained by the same procedure as used above for SSB-FM. The factor e_m^ of Eq. (7.6) is replaced by unity, and the bandwidth of the input noise is given by Eq. (7.19). Then the output signal-to-noise ratio becomes (S/N)0 = 3 62(6+l) (S/N)i (7.26) 85 when the input signal-to-noise ratio is large. It is also noted that (S/N)i = (C/N)j. (7.27) 7.1-5. Comparison of signal-to-noise ratios A comparison of the various modulation systems is now given by plotting (S/N)0/(S/N)^ as a function of the modulation index by use of Eqs. (7.2), (7.5), (7.23), and (7.26). This plot is shown in Figure 19. Likewise (S/N)0/(C/N)j as a function of the modulation index is shown in Figure 20, where Eqs. (7.3), (7.25), (7.26), and (7.27) are used. It is noted that in both of these figures the noise power band width was determined by the signal bandwidth. When systems are compared in terms of signal-to-noise ratios, a useful criterion is the output signal-to-noise ratio from the system for a given RF signal power in the channel--that is, (S/N)0/S-Â¡, This result can be obtained from (S/N)0/(S/N)-Â¡, which was obtained previously for each system, if the input noise, N-Â¡, is normalized to some convenient constant. This is done, for example, by taking only the noise power in the band 2oim (rad/s) for measurement purposes. (The actual input noise power of each system is not changed, just the measurement of it.) Then the normalized input noise power for all the systems is F0 [\| _ 2u)pi 2ir where the subscript I denotes the normalized power. Then the ratio (S/N)0/(S/N)j is identical to Nj[(S/N)0/Si] where Nj is the constant de fined above. Thus, to within the multiplicative constant Nj, comparison of (S/N)0/(S/N)j for the various systems is a comparison of the output 86 Figure 19 Output to Input Signal-to-Noise Power Ratios for Several Systems 87 Figure 20. Output Signal-to-No1se to Input Carrler- to-No1se Ratio for Several Systems 88 signal-to-noise ratios for the systems for a given RF signal power. This procedure is commonly used for system comparisons [23]. Likewise, a comparison of output signal-to-noise ratios for vari ous systems for a given aarrier power can be carried out by comparing (S/N)0/(C/N)j, where the subscript I denotes the normalized input noise power once again. The AM, SSB-AM-SC, SSB-FM, and FM systems will now be compared by using this procedure. For the AM system (S/N)i = (S/N)j so that Eq. (7.2) becomes (S/N)0 1 + (S/N)! (7.28) and, likewise, Eq. (7.3) becomes (S/N)0 = 2 (C/N)]. For the SSB-AM system Eq. (7.5) becomes (S/N)0 = 2(S/N)i. (7.29) (7.30) For the SSB-FM system, Eq. (7.23) becomes (7.31) and Eq. (7.25) becomes (S/N)0 662 10(2<$) + | 61,(26) (C/N)j (7.32) 89 For the FM system, Eo. (7.26) becomes (S/N)0 = 3 62(S/N)j and (S/N)Q = 3 62(C/N)j. (7.33) (7.34) A comparison of the output signal-to-noise ratios for the vari ous modulation systems can be made now for a given input signal or car rier power by using these equations. (S/N)0/(S/N)j as a function of modulation index is plotted for various systems in Figure 21. Likewise (S/N)0/(C/N)j is shown in Figure 22. From these figures, it is concluded that FM gives the greatest signal-to-noise ratio at the detector output for high index, followed by SSB-AM. For low index (6 < 1), SSB-AM is best, followed by SSB-FM and FM which have about the same (S/N)0, and AM gives the lowest (S/N)0. 7.2. Energy-Per-Bit of Information The concept of RF energy required per bit of received information is used by Raisbeck for comparing SSB-AM and FM systems [27]. This will be extended to AM and SSB-FM systems in this section. The (received) capacity of the system is given by [28] Cb > (b/2ir) log2 [1 + (S/N)Q] (7.35) where b is the baseband bandwidth (rad/s) (S/N)0 is the output signal-to-noise power ratio. Eq. (7.35) becomes an equality when the output noise is Gaussian. 90 Figure 21. Output SignaT-to-Noise to Input Signal-to-Normalized- Noise Power Ratio for Various Systems 91 Figure 22. Output Signal-to-Noise to Input Carrier-to-Normalized- Noise Power Ratio for Various Systems 92 Then the RF energy required per bit of received information is Sl (F0B/2tt)(S/N)1 f0b (S/N)1 Cb S Cb S b l0g2 [1 + (S/N)q] where F0 is the spectral density of the noise in the IF and B is the IF bandwidth (rad/s). In an ideal system the capacity of the IF is eaual to the capacity of the baseband even when (S/N).Â¡ -* 0. Therefore the ideal system has an energy-per-bit given by St Si r r lim Cb CB (S/N) -Â¡-K) FnB (S/N)i B log2 [1 + (S7N)i] log2e 0.693 Fn. (7.37) Then Eq. (7.36) may be written as S -L < (0.693 F0) Cb B (S/N)j 0.693 b Tog, [1 + (S/N)0] _4 Now the figure of merit will be defined as B (S/N)i 0.693 b log, [1 + (S/N)0] (7.38) which is the amount of energy required by the actual system over that of the ideal system in order to receive a bit of information, provided that the output noise is Gaussian. If the output noise is not Gaussian, the value of M will be somewhat larger than the ratio, energy-per-bit for the actual system to the energy-per-bit for the ideal system. H as a function of modulation index will be derived below for com parison of various systems. 93 7.2-1. AM system We now want to find M(s) for the AM system, described in Section 7.1-1, such that we will have an output signal-to-noise ratio of 27.5 db for 6=1. 27.5 db is an arbitrary value that is chosen here for com parison of systems using M as a figure of merit. This value is repre sentative of the (S/N)o requirement for actual communication systems. From Eq. (7.3) it follows that (C/N)-j = 27.5 db for 6 = 1. Also, for the AM system Eq. (7.38) becomes 2 )m [(1 + %62)(C/N)i3 M(6) = ; -- 0.693 log2 [1 + 62('C/N)i] For (C/N)j = 27.5 db, Eq. (7.39) reduces to 1620 (1 + %62) M{6) = log2 [1 +'560 62] (7.39) (7.40) The values of M(6) for the AM system, as given by Eq. (7.40), will be compared to those for other systems in Section 7.2-5. 7.2-2. SSB-AM-SC system To obtain M(6) for the SSB-AM-SC system, (S/N)0 = 27.5 db will be used once again. From Eq. (7.5) it follows that (S/N)j = 27.5 db. Also, for the SSB-AM system Eq. (7.38) becomes m (S/N)i M(6) = 0.693 com log2 [1 + (S/N)0] (7.41) For (S/N) 0 = (S/N)-f = 27.5 db, Eq. (7.41) reduces to M(6) = 19.5 db. (7.42) 94 7.2-3. SSB-FM system To obtain M(6) for the SSB-FM system with (S/N)Q = 27.5 db, it follows from Eq. (7.25) that (C/N)-f = 23.3 db for 6 = 1. Also, for the SSB-FM system Eq. (7.38) becomes M(6) _ / I12(26)1 -I 2(6+1) o)m /2 /l I (26)(C/N)i y I02(26) 0.693 com log 6 62 (6+1) /2 /l - 1 + T7(W In2(2.) -. (7.43) 10(26) + | 61^26) (C/N)i For (C/N)^ = 23.3 db, Eq. (7.43) reduces to 7.2-4. FM system To obtain M() for the FM system, for (S/N)0 = 27.5 db, it follows from Eqs. (7.26) and (7.27) that (C/N)-,* = 12 db (which is just above the threshold) for 6=2. Also, for the FM system, Eq. (7.38) becomes [2 (6+1) oom] (C/N) M(6) = : . 0.693 aw log2 [1+3 62(6+1)(C/N)i] (7.45) 95 For (C/N)-j = 12 db, this reduces to 46 (6+1) M(S) log2 [1 + 48 2(+l)] It is recalled that Raisbeck obtained this result [27]. (7.46) 7.2-5. Comparison of energy-per-bit for various systems It is recalled that M( when the output system noise is Gaussian. The output noise is Gaussian for the AM, SSB-AM-SC, and FM systems [for FM, (C/N)i = 12 db >> 0 db]. Also, from Eq. (7.10) it is seen that the noise out of the SSB-FM system is Gaussian for small index (say 6 < 1). Thus, for the systems that are analyzed above, M(6) represents the ratio of the energy-per-bit for the actual system to the energy-per-bit for the ideal system. Then in db, M() gives the energy-per-bit required above the ideal system. The systems are compared in terms of energy-per-bit (db) above the ideal system in Figure 23, where Eqs. (7.40), (7.42), (7.44), and (7.46) have beqn plotted for the AM, SSB-AM-SC, SSB-FM, and FM systems. From this figure it is seen that the FM system is best, followed by SSB-AM-SC, SSB-FM and AM. Furthermore, the FM system is about 12 db worse than the ideal system. These comparisons are valid for output signal-to-noise ratios of about 25 db. In addition, Figure 23 specifies the modulation index to use for each type of system in order to minimize the energy required to transmit one bit of information. Energy (db) Required Above Ideal System 35 1.0 2.0 Modulation Index () 3.0 4.0 Figure 23. Comparison of Energy-per-Bit for Various Systems 97 7.3. System Efficiencies The third figure of merit which will be used to compare systems is the system efficiency, defined by Transmitted power required for an ideal system 6 = Transmitted power required for an actual system = sj/S-j (7.47) where the ideal system is taken to be a system which has optimum trade off between predetection signal bandwidth and postdetection signal-to- noise ratio. This concept is used by Wright and doll iffe to compare SSB-AM-SC and FM systems [29]. Here, it will be extended to AM and SSB-FM systems. The trade-off between predetection signal bandwidth and post detection signal-to-noise ratio for an ideal system is obtained by equating the predetection capacity to the postdetection capacity since an ideal system does not lose information in the detection process [30]. Thus (B/2it) log2 [1 + (S'/N)i] = (b/2ir) log [1 + (S'/N)0] (7.48) where B is the IF bandwidth b is the baseband bandwidth (S'/N)-} is the input signal-to-noise ratio for the ideal system (S'/N)0 is the output signal-to-noise ratio for the ideal system The prime is used here to denote the ideal system. Eq. (7.48) reduces to (s7n)0 = [1 + (S'/N)]Y 1 (7.49) where y = B/b, the IF to baseband bandwidth ratio. The efficiency, e will now be calculated for various types of systems. 98 7.3-1, AM system For the AM system y = B/b = 2. Then setting Eq. (7.49) equal to Eq, (7.2) we have [1 + (S/N)i32 -1 1 + (S/N)i. (7.50) Substituting for S-Â¡ from Eq. (7.47), the efficiency for the AM system is obtained, and it is 3 = 2 1 1 + ig2 _(S'/N)i + 2_ (7.51) The AM efficiency will be compared to those for other systems in Section 7.3-5 as a function of (S'/N)i with the modulation index as a parameter. 7.3-2. SSB-AM-SC system For the SSB-AM-SC system y = B/b = 1. Then, equating Eq. (7.49) and Eq. (7.5), we have (S'/NJi = (S/N) -f (7.52) and substituting for Si using Eq. (7.47), the SSB-AM-SC efficiency is (7.53) 7.3-3. SSB-FM system For the SSB-FM system, using Eq. (7.20), = = 2 (6+1) /2 m ' 1^(26) 1 Iq2(26) Y (7.54) 99 Note that for SSB-FM y and s are uniquely related to each other (by Eq, (7.54)), unlike the AM and SSB-AM-SC cases. Equating Eq. (7.49) and (7.23), we have [i + (S7N),]Y l = Then, substituting for Sn- from Eg. (7.47), the efficiency for SSB-FM is 32y |_I02(2S) + | 6l0(2)I1(2) (S'/N)i [1 + (SVN)iF 1 (7.56) 362i I02(26) + | 6I0(26)I1(26) (S/N)- (7.55) where y and 6 are uniquely related by Eq. (7.54). 7.3-4. FM system For the FM system, using Eq. (7.19), Y = 7T = 2(6+1). (7.57) m Thus for FM, y and 6 are uniquely related, as was the case in SSB-FM. Eauating Eq, (7.49) and Eq. (7.26) we have [1 + (S'/N)i]T 1 = |y(J l)2 (S/N)-j. (7,58) Then, substituting for S-j from Eg. (7.47), the efficiency for FM is (7.59) This is identical to the result obtained by Panter [23]. 100 7,3-5. Comparison of system efficiencies Eqs. (7.51), (7.53), (7.56), and (7.59) are plotted in Figure 24 in order to compare the efficiencies for the AM, SSB-AM-SC, SSB-FM, and FM systems. The efficiency is given as a function of (S'/N)0 with the modulation index, 6, as a parameter. For example, from the figure it is seen that, for FM with 6=2 and (S'/N)i = 30 db, the FM system re quires about 135 db more power than an ideal system with the same IF-to- baseband bandwidth ratio and the same output signal-to-noise ratio. From Figure 24, it is seen that SSB-AM-SC is an ideal system in the sense of trading bandwidth for output signal-to-noise ratio. Also, AM is the next best system, and SSB-FM and FM are the poorest systems according to this criterion. 