Co-channel interference suppression for OFDM systems

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Co-channel interference suppression for OFDM systems
Won, Jonghyun ( Author, Primary )
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Supernova remnants ( jstor )
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Copyright 2003


Jonghyun Won

To my family.


First and foremost, I thank my adviser, Dr. Erik G. Larsson, for having con-

fidence in me and for allowing me to pursue this research. The work presented

in this thesis would not be possible without his continuous guidance. I would

like to thank my committee members, Dr. John M. Shea and Dr. Tan F. Wong,

for taking the time to read and critique my thesis. I also thank the professors in

Korea from Electrical Engineering Department at Dongguk University for helping

me to study abroad at University of Florida.

I thank my beloved parents for being there and believing in me for all the

years. They have provided me the strength and inspiration to continue my

studies at the University of Florida.

I want to thank all of my lovely friends for ahv--,- motivating me to be a

better person and for providing me with guidance and advice when I needed it.


ACKNOWLEDGMENTS ............................ iv

LIST OF FIGURES ................... ........ vii

ABSTRACT ...................... ........... ix


1 INTRODUCTION ............................. 1


2.1 Error Probability for Communication System ......... .. 3
2.2 Antenna Diversity ........... .............. 5
2.3 Interference . . . . . . . 7


3.1 Data Transmission and Detection in OFDM ...... ... .. 9
3.2 ML Detection for OFDM with White Noise ............ 11
3.3 ML Detection for OFDM with Colored Noise ........... 12

O FD M . ... . . . ..... . . 16

4.1 C('!t i ,, I Estimation ................... .... 16
4.2 Estimation of R(n) with Piecewise Constant Model ...... 18
4.3 Automatic Selection of Model Order ....... ... .... 20
4.4 Numerical Examples for the Piecewise Constant Model ..... 22
4.5 Estimation of R(n) with Moving Average Process ....... 26
4.6 Auto Selection of Model Order ..... . . ..... 28
4.7 Numerical Examples for Moving Average Process . ... 28


5.1 MIMO Systems and Space-Time Block Coding . ... 33
5.2 Space-Time OFDM .................. .... .. 35


6.1 C'!i i !,, I Estim ation .................. ..... .. 38
6.2 Estimation of R(n) with Moving Average Process ...... .. 40
6.3 Numerical Examples for the Moving Average Process ..... 41


REFERENCES ................... ..... ........ 47

BIOGRAPHICAL SKETCH .................. ........ .. 49

Figure page

3-1 Data transmission and reception scheme for an OFDM system 9

4-1 Example of the piecewise constant model for a SISO system . 19

4-2 Symbol-error-rate (SER) for the piecewise constant model when
there is only thermal noise present. ............... ..23

4-3 Symbol-error-rate (SER) for the piecewise constant model when
there are one strong interference signal (C/I 10dB) and thermal
noise present. .................. .. ...... 24

4-4 Autonomous selection of model order, SNR = 30dB, for piecewise-
constant model. .................. ..... 25

4-5 Comparison between detection (using piecewise constant model
and coherent detection using perfect knowledge of the channel
and the noise covariance). ............... .. ..26

4-6 Symbol-error-rate (SER) for the moving average process model
when there is only thermal noise present. ............. ..29

4-7 Symbol-error-rate (SER) for the moving average process model
when there are one strong interferer (C/I=10dB) and thermal
noise present. .................. ...... 30

4-8 Autonomous selection of model order, SNR = 30dB, for moving-
average process. .................. ..... 31

4-9 The comparison between detection using moving average process
and coherent detection using perfect knowledge of the channel
and the noise covariance. ................ ..... 32

6-1 Symbol-error-rate (SER) for the moving average process model
when there is only thermal noise is present. . ..... 42

6-2 Symbol-error-rate (SER) for the moving average process model
when strong interferer comes from a single antenna user. . 43

6-3 Symbol-error-rate (SER) for the moving average process model
when strong interferers comes from co-channel user's multiple
antennas. .................. ............ .. 44

Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science



Jonghyun Won

August 2003

C('! In: Erik G. Larsson
Major Department: Electrical and Computer Engineering

We propose new methods for co-channel interference suppression in orthog-

onal frequency division multiplexing (OFDM) systems with multiple receive

antennas. Our methods use a low-order time-domain model to parameterize

the matrix-valued spectral density of the noise and interference. A penalized

maximum likelihood-function approach is used to choose the proper model order

automatically in each received data frame. We show that these methods signif-

icantly improve the symbol-error-rate when interference exists in the received

data. Numerical examples are provided to illustrate the performance of our new

methods and compare with conventional methods. We also discuss the difficulties

associated with the application of our method to space-time OFDM (ST-OFDM).


Orthogonal frequency division multiplexing (OFDM) has become one of the

most promising transmission techniques for wireless local area networks because

of its low equalization complexity in frequency selective fading channels, and its

robustness to inter-symbol interference (ISI). OFDM transforms a frequ'-i-Tv-

selective fading channel into multiple independent flat fading subchannels. By

transmitting one symbol in each subchannel, the symbol duration is increased;

hence the effects of ISI caused by a frequency selective fading environment

are reduced. Therefore, an OFDM system can achieve a high data rate and a

reliable transmission in a fading channel. However, due to limited and congested

frequency bands in cellular radio environments, the system performance is often

limited by interference.

Interference is one of the major limiting factors on the performance of

wireless communication systems, as it typically increases the bit-error-rate

(BER). There are two 1i i, ri types of interference: so-called .I.1i i:ent channel

interference (ACI) and co-channel interference (CCI).

ACI comes from signals that are transmitted by a communication system

using a frequency band .,I.i ient to the desired user's frequency band in the

same area. This interference can be minimized by using a strictly restricted

signal bandwidth or a proper channel assignment. Guey et al. [1] proposes ACI

rejection methods for land-mobile radio systems.

CCI is caused by the frequency reuse scheme in wireless networks, for

example, when signals come from a co-channel cell which uses the same frequency

band as the desired user. As CCI consists of the same frequencies, it cannot

be rejected by simply increasing the transmit power or using narrow channel

filtering. There are many statistical CCI interference suppression methods for

radio communication networks. For example, Jongren et al. [2] proposed an

interference rejection method using low-rank features of the spatial covariance

matrix; Karlsson and Heinegard [3] -,-.- -1. .1 an "interference rejection" method

for the GSM system. These techniques have been quite successful (see Dam et al.


In this thesis, we propose new CCI suppression methods for OFDM and

space-time OFDM (ST-OFDM) with multiple receive antennas. The basic idea

of our algorithms is that the (matrix-valued) power spectrum of the interference

is smooth, and hence we can use a low-order time-domain model to parameterize

the spectral density of the noise and interference. One of our methods proposes

a simple model for the covariance of the interference and noise. This model

assumes that pairs of M .,I.i i'ent subchannels have the same spectral density.

Thus, the performance of our model depends on how we select the number of

subchannels, M. A penalized maximum-likelihood function approach is used to

choose M automatically in each received data frame. Another method uses a

multidimensional moving average (I! A) process to model the power spectrum

density of the interference and noise with a variable number of parameters,

M. Also here, a penalized maximum-Likelihood function approach is used to

select the proper number of parameters. We also apply these algorithms in the

ST-OFDM case and show numerical examples.


In a wireless environment, transmitted signals usually decrease in power or

become distorted in shape because of the time-variation of the channel and the

propagation environment. Communication in a moving car can be one example of

a fading channel, since the surrounding environment of the car varies so that the

propagation channel changes according to the velocity of the car.

2.1 Error Probability for Communication System

Let us consider a simple example of a wireless communication system to see

the effect of fading. We assume, for simplicity, that the propagation channel is

linear, time-invariant, and frequency flat; also assume that the system consists

of a single transmit and receive antenna. Such a system is called a single-input

single-output (SISO) system. Consider a transmitted symbol s from a finite

constellation S. Then the received signal can be expressed as

y=h s+e (2.1)

where h denotes the channel gain and e denotes zero-mean complex Gaussian

noise with variance a2. The probability of density function (p.d.f) for this linear

model is

p(y)= 2 exp hs 2 (2.2)

We can detect s using the maximum likelihood rule, which maximizes (2.2) and

is equivalent to

s = argmin ly hs 2 (2.3)

From Equation (2.3), we can derive the detection error probability for the
system. Suppose we detect a wrong symbol So from a finite constellation S
instead of the true symbol s. Then the pairwise probability for an incorrect
decision is derived as

P(s slh) =P(|y hs,2 < y hs 2)

P(ly|2 2Re(y*hs,) + |hs,12 < y12 2Re(y*hs) + Ihs 2) (2.4)

P(\h 2(Is2 2 _- 2) 2 Re{y*h(s, s)} < 0)

where (.)* denotes the complex conjugate. We divide both sides by ih 2 and
insert y = hs + e into the equation (2.4). Then we get

P(s sh) =P( ,2 + 2 2 -2Re{s*s + 2Re {(*(s )} < )