101 g(db) Figure 24. Efficiencies of Various Systems CHAPTER VIII SUMMARY In this work a new approach to SSB signal design and analysis for communication systems has been presented. The key to this approach is the philosophy of using a modulated-signal generating function--the generating function bing any entire function. It was hypothesized in Chapter I that SSB signals were of the third basic modulation class, the first two being AM and FM. In Chapter II a brief review of analytic signal theory was pre sented, and this theory was used in successive chapters to facilitate the derivations. In Chapter III it was shown that signals of the SSB class could be generated by use of entire generating functions and that these sig nals were truly SSB signals regardless of the modulating process. Generalized formulae were derived which may represent upper SSB or lower SSB modulated signals. These formulae are analogous to those representing AM and FM signals. However, it is noted that any SSB signal is a com bination of AM and FM. Chapter IV gave some examples of well-known SSB signals, using the appropriate entire generating function to obtain their mathematical representation and, consequently, their physical realization. The generating function concept, along with analytic signal theory, was used in Chapter V to obtain generalized formulae for the properties of SSB signals. The properties that were studied were: 102 103 1. Equivalent realizations for a given SSB signal 2. The condition for a suppressed-carrier signal 3. Autocorrelation function 4. Bandwidth (using various definitions) 5. Efficiency 6. Peak-to-average power ratio. The amplitude of the discrete carrier term was found to be equal to the absolute value of the entire function (associated with a particular SSB signal) evaluated at the origin and was not affected by the modulation. Furthermore, for suppressed-carrier SSB signals, the real and imaginary parts of the complex envelope are a unique Hilbert transform pair; otherwise, they are a Hilbert transform pair to within an additive con stant. In Chapter VI the properties for examples of various SSB signals were studied where stochastic modulation was assumed. The results were compared with those published in the literature where possible. In Chapter VII a comparison of AM, SSB-AM-SC, SSB-FM and FM systems was carried out. This was a comparison of the various modu lation schemes from the overall viewpoint of generation, transmission with additive Gaussian noise, and detection. Three figures of merit were used for comparison: 1. Output signal-to-noise ratios 2. Energy-per-bit of information 3. System efficiency. It was found that, for a given RF signal power, FM has the greatest post detection signal-to-noise ratio if the modulation index is large. For small index SSB-AM-SC is best, with SSB-FM and FM second, and AM is 104 is poorest. For the lease energy-per-bit of information, FM is best, followed by SSB-AM-SC, SSB-FM, and AM. When the systems are compared in terms of optimum trade-off between predetection bandwidth and post detection signal-to-noise ratio [i.e. system efficiency) SSB-AM-SC was found to be ideal, with AM second best, followed by SSB-FM and FM. In conclusion, the entire generating function concept should be helpful in obtaining new types of SSB signals, and the corresponding formulae for analyzing these signals will be helpful in classifying these signals according to their properties. However, one should also evaluate the overall system performance in the presence of noise to determine the usefulness of these signals. APPENDIX I PROOFS OF SEVERAL THEOREMS Theorem I If k(z) is analytic in the UHP, then the spectrum of k(t,0), denoted by F|<(w), is zero for all w < 0, assuming that k(t,0) is Fourier transformable. (This result is included in Theorem 95 of Titchmarsh [6] and in the work of Paley and Wiener [31].) Lemma to Theorem I If Wj(z) and W2(z) are analytic in the UHP, then W(z) W1(z)W2(z) is analytic in the UHP. Proof of the Lemma to Theorem I: Assume that W1(z) and W2(z) are analytic in the UHP, which implies that they are continuous. Then if W(z) satisifies the Cauchy-Riemann (C-R) relation for all z in the UHP, W(z) is analytic in the UHP. Given: Wx and W2 are analytic in UHP. Then W I = Uj + J'V1 => w2 = u2 + jV2 => 9U, 8V1 . 3U 1^ _ 1 in the UHP (I-la) 3X ay 3X 3y 8U2 3 V2 3V2 3U2 the (I-lb) = - in UHP 3X 3X 3y and these partial derivatives are continuous. To show: W = U + jV is analytic for all z in the UHP by showing in the UHP (I-2a) 3V __3U 3x --3y in the UHP (I-2b) 105 106 and that these partial derivatives are continuous. W = wrw2 = (Ux + j)(U2 + jV2) = (U^- v1v2) + j(v1u2 + V2U1) = U + jV. Then and aU = u, 3Uz + u, Mi. V, av2 - v, Ml ax 1 ax 2 ax 1 ax 2 ax av Vi 3U9 u2 aVi v2 aUi + u1 a V2 _ - + + 3y sy ay ay ay By substituting Eqs. (I-1 a) and (I-lb) into Eq. (1-4), 3V1 aV aU2 all = Un - + U0 i- + V, ay ax ax av2 ax + v. ax (1-3) (1-4) (1-5) But Eq. (1-5) is identical to Eq. (1-3) and the partial derivatives are continuous. Thus, the condition of Eq. (I-2a) is satisfied. Also, and 3V 3U2. + y Ml _l u Ml x 11 Ml = V, + V, + u, ax ax ax ax ax Then aU 3y = u, au2 ay + u. alii ay - v, av2 ay - Vc av1 ay ay \ ax ax ax ax (1-6) (1-7) and all the partial derivatives are continuous. By comparing Eq. (1-6) with Eq. (1-7) it is seen that the condition of Eq. (I-2b) is satisfied. Therefore W(z) is analytic in the UHP. 107 Proof of Theorem I: Given: k(z) is analytic in the UHP and eJ)Z is analytic in the UHP for 00 F(w) = | k(x,0)e"jwXdx = 0 Â¥ u> < 0. (1-8) By the Lemma k(z)eJuZ is analytic in the UHP for all k(z)e'ja)Zdz = 0 for c as shown in Figure 25 since k(z)e'J)Z is analytic in the UHP. Thus ^ k(z)e"JCl)Zdz k(x,0)eja)Xdx + lim f k(R sin e,R cos e)e R-* o jReJ6 RJeJ'6d6 . But for to < 0, lira I f k eR s1n V>Rcs Wed6| < lim f |k|eR s1" eRde R~**> R->o 108 and |k| < M, a constant, since k is analytic in the UHP, 1 im R-* ke'jwReje Rjejede| < M 1im f eR sin eRde = 0 R-x 'ft Therefore F(oj) I k(x,0)e"JuXdx = 0 u> < 0 Theorem II If Z(z) is an analytic function of z in the UHP and if g(W) is an entire function of W, then g[Z(z)] is an analytic function of z in the UH z-plane. Proof of Theorem II: The C-R relations will be used to show that g[Z(z)] is analytic in the UH z-plane. Given: Z(z) = Ux(x,y) + jV^x.y) is analytic in the UH z-plane. This implies that 3U-, - ?h. . = 3 U, 3X 3y 3X sy (I-9a,b) in UHP and these derivatives are continuous there. g(W) U2(U1,V1) + JV2(Ux,Vx) is analytic in the finite W-plane. This implies that 3U0 3 V 2 *2 . 3 U i 3V1 3V2 -3U2 3tj7 dV1 (I-10a,b) in the finite W-plane and these derivative are continuous there, To show: That 3U2 3V2 3V2 -3U2 ay 3x 3y 3X (1-1la,b) 109 in the UH z-plane and that they are continuous in the UHP also. Now, 3V2 3V2 3Ux aV2 3V (1-12) 3y 3U1 3y 3 V1 3y Substituting Eqs. (1-9) and (1-10) into Eq. (1-12), Eq. (1-12) becomes (1-13) Thus the condition given by Eq. (I-11 a) is satisfied in the region where these derivatives exist and are continuous. Similarly, 3V2 3V2 311} 3V2 3Vi (1-14) 3X 3 U x 3X 3 V i 3X Substituting Eqs. (1-9) and (I-10) into Eq. (1-14), Eq. (1-14) becomes Thus the condition given by Eq. (I-llb) is satisfied in the region where these derivatives exist and are continuous. It is now argued that the derivatives exist and are continuous for z in the UHP. This is true because for any z in the UHP, including UH , z(z) may take on any value in the finite W plane. Also, the derivatives of Ux and V1 with respect to x and y exist and are continuous for z in the UHP, and the derivatives of U2 and V2 with respect to U} and \1 exist and are continuous anywhere in the finite W plane. Thus the con ditions given by Eqs. (I-11 a) and (I-llb) are satisfied, and g[Z(z)] is analytic in the UH z-plane. no Theorem III If h(z) is analytic for all z in the UHP and if F[h(x,0)] = (oo), then for u0 > 0 i{Re[h(xtO)eJwox]} . h F^u-uq) u > O) < Ur jg Fh ( -u-Uq ) u < -uc (1-16) Lemma to Theorem III If h(z) is analytic for all z in the UHP and if F[h(x,0)] = Fh(u), then FReh(u) = ^tRe[h(x,0)]} % Fh(u) u > 0 [F^uJ-j^TTk^u)! = [Fj!|(-u)+j2Trk1(-u)] u = 0 h Fh(-u) u < 0 (1-17) Proof of Lemma to Theorem III: From Eq. (1-23) h(x,0) = U(x,0) + j[G(x,0)+k1]. Thus Fh(u) r J U(x,0) + j [ (x, 0) + kjle'^dx U(x,0) e~JuXdx + j f U(xiO)e"JwXdx + jk, / e"j,xdx, Using Eg. (2.2) and the difinition for the Fourier transform, we F^(w) = FReh(a)) + j[-j sgn (w)] FReh(w) + j^irk^U) or FRehM *sFh(w) a) > 0 Ff-i(a))-j2-rrki6(ai) w s 0 Also, it is recalled that FReh(-) = FReh() This is seen from Re[h(x Thus from Eqs. (1-18) and (1-19) we obtain ,0)]eJtXdx Re[h(x,0)]e"J)Xdx FReh() JgF^(a)) a) > 0 [F^uO-J^irk^w)] = [F^i-wJ+jZTrk^i-w)] w = 0 %Ffi(-o)) a) < 0 Proof of Theorem III: By aid of Eg. (1-23) we have 111 obtain (1-18) (1-19) h(x,0)ejox = {U(x,0)+j[U(x,0)+k1]}ejx . 112 Then ReCh(x90)e^a>x3> = f (U(x,0) cos to0x [Uix.Oj+k^sin wox}eja)Xdx U(x.O) >s(ej,oX + e1"x) + J[0{x,O)+k1]yejx-e-;ilX)>e-jxdx 00 CO = h J U(x,0)e"J^"w^xdx + % f U(x,0)e"J'^a)+)^xdx 00 CO 00 00 + 3h C U(x,O)e'J^"w0^xdx jig f U(x,0)e'^w+w^xdx v/ J 00 oo 00 00 + Jhk1 f ej("w+)o)xdx jj5k1 f ej(_)"a)o)xdx and by using Eq. (2.2) and the Fourier transform of U(x,0), '{Re[h(x,0)e,;)oX]}= FRe^()-)0) + % FR(ahU+o>0) Reh' + 3h [-J sgn (o-jq)3 FReh(c-)0) j% [-j sgn (w+a)0)]FReh(cD+iD0) + 3h 2Trk1(-to+Jo) 3h 2-rrk16(-cj-u)o). (1-20) Using Eq. (1-17) from the Lemma to Theorem III to evaluate FReh(*) in Eq. (1-20), Eq. (1-20)becomes %F^(t->o) w > wq 0 |o)| < uo %F^(-o)-w0) w -w0 F{Re[h(x,0)eja)x]} = 113 Theorem IV If h(z) is analytic for all z in the UHP and i[h(x,0)] 5 F^U) where Fh(^) = 0 for all n ud, then for to0 > 0 F{Re[h(x50)ej)x]} %Fp,(-a)+u0) 0 0 < ) < 0)Q 10)1 > coo %Fh(co+co0) 0 > ) > aiQ (1-21) Proof of Theorem IV: The proof for Theorem IV is very similar to that for Theorem III. By the aid of Eq. (1-23) we have h(x,0)e'ja)x = {U(x,0) + j[U(x.0)+k1]}eJwx . Then F{Re[h(x,0)eJuoX]} = {U(x,0) cos co0x + [GU.CO+kJ sin + e~ja)x) jtUx.Oj+kjMe^-e"^)} e'j)Xdx 00 = % 00 U(xt0)e^u"u^xdx + % 00 00 U(x,O)e"^)+a>0^xdx CO U(x,0)e"^")xdx + jk J~ (xs0)e'^)+)^xdx CO - 3k KieJ(-+o)xdx + kjej(--.o)xdx 00 00 114 and by using Eq. (2.2) and the Fourier transform of U(x,0) i,{Re[h(x,0)e Ja)ox]} = ^ FReh^-^o) + % F^U+wo) - zh C-j sgn (w-wo)] FReh(oo-a)0) + zh [-j sgn(w+w0)]FRe(u)+u)0) - j% 2ttk^ (~c+jo) + zh 2?rki6( uju)q) (1-22) Using Eq. (1-17) from the Lemma to Theorem III to evaluate FRe^(*) in Eq. (1-22) and noting that = 0 for Q > u0 Eq. (1-22) becomes {Re[h(xs0)e"j)x]} = %F^(-)+Wo) 9 0 < ) < )Q ^F^w+coo) j |w| > u)g 0 > o) > JQ Theorem V If h(x,y) = U(x,y) + jV(x,y) is analytic in the UHP (including UH ) then h(t,0) = U(t,0) + jHU.Oh^] (1-23) or h(t,0) = [-V(t,0)+k2] + jV(t,0) (1-24) or h(t,0) = [-V(t,0)+k2] + j[U(t,0)+k1]- (1-25) where TT ki = -lim ( V(R cos e,R sin e)de a real constant (1-26) n R^ J o 115 and k2 = 11m [ U(R cos 0,R sin e)de a real constant. (1-27) ^ R-x > 0 Proof of Theorem V; By Cauchy's Theorem h(x) dz = 0 (1-28) Jc z-t for c as shown in Figure 26 since h(z) is analytic in the UHP, where t is real and finite. Figure 26. Contour of Integration Thus for e > 0 0 = lim Â£-0 t-e h(x,0) x-t dx + h(t+ee^e)ee'-l'ej Je eec d9 + I dx t+e 11. I h(Re^)RjeJe de R-x ReJ0-t or 0 = P J rh(x-P>. dx jirh(t.O) + lim f MReJ' ^R-J-e~- de X-t R^oo J DoJ0. ReJ -t 116 or for h(z) = U(x,y) + jV(x,y) 0 = P U(x,0) + jV(x,0) x-t dx Jtt[U(t,0) + jV(t,0)] + lim R-+00 [U(R cos e,R sin 0) + jV(R cos 0,R sin R(cos 0 + j sin 0) t Aside: calculate the term: t f R[U+jV][-sin 0 + j cos 0] I im j -- R-x J (R cos e-t) + jR sin 0 d0 1 im Â¡ R-* ~ R[U + jV][t sin 0 + j(R-t cos 0)] (R cos 0-t)2 + R2 sin2 0 d0 lim R400 {[Ut sin 0 + V(t cos 0-R)] + j[Vt sin 0 + U(R-t R 2t cos 0 + t2/R For finite t, lim (t cos 0 R) = -R, lim (R-t cos 0) = R, and R-Kjo R--K lim [R 2t cos 0 + t2/R] = R, Thus Eq. (1-30) becomes R-x 1 im R->oo {[Ut sin 0 VR] + j[Vt sin 0 + UR]} R d0 1 im * R-* 'Ut sin 0 R - V + j Vt sin 0 R + U de 0)]RjeJed0 (1-29) COS 0)]} de. (1-30) (1-31) Since U and V are real and imaginary