=P(|s S2 + 2Re {()*( s) < 0) (2.5)

_Q 20_ 2

The Q-function in the last step is a consequence of the fact that 2Re{(e/h)*(s -
So)} is a Gaussian random variable with zero mean and variance 2s So 2 2/lh12
The above result is the error probability for a constant channel gain. If the
channel suffers from Rayleigh fading, h can be modeled as a complex Gaussian
random variable with zero-mean and variance p2. We can compute an average
error probability for the fading channel by using the C'I, i ir.!.1 bound. Due to the
Rayleigh distribution assumption, the probability density function of the channel
h can be expressed as

p(| 2) exp ( 12 (2.6)
p Ipli

Averaging (2.5) over h using (2.6), we get

Eh[P(s -+ slh)] = P(s sh)P(ht2dh 2

< exp(- S s- 2 2h1 exp(- I)dlh12 (2.7)
(s- s2 52 1 02)12 2-1
4 2 42 2
The signal-to-noise (SNR) for the received signal is p2/a2, and the last step of

(2.7) shows that the average error probability for the fading channel behaves as

P(error) o SNR-1 (2.8)

We can compare this result to that of a constant channel. If we use the fact that

the SNR equals p2/o2 and the ('I, i i, ll' bound in (2.5), then the error probability

for a deterministic channel behaves as

P(error) o exp(-SNR) (2.9)

2.2 Antenna Diversity

Since the fading is Rayleigh distributed and time-variant, sometimes the

channel can be useless. One of the most popular methods for combating the

fading effect is antenna diversity. By employing multiple antennas on the

transmitter or the receiver side, the communication system can transmit the

same information over multiple channels, which are independent of each other.

Such a system is called a multiple-input multiple-output (\MI \ O) system. In

a MIMO system, the fading effect can be significantly mitigated and also the

channel capacity can be increased as another advantage. Equipping multiple

antennas at the base station, which gives receive diversity for the uplink, is done

in many communication systems. Considering n, receive antennas instead of one

antenna in a SISO system, we can see how receive antenna diversity counteracts

the fading effect. In this case, the channel h becomes an 1 x n, vector and thus,
we let h = [hi .. h,]T be the propagation channel vector in which each value
corresponds to the channel gain between the transmit antenna and each receive
antenna. Here (.)T denotes the transpose. Then, with the same assumptions as
for the SISO system example, the received data can be written as

y h s + e (2.10)

where y = [ylr y,J]T is an received data vector and e = [e ... en]T is a
complex Gaussian noise vector with zero mean and covariance matrix a2j ,. The
pairwise error probability, obtained by a similar calculation as Section 2.1, is

P(s-I sh) 2 Q o2 ) (2.11)

If we assume that the n, channels are independent and complex Gaussian
random variables with zero mean and variance p2, respectively, the probability
density function for h can be expressed as

P(h) f J 2exp(- 2 (2.12)
n-1 Pn n92
Again, using the ('!. i 11l' bound, (2.11) and (2.12), we can derive a bound on the
average error probability for MIMO fading channels as follows

Eh[P(s solh)] f= J J 1 P(s soh)P(h)dlhl12 ... dlh 2
| 2 0_ 2 01 | | 2

|s s, 2|? 2\ \h, 2 2 (2.13)
< exp ( IS So121h, 1 1 t ( hi12 \j\ 2..

422 2
1( 1 2 2 1) r 1 it -
S4 +t 4
ni1 ni1

The last part of Equation (2.13) shows that the more antennas there are on the

receiver side, the more mitigation of fading effect is achieved. Also, it shows that

the average probability of a detection error over the fading channel behaves as

-(diversity order)
P(error) oc SNR =- (2.14)

Also, a diversity effect as well as other benefits can be obtained by equipping

multiple antennas at transmitter side (this is called transmit diversity). Accord-

ing to recent research, it is possible to reproduce the diversity from multiple

transmit antennas while preserving the transmission data rate and a simple

decoding complexity. This method, which is called Space-Time Coding, will be

introduced in ('! Ilpter 5.

2.3 Interference

Recently, the population of mobile users has dramatically increased due

to the fast development of wireless communication technologies; hence, using

carefully chosen channel reuse schemes has become more important in cellular

communication systems. Because of channel reuse, interference between signals

from the same frequency band is often generated. This interference, which is

called co-channel interference (CCI), is the one of the n ii' ,r limiting factors for

the performance of wireless communication systems. Unlike thermal noise which

can be overcome by increasing the transmit power, CCI cannot be overpowered

by increasing the carrier power. Moreover, an increase in carrier transmit power

can augment the interference. However, the effect of interference can often be

reduced by suitable signal processing method at receiver side.


As wireless communications have experienced great success and are widely

used everywhere, the demands for good quality and variety of service have

increased. To fulfill these desires, development for high speed wireless commu-

nication systems has been investigated. One successful approach to achieving

reliable and efficient high speed data transmission scheme is OFDM.

The main idea of OFDM is that by dividing the entire channel bandwidth

into independent multiple narrow bands, each sub-band becomes relatively

flat. We can compare OFDM with a conventional single carrier system. A con-

ventional system transmits each data symbol one by one in a whole occupied

channel. One of the disadvantages using a single carrier is that if the channel suf-

fers from frequency selective f ,lii- the channel equalization usually is complex.

On the other hand, in an OFDM system, each subchannel occupies only a small

fraction of the total bandwidth, and OFDM transmits one data symbol in each

subchannel in parallel. Since a small bandwidth of each subchannel makes the

symbol durations much longer than the memory of the channel, OFDM can use

simple channel equalization and also obtain the same data rate as a conventional

system. This property allows OFDM to support high speed rates. A detailed

explanation of OFDM can be found in Larsson and Stoica [5], Heiskala and Terry

[6] and Li et al. [7]. Here, we introduce the fundamental principles of an OFDM

system following Larsson and Stoica [5].


Pre (equal)

I 0 Data NO-Npre-1 N0-1 I


Figure 3 1: Data transmission and reception scheme for an OFDMI) ---- (I:
Larsson and Stoica [5], '. ,ter5)

3.1 Data Transmission and Detection in OFDM

Consider a general OFDM system with a single transmit antenna and n,

receive antennas. Suppose we have s(n) symbols for n = 0,..., N 1 to be

transmitted. After taking inverse Fourier transform for the symbols, the encoded

data can be expressed as
x(n) =k)e, = 0,..., N 1 (3.1)

Prior to transmission, a cyclic prefix (CP) is added at the beginning of the

sequence x(n). The CP is a copy of the last Npre symbols, where Npre > L and L

is the channel delay spread. The CP makes the transmitted signal periodic with

period N so that the linear convolution induced by the propagation channel is

transformed into a circular convolution, which corresponds to multiplication of

channel gains and symbols in the frequency domain. We can express the encoded

data with a cyclic prefix as follows

x(n) =x(n+N), n Np e... -1


Figure 3-1 illustrates the transmission scheme. From the encoded data x(n), we

can observe that each symbol s(k) is carried on each subcarrier c which are

orthogonal each other. The encoded data are transmitted through a channel,

which has L d 1 iv taps, and after discarding CP, the received data vector from

the multiple antennas can be expressed as linear convolution of the channel and

the encoded data plus thermal noise and interference

y(n) = hkx(n k)+ e(n)
k0 L N- (3.3)

s(m)hk exp 7(n k)m + e(n)
k= 0 m= 0

where hk denotes kth vector-valued channel tap. After taking the Fourier

transform of the received data, we get the multiplication of each frequency flat

fading transformed channel and data (see, e.g., Larsson and Stoica [5], ('! ipter

z () -= Y Y() e -

N-1N-1 L N-1
^jEEE8snrnk "C6 15e(l)eC

1=0 m=0 k=0 =0
1 +1 m in
Sn)hke- (n m)l k e(l)e- (3.4)
1=0 m= k= 0 0 0
L N-1

L e(l)e=0
kc-O 0 1- 1=0
(n)( -h n) +)e


where we have defined the channel transfer function

h(2n h Cke- (3.5)
k 0

From Equation (3.4), we can detect the symbols s(n) using simple equalization.

3.2 ML Detection for OFDM with White Noise

First, we need to consider the noise term before deriving the ML detector

for OFDM. We assume for simplicity that the noise vectors e(n) are spatially

white circular Gaussian random variables with zero mean and variance a2. This

assumption is proper when only thermal noise exist. Let consider the noise

covariance in Equation (3.4) as follows

N-1 N-1
E[e(ienH(k)] = E -- e() e- i e(m)e- k HI
0o m o (3.6)
SlSE[e(l)eH(m)]e- e-
1=0 m=0

From the assumption on e(n),

E[e(n)eH(k)] = Y E[e(l)eH(1)]- -

a2I, n k (3.7)

0, n/k

Equation (3.7) shows that the noise term in (3.4) is white Gaussian noise with

covariance a21, and thus, z(n) is a sufficient statistic for the detection of s(n).

Hence, ML detection for OFDM is equivalent to minimizing the following metric

Iz(n) s(n) h(n) 12 (3.8)

h(n)= hie--' (3.9)

3.3 ML Detection for OFDM with Colored Noise

A more general assumption on the noise e(n) is to ,-v it is circularly

symmetric Gaussian with the following covariance function

R, = E[e(n)eH(n 1)1 (3.10)

where (.)H denotes the conjugate transpose. Then the power spectrum of e(n) is

R(n) = Rie~ (3.11)

Such colored noise can be used to model co-channel interference. 1

If N is reasonably large in (3.4), we show below that e(n) and e(m) are

independent for n / m and hence the ML detection for an OFDM system is

equivalent to minimizing the following metric

mmin R-12(n)(z(n)- h()s(n)) (3.12)
s(n) ES

We now show that R(n) is the covariance of the noise term in Equation

(3.4) and that e(n) is independent of e(m), n / m. First we consider the

auto-covariance function of e(n) as follows

N-1 N-1 H
E[e(n)eH(n)] = E 1 e(k)e 1 e(k)e-
kL O (3.~3)
N-1 N-1(k k)
Y Y E[e(k)eH(k)1e- k-)
k=0 k=o

Note that, due to the insertion of the prefix, a signal transmitted in an
OFDM ---.---- is not statisti, : iy white even if the source signal is white.
.* 'ct this effect in this thesis.

Let k- k then

E[e(n)eH (n1=i R Ie-
k=0 l=k-N+1
/ 0 l+N-1 N-lN-
( le-i2n i2 n)
l=-N+1 k 0 l=1 k=
( )Ni (3.14)
S(N + 1)Rle- + (N +)Re-
l -N+I l1-
N-1 N-1

l -(N-1) l=-(N-1)

Under the mild assumption that the covariance sequence decays sufficiently

rapidly, the last term in (3.14) goes to zero when N -0 oc. Hence

E[e(n)eH() R Ifn (3.15)

Hence, in an OFDM system, the auto-covariance of the noise term after taking

the Fourier transform of the received data is the same as the power spectral

density of the noise in the received data. Next, we calculate the cross-covariance

of e(n)

1 N-1 N-I kH-
E[e(n)eH(m)] = E 1 e(k)e-i k k e()k-ei

Z E[e(k)eH()]ei V-(nk-
k=0 k=o

If we let k k = then Equation (3.16) can be expressed as follows

N-1 k
E[e(n)eH(m = RIC Rm-l)
k 0 l=k-N+l
0 l+N-1
l=-N+1 k=0

+ N iN N-1 (n- m)kf-m


Note that

(n-m)k 0


k C


If we take the norm of (3.17),

can be expressed as follows

(n-m)k (n-m)k (3.19)
k 1
the norm of first summation term, ||A(n m)}\,

I|A(n -m)|| 1


|R111 le -- )k
< 111^^ ^^^ ^^ ^ ^


N 1NR1
-N +



Also, IB(n m)| can be expressed as follows with (3.19)

N-1 N-1
IIB(n n)ll NR e (n m)
l=1 k=l
N-1 l-1

l11 kO0
1 R
N-I 1-1I (3.21

l=1 k=-O

< Y 111111
l -N+I

Under the mild assumption that the covariance sequence decays sufficiently

rapidly, when N -- oc, (3.20) and (3.21) go to zero. As a consequence, the

Fourier transformed noise is temporally white and has covariance R(n).


When there is strong interference in the received signals, it can be beneficial

to suppress the interference, so that correct signal decisions can be made. One

approach to estimate the covariance of the interference is to instantly estimate

the covariance of each subchannel, but normally such instantaneous covariance

estimates do not work well if there is only a short training sequence. One

approach to cancel the co-channel interference is -, i-.- -1. 1 in Li and Sollenberger

[8], which uses a minimum mean-square error (\1\ SE) diversity combiner (DC).

They estimate the channels and the noise covariance in the frequency domain,

and use these estimated values to suppress the interference. In this chapter, we

introduce novel methods for interference suppression in an OFDM system. These

algorithms are based on a low-order time domain model to parameterize the

matrix-valued spectral density of the interference.

4.1 ('i! ,i ,, I Estimation

Let us assume that we received more than one training sequence (T > 1).

After taking the Fourier transform of the received training data, they can be

expressed as

L N-1
t(n) = h exp i ) s(n) + e()e (4.1)
1=0 k O

for n = 0,... 1, t = 1,... T, where zt(n) stands for Fourier-transformed

known data. We can estimate the channel from (4.1), assuming known training

data, using various methods.

The ]' i-ii, ilr: principle [9], explains that among models describing data

well, the model with the smallest number of parameters is a better choice than

others. For this reason, it is more desirable to estimate time domain channels

instead of frequency domain channels, because the former consist of n,(L + 1)

parameters; however in the frequency domain, we would need to estimate nN

parameters. ('!i ,ii!, I estimation can be done by the following procedure.

Denote the transmitted and received training data before Fourier trans-

formation by xt(n) and yt(n). Using Toeplitz matrices, the training data Xt,

channel H, noise Et and received data sequence Yt can be expressed by

Yt = XtH + Et (4.2)


xt(0) xt(-1) ... .. xt(-L)

Xt() Xt(0) xt( L) (4.3)

xt(N 1) xt(N 2) ... ... xt(N 1 L)

SeL(0) y (0)
H = Et= Yt (4.4)

h e(N -1) y(N 1)

The negative log likelihood function for (4.2) becomes, assuming the noise is

min IYt- XtH| 2 (4.5)

and the ML estimate of channel H is determined by
H (= XHX,) XHYY (4.6)
t=1 t=1

After estimating the channel H, the frequency domain h(n) channels can be

easily calculated by using Fourier transformation
h(n) hle-i2WNnl (4.7)

4.2 Estimation of R(n) with Piecewise Constant Model

The interference signal in e(n) consists of the same frequencies as desired

signal, and it has propagated over a channel with similar properties as the

channel h(n). Hence, we can expect that the power spectrum of e(n) is smooth

and this characteristics can be used to estimate R(n). Following this reasoning,

we can expect that the covariance in a portion of the frequency band consist of

approximately the same values. The piecewise constant model assumes that the

instant covariance R(n) of M .il1i i:ent subchannels consists of the same value or

approximately the same value. With this model, we partition the frequency band

in N/M segments, calculate R(n) for each subchannel, take an average of R(n)

values in each segment which consists of M .,Ii] ient subchannels and apply each

averaged value R(n) to M subchannels in each segment instead of R(n). More

detailed explanations are as follows Using an estimated channel H from (4.6), we

calculate residuals from received training data

Yt = (Yt XtH)T (4.8)

where Yt consists of

Y = t(0) .. yt(N l) (4.9)



1.5- I
I\ I

/ I

0.5- / "

10 20 30 40 50 60

Figure 4 1: of the constant model for a SISO .-em.

We take the Fourier transform of yt(n) as follows

rt(n) =- yt(k)e- (4.10)

We next estimate R(n) for each subchannel by

R(n) j_ rt(n)rt"(f) (4.11)
t= 1

and then, take the average in each segment i = 0,... N/M 1, according to the

parameter M
RM(t) =M R(Mi + m) (4.12)

We use RM(i) for segment i to suppress the interference instead of the instan-

taneous moment based estimate R(n). Figure 4-1 shows an example of the

piecewise constant model for a SISO system. The dash-dotted line is the true

covariance of the interference and the thermal noise, and the solid line is the es-

timated covariance using our method. Hence, according to (3.12), the maximum

likelihood detection for information data is expressed as

-1/2 2
min M (i)(z(Mi + m) h(Mi + m)s(Mi + m)) (4.13)

for i 0,...,N/M- 1 and m 0,...,M- 1.