parts of a function which is analytic 117 in the open UHP, U and V are finite as R for 0 < 0 < rr. Thus 1 im R-x Ut sin 6 R 0 and 1 im R-* Vt sin e R = 0 for 0 < e < tt, and Eq. (1-31) becomes 11m {-V + jU} de R-" 0 (1-32) Substituting the right side of Eq. (1-32) for the right-hand term on the right side of Eq. (1-29), Eq. (1-29) becomes O P / dx + j P V(x,0) d.. x-t ax - jirU(t.O) + irV(t,0) - 1im / Vde + j Tim I Ude R-x J R-^> J (1-33) Setting the real and imaginary parts of Eq. (1-33) equal to zero we get 0 = P U(x,0) x-t dx + ttV (t ,0) 1 im R-* Vde (I-34a) and 0 = P V(x,0) d.. x-t ax TT TrU(t,0) + lim f Ude R-* vi (I-34b) Thus and V(t,0) = U(t,0) + lim \ Vde 11 R-*= x U(t,0) = -V(t,0) + lim f Ude 17 R-x J (I-35a) (I-35b) 0 118 Now show that the integrals in Eq, (1-35) are bounded. Using Schwarz's inequality, 1 im R-* l TT IT l2de TT f (lU)2"-7lvU>2 0 where |V|max = max 11 im V(R cos e,R sin e)| for 0 < e < ir, which is R-x finite since h(z) is analytic in the UHP. Similarly 1 im R-x 1 IT TT f U de J o is bounded. Thus using Eqs. (I-35a) and (I-35b) h(t,0) 5 U(t,0) + jV(t,0) = U(t,0) + Â¡[UU.O+k!] = [-v(t,0)+k2] + jV(t,0) = [-v(ts0)+k2] + jtuit.oj+kj where and ki 1 im R-* r J 0 V de a finite real constant k2 1 TT 1 im R-**> TT de o a finite real constant. APPENDIX II EVALUATION OF eo(x + J'y) Assume: x and y are joint Gaussian random variables, bbth having zero mean values. To show: e^x ^ = e"^x +J2uxy~^y2l when x and y are joint Gaussian random variables. The joint density function is 1 p(x,y) 2lTaXCTy(l "P T- e 2ax2-ay2^p2) [ay2x2-2axavpXy+0x2y2] Then J[x+jy] _2TTaxay(l-p2)^ 00 00 r ,j(x+jy)e 2ax2ay2(1-p2) [ay2x2*2ax0ypxy+ax2y2] dxdy oo oo r - 2ax2(l-p2) x2-2(^~ py+jax2(l-p2)j x+ CTy2 u -oo co y2+y2a/(l-p2)) dxdy oo oo r' J 00 00 2ax2(l-p2) [x-k(y| [2 - 2ax2(l-p2) L | y k^y)+^y2+y2ax2(l-p2l| V dxdy where k(y) = ~ py + jox2(l-p2) ay 119 120 Then J[x+jy] 27raxcjy(l-p2)^ [i/ZtT ax(l-p2)^] 2ax2(l-p2) *- [-k2(y)+^2+y2 1 [y+L]2 E-L2+ay2ox2(l-p2)] /2ir e 2ay2 e 2cry2 dy where L = av2(l-j p) 7y Jy 0v Thus e J[x+jy] ___ (/27 oy) e 2oy2 /2t CTy or eJ (x+jy) = e-Js(ax2+j 2vxy-oy2} where and x2o-p2 yxy = xy . REFERENCES 1. J.R.V. Oswald, "The Theory of Analytic Band Limited Signals Applied to Carrier Systems," IRE' Trans. on Circuit Theory, vol. CT-3, December 1956, 2. F.F. Kuo and S.L. Freeny, "Hilbert Transforms and Modulation Theory," Proa. NEC, vol. 18, 1962. 3. E. Bedrosian, "The Analytic Signal Representation of Modulated Waveforms" Proa. 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Powers, "The Compatibility Problem in Single-Sideband Trans mission," Proc. IRE, vol. 48, August 1960. Comment: L.R. Kahn, same issue, p. 1504. 13. T.S. George, "Correlation Estimation in Noise-Modulation Systems by Finite Time Averages," IEEE Trans, on Instrumentation and Measurement3 vol. IM-14, March/June 1965. 14. F. Haber, "Signal Representation," IEEE Trans, on Communication Technology, vol. COM-13, June 1965. 121 15. 122 D. Middleton, Introduction to Statistical Communication Theory3 New York: McGraw-Hill, 1960. 16. R.E, Kahn and J.B. Thomas, "Some Bandwidth Properties of Simultaneous Amplitude and Angle Modulation," IEEE Trans, on Information Theory3 vol. IT-11, October 1965. 17. L. A. Wainstein and V.D. Zubakov, Extraction of Signals from Noise3 Englewood Cliffs, N.J.: Prentice-Hall, 1962. 18. H. Voelcker, "Toward a Unified Theory of Modulation-Part II: Zero Manipulation," Proo. IEEE3 vol. 54, May 1966. 19. J.E. Mazo and J. Salz, "Spectral Properties of Single-Sideband Angle Modulation," IEEE Trans, on Communication Technology3 vol. COM-16, February 1968. 20. R.E. Kahn and J.B. Thomas, "Bandwidth Properties and Optimum Demodu lation of Single-Sideband FM," IEEE Trans, on Communication Tech nology 3 vol. COM-14, April 1966. 21. A.H Nuttall and E. Bedrosian, "On the Quadrature Approximation to the Hilbert Transform of Modulated Signals," Proc. IEEE3 vol. 54, October 1966. 22. H.E. Rowe, Signals and Noise in Communication Systems3 Princeton, N. J.: Van Nostrand, 1965. 23. P.F. Panter, Uodulation3 Noise3 and Spectral Analysis3 New York: McGraw-Hill, 1965. 24. S.O. Rice, "Mathematical Analysis of Random Noise," B.S.T.J., vols. 23 and 24, 1944 and 1945 (Reprinted in Selected Papers on Noise and Stochastic Processes3 N. Wax, Dover Paperback, 1954). 25. S.O. Rice, "Statistical Properties of a Sine-Wave plus Random Noise," b.s.t.j.j vol. 27, January 1948. 26. S.O. Rice, "Noise in FM Receivers," Time Series Analysis3 M. Rosen blatt, Editor, New York: Wiley, 1963. 27. G. Raisbeck, Information Theory3 Cambridge, Mass.: M.I.T. Press (paperback), 1963. 28. C.E. Shannon, "The Mathematical Theory of Communciation," B.S.t.j.3 vol. 27, July and October 1948 (Also in paperback, Univ. of Ill. Press, 1963). 29. N.L. Wright and S.A.W. Jolliffe, "Optimum System Engineering for Satellite Communication Links with Special Reference to the Choice of Modulation Method," J. Brit. IRE3 May 1962. 123 30. W.G. Tuller, "Theoretical Limits on the Rate of Information," Proa, IREj vol. 37, 1949. 31. R.E.A.C. Paley and N. Wiener, "Fourier Transforms in the Complex Domain," Am. Math, Soa. Colloq. Publ. vol. 10, 1934. BIOGRAPHICAL SKETCH Leon Worthington Couch, II was born on July 6, 1941, in Durham, North Carolina, In June, 1959, he was graduated from Goldsboro High School, Goldsboro, North Carolina. The author received a degree of Bachelor of Science in Electrical Engineering from Duke University, Durham, North Carolina, in June, 1963; and in the following fall, he entered the University of Florida where he received a degree of Master of Engineering in August, 1964. In September, Mr. Couch continued his studies at the University of Florida, Department of Electrical Engineering, working toward the degree of Doctor of Philosophy. During his time of study at the University of Florida, the author held a Graduate Teaching Assistantship until August, 1966. At that time-he accepted a NASA Traineeship which he resigned in January, 1967, to accept the position of Research Associate in the Department of Electrical Engineering. Mr. Couch is married to the former Margaret Elizabeth Wheland. He is a member of Tau Beta Pi and Eta Kappa Nu and a student member of the Institute of Electrical and Electronics Engineers. In addition, the author holds a First Class Radiotelephone license and an Amateur Radio license as issued by the Federal Communications Commission. 124 This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial ful fillment of the requirements for the degree of Doctor of Philosophy. June, 1968 Dean, College of Engineering Dean, Graduate School Supervisory Committee: Chairman f//7aj Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: Couch, Leon TITLE: Synthesis and Analysis of Real Single Side Band... PUBLICATION 1968 DATE: I, L^>ov\ CoucM as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees-of the-University f-Elafida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant ol permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be llmittPtb those specifically allowed by "Fair Use" as prescribed by the teams,nf United States copyright legislatiop-fcf. Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate image- andTxf:;basfi"'versions'as appropriate and to provide and enhance access using search-software. cU-1 This grant of permissions prohibits use of the digitized versions for commercial use or profit. Signature of Copyright Holder Loon Gwc^i Printed or Typed Name of Copyright Holder/Licensee Personal Information Blurred Date of Signature 84 Using Eo. (7.16) and Eq. (7.22), we have 6 6 2(6+1) /2 / 1 (S/N)0 = Ij (26) TTT In2(2)+| 610(26)1,(26) (S/N) -f (7.23) for the case of SSB-FM plus Gaussian noise into a FM detector. The signal-to-noise output can also be obtained in terms of the unmodulated-signal-to-noise ratio [i.e. the carrier-to-noise power at the input). From Eq. (7.6) we obtain (S/N)-f = 10(26) (C/N)i (7.24) and Eq. (7.23) becomes (S/N)o = Il2(26) 6 62(6+1) /2 1 _ 10 2 (^ 6) Iq(26) + | 6I1(26) (C/N)i where (C/N)^ is the carrier-to-noise power ratio. (7.25) 7.1-4. FM system The signal-to-noise ratio at the output of a FM receiver for a FM signal plus narrow-band Gaussian noise at the input can be obtained by the same procedure as used above for SSB-FM. The factor e_m^ of Eq. (7.6) is replaced by unity, and the bandwidth of the input noise is given by Eq. (7.19). Then the output signal-to-noise ratio becomes (S/N)0 = 3 62(6+l) (S/N)i (7.26) 15. 122 D. Middleton, Introduction to Statistical Communication Theory3 New York: McGraw-Hill, 1960. 16. R.E, Kahn and J.B. Thomas, "Some Bandwidth Properties of Simultaneous Amplitude and Angle Modulation," IEEE Trans, on Information Theory3 vol. IT-11, October 1965. 17. L. A. Wainstein and V.D. Zubakov, Extraction of Signals from Noise3 Englewood Cliffs, N.J.: Prentice-Hall, 1962. 18. H. Voelcker, "Toward a Unified Theory of Modulation-Part II: Zero Manipulation," Proo. IEEE3 vol. 54, May 1966. 19. J.E. Mazo and J. Salz, "Spectral Properties of Single-Sideband Angle Modulation," IEEE Trans, on Communication Technology3 vol. COM-16, February 1968. 20. R.E. Kahn and J.B. Thomas, "Bandwidth Properties and Optimum Demodu lation of Single-Sideband FM," IEEE Trans, on Communication Tech nology 3 vol. COM-14, April 1966. 21. A.H Nuttall and E. Bedrosian, "On the Quadrature Approximation to the Hilbert Transform of Modulated Signals," Proc. IEEE3 vol. 54, October 1966. 22. H.E. Rowe, Signals and Noise in Communication Systems3 Princeton, N. J.: Van Nostrand, 1965. 23. P.F. Panter, Uodulation3 Noise3 and Spectral Analysis3 New York: McGraw-Hill, 1965. 24. S.O. Rice, "Mathematical Analysis of Random Noise," B.S.T.J., vols. 23 and 24, 1944 and 1945 (Reprinted in Selected Papers on Noise and Stochastic Processes3 N. Wax, Dover Paperback, 1954). 25. S.O. Rice, "Statistical Properties of a Sine-Wave plus Random Noise," b.s.t.j.j vol. 27, January 1948. 26. S.O. Rice, "Noise in FM Receivers," Time Series Analysis3 M. Rosen blatt, Editor, New York: Wiley, 1963. 27. G. Raisbeck, Information Theory3 Cambridge, Mass.: M.I.T. Press (paperback), 1963. 28. C.E. Shannon, "The Mathematical Theory of Communciation," B.S.t.j.3 vol. 27, July and October 1948 (Also in paperback, Univ. of Ill. Press, 1963). 29. N.L. Wright and S.A.W. Jolliffe, "Optimum System Engineering for Satellite Communication Links with Special Reference to the Choice of Modulation Method," J. Brit. IRE3 May 1962. 45 5*4-3. Equivalent-noise bandwidth the equivalent-noise bandwidth, Aw* for the continuous part of the power spectrum is defined by (2aw) 27 pg-scC0) 2tt Pg-SC(w)du = Rg_sc() (5.49) But Thus 00 Pg-Sc(O) = f Rg-Sc(T>dT on 00 (Aco) = g-sc (0) Rg-SC(-r)di Substituting for Rg_sc(T) by using Eq. (5.41) or Eq. (5.42) we obtain (noting once again that R^Ct) is even and Ryy(-r) is odd) (5.50) 5.5. Efficiency A commonly.Used definition of efficiency for modulated signals is [18] n = Sideband Power/Total Power. 5.51 table of contents Page ACKNOWLEDGMENTS iv LIST OF FIGURES viii KEY TO SYMBOLS x ABSTRACT xiii CHAPTER I. INTRODUCTION 1 II. MATHEMATICAL PRELIMINARIES 4 III. SYNTHESIS OF SINGLE-SIDEBAND SIGNALS 9 IV, EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN 18 4.1. Example 1: Single-Sideband AM with Suppressed-Carrier 18 4,2 Example 2: Single-Sideband PM . 19 4.3. Example 3: Single-Sideband FM 21 4.4. Example 4: Single-Sideband a 23 V.ANALYSIS OF SINGLE-SIDEBAND SIGNALS .... 