4.3 Automatic Selection of Model Order

To get the best performance, we need to choose the model order, the

parameter M, autonomously in each sequence of data. As a sub-optimal way,

we use a maximum-likelihood (\! I) approach for selecting M. After estimating

the covariance for different M, we insert those values into the negative log-

likelihood function with training symbols, compare and select the minimum one

as the proper model order for the information data sequence. It is desirable to

choose the model, that explains the data (interference and noise) well and has a

parametrization as simple as possible. But ML .i.- li- chooses the highest model

order, when there is interference in the training sequence. To counterbalance this

tendency, we add a penalty term inspired by the minimum description length

(\!I)L) method (more detailed discussion can be found in Kay [10], Section 6.8)

to the ML rule

MDL(M) InP(Yt;RM H) + N (4.4)

where nM is the number of estimated parameters in RM. Thus, this method

finds the model order M which minimizes

T N/M-1 M-1
mn {m Y R 2(i)(zt(Mi + m) (Mi+ m)st(M ))
t=1 i=0 m=0 (4.15)
N/M-1T M n
+ TM- log RM(i + 2 In

where st(n) is the known received data. Note that RM(i) is the average value of

the moment based covariance matrices of M .il.i i:ent subchannels and

zt(Mi + m) h(Mi + m)st(Mi + m) = rt(Mi + m) (4.16)

Note that the norm square term of equation (4.15) can be written

T N/M-1 M-1
S R-1 2(i)(z,(Mi + m) h(Mi + m)st (Mi + mn))
t=l i=0 m=0
T N/M-1 M-1
-1/2 -1H/2
E E M Tr{RM/(i)t(Mi + m) (Mi + m)R )}
t=l i=0 m=0
T N/M-1M-1
E E Tr{RM 2()R /(i)rt(Mli+m)rHMi+m)}
t=l i=0 m=0
N/M-1M-1 T
= E Tr{RMI(i)YRt(Mi+ m)}H(M + )}
i=O m=O t=l (4.17)
N/M-1 M-
T Y yTr{RM ()R(Mi +nnm)}
i=O m=0
NIM- 1
T Tr{RM (n)(R(Mk)+R(Mk+ 1)+..-+R(Mk + M- ))}
T Y M- T{RM(i)RM()

= TN Tr{I}

where Tr{-} stands for the trace of a matrix. Following this result, Equation

(4.15) can be rewritten in the more simplified form

minM log RM(i) + n nN (4.18)
Mr 2

We use this simplified method with training sequences to select the proper

value of M and apply the chosen model to suppress the interference in the data


4.4 Numerical Examples for the Piecewise Constant Model

In this section, we show some simulation examples for our new algorithm for

an OFDM system. In our experiment, N = 64 subcarriers, 1 training sequence

and n, = 2 receive antennas are used. The channel consists of L + 1 = 5

independent Rayleigh fading channel taps, and has an exponentially decaying

power delay profile: E[l 112] O2 1.5-, 0, ... L. The training sequence is

an impulse to estimate channel, and the constellation size for our simulation is


Figure 4-2 shows the estimated Symbol-error-rate (SER) versus the Carrier-

to-Noise ratio (C/N) when the received signals are not affected by interference,

and when only thermal noise is present. We present the result for the cases when

M = 2, 4, 8, and 64, and with the autonomous selection. we also compare the

result to coherent detection and conventional detection ( In this case, we use

noise covariance R(n) = I for all n. ) The non-coherent detection and our

method with M = 64 show almost same result. The reason for this is that setting

M = 64 implies that the noise spectrum is independent of n. With the other

values of M, we see a performance loss relative to conventional detection. Our

auto-selection method chooses M = 64 model order for each sequence as we


I I a I .

Est. channel
I \.

10 -15
F) s

E Coherent \v
0 -- Est. channel (Conventional) +
,.o,. Est. channel and noise cov. M=64 e da a
p* a. Est. channel and noise cov. M=8 ". .
f0-3. s oo Est. channel and noise cov. M=4 ad ..
+. Est. channel and noise cov. M=2 \ 0.
.. x Est. channel and noise cov. auto '

0 5 10 15 20
Signal-to-noise ratio (SNR) [dB]

Figure 4-2: 'mbol-error-rat.e (SER) for the piece@wise constant model when
there is ,'.--! thermal noise present.

In Figure 4-3, we present the case when there is one strong interferer

present. The co-channel user has the same channel delay profile and power delay

profile as the desired user has, and the signal to interference ratio is 10dB. This

figure shows poor performances of coherent detection and conventional detection

without estimation of interference, but our method successfully mitigates the

co-channel interference. For high SNR, the M = 4 model order outperforms the

other models, but in the region of less than 20dB, the M = 8 model order shows

somewhat better performance than M = 4. For each SNR, our auto-selection


10 I "o I

a) E. .. .
S Coherent

.o.. Est. channel and noise cov. M=64
.3 Est. channel and noise cov. M=8.
.. Est. channel and noise cov. M=4 i "'....

10 15 20 25 30 35
Signal -to-nnoise ratio (SNR) dB]

Figure 4-3: bol-error-rat e (SER) anr the piecewise constant model when
there are one strong interference signal (Cl/I 0dB) and thleral noise present.

method does not al-- t. choose the best model, which has the lowest symbol-

error-rate. However, the auto-selection method significantly outperfrms the

coherent method without estimating the covariance of interference.

Figure 4-4 shows how our autonomous selection method performs when
the signal-to-noise ratio is 30dB. Our model a-- chooses M 64 when only

thermal noise is present as we expect. When one strong interferer and thermal

noise are present, our method usually selects M = 8 and M = 4.

Noise only when SNR = 30dB





0 II I
24 8 64

interference with no!e when SNR = 30dB





24 8 64
2 4 8 64

Figure 4-4: Autonomous selection of model order, SNR
constant model.

'dB, .: piecewise-

We summarize the comparison between detection using our method and

coherent detection ( using perfect knowledge of the channel and the noise

covariance ) in Figure 4-5. In the case when the received signals are only affected

by thermal noise, Figure 4-5 indicates that conventional detection and our

new method have the same performance. In the case when received signals are

affected by strong interference, our method shows a significant improvement

compared to conventional detection.


a. ,



,. ..
, 0.
* *

Coherent (C/I=-)
Conventional (C/I=o)
New method (C/I=o)
Coherent (C/l=10 dB)
Conventional (C/1=10 dB)
New method (C/I=10 dB)

0... 01 1.. 0 ...... 0 .0...0. 0

x ********o ..., ..,,*.,*o





15 20 25 30 35
Carrier-to-noise ratio (C/N) [dB]

Figure 4-5:

Comparison betw een detection (using pieceNwise constant model and
coherent detection using ...r.. knowledge of the channel and the
noise covariance).

4.5 Estimation of R(n) with Moving Average Process

In this section, we introduce an alternative new method for interference

suppression. This method uses a low-order time-domain model to parameterize

the matrix-valued spectral density of the noise and interference. We still use

the channel estimation technique described in Section 4.1. Then, we calculate

residuals from the training sequence using estimates of the channel, H, in (4.6)



as follows

Yt (Yt- XtH)T (4.19)

From these residuals, we estimate RI using the biased auto-covariance estimates

in ( see, e.g., Soderstrom and Stoica [9] and Stoica and Moses [11])

S NT y+ (YL )H (4.20)

for = 0,... where Y+I is the matrix Y with the first I columns removed, and

Y-f is the equal to Y with the last I columns removed. We next estimate R(n)

via a truncated weighted Fourier sum as follows

R(n) 1-m )Re- (.21)
m= -M
Because the covariance function is conjugate symmetric we can set R-m Rm

.Truncating the sum amounts to assuming that the power spectrum of e(n) is

smooth, which is probably a reasonable assumption in practice. To use R for

detection in (3.12), it is necessary that R is positive definite for n = 0,..., N 1.

Fortunately, this is the case. To see why this is so, note first that

1 H
R(n) fYtEX()Yf (4.22)

where E(n) is an N x N matrix whose (k,/)th element is equal to Ek,l(n)

(1 Ik 1//(M + 1))e-i2(k-1)n/N when Ik I < M and zero otherwise. But

(1 Ik l/(M + 1)) e-"" is the covariance function of a moving average process

with coefficients {1, e- ,..., e-i(M-1)} and hence E(n) is positive semi-definite.

Because of the noise in Yt, R(n) is positive definite with probability one.

4.6 Auto Selection of Model Order

Also for the method using a moving average process, it is necessary to find

the proper model order M. As for the piecewise constant model, we can apply

penalized ML to obtain an autonomous selection method. But, unfortunately,

this does not appear to work that well for the moving average process model. We

do not know the exact reason for why this does not work for the moving average

process model (in our simulation, the value of the ML metric is much 'i. -.--r

than that of the penalty term): but it is probably related to the fact that the

estimates R(n) and h(n) we use are not maximum likelihood. Instead, we have

used the detected data sequence to determine the model order. For example,

if we transmit one training sequence and five data sequences, we estimate the

channel and covariance using the training sequence and then detect first data

sequence. Then, we use the detected data sequence to select model order, and

the chosen model order is used to detect the other data sequences. Thus, our

autonomous selection can be expressed as follows

min y log |(n)|+ (n) (n) h(n)s(n) + log N (4.23)

where s(n) is the detected data.

4.7 Numerical Examples for Moving Average Process

In this section, we show some simulation examples using the new algorithm

with the moving average process. This experiment was done under the same

conditions as described in Section 4.4.

Figure 4-6 shows the estimated Symbol-error-rate (SER) versus the Carrier-

to-Noise ratio (C/N) when only thermal noise is present. We demonstrate the

results for the model orders M = 0, 2, 4, 6, 8 and for autonomous selection among

these models (in contrast to the piecewise constant method, increasing the model


1 0 -..

10 -.

L o___________

o .- Coherent

** Est. channel and noise cov. M=2
*-- Est. channel and noise cov. M=4
10-3 o n+ Est. channel and noise cov. M=6
S-o- Est. channel and noise cov. M=8
... Auto

0 5 10 15 20
Carrier-to-noise ratio (C/N) [dB]

Figure 4-6: 'mbol-error-rate (SER) for the moving average process model when
there is (h ce.- thermal noise present.

order here means increasing the number of parameters we estimate). Also, to

compare our method with a conventional detection scheme, we show results for

coherent detection and conventional detection. This figure shows that there

is not much difference between conventional detection and the method using

moving average process.

Figure 4-7 shows the case when there is one strong interferer present (

C/I = 10dB ). From this figure, we see that our method successfully mitigates

co-channel interference. Like the piecewise constant model, this moving average







S. .... . .

.... [3 ........ E3 .... .. .

+ .
., "4 ........ ; ': ,
I ........ 0. .. i '

10 15 20 25 30 35
Carrier-to-noise ratio (C/N) [dB]

Figure 4-7: 1m-bol-error-rate (SER) for the moving average process model when
there are one strong interfere (C/Il10dB) and thermal noise present.

process model suppresses the interference more effectively by increasing the

number of parameters. Autonomous selection does not ah v--i- choose the model

order which shows the best SER.

Figure 4-8 di pl- how our autonomous selection model performs when the

signal-to-noise ratio is 30dB. Our method chooses M = 0 most of the time when

only thermal noise is present. When one strong interferer and thermal noise are

present, our model selects mostly M = 6 but also M = 8.


Est. channel (Conventional)
Est. channel and noise cov. M=0
Est. channel and noise cov. M=2
Est. channel and noise cov. M=4
Est. channel and noise cov. M=6
Est. channel and noise cov. M=8

.. ] .

0 --.
"O '




Noise only when SNR = 30dB

1u I I I I





0 I I I
0 2 4 6 8

interference with no!e when SNR = 30dB
601 1 I I I I I


2 20

Figure 4-8: Autonomous selection of model order, SNR
average process.

l:JB, 0: : moving-

Figure 4-9 shows a comparison between detection using our method and co-

herent detection using perfect knowledge of the channel and the noise covariance.

Note the similarity between Figure 4-9 and Figure 4-5.

rI I -J-- b __ LI ___ I r' I



.x -.


-, x

| |

Coherent (C/I=-)
-e- Conventional (C/I=o)
New method (C/I=-)
..... Coherent (C/I=10 dB)
.o' Conventional (C/I=10dB)
New method (C/I=10 dB)

S.. .... o.... 0 ........ .. .... 0 ...


15 20 25
Carrier-to-noise ratio (C/N) [dB]

T' F comparison between detection using moving average process
and coherent detection using : knowledge t he channel and
the noise covariance.






Figure 4 9:

*" 'X....... x,


In C'!i pter 2, we introduced receive diversity, which is a popular method

to combat fading in wireless communications. By installing multiple antennas

at receiver side (usually at the base station), we can mitigate channel fading as

well as increase the channel capacity. In recent years, researchers have found

that it is possible to obtain the same advantage as receive diversity by using

multiple antennas at the transmitter side (this is called transmit diversity).

Using multiple antennas at both sides, we can get much more benefits than with

receive diversity or transmit diversity alone. However, to exploit these benefits,

we need a coding scheme, which is called Space-Time Coding. In this chapter,

we briefly introduce a MIMO system, space-time block coding (STBC) and

space-time OFDM (ST-OFDM), which combines space-time coding and OFDM.

5.1 MIMO Systems and Space-Time Block Coding

Consider a MIMO system which consists of nt transmit antennas and n,

receive antennas. We assume that the propagation channels are linear, time-

invariant, and frequency flat. Let X and Y be nt x N transmitted data and

n, x N received data matrices, respectively. Then, the received data can be

expressed in a linear model as follows

Y HX+E (5.1)

where H is an n, x nt channel matrix in which the (m, n)th element corresponds

to the channel gain between the receive antenna m and the transmit antenna n;

E is an n, x N noise matrix in which each element is independent and complex
Gaussian with zero mean and variance a2. Then, given Y, ML detection of the
code matrix X from a finite constellation X is equivalent to

X =argmin mY HX 12 (5.2)

Like for the SISO system in Section 2.1, we can derive the detection error
probability from Equation (5.2). By a similar calculation as in Section 2.1

P(X, |H(X xo)
P(XX-+XH) =2a2 ) (5.3)

Next, we assume, for simplicity, that all channels in H are independent and
complex Gaussian with zero mean and variance p2; hence, the channel is Rayleigh
fading. The probability density function for the Rayleigh fading channels can
then be expressed as

p(H) f= l2t exp ( H 2 (5.4)
pnnt W 2

Using the C'!l ~i i l' bound, we get the average error probability as follows (a more
detailed derivation can be found in [5, Section 4.3])

EH[P(X XoH)] j P(X -X XoH)p(H)dH
42(X Xo)(X Xo)" + I- (5.5)
< (X- Xo)(X- Xo)lH I (492)

The above equation shows that the average error probability for a MIMO system
behaves as

P(error) oc SNR-"'"


which means that receive diversity of order n, and transmit diversity of order

nt are obtained, as long as X Xo has full rank. Hence, the length of the

transmission N has to be longer than the number of transmit antennas, nt.

It is not easy to achieve transmit diversity without losing transmission data

rate and increasing the decoding complexity. Space-time coding is a way to

achieve high data rate and low decoding complexity. One of well known space-

time coding schemes is the Alamouti code [12], which is a transmission scheme

for two transmit antennas. Another case is the linear orthogonal space-time block

code (OSTBC), which has a simple structure. Consider the transmission of n8

symbols {si, s s,}. Then, encoded linear STBC has following structure
X = (snA + isnB,) (5.7)

where An and B, are fixed code matrices, and s, and s9 are the real and

imaginary parts of the symbol s,, respectively. OSTBC has the special merit

that the MIMO channel decouples into n8 independent scalar channels, because

of the following orthogonality property of OSTBC

XXH Sn 2 I (5.8)

A more detailed explanation for OSTBC can be found in Larsson and Stoica [5],

C!i lpter 7, Tarokh et al. [13] and Larsson et al. [14].

5.2 Space-Time OFDM

ST-OFDM is a combination of space-time block coding and OFDM to obtain

transmit diversity for an OFDM system. The main idea of ST-OFDM is identical

to that of OFDM. Because a ST-OFDM system uses multiple transmit antennas,

it sends No encoded nt x N matrices in which n, symbols are encoded. Hence,

nNo symbols are transmitted during NNo symbol transmit time intervals. The

transmission and detection procedures for ST-OFDM are as follows. First, from

nsNo symbols to be transmitted, encode each group of n, symbols to No code

matrices G(n) for n = 0, No 1. Second, take the inverse Fourier transform

of the code matrices as follows

X(n)= (G(k) exp ( nk n = 0,...,N 1 (5.9)

Then, add the CP which is taken from Np,, last code matrices of the transmitted

data matrices, and select the first columns of all code matrices to transmit them

sequentially through channels, assuming that each channel has L dl 1 i taps.

Next, select the next columns for the transmission until all the columns of the

code matrices have been sent out. At the receiver, discard the CP. Then the

received code matrices can be expressed as follows

Y(n) = HkX(n k) + E(n)
k=o (5.10)

1 E N HkG(k'. -'" +E(n)
k=-0 m=0

After taking Fourier transform of (5.10) and following similar derivation as

in (3.4), the received data become

Z(n) = 1 Y(l)e
l =0
H( G() + E(511)
1 No- 1
H(n) G(n) + 1- E(1)e-- i


H(n) >= Hke- (5.12)

Hence, ML detection for ST-OFDM is equivalent to minimizing the following


IIZ(n)- H(n)G(n) 12 (5.13)

More about ST-OFDM can be found in Larsson and Stoica [5], ('!i ilter 8 and Li

et al. [15].

In this chapter, we extend our interference suppression algorithm to the
ST-OFDM case.
6.1 ('CI ,i,, I Estimation

Because a ST-OFDM system uses multiple transmit antennas and transmits
an nt x N inverse Fourier transformed code matrix X, the number of training
sequences, which consist of No symbols each, is at least N. Let us assume that
we have received NT training sequences (T > 1) and that each channel has L
d.1 iv taps. Denote transmitted and received code matrices by Xt(n) and Yt(n)

for n = 0, No 1, t = 1, T. Then, the received training data can be
expressed as follows
Yt(n) = HiXt(n-1) + E(n), n 0,...,No 1 (6.1)
where Hi stands for Ith matrix valued delay channel tap. By using a block-
Toeplitz matrix, Equation (6.1) can be rewritten as

[Y,() Yt(No-) (6.2)

=Ho ... HL Xt + Et(0) ... Et(No- 1)

.. ... Xt(N L- )

Xt(No L)

Xt(-L) Xt(-L- ) ... X,(O) ...