28 51. Three Additional Equivalent Realizations 28 5.2. Suppressed-Carrier Signals 30 5.3. Autocorrelation Functions 38 5.4. Bandwidth Considerations 42 5.4-1. Mean-type bandwidth 43 5.4-2, RMS-type bandwidth 44 5.,4-3. Equivalent-noise bandwidth 45 5 5 Efficiency 45 5,6 Peak-to-Average Power Ratio 46 v 3 general for the whole set, and some outstanding members of the set will be chosen for examples to be examined in detail. It should be noted that Bedrosian has classified various types of modulation in a similar manner; however, he does not give a general repre sentation for single-sideband signals [3], 13 \F[q(l(t))e^^} i Figure 4 Voltage Spectrum of the Positive Frequency- Shifted Entire Function of the Analytic Signal The real upper single-sideband signal can now be obtained from the complex single-sideband signal, g[Z(t)]e'-*a,ot, by taking the real part. This is seen from Theorem III Theorem III, If h(z) is analytic for all z in the UHP and F[h(x,0)] e Fh(w), then for > 0, F{Re[h(x,0)eJwox]} ^sFp-! (oCQ ) O) > 0)g o 0) < 0),. J O),. This theorem is proved in Appendix I. Thus the upper single-sideband signal for a given entire function is XUSSB(t) = ReigCZUne^} = Re{[U(ReZ(t) ,lmZ(t)) + jV^ReZ(t),lmZ(t))]eJwot} = Re{[U(m(t) ,m(t)) + jV(mU) *m(t J)]eJa)^t} Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SYNTHESIS AND ANALYSIS OF REAL SINGLE-SIDEBAND SIGNALS FOR COMMUNICATION SYSTEMS By Leon Worthington Couch, II June, 1968 Chairman: Professor T. S. George Major Department: Electrical Engineering A new approach to single-sideband (SSB) signal design and ana lysis for communications systems is developed. It is shown that SSB signals may be synthesized by use of the conjugate functions of any entire function where the arguments are the real modulating signal and its Hilbert transform. Entire functions are displayed which give the SSB amplitude-modulated (SSB-AM), SSB frequency-modulated (SSB-FM), SSB envelope-detectable, and SSB square-law detectable signals. Both upper and lower SSB signals are obtained by a simple sign change. This entire generating function concept, along with analytic signal theory, is used to obtain generalized formulae for the properties of SSB signals Formulae are obtained for (1) equivalent realizations for a given SSB signal, (2) the condition for a suppressed-carrier SSB signal, (3) autocorrelation function, (4) bandwidth (using various-de finitions), (5) efficiency of the SSB signal, and (6) peak-to-average power ratio. The amplitude of the discrete carrier term is found to be xi i i BIOGRAPHICAL SKETCH Leon Worthington Couch, II was born on July 6, 1941, in Durham, North Carolina, In June, 1959, he was graduated from Goldsboro High School, Goldsboro, North Carolina. The author received a degree of Bachelor of Science in Electrical Engineering from Duke University, Durham, North Carolina, in June, 1963; and in the following fall, he entered the University of Florida where he received a degree of Master of Engineering in August, 1964. In September, Mr. Couch continued his studies at the University of Florida, Department of Electrical Engineering, working toward the degree of Doctor of Philosophy. During his time of study at the University of Florida, the author held a Graduate Teaching Assistantship until August, 1966. At that time-he accepted a NASA Traineeship which he resigned in January, 1967, to accept the position of Research Associate in the Department of Electrical Engineering. Mr. Couch is married to the former Margaret Elizabeth Wheland. He is a member of Tau Beta Pi and Eta Kappa Nu and a student member of the Institute of Electrical and Electronics Engineers. In addition, the author holds a First Class Radiotelephone license and an Amateur Radio license as issued by the Federal Communications Commission. 124 34 for the case of FM Gaussian noise [13]. Continuing with our SSB signals, it will now be shown that kx and k2 depend only on the entire function associated with the SSB signal and not on the n]odulation. From Theorem IV we have IT k = lim V[m,(R cos e,R sin e) m. (R cos e,R sin e)]de 1 IT D J 1 i R-* o and IT k, = lim / U[m.,(R co$ e,R sin e) m(R cos e,R sin )]de 17 R^ o where U and V are the real and imaginary parts of the entire function l1{z) = m^z) + jm^z) is the analytic function associated with the analytic signal Z(t) of m(t). It is seen that if lim m (R cos e,R sin e) = 0 0 < e < v (5.18a) R-x and lim mJR cos e,R sin e) = 0 0 < e < tt (5.18b) R-* then kx and k2 depend only on U and V of the entire function and not on m. Thus we need to show that Eq. (5.18a) and (5.18b) are valid. By the theory of Chapter III there exists a function Z^z) = m^z) + jm^z) which is analytic in the UHP such that (almost everywhere) 11^ Zx(t + jy) y = Z(t) = m(t) + jm(t) where the Fourier transform of Z(t), F(ai), is l2(-, ). Then we have 51 The peak-to-average power ratio for the SSB-AM-SC signal follows from Eq. (5.56c), and it is f[m(t)]2 + [ii(t)]2} tpeak SC-SSB-AM 2 (6.10) m 6.2. Example 2: Single-Sideband PM The SSB-PM signal has a discrete carrier term. This is shown by calculating the constants k1 and k2. Substituting Eq. (4.5b) into Eq. (5.4) we have K .-Inin f cos e-R sin e)s1n [m,(R cos e.R sir, e)]de. 1 7r R-K. J 1 But from Eqs. (5.18a) and (5.18b) lim m^R cos e,R sin e) = 0 for R-X 0 < e < Ti and lim m.(R cos e, R sin e) = 0 for 0 < e < tt. Thus R-**> kl = 0. (6.11) Likewise, substituting Eq. (4.5a) into Eq. (5.5) we have IT k = C e" cos 0 de = 1. (6.12) 2 TT J 0 Thus the SSB-PM signal has a discrete carrier term since k2 = 1 f 0. There are equivalent representations for the SSB-PM signal since k and k are not both zero. For example, for the upper sideband signal, 1 2 equivalent representations are given by Egs. (5.7) and (5.8). It is noted that Eq. (5.8) is identical to Eq. (5.9) for the USSB-PM signal 49 since 1 im m(R cos e,R sin e) = 0 for 0 < e < ir from Eq (5.18a). Further- R-* more, since both k and k2 are zero, the equivalent realizations for the SSB signals, as given by the equations in Section 5.1, reduce identically to the phasing method of generating SSB-AM-SC signals (which was given previously in Figure 8). The autocorrelation for the SSB-AM-SC signal is readily given by use of Eq. (4.2a) and Eq (5.20). Thus y-(m(t) ,m(t)) = m(t). (6.3) Then the autocorrelation of the suppressed-carrier USSB-AM signal is given via Eq. (5.43b), and it is RXU-SC-SSB-AM^ Rmm^ cos WT Rmm^x^ sin oT* (6.4) Likewise, by use of Eq. (5.44b) the autocorrelation for the suppressed- carrier LSSB-AM signal is RXL-SC-SSB-AM^ WT) cos oT r(t) mm sin From Eq. (6.4) it follows that the spectrum of the USSB-AM-SC signal is just the positive-frequency spectrum of the modulation shifted up to on and the negative-frequency spectrum of the modulation shifted down to oj0. That is, there is a one-to-one correspondence between the spectrum of this SSB signal and that of the modulation. This is due to the fact that the corresponding entire function for the signal, g(W) = W, is a linear function of W. Consequently, the bandwidths for this SSB 36 For e = 0 or 0 = tt Z (+>) ,0 = 0 Tim Â¡Z (Reje) R->~ Z(-oo) = 0 since Z(t) e L (-*>, ). Then lim |Z1(ReJ0)| = 1 im |Z?L(R cos 0, R sin 0) | = 0 0 < 0 < rr R-x Rx which shows that Eqs. (5.18a) and (5.18b) are valid statements. Thus, the presence (kx and k2 not both zero) or the absence (kx = k2 = 0) of a discrete carrier depends only on the entire funtion associated with the SSB signal and not on the modulation. Furthermore, it is seen that the amplitude of the discrete carrier is given by the magnitude of the entire function evaluated at the origin (of the W plane), and the power in the discrete carrier is one-half the square of the magnitude. For every generalized USSB signal represented by Eq. (3.5), there exists a corresponding sppressed-carrier USSB signal: XUSSB-SC^ = ,^Km('t) ,m(t)) cos w0t Â¥-(m(t) ,m(t)) sih wot (5.19) 43 that the suppressed-carrier formulae are needed' instead of the "total sig nal" formulae since, from the engineering point of view, the presence or absence of a discrete carrier should not change the bandwidth of the sig nal Various definitions of bandwidth will be used [16, 17], 5.4-1. Mean-type bandwidth Since the spectrum of a SSB signal is one-sided about the carrier frequency, the average frequency as measured from the carrier frequency is a measure of the bandwidth of the signal: f wPg_ScU)du j Rg_sc() oo - = *' (5 45) CO v ' f Pg-SC^^ Rg-SC(O) 00 where Pg_^c(w) is the power spectral density of g$c(m(t) ,m(t))and the prime indicates the derivative with respect to t. The relationship is valid whenever Rg_sg(0) and Rg_$c^ exist. Substituting Eg. (5.41) into Eq. (5.45) we have f[R' (0) + JR' (0)] j trtr trtr " a'1 o 2[RW(0) + jRyyjO)] But it recalled that Ryy(x) is an even function of t and, from Chapter II, RyU(t) is an odd function of t. Then Ryy(O) = Ryy(O) = 0 and it follows that Ryy.(0) Rw(o) Ryy-(o) Rw(o) (5.46) 72 Likewise, substituting Eq. (4.10a) into Eq. (5.5), we have k2 =1. (6.53) Thus the SSB-a signal has a discrete carrier term. It follows that equivalent representations for the SSB-a signal are possible since k2 f 0. This is analogous to the discussion on equiva lent representations for SSB-PM signals (Section 6.2) so this subject will not be pursued further. The autocorrelation function for the SSB-a signal will now be ob tained in terms of Rmm(T) Using Eq. (5.21) and Eq. (4.10b) we have Rw(t) = [eam(t) sin am(t)][eam(t"T) sin am(t-t)] or r (t) = 53{ea[m(t)+m(t-T)]} {eja[m(t)-m(t-T)] _eja[m(t)+m(t-T)]} + %{ea^(t)+m(t-r)]} {_eja[-m(t)-m(t-T)] + eja[-m(t)+m(t-r)]}. (6.54) The density function of m(t) has to be specified in order to carry out this average. It is recalled that m(t) is related to the modulating signal e(t) by the equation: m(t) In [1+(t)H . Now assume that the density function of the modulation is chosen such that m(t) is a Gaussian random process of all orders. Eq. (6.54) can then be evaluated by the procedure that was used to evaluate Eo. (6.16). 30 Using Eq. (5.2) the second equivalent representation is XUSSB^ = t-V(m(t),m(t))+k2] cos w0t V(m(t),m(t)) sin w0t. (5.8) Using Eq. (5.3) the third equivalent representation is XUSSB^ = iv(m(t;) iti(t))+k23 cos (o0t [U(m(t) ,m(t))+k1], sin u)Qt (5.9) Likewise the three lower SSB signals, which are equivalent to Eq. (3.8), are It should be noted, however, that if for a given entire function k: and k2 are both zero, then all four representations for the USSB or the LSSB signals are identical since by Theorem V, U = -V and V = 0 under these conditions. 5.2. Suppressed-Carrier Signals The presence of a discrete carrier term appears as impulses in the (two-sided) spectrum of transmitted signal at frequencies w0 and - depending on whether the carrier term is cos a)0t, sin u)0t, or a com bination of the two. Thus the composite voltage spectrum of the modulated 63 This has the same numerical value as that obtained from the result given by Mazo and Salz [19]. The result may also be compared to that given by Kahn and Thomas for the SSB-PM signal with sinusoidal modulation [20]. From Eg. (19) of their work ^rms)$SB-,PM-S = ua^ (6.30) where wa is the frequency of the sinusoidal modulation and 6 is the modu lation index. For comparison purposes, equal power will be used for m(t) in Eq. (6.30) as was used for m(t) in Eq. (6.29). Then Eq. (6.30) becomes (k>rms)ssB-PM-S = ^ 3 *^o (6.31) Thus it is seen that for Gaussian modulation the rms bandwidth is propor tional to the power in m(t) when the power is large (ip0 > > tt/4), and for sinusoidal modulation the rms bandwidth is proportional to the square root of the power m(t). The efficiency for the SSB-PM signal with Gaussian modulation will now be obtained. Substituting Eo. (6.22) into Eq. (5.54) we have e2+ra-i "SSB-PM-GN + or - i p"2^m nSSB-PM-GN (6.32) where is the noise power of m(t). The peak-average to average power ratio for Gaussian m(t) is given 25 and V3(m(t)9m(t)) = em^ sin (am(t)). (4,10b) Substituting Eqs. (4.10a) and (4.10b) into Eq. (3.5), the SSB-a signal is XUSSB-a^ eam^ cos (om(t)) cos w0t - eam^ sin .(am(t)) sin ajQt or XUSSB-a^ ~ cos (0t + (4.11) In terms of the input audio waveform, Eq. (4.11) becomes XUSSB-a(t) = ealn[1+e(t)] cos ()Qt + an[l+e(t)]) or XUSSB-a^ = [l+e(t)]a cos u0t + a1n[1+e(t)]). (4.12) For a = 1 we have an envelope-detectable SSB signal, as is readily seen from Eq. (4.12). Voelcker has recently published a paper demon strating the merits of the envelope-detectable SSB signal [11]. The real ization of Eq. (4.12) is shown in Figure 11. For a = 1/2 we have a square-law detectable SSB signal. This type of signal has been studied in detail by Powers [12]. Figure 12 gives the block-diagram realization for the square-law detectable SSB exciter. 71 consequently, infinite pwer. In other words, the system does not have a steady-state output condition if the input has a power around oi =0. Thus, this system ia actually conditionally stable, the output being bounded only if the input power spectrum has a slope greater than or equal to +6 db/octave near the origin (and, consequently, zero at the origin) as seen from Eq. (6,50). It is interesting to note that for the case of FM, ejm^ is stationary regardless of the shape of the spectral density Pqq(ua). This is due to the fact that ejm^ is bounded regardless of whether m(t) is bounded or not. From Eq. (6.50) we can readily obtain Rmm(ir) for any input process e(t) which has a bounded output process m(t). Thus oo WO = J- f ^-ej3T du (6.51) IT J tl) 00 Furthermore, R^m(0), Rmm(T), and Rmm(0) may be obtained in terms of Pee(w). By substituting for these quantities in the equations of Section 6.2, the properties of a SSB-FM signal can be obtained in terms of the spectrum of the modulating process. 6.4. Example 4: Single-Sideband g The SSB-a signal has a discrete carrier term. This is readily shown by calculating the constants kT and k2. Substituting Eq. (4.10b) into Eq. (5.4) we have IC 1 11m f e"MR cos e*R s1n 6> sin am, (R cos 9.R sin 8)d8 . 1 n R-Mo J 1 .0 But lim m^R cos e, R sin e) = 0, for 0 < e s: tt and lim m^R cos e, R-x R-x R sin e) = 0 for 0 Â£ e $ it. Thus kx =0. (6.52) CHAPTER VIII SUMMARY In this work a new approach to SSB signal design and analysis for communication systems has been presented. The key to this approach is the philosophy of using a modulated-signal generating function--the generating function bing any entire function. It was hypothesized in Chapter I that SSB signals were of the third basic modulation class, the first two being AM and FM. In Chapter II a brief review of analytic signal theory was pre sented, and this theory was used in successive chapters to facilitate the derivations. In Chapter III it was shown that signals of the SSB class could be generated by use of entire generating functions and that these sig nals were truly SSB signals regardless of the modulating process. Generalized formulae were derived which may represent upper SSB or lower SSB modulated signals. These formulae are analogous to those representing AM and FM signals. However, it is noted that any SSB signal is a com bination of AM and FM. Chapter IV gave some examples of well-known SSB signals, using the appropriate entire generating function to obtain their mathematical representation and, consequently, their physical realization. The generating function concept, along with analytic signal theory, was used in Chapter V to obtain generalized formulae for the properties of SSB signals. The properties that were studied were: 102 107 Proof of Theorem I: Given: k(z) is analytic in the UHP and eJ)Z is analytic in the UHP for 00 F(w) = | k(x,0)e"jwXdx = 0 Â¥ u> < 0. (1-8) By the Lemma k(z)eJuZ is analytic in the UHP for all k(z)e'ja)Zdz = 0 for c as shown in Figure 25 since k(z)e'J)Z is analytic in the UHP. Thus ^ k(z)e"JCl)Zdz k(x,0)eja)Xdx + lim f k(R sin e,R cos e)e R-* o jReJ6 RJeJ'6d6 . But for to < 0, lira I f k eR s1n V>Rcs Wed6| < lim f |k|eR s1" eRde R~**> R->o CHAPTER IV EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN Specific examples of upper single-sideband signal design will now be presented* Entire functions will be chosen to give signals which have various distinct properties. In Chapter VI these properties will be ex amined in detail. Only upper sideband examples are presented here since the corresponding lower sideband signals are given by the same equation except for a sign change (Eq. (3.5) and Eq. (3.8)). 4.1. Example 1: Single-Sideband AM With Suppressed-Carrier This is the conventional type of single-sideband signal that is now widely used by the military, telephone companies, and amateur radio operators. It will be denoted here by SSB-AM-SC, Let the entire function be 9l(W)=W (4.1) and let m(t) be the modulating signal. Then substituting the corresponding analytic signal for W gx(Z(t)) = m(t) + jm(t) or (m(t),iii(t)) = m(t) and Vx(m(t),m(t)) = m(t). (4.2 a,b) 18 100 7,3-5. Comparison of system efficiencies Eqs. (7.51), (7.53), (7.56), and (7.59) are plotted in Figure 24 in order to compare the efficiencies for the AM, SSB-AM-SC, SSB-FM, and FM systems. The efficiency is given as a function of (S'/N)0 with the modulation index, 6, as a parameter. For example, from the figure it is seen that, for FM with 6=2 and (S'/N)i = 30 db, the FM system re quires about 135 db more power than an ideal system with the same IF-to- baseband bandwidth ratio and the same output signal-to-noise ratio. From Figure 24, it is seen that SSB-AM-SC is an ideal system in the sense of trading bandwidth for output signal-to-noise ratio. Also, AM is the next best system, and SSB-FM and FM are the poorest systems according to this criterion. e(t) Figure 12. Square-Law Detectable USSB Signal Exciter 15 as in the upper single-sideband synthesis. Then the Fourier transform of the down-shifted complex baseband signal is F[g(St))e-ot] F[g(Zt))] * - Fn {W ) (u>+\on) or [g(Z(t))ej)ot] = Fg(u+u0) a), > 0. (3,7) This spectrum is illustrated in Figure 6. Figure 6. Voltage Spectrum of the Negative Frequency- Shifted Entire Function of an Analytic Signal Theorem IV: If h(z) is analytic for all z in UHP and F[h(x,0)] e F^(oj) where Fh(fi) = 0 for all n > go0, then for oj0 > 0 y{Re[h(x,0)e'JUJx] ^(-W+uJg) 0 0 < j < )q M > uo JgFh (tD+u)0} 0 > oo > - 76 tern is taken to be a system which has optimum trade-off between predetection signal bandwidth and postdetection signal-to-noise ratioo) Comparison of AM, FM, SSB-AM-SC, and SSB-FM systems will be made using these three figures of merit. It is clear that these comparisons are known to be valid only for the conditions specified; that is, for the given modulation density function, and detection schemes which are used in these comparisons. 7.1, Output Signal-to-Noise Ratios 7.1-1. AM system Consider the coherent receiver as shown in Figure 16 where the input AM signal plus narrow-band Gaussian noise is given by X(t) + n.Â¡(t) = {A0[l + 6 sin tmt] cos co0t} + ixc(t) cos )Qt xs(t) sin cdQt} (7.1) where X(t) is the input signal, n-Â¡(t) is the input noise with a flat spec trum over the bandwidth 2wm> and 6 is the modulation index. X(t)+nj(t) Low Pass Fi 1 ter AC Couple 2k cos wqL Figure 16. AM Coherent Receiver Output Then the output signal-to-noise power ratio, where A0k6 sin ojmt is the output signal, is given by (S/N) 0 = 6 2 1 + %62 (S/N)1 (7.2) 79 The input signal plus noise is given by X(t) + n^(t) A0e"^^^ cos [o>0t + m(t)] + n-j(t) (7.6) where A0 ~ The amplitude of carrier u>o The radian frequency of the carrier m(t) = D /t v(t) dt m(t) = m(t) ~= The Hilbert transform of m(t) nj(t) Narrow-band Gaussian noise with power spectral density F0 over the (one-sided spectral) IF band and v(t) is the modulation on the upper SSB-FM signal. The independent narrow-band Gaussian noise process may be represented by n^(t) = R(t) cos [w0t + 4>(t)J = xc(t) cos w0t x$(t) sin w0t where xs(t) = xc(t) since the IF passes only the frequencies on the upper sideband of the carrier frequency. Then the phase of the detector output is obtained from Eq. (7.6) and is ;p(t) = k tan which reduces to A0e~^ sin m(t) + R(t) sin A0e^ cos m(t) + R(t) cos (7.7) xp(t) km(t) + k tan^ R(t) sin [ A0e-^ + R(t) cos [m(t) (7.8) where k is a constant due to the detector. The detector output voltage is given by Eq. (7.8) is identical to the phase output when the input is conventional FM plus noise except for the factor e_n1' . 38 n(t) = s(t) cos wot s(t) sin wot. (5.24) Thus Eq. (5.22) checks with Eq, (5.24) where it = s(t), and Eq. (5.23) checks also where -V e s(t). The corresponding representations for LSSB suppressed-carrier signals are given by XLSSB-SC^ = -y-(m(t),rn(t)) cos w0t + tt(m(t),m{t)) sin w0t (5.25) and XLSSB-SC^ = "'^'(m(t) ,m(t)) cos w0t + Â¥(m(t) ,m(t)) sin w0t (5.26) where it and-V-are given by Eq. (5.20) and Eq. (5.21). This representation also checks with that given by Haber for pro cesses with spectral components only for |w| < wo which is n(t) = s(t) cos wot + Â§(t) sin w0t. (5.27) 5.3. Autocorrelation Functions The autocorrelation function for the generalized SSB signal and the corresponding suppressed-carrier SSB signal will now be derived. Using the result of Chapter III, it is known that the generalized upper SSB signal can be represented by XUSSB j(w0t+ (t) = Re{g(m(t),m(t))e' (5.28) 103 1. Equivalent realizations for a given SSB signal 2. The condition for a suppressed-carrier signal 3. Autocorrelation function 4. Bandwidth (using various definitions) 5. Efficiency 6. Peak-to-average power ratio. The amplitude of the discrete carrier term was found to be equal to the absolute value of the entire function (associated with a particular SSB signal) evaluated at the origin and was not affected by the modulation. Furthermore, for suppressed-carrier SSB signals, the real and imaginary parts of the complex envelope are a unique Hilbert transform pair; otherwise, they are a Hilbert transform pair to within an additive con stant. In Chapter VI the properties for examples of various SSB signals were studied where stochastic modulation was assumed. The results were compared with those published in the literature where possible. In Chapter VII a comparison of AM, SSB-AM-SC, SSB-FM and FM systems was carried out. This was a comparison of the various modu lation schemes from the overall viewpoint of generation, transmission with additive Gaussian noise, and detection. Three figures of merit were used for comparison: 1. Output signal-to-noise ratios 2. Energy-per-bit of information 3. System efficiency. It was found that, for a given RF signal power, FM has the greatest post detection signal-to-noise ratio if the modulation index is large. For small index SSB-AM-SC is best, with SSB-FM and FM second, and AM is xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008248500001datestamp 2009-02-09setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Synthesis and analysis of real single-sideband signals for communication systems.dc:creator Couch, Leon Worthington, IIdc:publisher Leon Worthington Couch, IIdc:date 1968dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082485&v=0000116961426 (oclc)000955703 (alephbibnum)dc:source University of Floridadc:language English 62 Thus Rmm(T) = "J i^o e 2 T j 0T or Rmm(T) ~ " Rmm(T) a n . From Ea. (6.27) it follows that Rmm() = -Vo2 and from Ea. (6.28) we have Rmm(O) s ~pzr /2tt Substituting these two equations into Ea. (6.25) we get ( ta rms 2 1 e~Z>po Thus if m(t) has a Gaussian spectrum and if the modulation has density function, the SSB-FM signal has the rms bandwidth: (wrms^sSB-PM-GN 2i|>0az[4(ip0/;ir) + 1] 1 e where t|>0 is the total noise power in m(t) a4is the "variance" in the spectrum of m(t). (6.28) Gaussian (6.29) Page VIII. SUMMARY 102 APPENDIX L PROOFS OF SEVERAL THEOREMS 105 II. EVALUATION OF e j(x + Jy^ 119 REFERENCES 121 BIOGRAPHICAL SKETCH 124 vi i 97 7.3. System Efficiencies The third figure of merit which will be used to compare systems is the system efficiency, defined by Transmitted power required for an ideal system 6 = Transmitted power required for an actual system = sj/S-j (7.47) where the ideal system is taken to be a system which has optimum trade off between predetection signal bandwidth and postdetection signal-to- noise ratio. This concept is used by Wright and doll iffe to compare SSB-AM-SC and FM systems [29]. Here, it will be extended to AM and SSB-FM systems. The trade-off between predetection signal bandwidth and post detection signal-to-noise ratio for an ideal system is obtained by equating the predetection capacity to the postdetection capacity since an ideal system does not lose information in the detection process [30]. Thus (B/2it) log2 [1 + (S'/N)i] = (b/2ir) log [1 + (S'/N)0] (7.48) where B is the IF bandwidth b is the baseband bandwidth (S'/N)-} is the input signal-to-noise ratio for the ideal system (S'/N)0 is the output signal-to-noise ratio for the ideal system The prime is used here to denote the ideal system. Eq. (7.48) reduces to (s7n)0 = [1 + (S'/N)]Y 1 (7.49) where y = B/b, the IF to baseband bandwidth ratio. The efficiency, e will now be calculated for various types of systems. 