Xt(No- 1)

vec(ABC) (CT

A) vec(B)


where 0 denotes the Kronecker product and vec(-) is the vectorization of a

matrix. By vectorizing both sides of (6.3), we get




vec(Yt(No 1))

Maximum likelihood estimate of h

following metric

vec(Ho) vec(Et(0))
S et (6.6)

vec(HL)_ vec(Et(N 1))

in (6.5) is equivalent to minimizing the

minE Yt -(XT Il,)h 2

and the ML estimate of the channel h is determined by

h =vec(Ho)T .. vec(H)T

= T [((X,)(XtX,) )T





Note that



In,] Yt


Yt = (X & I,,)h + et

Also, note that after estimating the channel h and reconstructing H,
H(n) for the flat fading subchannels can be calculated by taking the Fourier

H(n) = Hle-i2-F Nl (6.9)

6.2 Estimation of R(n) with Moving Average Process
First, note that the received code matrix, Yt(n), consists of N number of
n, x 1 column vectors. Vectorization of Yt(n) gives

Yt (n)
vec(Yt(n)) = yt(n) = (6.10)


Next, we calculate the residuals from the vectorization of the received data using
the estimated channel h and the block-Toeplitz matrix (6.3) as follows

9 t (XT & I,n)h (6.11)

Following (6.10), we can express it as

it = yT(0) ... y[(N 1)

= y(0)) T -. ( ( )) T -. (N.- 1)) .. ( N(N 1))


Then, we reconstruct the matrix Yt(n) as follows

Yt(n) = ~r(0) ... YT( 1) = 0,..., N- 1 (6.13)

Following the same procedures as (4.20), we estimate R, using the biased auto-

covariance estimates
-NNo E (k) (6.14)
t= 1 k=

for 1 0,... where Y+(k) is the matrix Y(k) with the first I columns removed,

and Y -(k) is the equal to Y(k) with the last I columns removed. We next

estimate R(n) via a truncated weighted Fourier sum as follows

R(n)= (i M- l)me -, mn (6.15)
= -- M

Using R(n), the maximum likelihood detection rule for the information symbols

is expressed as
argmin R(n)(Z(n) H(n)G(n)) (6.16)

To choose a proper model order, the autonomous selection method can be

expressed as follows

min {log |R()|+ R-1/2(Z() ()G()} + logN (6.17)

where G(n) is the detected code matrix.

6.3 Numerical Examples for the Moving Average Process

We show simulation results for our algorithm for an ST-OFDM system

with N = 64 subcarriers, nt = 2 transmit and n, = 2 receive antennas. The

Alamouti code matrix is used in our experiment, and the channel has the same

d,1 iv taps and the same average power delay profile as in Sections 4.4 and 4.7.

Two training sequences, which each sequence being an impulse, are used, and the

constellation is QPSK.

S; ,.+ Est. channel and noise cov. M=6
W -2 o0 Est. channel and noise cov. M=8
10 Auto

2 -3
10 -


0 5 10 15 20
Carrier-to-noise ratio (CN) [dB]

Figure 6 -1: m" bol-error-rate (SER) for the moving average process model when
there is -. thermal noise is present.

Figure 6-1 shows that the Symbol-error-rate performance versus the Carrier-

to-Noise ratio (C/N) for an ST-OFDM system when only thermal noise is

present. This figure shows that transmit diversity effectively combats the channel

fading compared to Figure 4-6.

Figure 6-2 shows results for the case when one strong interferer comes

from a single antenna user. CCI has the same channel taps and average power

d. 1liv profile as the desired user has, and the signal to interference ratio is 10dB.

The interference suppressing algorithm outperforms the one that does not

10 Est. channel and noise cov. M=U
*o: Est. channel and noise cov. M=2
S-0.- Est. channel and noise cov. M=4
.. -+ Est. channel and noise cov. M=6
S-2 \ o-- Est. channel and noise cov. M=8
S10 '

U -2.. Est..hanne an noiex .coy. M, =
...... Auto

2 -3

SE I ",' K.....
CO 10 -

10-5 -

0 5 10 15 20 25 30 35
Carrier-to-noise ratio (C/N) [dB]

Figure 6-2: "'-mbol-error-rate (SER) for the moving average process model when
strong i -..:.... comes from a single antenna user.

take interference into account. Also, the algorithm suppresses the interference

more effectively by increasing the number of parameters we estimate. However,

Figure 6-3 shows that when strong interferers comes from multiple transmit

antennas, our method can not mitigate interference well. Also, in this case,

increasing the model order does not have a significant effect on the interference


0 5 10 15 20 25
Carrier-to-noise ratio (C/N) [dB]

30 35

Symbol-error-rate (SEl) for the moving average process model when
strong interferers comes from co-channel user's multiple antennas.

Figure 6 3:


Co-channel interference is a major limiting factor for cellular communication

systems, and it is expected to become a major bottleneck also for wireless local

area networks. In this thesis, we have presented a conceptually very simple and

computationally cheap method for suppression of co-channel interference in

systems that use OFDM modulation and which have multiple receive antennas.

Our method comes in two versions: one that parameterizes the interference and

noise spectrum via a piecewise-constant model, and one that uses a low-order

moving-average model. In either case, a technique inspired by MDL is used to

automatically select the model order, hence avoiding any user parameters to be

selected prior to the application of the algorithm. Simulation results show that

our method can substantially improve the performance compared to conventional

training-based detection.

Several problems remain which could be the topic of future studies. First,

the model order selection strategy used is by no means optimal (optimal such

selection methods in general do not exist, see Lanterman [16]) and it is possible

that an alternative methods for choosing the model order may outperform ours.

Second, the estimates of the channel and noise covariance that we use are not

maximum-likelihood, but nevertheless they are computationally simple to obtain.

It is likely that using maximum-likelihood estimates in lieu of our proposed

estimates may lead to a method that works better than ours. The maximum-

likelihood estimates generally do not exist in closed form, and obtaining them

numerically is computationally burdensome. Yet, some low-complexity approx-

imations to maximum-likelihood for the problem under study can be found in

Jeremic et al. [17]; using the estimates proposed therein instead of our subopti-

mal estimates might improve performance. Finally, as the interference signals are

Gaussian only to within a certain degree of approximation, using a model that

allows for non-Gaussian interference may also improve the error rate.

We also studied the application of our algorithm in a system that uses space-

time OFDM. This scenario is more difficult, because interference signals that

originate from a user with multiple transmit antennas have a richer correlation

structure than signal originating from a single-antenna transmitter. For space-

time OFDM, our algorithm was able to suppress co-channel interference only to a

small degree. It is possible that the use of a more accurate model, perhaps along

with a larger number of receive antennas, may improve the situation. This topic

is also left for future work.


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12 G. Jongren, D. Astily, and B. Ottersten, "Structured spatial interference
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Jonghyun Won was born in Buchon, Korea. He received his B.Sc. degree

in 2000 from the Department of Electrical Engineering, Dongguk University,

Seoul, Korea. Since 2001, he has been pursuing a Master of Science degree in

the Department of Electrical and Computer Engineering, University of Florida,

Gainesville, FL.

Full Text






TABLEOFCONTENTS page ACKNOWLEDGMENTS............................ iv LISTOFFIGURES............................... vii ABSTRACT................................... ix CHAPTER 1INTRODUCTION............................. 1 2WIRELESSCOMMUNICATIONSOVERFADINGCHANNELS... 3 2.1ErrorProbabilityforCommunicationSystem........... 3 2.2AntennaDiversity.......................... 5 2.3Interference............................. 7 3ORTHOGONALFREQUENCYDIVISIONMULTIPLEXING(OFDM) 8 3.1DataTransmissionandDetectioninOFDM........... 9 3.2MLDetectionforOFDMwithWhiteNoise............ 11 3.3MLDetectionforOFDMwithColoredNoise........... 12 4NEWALGORITHMFORINTERFERENCESUPPRESSIONIN OFDM.................................. 16 4.1ChannelEstimation......................... 16 4.2Estimationof R ( n )withPiecewiseConstantModel....... 18 4.3AutomaticSelectionofModelOrder................ 20 4.4NumericalExamplesforthePiecewiseConstantModel..... 22 4.5Estimationof R ( n )withMovingAverageProcess........ 26 4.6AutoSelectionofModelOrder................... 28 4.7NumericalExamplesforMovingAverageProcess........ 28 5INTRODUCTIONTOSPACETIME-OFDM(ST-OFDM)...... 33 5.1MIMOSystemsandSpace-TimeBlockCoding......... 33 5.2Space-TimeOFDM........................ 35 6APPLICATIONOFOURALGORITHMTOST-OFDM....... 38 v


6.1ChannelEstimation......................... 38 6.2Estimationof R ( n )withMovingAverageProcess....... 40 6.3NumericalExamplesfortheMovingAverageProcess..... 41 7CONCLUSIONANDFUTUREWORK................ 45 REFERENCES.................................. 47 BIOGRAPHICALSKETCH........................... 49 vi