114 and by using Eq. (2.2) and the Fourier transform of U(x,0) i,{Re[h(x,0)e Ja)ox]} = ^ FReh^-^o) + % F^U+wo) - zh C-j sgn (w-wo)] FReh(oo-a)0) + zh [-j sgn(w+w0)]FRe(u)+u)0) - j% 2ttk^ (~c+jo) + zh 2?rki6( uju)q) (1-22) Using Eq. (1-17) from the Lemma to Theorem III to evaluate FRe^(*) in Eq. (1-22) and noting that = 0 for Q > u0 Eq. (1-22) becomes {Re[h(xs0)e"j)x]} = %F^(-)+Wo) 9 0 < ) < )Q ^F^w+coo) j |w| > u)g 0 > o) > JQ Theorem V If h(x,y) = U(x,y) + jV(x,y) is analytic in the UHP (including UH ) then h(t,0) = U(t,0) + jHU.Oh^] (1-23) or h(t,0) = [-V(t,0)+k2] + jV(t,0) (1-24) or h(t,0) = [-V(t,0)+k2] + j[U(t,0)+k1]- (1-25) where TT ki = -lim ( V(R cos e,R sin e)de a real constant (1-26) n R^ J o i o Since m(t) is generated by a physically realizable process, it con tains finite power for a finite time interval This, of course, is equiva lent to saying that m(t) is a finite energy signal or, in mathematical terms, it is a L2 function. By Theorem 91 of Titchmarsh, m(t) is also a member of the L2 class of functions almost everywhere [6]. Now the complex signal Z(t) is formed by Z(t) = m(t) + jm(t). (3,1) It is recalled that Z(t) is commonly called an analytic signal in the literature. By Theorem 95 of Titchmarsh there exists an analytic func tion (regular for y > 0), Z^z), such that as y -* 0 Zj(x + jy) Z(t) = m(t) + jm(t) x = t for almost all t and, furthermore, Z(t) is a L2 (-*>, function [6] It follows by Theorem 48 of Titchmarsh that the Fourier transform of Z(t) exists [6], Theorem I: If k(z) is analytic in the UHP then the spectrum of k(t,0), denoted by F^oj), is zero for all to < 0, assuming that k(t,0) is Fourier trans formable. For a proof of this theorem the reader is referred to Appendix I. Thus the voltage spectrum of Z(t) is zero for ui < 0 by Theorem I since Z(t) takes on values of the analytic function Z,(z) almost every where along the x axis. Furthermore, since Z(t) is an analytic signal-- that is, it is defined by Eq. (3=1)its voltage spectrum is given by CHAPTER III SYNTHESIS OF SINGLE-SIDEBAND SIGNALS Eq. (1.4), which specifies the set of single-sideband signals that can be generated from a given modulating waveform or process, will be derived in this chapter. The equation must be a real function of a real input waveform, m(t), since it represents the generating function of a physically realizable system--the single-sideband transmitter--and, in general, it is non-linear. Analytic signal techniques will be used in the derivation. It will be shown that if we have a complex function k(z) where k(z) is analytic for z = x + jy in the upper half-plane (UHP), then the voltage spectrum of k(x,0) k(t) is zero for w < 0. In order to synthesize real SSB signals from a real modulating waveform, an UHP analytic generating function of the complex time veal modulating process must be found regardless of the particular (physically realizable) wave form that the process assumes. Let m(t) be either the real baseband modulating signal or a veal function of the baseband modulating signal e(t), Then the amplitude of the voltage spectrum of m(t) is double sided about the origin, for ex ample, as shown by Figure 1. 9 60 The equivalent-noise bandwidth is obtained by substituting Eq. (6.18) into Eq. (5.50): (Ato) %{e2Rmm(r) cos [26mrT1(T)] 1} dt %[e^m-l] ^ or (Aw) ;(e2^i 1) SSB-PM-GN {e2Rmm(T) co$ [2Rrnm(T)] 1 }di (6.26) It is noted that the equivalent-noise bandwidth may exist when the formu lae for the other types of bandwidth are not valid because of the non existence of derivatives of Rmm(i:) at t = 0. It is obvious that the actual numerical values for the bandwidths depend on the specific autocorrelation function of the Gaussian noise. For example, the rms bandwidth of the SSB-PM signal will now be calculated for the particular case of Gaussian modulation which also has a Gaussian spectrum. Let 2 -0) 0 where Pm(u) is the spectrum of m(t) 4>0 = is the total noise power in m(t) o2 is the "variance" of the spectrum. The autocorrelation corresponding to this spectrum is Rmm(r) ~ Vr 1 2 2 -Ho t (6.27) 77 or (7.3) where (S/N)-j The input signal-to-noise power ratio (C/N)i = The input carrier-to-noise power ratio and the spectrum of the noise is taken to be flat over the IF bandpass which is 2a)m(rad/s). 7.1-2. SSB-AM-SC system Consider the coherent receiver (Figure 16) once again, where the input is a SSB-AM-SC signal plus narrow-band Gaussian noise. Then the input signal plus noise is X(t) + ni(t) = iA0[m(t) cos w0t m(t) sin o>0t]} + [xc(t) cos (O0t xs(t) sin oj0t] (7.4) where m(t) = 6 sin wt m and xs(t) = xc(t) if the IF passes only upper sideband components. The input noise is assumed to have a flat spectrum over the bandwidth )m. Then the output signal-to-noise power ratio, where AQk6 sin wmt is the output signal, is given by [23] (7.5) (S/N)Q = (S/N)i where the spectrum of the noise is taken to be flat over the IF bandpass which is cjjm(rad/s). It is interesting to note that the same result is obtained from a 61 The Hilbert transform of Rmm(T) is also needed and is obtained by the frequency domain approach. It is recalled from Chapter II that P*(o)) = mm j PmmU) > w > 0 , cu 0 j Pmn ) < 0 Then Wt) = ^im 2tr -U)2 t -00 iaiS ei#* ejwTdu> f e2^ eJ<*Tdw which reduces to fynm^) /2rrcr f St ~coz ... sin oox dw . This integral 1s evaluated by using the formula obtained from page 73, #18, of the 8ateman Manuscript Project, Tab1es of In teg ral fra ns forms, vol. 1 [5]: 00 / 1 ..2 eaX sin xy dx e ^ Erf(~4=. y) Z/a \2/a / Re a > 0 where Erf (x) * Jr f -t e z dt. equal to the absolute value of the entire generating function evaluated at the origin provided the modulating signal is AC coupled. Examples of the use of these formulae are displayed where these properties are evaluated for stochastic modulation. The usefulness of a SSB signal depends not only on the pro perties of the signal but on the properties of the overall system as well. Consequently, a comparison of AM, SSB-AM, SSB-FM, and FM systems is made from the overall viewpoint of generation, transmission with additive Gaussian noise, and detection. Three figures of merit are used in these comparisons: (1) Output signal-to-noise ratios, (2) Energy-per-bit of information, and (3) System efficiency. In summary, the entire generating function concept is a new tool for synthesis and analysis of single-sideband signals. xiv 12 Figure 3. Voltage Spectrum of an Entire Function of an Analytic Signal Now multiply the complex baseband signal g[Z(t)] by eJuJot to translate the signal up to the transmitting frequency, oi0. It is noted that g[Zj(z)] and eju)z for ojq > 0 are both analytic functions in the UH z-plane By the Lemma to Theorem I in Appendix I, g[Z; (z)]ev,l'Uo2 is ana lytic in the UH z-plane. Therefore, by Theorem I the voltage spectrum of g[Z(t)]eJU)ot is one sided about the origin. Furthermore, FCgUltiJeK4] = -LF[g(z(t))] * (- 7T = Fg (oo) 6(w-o)0) or F[gU(t))e^ot] = Fg(ld-u)q) o>0 > 0 (3.4) This spectrum is illustrated in Figure 4. CHAPTER V ANALYSIS OF SINGLE-SIDEBAND SIGNALS The generalized SSB signal, that was developed in Chapter III, will now be analyzed to determine such properties as equivalent gener alized SSB signals, presence or absence of a discrete carrier term, autocorrelation functions, bandwidths, efficiency, and peak-to-averajge power ratio. Some of these properties will depend only on the entire function associated with the SSB signal, but most of the properties will be a function of the statistics of the modulating signal as well. 5.1. Three Additional Equivalent Realizations Three equivalent ways (in general) for generating an upper SSB signal will now be found in addition to the realization given by Eq. (3.5). Similar expressions will also be given for lower SSB signals which are equivalent to Eq. (3.8). It is very desirable to know as many equivalent realizations as possible since ally orle of them might be the most econom ical to implement for particular SSB signal. Theorem V: If h(x,y) = U(x,y) + jV(x,y) is analytic in the HP (including UH) then h (t ,0) = U(t,0) + jiOU^+kj] (5.1) or h(t,0) = [4(t,0)+k2] + jV(t,0) (5.2) 28 39 where a uniformly distributed phase angle 4> has been included to account for the random start-up phase of the RF oscillator in the SSB exciter. Then, using Middleton's result [15], the autocorrelation of the USSB sig nal is RXU^ XUSSB^t+T^XUSSB^t^ %Re{eJa)TRg(T)} (5.29) where Rg(t) = g(m(t+x),m(t+T)) g*(m(t),m(t)) (5.30) and g(m(t),m(t)) = U(m(t),m(t)) + jV(m(t) ,m(t)). (5.31) The subscript XU indicates the USSB signal. For the generalized LSSB signal the corresponding formulae are XL$SB(t) = Re{g(m(t),m(t))e"J^t+
(5.32)and Rxl(t) = %Re{e"J"wTRg(T)}. (5.33) These equations can be simplified if we consider the autocorre lation for the continuous part of the spectrum of the SSB signal. The suppressed DC carrier version of g, denoted by g$c> will first be found in terms of g, and then the corresponding autocorrelation function Rg_sc(T) APPENDIX I PROOFS OF SEVERAL THEOREMS Theorem I If k(z) is analytic in the UHP, then the spectrum of k(t,0), denoted by F|<(w), is zero for all w < 0, assuming that k(t,0) is Fourier transformable. (This result is included in Theorem 95 of Titchmarsh [6] and in the work of Paley and Wiener [31].) Lemma to Theorem I If Wj(z) and W2(z) are analytic in the UHP, then W(z) W1(z)W2(z) is analytic in the UHP. Proof of the Lemma to Theorem I: Assume that W1(z) and W2(z) are analytic in the UHP, which implies that they are continuous. Then if W(z) satisifies the Cauchy-Riemann (C-R) relation for all z in the UHP, W(z) is analytic in the UHP. Given: Wx and W2 are analytic in UHP. Then W I = Uj + J'V1 => w2 = u2 + jV2 => 9U, 8V1 . 3U 1^ _ 1 in the UHP (I-la) 3X ay 3X 3y 8U2 3 V2 3V2 3U2 the (I-lb) = - in UHP 3X 3X 3y and these partial derivatives are continuous. To show: W = U + jV is analytic for all z in the UHP by showing in the UHP (I-2a) 3V __3U 3x --3y in the UHP (I-2b) 105 95 For (C/N)-j = 12 db, this reduces to 46 (6+1) M(S) log2 [1 + 48 2(+l)] It is recalled that Raisbeck obtained this result [27]. (7.46) 7.2-5. Comparison of energy-per-bit for various systems It is recalled that M( when the output system noise is Gaussian. The output noise is Gaussian for the AM, SSB-AM-SC, and FM systems [for FM, (C/N)i = 12 db >> 0 db]. Also, from Eq. (7.10) it is seen that the noise out of the SSB-FM system is Gaussian for small index (say 6 < 1). Thus, for the systems that are analyzed above, M(6) represents the ratio of the energy-per-bit for the actual system to the energy-per-bit for the ideal system. Then in db, M() gives the energy-per-bit required above the ideal system. The systems are compared in terms of energy-per-bit (db) above the ideal system in Figure 23, where Eqs. (7.40), (7.42), (7.44), and (7.46) have beqn plotted for the AM, SSB-AM-SC, SSB-FM, and FM systems. From this figure it is seen that the FM system is best, followed by SSB-AM-SC, SSB-FM and AM. Furthermore, the FM system is about 12 db worse than the ideal system. These comparisons are valid for output signal-to-noise ratios of about 25 db. In addition, Figure 23 specifies the modulation index to use for each type of system in order to minimize the energy required to transmit one bit of information. Figure 11. Envelope-Detectable USSB Signal Exciter no Theorem III If h(z) is analytic for all z in the UHP and if F[h(x,0)] = (oo), then for u0 > 0 i{Re[h(xtO)eJwox]} . h F^u-uq) u > O) < Ur jg Fh ( -u-Uq ) u < -uc (1-16) Lemma to Theorem III If h(z) is analytic for all z in the UHP and if F[h(x,0)] = Fh(u), then FReh(u) = ^tRe[h(x,0)]} % Fh(u) u > 0 [F^uJ-j^TTk^u)! = [Fj!|(-u)+j2Trk1(-u)] u = 0 h Fh(-u) u < 0 (1-17) Proof of Lemma to Theorem III: From Eq. (1-23) h(x,0) = U(x,0) + j[G(x,0)+k1]. Thus Fh(u) r J U(x,0) + j [ (x, 0) + kjle'^dx U(x,0) e~JuXdx + j f U(xiO)e"JwXdx + jk, / e"j,xdx, 33 XPM(t) = tAn cos (zr cos wat)] COS u)nt '0 'Wa a*'-1 -0' - [An sin (cos to t)] sin wnt. (5.16) To identify Eq. (5.16) with Eq (5.13) we have to find the DC terms of and f. (t) + c. e An cos (cos coat) i i o wa a and f0(t) + c2 = A0 sin (- cos u)at). wa These are c, = An cos ( cos w,t) i u M- a Ao C D / cos ( cos wat)dt v/ 0) a Vo<Â£> (5.17a) c. = A sin ( cos wat) 2 o>a _ A0 T o 0 J sin (^~ cos aiat)dt (5.17b) Then for sinusoidal frequency-modulation it is seen that the discrete carrier term has an amplitude of AQJ0(D/ul^) which may or may not be zero depending on the modulation index D/wa. Consequently, for FM it is seen that the discrete carrier term may or may not exist depending on the modulation. Prof. T. S. George has given the discrete carrier condition 69 where L is a linear time-invariant operator, then y(t) is strict-sense stationary if x(t) is strict-sense stationary and that y(t) is wide-sense stationary if x(t) is wide-sense stationary [4]. Since the integral is a linear operator, we need to show only that it is time-invariant, that is to show that y(t+e) = L Jx(t+E)j or t+e e(t1)dt1 = j e(t2+e)dtr This is readily seen to be true by making a change in the variable, letting t = t2 + e. Thus, if e(t) is stationary, then m(t) is stationary. Moreover, in the same way it is seen that if m(t) had been defined by t mj (t) = D 1 (t')dt' (6.47) tQ then m (tj is not necessarily stationary for e^t) stationary since the system is time-varying (i.e. it was turned on at tQ). But this should not worry us because, as Middleton points out, all physically realizable systems have non-stationary outputs since no physical process could have started out at t = - and continued without some time variation in the parameters D5]. However, after the "time-invariant" physical systems have reached steady-state we may consider them to be stationary processes provided there is a steady state. Thus by letting tQ -> - we are con sidering the steady-state process m(t) which we have shown to be stationary Now the autocorrelation of m(t) can be obtained by using power-spectrum techniques since m(t) has been shown to be stationary. From Eq. (6.46) 118 Now show that the integrals in Eq, (1-35) are bounded. Using Schwarz's inequality, 1 im R-* l TT IT l2de TT f (lU)2"-7lvU>2 0 where |V|max = max 11 im V(R cos e,R sin e)| for 0 < e < ir, which is R-x finite since h(z) is analytic in the UHP. Similarly 1 im R-x 1 IT TT f U de J o is bounded. Thus using Eqs. (I-35a) and (I-35b) h(t,0) 5 U(t,0) + jV(t,0) = U(t,0) + Â¡[UU.O+k!] = [-v(t,0)+k2] + jV(t,0) = [-v(ts0)+k2] + jtuit.oj+kj where and ki 1 im R-* r J 0 V de a finite real constant k2 1 TT 1 im R-**> TT de o a finite real constant. 56 and yxy 25 x(t)y(t) . Thus oxj ~ [m(t)-m(t-t )F = 2[am -R^ir)] cx2 [m(t)+m(t-i)]T 2[orn2+Rmm( r)] ox? [-m(t)-m(t-x )]2 = 2[am2+Rmm(x)] axl C-nri(t)+m(t-r)]2- 2[am2-RtTim(x)] and a/ [l(t)+l(t-,)]2 2[am2+Rnlm(r)] . From Chapter II it is recalled that Rm^(0) = 0 and R^r) = -R-m(x) - -Rmm(x) so that the y averages are yx v = [m(t)-m(t- r )][m(t)+m(t-i)] -2Rmm(t) iy yx v = [m(t)-m(t-x)][m(tT+m(t-x)] = 0 2J Mv = -[m(t)+m(t-r)][m(t)+m(tX)] = 0 and %y = -[m(t)-m(t-x)][m(t)+m(t-x)] = 2Rmrtl(x) . 104 is poorest. For the lease energy-per-bit of information, FM is best, followed by SSB-AM-SC, SSB-FM, and AM. When the systems are compared in terms of optimum trade-off between predetection bandwidth and post detection signal-to-noise ratio [i.e. system efficiency) SSB-AM-SC was found to be ideal, with AM second best, followed by SSB-FM and FM. In conclusion, the entire generating function concept should be helpful in obtaining new types of SSB signals, and the corresponding formulae for analyzing these signals will be helpful in classifying these signals according to their properties. However, one should also evaluate the overall system performance in the presence of noise to determine the usefulness of these signals. 59 where is the noise power of m(t) It is seen that Eq, (6,23) may or may not exist depending on the autocorrelation of m(t). The rms-type bandwidth can be obtained with the help of the second derivative pf Eq. (6.18): C(i) i-e2R(t) sin [2Rmm(T)]> 2[Rm(,)]2 *VV- + (-e2R(r) cos [2Rmm(T)]) 2[R^(t)]2 + t-e2R(l) sin [2Rmm(T)]> R^(t) + t e2R(T> cos [2Rmnl(-r)]> 2[R(Of + <-e2R"(T) sin [2R(t)]} 2R^(T)t(,) + { e2R" Rw() = e 2^m {RÂ¡>) 2C4(0)]2> (6.24) Substituting Eq. (6.24) and Eq. (6.22) into Eq. (5.48) we have for the rms-type bandwidth of the SSB-PM signal with Gaussian noise modulation: (wrms) /2{2[RmlO)f R"(0)} SSB-PM-GN 1 e -2iPm (6.25) This expression for the rms bandwidth may or may not exist depending on the autocorrelation of m(t). It is interesting to note that Mazo and Salz have obtained a formula for the rms bandwidth in terms of different para meters [19] However both of these formulae give the same numerical re sults, as we shall demonstrate by Eq. (6.29). 98 7.3-1, AM system For the AM system y = B/b = 2. Then setting Eq. (7.49) equal to Eq, (7.2) we have [1 + (S/N)i32 -1 1 + (S/N)i. (7.50) Substituting for S-Â¡ from Eq. (7.47), the efficiency for the AM system is obtained, and it is 3 = 2 1 1 + ig2 _(S'/N)i + 2_ (7.51) The AM efficiency will be compared to those for other systems in Section 7.3-5 as a function of (S'/N)i with the modulation index as a parameter. 7.3-2. SSB-AM-SC system For the SSB-AM-SC system y = B/b = 1. Then, equating Eq. (7.49) and Eq. (7.5), we have (S'/NJi = (S/N) -f (7.52) and substituting for Si using Eq. (7.47), the SSB-AM-SC efficiency is (7.53) 7.3-3. SSB-FM system For the SSB-FM system, using Eq. (7.20), = = 2 (6+1) /2 m ' 1^(26) 1 Iq2(26) Y (7.54) Figure 13* USSB-PM Signal ExciterMethod II Alo = Equivalent-Noise Bandwidth nr = Mean-Type Bandwidth * = The Convolution Operator ()* = The Conjugate of () () = The Hilbert Transform of () () = The Averaging Operator XI 1 REFERENCES 1. J.R.V. Oswald, "The Theory of Analytic Band Limited Signals Applied to Carrier Systems," IRE' Trans. on Circuit Theory, vol. CT-3, December 1956, 2. F.F. Kuo and S.L. Freeny, "Hilbert Transforms and Modulation Theory," Proa. NEC, vol. 18, 1962. 3. E. Bedrosian, "The Analytic Signal Representation of Modulated Waveforms" Proa. IRE, vol. 50, October 1962. 4. A. Papoulis, Probability,Random Variables,and Stochastic Processes3 New York: McGraw-Hill, 1965. 5 Bateman Manuscript Project, Tables of Integral Transforms, vols, 1 and 2, New York: McGraw-Hill, 1954. 6. E.C. Titchmarsh, Theory of Fourier Integrals, Second Edition, London: Oxford, 1948. 7. F.E. Terman, Electronic and Radio Engineering3 Fourth Edition, New York: McGraw-Hill, 1955. 8. DoE. Norgoard, "The Phase-Shift Method of Single-Sideband Signal Generation," Proc. ire, vol. 44, December 1956. 9. J.L. Dubois and J.S. Aagaard, "An Experimental SSB-FM System," IEEE Trans, on Communication Systems3 vol. CS-12, June 1964. 10. R.M. Glorioso and E.H. Brazeal, Jr., "Experiments in SSB-FM Communi cation Systems, IEEE Trans, on Communication Technology, vol. COM-13, March 1955. 11. H. Voelcker, "Demodulation of Single-Sideband Signals via Envelope Detection," IEEE Trans, on Communication Technology3 vol. COM-14 February 1966. 12. K.H. Powers, "The Compatibility Problem in Single-Sideband Trans mission," Proc. IRE, vol. 48, August 1960. Comment: L.R. Kahn, same issue, p. 1504. 13. T.S. George, "Correlation Estimation in Noise-Modulation Systems by Finite Time Averages," IEEE Trans, on Instrumentation and Measurement3 vol. IM-14, March/June 1965. 14. F. Haber, "Signal Representation," IEEE Trans, on Communication Technology, vol. COM-13, June 1965. 121 57 Therefore, using Eq. (6.17), Eq, (6.16) becomes -%{2[am -Rmm(r)] + j2[-2Rmm(-t)] 2[am2+Rmm(T)]} u e-^i2[om +Rmm(t)] + j20 2[am2+Rmm(T)]} _ ^ e-%{2[am2+Rmt11(T)] + j20 2[am2+Rrnm(t)]} + ^ eJs{2[o|T| -Rmm(T)3 + j2[2Rmm(x)] 2[am2+Rmm(i:)]} which reduces to (6.18) where W -SSB-PM-GN denotes the autocorrelation of the imaginary part of the entire function which is associated with the suppressed-carrier SSB- PM signal with Gaussian noise modulation. It is noted that Eq. (6.18) is an even function of t, as it should be, since it is the autocorrelation of the real function -V-(m(t) ,m(t)) Furthermore Ryy(O) is zero when Rmii)(0) 0, as it should be, since the power in any suppressed-carrier signal should be zero when the modulating power is zero. The autocorrelation of the USSB-PM signal is now readily obtained for the case of Gaussian noise modulation by substituting Eq. (6.18) into Eq. (5.42) and using Eq. (537): R XU-SSB-PM-GN (t) = % Re ejT{[e2R>(x) Cos (2Rmm(t))] (6.19) 42 form is relatively easy to obtain On the other hand Rg(x) may be calcu lated directly from g(m(t),m(t)) or indirectly by use of Ry^t), Rvv(x), Ruv(t), and Rvu(t). The autocorrelation functions for the generalized USSB and LSSB signals having a suppressed-carrier are readily given by Eq. (5.37) and Eg. (5.38) with kj = k, 0: (5.43a) (5.43b) (5.43c) and (5.44a) (5.44b) (5.44c) It follows that the power spectral density of any of these SSB signals may be obtained by taking the Fourier transform of the appro priate autocorrelation function presented above. 5.4. Bandwidth Considerations The suppressed-carrier autocorrelation formulae developed above will now be used to calculate bandwidths of SSB signals. It is noted RXL-SC Rw(t) cos O)0x + R^t) sin 000 W Rw(t) cos UQT + Rw(x) sin CjOq ' RXU-SC^ %Re{eJTRg_sc(T)} - Ryy.(t) cos o>qT Ryy( :) sin coot w t) cos wot Ryy(x) sin u)0t 81 or dn0(t) dt kem(t) % r___ ~Yo J~2F(con) K-w0) COS [(wn-w0)t + 0n] + ke(t) Ao CO z n=l ^2F{n) 2. sin C(n-u.0)t + 8]. Noting that {en} are independent as well as uniformly distributed and that the noise spectrum is zero below the carrier frequency, the output noise power is N o dn0(t) dt m F^d. + -- e2S dm (t) dt 2-it ^m F0dw o k2 e2m(t)' ^ m3 + k2 e2m(t) dm(t) Ao2 J 2 rr 3 /\ 2 no dt - _ (7.11) where () is the averaging operator and wm is the baseband bandwidth (rad/s) Now let v(t) = -Am cos tomt then, averaging over t, we have 2m(t) w, 2u/c| m 2tt m e26 cos dt = In(26) and 2m(t) dm(t) dt I (m6)2 [10(26) Iz(2)] 1^6 1,(26) (7.12) (7.13) 90 Figure 21. Output SignaT-to-Noise to Input Signal-to-Normalized- Noise Power Ratio for Various Systems DEDICATION The author proudly dedicates this dissertation to his parents, Mrs,. Leon Couch and the late Rev, Leon Couch, and to his wife, Margaret Wheland Couch, 66 The terms in the brackets are the USSB and LSSB parts of the suppressed- subcarrier signal m(t). But these USSB and LSSB parts are recognized as the well-known representation for a narrow-band Gaussian process. Thus m(t) is a narrow-band Gaussian process. Now the previous expressions for bandwidth, which assume that m(t) is Gaussian, may be used. The mean-type bandwidth for the multi plexed SSB-PM signal is then readily given via Eq. (6.23), and it is (<*>) M-SSB-PM-GN oo f eRaa(x) cos waA cos[RaaU) sin coaA]dA 00 e^a 1 (6.39) where tpa is the average power of the Gaussian distributed subcarrier modulation a(t). Obtained in a similar manner, the rms bandwidth is (rms^.ssB.pM. GN a W1) Raa(0) 1 e^a (6.40) and the equivalent-noise bandwidth is (aw) r[e2*a-l] M-SSB-PM-GN J' eRaa^T^ cos aT Cos[Raa(x) sin toax] (6.41) Thus, it is seen once again that the bandwidth depends on the spectrum of the modulation, actually the subcarrier modulation a(t). To obtain a numerical value for the rms bandwidth of the multi plexed SSB-PM signal assume that the spectrum of a(t) is flat over |o)| < w0 < )a. 86 Figure 19 Output to Input Signal-to-Noise Power Ratios for Several Systems 99 Note that for SSB-FM y and s are uniquely related to each other (by Eq, (7.54)), unlike the AM and SSB-AM-SC cases. Equating Eq. (7.49) and (7.23), we have [i + (S7N),]Y l = Then, substituting for Sn- from Eg. (7.47), the efficiency for SSB-FM is 32y |_I02(2S) + | 6l0(2)I1(2) (S'/N)i [1 + (SVN)iF 1 (7.56) 362i I02(26) + | 6I0(26)I1(26) (S/N)- (7.55) where y and 6 are uniquely related by Eq. (7.54). 