LISTOFFIGURES Figure page 3{1DatatransmissionandreceptionschemeforanOFDMsystem... 9 4{1ExampleofthepiecewiseconstantmodelforaSISOsystem..... 19 4{2Symbol-error-rate(SER)forthepiecewiseconstantmodelwhen thereisonlythermalnoisepresent.................. 23 4{3Symbol-error-rate(SER)forthepiecewiseconstantmodelwhen thereareonestronginterferencesignal(C/I=10dB)andthermal noisepresent.............................. 24 4{4Autonomousselectionofmodelorder,SNR=30dB,forpiecewiseconstantmodel............................. 25 4{5Comparisonbetweendetection(usingpiecewiseconstantmodel andcoherentdetectionusingperfectknowledgeofthechannel andthenoisecovariance)...................... 26 4{6Symbol-error-rate(SER)forthemovingaverageprocessmodel whenthereisonlythermalnoisepresent............... 29 4{7Symbol-error-rate(SER)forthemovingaverageprocessmodel whenthereareonestronginterferer(C/I=10dB)andthermal noisepresent.............................. 30 4{8Autonomousselectionofmodelorder,SNR=30dB,formovingaverageprocess............................. 31 4{9Thecomparisonbetweendetectionusingmovingaverageprocess andcoherentdetectionusingperfectknowledgeofthechannel andthenoisecovariance....................... 32 6{1Symbol-error-rate(SER)forthemovingaverageprocessmodel whenthereisonlythermalnoiseispresent............. 42 6{2Symbol-error-rate(SER)forthemovingaverageprocessmodel whenstronginterferercomesfromasingleantennauser..... 43 vii


6{3Symbol-error-rate(SER)forthemovingaverageprocessmodel whenstronginterfererscomesfromco-channeluser'smultiple antennas................................ 44 viii


Weproposenewmethodsforco-channelinterferencesuppressioninorthog-onalfrequencydivisionmultiplexing(OFDM)systemswithmultiplereceiveantennas.Ourmethodsusealow-ordertime-domainmodeltoparameterizethematrix-valuedspectraldensityofthenoiseandinterference.Apenalizedmaximumlikelihood-functionapproachisusedtochoosethepropermodelorderautomaticallyineachreceiveddataframe.Weshowthatthesemethodssignif-icantlyimprovethesymbol-error-ratewheninterferenceexistsinthereceiveddata.Numericalexamplesareprovidedtoillustratetheperformanceofournewmethodsandcomparewithconventionalmethods.Wealsodiscussthedicultiesassociatedwiththeapplicationofourmethodtospace-timeOFDM(ST-OFDM). ix


Orthogonalfrequencydivisionmultiplexing(OFDM)hasbecomeoneofthemostpromisingtransmissiontechniquesforwirelesslocalareanetworksbecauseofitslowequalizationcomplexityinfrequencyselectivefadingchannels,anditsrobustnesstointer-symbolinterference(ISI).OFDMtransformsafrequency-selectivefadingchannelintomultipleindependentatfadingsubchannels.Bytransmittingonesymbolineachsubchannel,thesymboldurationisincreased;hencetheeectsofISIcausedbyafrequencyselectivefadingenvironmentarereduced.Therefore,anOFDMsystemcanachieveahighdatarateandareliabletransmissioninafadingchannel.However,duetolimitedandcongestedfrequencybandsincellularradioenvironments,thesystemperformanceisoftenlimitedbyinterference. Interferenceisoneofthemajorlimitingfactorsontheperformanceofwirelesscommunicationsystems,asittypicallyincreasesthebit-error-rate(BER).Therearetwomajortypesofinterference:so-calledadjacentchannelinterference(ACI)andco-channelinterference(CCI). ACIcomesfromsignalsthataretransmittedbyacommunicationsystemusingafrequencybandadjacenttothedesireduser'sfrequencybandinthesamearea.Thisinterferencecanbeminimizedbyusingastrictlyrestrictedsignalbandwidthoraproperchannelassignment.Gueyetal.[1]proposesACIrejectionmethodsforland-mobileradiosystems. CCIiscausedbythefrequencyreuseschemeinwirelessnetworks,forexample,whensignalscomefromaco-channelcellwhichusesthesamefrequency 1


bandasthedesireduser.AsCCIconsistsofthesamefrequencies,itcannotberejectedbysimplyincreasingthetransmitpowerorusingnarrowchannelltering.TherearemanystatisticalCCIinterferencesuppressionmethodsforradiocommunicationnetworks.Forexample,Jongrenetal.[2]proposedaninterferencerejectionmethodusinglow-rankfeaturesofthespatialcovariancematrix;KarlssonandHeinegard[3]suggestedan"interferencerejection"methodfortheGSMsystem.Thesetechniqueshavebeenquitesuccessful(seeDametal.[4]). Inthisthesis,weproposenewCCIsuppressionmethodsforOFDMandspace-timeOFDM(ST-OFDM)withmultiplereceiveantennas.Thebasicideaofouralgorithmsisthatthe(matrix-valued)powerspectrumoftheinterferenceissmooth,andhencewecanusealow-ordertime-domainmodeltoparameterizethespectraldensityofthenoiseandinterference.Oneofourmethodsproposesasimplemodelforthecovarianceoftheinterferenceandnoise.ThismodelassumesthatpairsofMadjacentsubchannelshavethesamespectraldensity.Thus,theperformanceofourmodeldependsonhowweselectthenumberofsubchannels,M.Apenalizedmaximum-likelihoodfunctionapproachisusedtochooseMautomaticallyineachreceiveddataframe.Anothermethodusesamultidimensionalmovingaverage(MA)processtomodelthepowerspectrumdensityoftheinterferenceandnoisewithavariablenumberofparameters,M.Alsohere,apenalizedmaximum-Likelihoodfunctionapproachisusedtoselectthepropernumberofparameters.WealsoapplythesealgorithmsintheST-OFDMcaseandshownumericalexamples.


Inawirelessenvironment,transmittedsignalsusuallydecreaseinpowerorbecomedistortedinshapebecauseofthetime-variationofthechannelandthepropagationenvironment.Communicationinamovingcarcanbeoneexampleofafadingchannel,sincethesurroundingenvironmentofthecarvariessothatthepropagationchannelchangesaccordingtothevelocityofthecar. 2.1 Error Probability for Communication System Letusconsiderasimpleexampleofawirelesscommunicationsystemtoseetheeectoffading.Weassume,forsimplicity,thatthepropagationchannelislinear,time-invariant,andfrequencyat;alsoassumethatthesystemconsistsofasingletransmitandreceiveantenna.Suchasystemiscalledasingle-inputsingle-output(SISO)system.ConsideratransmittedsymbolsfromaniteconstellationS.Thenthereceivedsignalcanbeexpressedasy=hs+e(2.1) wherehdenotesthechannelgainandedenoteszero-meancomplexGaussiannoisewithvariance2.Theprobabilityofdensityfunction(p.d.f)forthislinearmodelisp(y)=1 Wecandetectsusingthemaximumlikelihoodrule,whichmaximizes(2.2)andisequivalentto^s=argmins2Sjyhsj2(2.3) 3



Averaging(2.5)overhusing(2.6),wegetEh[P(s!sojh)]=Zjhj2P(s!sojh)P(jhj2)djhj2Z10expjssoj2jhj2 Thesignal-to-noise(SNR)forthereceivedsignalis2=2,andthelaststepof(2.7)showsthattheaverageerrorprobabilityforthefadingchannelbehavesasP(error)/SNR1(2.8) Wecancomparethisresulttothatofaconstantchannel.IfweusethefactthattheSNRequals2=2andtheChernoboundin(2.5),thentheerrorprobabilityforadeterministicchannelbehavesasP(error)/exp(SNR)(2.9) 2.2 Antenna Diversity SincethefadingisRayleighdistributedandtime-variant,sometimesthechannelcanbeuseless.Oneofthemostpopularmethodsforcombatingthefadingeectisantennadiversity.Byemployingmultipleantennasonthetransmitterorthereceiverside,thecommunicationsystemcantransmitthesameinformationovermultiplechannels,whichareindependentofeachother.Suchasystemiscalledamultiple-inputmultiple-output(MIMO)system.InaMIMOsystem,thefadingeectcanbesignicantlymitigatedandalsothechannelcapacitycanbeincreasedasanotheradvantage.Equippingmultipleantennasatthebasestation,whichgivesreceivediversityfortheuplink,isdoneinmanycommunicationsystems.ConsideringnrreceiveantennasinsteadofoneantennainaSISOsystem,wecanseehowreceiveantennadiversitycounteracts


thefadingeect.Inthiscase,thechannelhbecomesan1nrvectorandthus,weleth=[h1hnr]Tbethepropagationchannelvectorinwhicheachvaluecorrespondstothechannelgainbetweenthetransmitantennaandeachreceiveantenna.Here()Tdenotesthetranspose.Then,withthesameassumptionsasfortheSISOsystemexample,thereceiveddatacanbewrittenasy=hs+e(2.10) wherey=[y1ynr]Tisanreceiveddatavectorande=[e1enr]TisacomplexGaussiannoisevectorwithzeromeanandcovariancematrix2Inr.Thepairwiseerrorprobability,obtainedbyasimilarcalculationasSection2.1,isP(s!sojh)=Qs IfweassumethatthenrchannelsareindependentandcomplexGaussianrandomvariableswithzeromeanandvariance2nr,respectively,theprobabilitydensityfunctionforhcanbeexpressedasP(h)=nrYn=11 Again,usingtheChernobound,(2.11)and(2.12),wecanderiveaboundontheaverageerrorprobabilityforMIMOfadingchannelsasfollowsEh[P(s!sojh)]=Zjh1j2Zjhnrj2P(s!sojh)P(h)djh1j2djhnrj2Z10expjssoj2jh1j2