7.3-4. FM system For the FM system, using Eq. (7.19), Y = 7T = 2(6+1). (7.57) m Thus for FM, y and 6 are uniquely related, as was the case in SSB-FM. Eauating Eq, (7.49) and Eq. (7.26) we have [1 + (S'/N)i]T 1 = |y(J l)2 (S/N)-j. (7,58) Then, substituting for S-j from Eg. (7.47), the efficiency for FM is (7.59) This is identical to the result obtained by Panter [23]. 19 Substituting Eqs. (4.2a) and (4.2b) into Eq. (3.5) we obtain the upper single-sideband signal: XUSSB-AM-SC^ cos o1 m(t) sin Jot (4.3) where m(t) is the modulating audio or video signal and m(t) is the Hil bert transform of m(t). It is assumed that m(t) is AC coupled so that it will have a zero mean. The upper single-sideband transmitter corresponding to the gene rating function given by Eq, (4.3) is illustrated by the block diagram in Figure 8. It is recalled that this is the well-known phasing method for generating SSB-AM-SC signals [7, 8], 4.2. Example 2: Single-Sideband PM Single-sideband phase-modulation was described by Bedrosian in 1962 [3]. To synthesize this type of signal, denoted by SSB-PM, use the entire function: (4.4) Let m(t) be the modulating audio or video signal. Then g2(Z(t)) = eJ(m^ + ^(t)) = or U2(m(t),iii(t)) = e-"i(t) cos m(t) (4.5a) 113 Theorem IV If h(z) is analytic for all z in the UHP and i[h(x,0)] 5 F^U) where Fh(^) = 0 for all n ud, then for to0 > 0 F{Re[h(x50)ej)x]} %Fp,(-a)+u0) 0 0 < ) < 0)Q 10)1 > coo %Fh(co+co0) 0 > ) > aiQ (1-21) Proof of Theorem IV: The proof for Theorem IV is very similar to that for Theorem III. By the aid of Eq. (1-23) we have h(x,0)e'ja)x = {U(x,0) + j[U(x.0)+k1]}eJwx . Then F{Re[h(x,0)eJuoX]} = {U(x,0) cos co0x + [GU.CO+kJ sin + e~ja)x) jtUx.Oj+kjMe^-e"^)} e'j)Xdx 00 = % 00 U(xt0)e^u"u^xdx + % 00 00 U(x,O)e"^)+a>0^xdx CO U(x,0)e"^")xdx + jk J~ (xs0)e'^)+)^xdx CO - 3k KieJ(-+o)xdx + kjej(--.o)xdx 00 00 Energy (db) Required Above Ideal System 35 1.0 2.0 Modulation Index () 3.0 4.0 Figure 23. Comparison of Energy-per-Bit for Various Systems 108 and |k| < M, a constant, since k is analytic in the UHP, 1 im R-* ke'jwReje Rjejede| < M 1im f eR sin eRde = 0 R-x 'ft Therefore F(oj) I k(x,0)e"JuXdx = 0 u> < 0 Theorem II If Z(z) is an analytic function of z in the UHP and if g(W) is an entire function of W, then g[Z(z)] is an analytic function of z in the UH z-plane. Proof of Theorem II: The C-R relations will be used to show that g[Z(z)] is analytic in the UH z-plane. Given: Z(z) = Ux(x,y) + jV^x.y) is analytic in the UH z-plane. This implies that 3U-, - ?h. . = 3 U, 3X 3y 3X sy (I-9a,b) in UHP and these derivatives are continuous there. g(W) U2(U1,V1) + JV2(Ux,Vx) is analytic in the finite W-plane. This implies that 3U0 3 V 2 *2 . 3 U i 3V1 3V2 -3U2 3tj7 dV1 (I-10a,b) in the finite W-plane and these derivative are continuous there, To show: That 3U2 3V2 3V2 -3U2 ay 3x 3y 3X (1-1la,b) 58 Likewise, the autocorrelation of the LSSB-PM signal may be obtained by using Eq. (5.38). The autocorrelation of the suppressed-carrier USSB-PM signal with Gaussian modulation is given by using Eq. (543a): RXU-SC-SSB-PM-GN^ = ** Re e^oT{[e2Rmm(0 cos (2Rmm(T)) 1] + j[e2lWT) cos (2Rmm(T)]: (6.20) Similarly, the autocorrelation of the suppressed-carrier LSSB-FM signal may be obtained by using Eq. (5.44a). The mean-type bandwidth will now be evaluated for the SSB-PM signal assuming Gaussian noise modulation. From Eq. (6.18) we obtain R^(t) = 1 r e2R"(x) cos (2Rmm(i)di 2J (t-x)2 Rvv(0) 2 it ,2Rmm(x) cos [2Rmm(x)]d/ (6.21) and from Eq. (6.18) Rw(0) = %[e2^ l] (6.22) where = om2 is the average power of m(t). Substituting Eqs. (6.21) and (6.22) into Eq. (5.46) we have the mean-type bandwidth for the Gaussian noise modulated SSB-PM signal: oo P J~ ~T e2FWA) cos[2Rmm(x)]dx A () SSB-PM-GN 32^m (6.23) 120 Then J[x+jy] 27raxcjy(l-p2)^ [i/ZtT ax(l-p2)^] 2ax2(l-p2) *- [-k2(y)+^2+y2 1 [y+L]2 E-L2+ay2ox2(l-p2)] /2ir e 2ay2 e 2cry2 dy where L = av2(l-j p) 7y Jy 0v Thus e J[x+jy] ___ (/27 oy) e 2oy2 /2t CTy or eJ (x+jy) = e-Js(ax2+j 2vxy-oy2} where and x2o-p2 yxy = xy . This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial ful fillment of the requirements for the degree of Doctor of Philosophy. June, 1968 Dean, College of Engineering Dean, Graduate School Supervisory Committee: Chairman 74 and RXL-SSB-a-GN^ ~ Re e"JuoT{[e2a2Ree(T) cos (2a2fr0e(r))] + J [e 2a2Ree(t) cos (2a2tee(r))]} (6.58) The efficiency is readily obtained by substituting Eq. (6.56) into Eq. (5.54): 'SSB-a-GN = 1 5-2a2(J>m (6.59) where is the power in the Gaussian m(t) and |e(t)| < < 1. This result may be compared for a = 1 to that given by Voelcker for envelope-detectable SSB(a = 1) [11]. That is, if e(t) has a small variance, then m(t) = e(t); and Eq. (6.59) becomes nSSB-oi-GN : 1 e 00 ~ 2o0/. (6.60) This agrees with Voelcker"s result (his Eq. (38)) when the variance of the modulation is small-~the condition for Eq. (6.60) to be valid. The expressions for the other properties of the SSB-ct signal, such as bandwidths and peak-to-average power ratio, will not be examined further here since it was shown above that these properties are the same as those obtained for the SSB-PM signal when Um)sSB-PM ~ a2(iJm^sSB-a m(t) is Gaussian. as long as 82 where 6 DAm/tom, the modulation index. Substituting Eq, (7,12) and Eq (7,13) into Eq, (7,11) we obtain for the output noise power k2F0i%3 2ttA02 ll0(2i) +isl1(26) Referring to Eq. (7.9), the output signal power is 2 = y Then the output signal-to-noise ratio is dkm(t) dt (7,14) (7.15) (S/N)0 1 1 - 1.(26) + 61,(26) 3 0 2 1 (S/N)n - A0262 2 <%, 1 Io(26) + I 01,(26)1 2ir 3 2 (7.16) Referring to Eq. (7.6), the signal power into the detector is Si = A02 e2"1^ cos2 [o)0t + m(t)] = A02 e2^^^ jA02 I0(26) (7.17) Kahn and Thomas have given the ratio of the rms bandwidths (taken about 80 For large input signal-to-noise ratios {i.e. A0e-m^ > > R(t) most of the time), Eq (7o8) becomes kR(t) ip(t) km(t) + sin [ (7.9) dn0(t) Then the noise output voltage is where n (t) = R(t) sin [ (7.10) Now the phase (t) is uniformly distributed over 0 to 2n since the input noise is a narrow-band Gaussian process. Then for m(t) deterministic, U(t) m(t)J is distributed uniformly also. Furthermore, R(t) has a Rayleigh density function. Then it follows that R(t) sin [ is Gaussian (at least to the first order density) and, using Rice's formulation [24, 25], where F(u>) = F0 is the input noise spectrum and {n} are independent random variables uniformly distributed over 0 to 2-rr. Actually it is known that the presence of modulation produces some clicks in the out put [26], but this effect is not considered here. Eq. (7.10) then be comes 32 Likewise V has a zero mean value. Then, identifying Eg. (5.13) with Eq. (5.9), it is seen that fx(t) S -V(m(t),m(t)) (5.14a) f2(t) = U(m(t),m(t)) (5.14b) 1 = k2 and c2 = k . (5.14c ,d) Similarily, for lower SSB signals Eq. (5.13) can be identified with Eq. (5.12). Thus the SSB signal has a discrete carrier provided that kx and k2 are not both zero. As an aside, it is noted that the criterion for a discrete car rier, given by Eq. (5.13), is not limited to SSB signals; it holds for all modulated signals. For AM Eq. (5.13) can be identified with Eq. (1.1) Here fj(t) = A0m(t) (5.15a) f2(t) 5 0 (5.15b) 5 Ao and c2 = 0 (5.15c,d) because m(t) has a zero mean due to AC coupling in the modulator of the transmitter. Thus for AM it is seen that there is a discrete carrier term of amplitude c1 = AQ which does not depend on the modulation. For FM Eq. (1.2) can be expanded. Then, assuming sinusoidal modulation of fre quency wa, we obtain 5 The Fourier transform of m(t) is given by FfiU) = [-j sgn (o)] Fm(to) (2.2) where sgn (u) + 1 u) > 0 0 a) = 0 - 1 to < 0_ (2.3) and Fm(j) is the Fourier transform of m(t). In other words, the Hilbert transform operation is identical to that performed by a -90 all-pass linear (ideally non-realizable) network. From Eq. (2.2), it follows that F*(u>) = [-j sgn (w)]2 Fm(u>) -Fm(co) (2.4) or m(t) = -m(t). (2.5) The (complex) analytic signal associated with the real signal m(t) is defined by Z(t) = m(t) + jm(t). (2.6) The Fourier transform of Z(t) follows by the use of Eq. (2.2), and it is Fz(<>) = Fm(o>) + j[-j sgn (w)] Fm(w) 52 since k1 = 0. Thus the two equivalent representations are: X USSB PM^ = [e"m^cos m(t)] cos a)0t [_e '"'"'cos mQt;J sin w0t (6.13) and XUssB_p|v|(t) = [-^;sin m(t))+l]cos oi0t [e_r"^sin m(t)]sin w0t. (6.14) The USSB-PM exciters corresponding to these equations are shown in Figure 13 and Figure 14. They may be compared to the first realization method given in Figure 9. The autocorrelation function for the SSB-PM signal will now be examined. In Section 6.1 the autocorrelation for the SSB-AM-SC signal was obtained in terms of the autocorrelation function of the modulation. This was easy to obtain since 44 = m(t). However, for the SSB-PM case 44 and -V-are non-linear functions of the modulation m(t). Consequently, the density function for the modulation process will be needed in order to obtain the autocorrelation of the SSB-FM signal in terms of Rrnm(T) To calculate the autocorrelation function for the SSB-PM signal, first Ryy(t) will be obtained in term of R^fx). Using k;L = 0, Eq. (5.21), and Eq. (4.5b) we have V-(m(t) ,m(t)) a V(m(t),m(t)) = e_m(t) sin m(t). (6.15) Then 91 Figure 22. Output Signal-to-Noise to Input Carrier-to-Normalized- Noise Power Ratio for Various Systems 92 Then the RF energy required per bit of received information is Sl (F0B/2tt)(S/N)1 f0b (S/N)1 Cb S Cb S b l0g2 [1 + (S/N)q] where F0 is the spectral density of the noise in the IF and B is the IF bandwidth (rad/s). In an ideal system the capacity of the IF is eaual to the capacity of the baseband even when (S/N).Â¡ -* 0. Therefore the ideal system has an energy-per-bit given by St Si r r lim Cb CB (S/N) -Â¡-K) FnB (S/N)i B log2 [1 + (S7N)i] log2e 0.693 Fn. (7.37) Then Eq. (7.36) may be written as S -L < (0.693 F0) Cb B (S/N)j 0.693 b Tog, [1 + (S/N)0] _4 Now the figure of merit will be defined as B (S/N)i 0.693 b log, [1 + (S/N)0] (7.38) which is the amount of energy required by the actual system over that of the ideal system in order to receive a bit of information, provided that the output noise is Gaussian. If the output noise is not Gaussian, the value of M will be somewhat larger than the ratio, energy-per-bit for the actual system to the energy-per-bit for the ideal system. H as a function of modulation index will be derived below for com parison of various systems. 93 7.2-1. AM system We now want to find M(s) for the AM system, described in Section 7.1-1, such that we will have an output signal-to-noise ratio of 27.5 db for 6=1. 27.5 db is an arbitrary value that is chosen here for com parison of systems using M as a figure of merit. This value is repre sentative of the (S/N)o requirement for actual communication systems. From Eq. (7.3) it follows that (C/N)-j = 27.5 db for 6 = 1. Also, for the AM system Eq. (7.38) becomes 2 )m [(1 + %62)(C/N)i3 M(6) = ; -- 0.693 log2 [1 + 62('C/N)i] For (C/N)j = 27.5 db, Eq. (7.39) reduces to 1620 (1 + %62) M{6) = log2 [1 +'560 62] (7.39) (7.40) The values of M(6) for the AM system, as given by Eq. (7.40), will be compared to those for other systems in Section 7.2-5. 7.2-2. SSB-AM-SC system To obtain M(6) for the SSB-AM-SC system, (S/N)0 = 27.5 db will be used once again. From Eq. (7.5) it follows that (S/N)j = 27.5 db. Also, for the SSB-AM system Eq. (7.38) becomes m (S/N)i M(6) = 0.693 com log2 [1 + (S/N)0] (7.41) For (S/N) 0 = (S/N)-f = 27.5 db, Eq. (7.41) reduces to M(6) = 19.5 db. (7.42) |