ThelastpartofEquation(2.13)showsthatthemoreantennasthereareonthereceiverside,themoremitigationoffadingeectisachieved.Also,itshowsthattheaverageprobabilityofadetectionerroroverthefadingchannelbehavesasP(error)/SNR(diversityorder| {z }=nr)(2.14) Also,adiversityeectaswellasotherbenetscanbeobtainedbyequippingmultipleantennasattransmitterside(thisiscalledtransmitdiversity).Accord-ingtorecentresearch,itispossibletoreproducethediversityfrommultipletransmitantennaswhilepreservingthetransmissiondatarateandasimpledecodingcomplexity.Thismethod,whichiscalledSpace-TimeCoding,willbeintroducedinChapter5. 2.3 Interference Recently,thepopulationofmobileusershasdramaticallyincreasedduetothefastdevelopmentofwirelesscommunicationtechnologies;hence,usingcarefullychosenchannelreuseschemeshasbecomemoreimportantincellularcommunicationsystems.Becauseofchannelreuse,interferencebetweensignalsfromthesamefrequencybandisoftengenerated.Thisinterference,whichiscalledco-channelinterference(CCI),istheoneofthemajorlimitingfactorsfortheperformanceofwirelesscommunicationsystems.Unlikethermalnoisewhichcanbeovercomebyincreasingthetransmitpower,CCIcannotbeoverpoweredbyincreasingthecarrierpower.Moreover,anincreaseincarriertransmitpowercanaugmenttheinterference.However,theeectofinterferencecanoftenbereducedbysuitablesignalprocessingmethodatreceiverside.


Aswirelesscommunicationshaveexperiencedgreatsuccessandarewidelyusedeverywhere,thedemandsforgoodqualityandvarietyofservicehaveincreased.Tofulllthesedesires,developmentforhighspeedwirelesscommu-nicationsystemshasbeeninvestigated.OnesuccessfulapproachtoachievingreliableandecienthighspeeddatatransmissionschemeisOFDM. ThemainideaofOFDMisthatbydividingtheentirechannelbandwidthintoindependentmultiplenarrowbands,eachsub-bandbecomesrelativelyat.WecancompareOFDMwithaconventionalsinglecarriersystem.Acon-ventionalsystemtransmitseachdatasymbolonebyoneinawholeoccupiedchannel.Oneofthedisadvantagesusingasinglecarrieristhatifthechannelsuf-fersfromfrequencyselectivefading,thechannelequalizationusuallyiscomplex.Ontheotherhand,inanOFDMsystem,eachsubchanneloccupiesonlyasmallfractionofthetotalbandwidth,andOFDMtransmitsonedatasymbolineachsubchannelinparallel.Sinceasmallbandwidthofeachsubchannelmakesthesymboldurationsmuchlongerthanthememoryofthechannel,OFDMcanusesimplechannelequalizationandalsoobtainthesamedatarateasaconventionalsystem.ThispropertyallowsOFDMtosupporthighspeedrates.AdetailedexplanationofOFDMcanbefoundinLarssonandStoica[5],HeiskalaandTerry[6]andLietal.[7].Here,weintroducethefundamentalprinciplesofanOFDMsystemfollowingLarssonandStoica[5]. 8


9 Figure3{1:DatatransmissionandreceptionschemeforanOFDMsystem(from LarssonandStoica[5],chapter5) 3.1 Data Transmission and Detection in OFDM ConsiderageneralOFDMsystemwithasingletransmitantennaand n r receiveantennas.Supposewehave s ( n )symbolsfor n =0 ;:::;N 1tobe transmitted.AftertakinginverseFouriertransformforthesymbols,theencoded datacanbeexpressedas x ( n )= 1 p N N 1 X k =0 s ( k )exp i 2 N nk ;n =0 ;:::;N 1(3.1) Priortotransmission,acyclicprex(CP)isaddedatthebeginningofthe sequence x ( n ).TheCPisacopyofthelast N pre symbols,where N pre L and L isthechanneldelayspread.TheCPmakesthetransmittedsignalperiodicwith period N sothatthelinearconvolutioninducedbythepropagationchannelis transformedintoacircularconvolution,whichcorrespondstomultiplicationof channelgainsandsymbolsinthefrequencydomain.Wecanexpresstheencoded datawithacyclicprexasfollows x ( n )= x ( n + N ) ;n = N pre ;:::; 1(3.2)


Figure3{1illustratesthetransmissionscheme.Fromtheencodeddatax(n),wecanobservethateachsymbols(k)iscarriedoneachsubcarrierei2 Nnwhichareorthogonaleachother.Theencodeddataaretransmittedthroughachannel,whichhasLdelaytaps,andafterdiscardingCP,thereceiveddatavectorfromthemultipleantennascanbeexpressedaslinearconvolutionofthechannelandtheencodeddataplusthermalnoiseandinterferencey(n)=LXk=0hkx(nk)+e(n)=1 N(nk)m+e(n)(3.3) wherehkdenoteskthvector-valuedchanneltap.AftertakingtheFouriertransformofthereceiveddata,wegetthemultiplicationofeachfrequencyatfadingtransformedchannelanddata(see,e.g.,LarssonandStoica[5],Chapter5)z(n)=1 Nln=1 N(lk)mei2 Nln+1 Nln=1 N(nm)lei2 Nkm+1 Nln=s(n)LXk=0hkei2 Nkn+1 Nln=s(n)h2 Nn+1 Nln| {z }~e(n)(3.4) wherewehavedenedthechanneltransferfunctionh2 Nn=LXk=0hkei2 Nkn(3.5)


FromEquation(3.4),wecandetectthesymbolss(n)usingsimpleequalization. 3.2 ML Detection for OFDM with White Noise First,weneedtoconsiderthenoisetermbeforederivingtheMLdetectorforOFDM.Weassumeforsimplicitythatthenoisevectorse(n)arespatiallywhitecircularGaussianrandomvariableswithzeromeanandvariance2.Thisassumptionisproperwhenonlythermalnoiseexist.LetconsiderthenoisecovarianceinEquation(3.4)asfollowsE[~e(n)~eH(k)]=E1 Nln1 NmkH=1 Nlnei2 Nmk(3.6) Fromtheassumptionone(n),E[~e(n)~eH(k)]=1 Nl(nk)=8>><>>:2I;n=k0;n6=k(3.7) Equation(3.7)showsthatthenoisetermin(3.4)iswhiteGaussiannoisewithcovariance2I,andthus,z(n)isasucientstatisticforthedetectionofs(n).Hence,MLdetectionforOFDMisequivalenttominimizingthefollowingmetrickz(n)s(n)h(n)k2(3.8) whereh(n)=LXl=0hlei2 Nln(3.9)


3.3 ML Detection for OFDM with Colored Noise Amoregeneralassumptiononthenoisee(n)istosayitiscircularlysymmetricGaussianwiththefollowingcovariancefunctionRl=E[e(n)eH(nl)](3.10) where()Hdenotestheconjugatetranspose.Thenthepowerspectrumofe(n)isR(n)=1Xl=Rlei2 Nnl(3.11) Suchcolorednoisecanbeusedtomodelco-channelinterference.1 WenowshowthatR(n)isthecovarianceofthenoiseterminEquation(3.4)andthat~e(n)isindependentof~e(m),n6=m.Firstweconsidertheauto-covariancefunctionof~e(n)asfollowsE[~e(n)~eH(n)]=E"1 Nkn1 Nnk!H#=1 N(kk)n(3.13)


Letkk=l,thenE[~e(n)~eH(n)]=1 Nln=1 Nln+N1Xl=1N1Xk=lRlei2 Nln!=1 Nln+N1Xl=1(N+l)Rlei2 Nln)=N1Xl=(N1)Rlei2 NlnN1Xl=(N1)jlj Nln(3.14) Underthemildassumptionthatthecovariancesequencedecayssucientlyrapidly,thelasttermin(3.14)goestozerowhenN!1.HenceE[~e(n)~eH(n)]N1Xl=(N1)Rlei2 Nln(3.15) Hence,inanOFDMsystem,theauto-covarianceofthenoisetermaftertakingtheFouriertransformofthereceiveddataisthesameasthepowerspectraldensityofthenoiseinthereceiveddata.Next,wecalculatethecross-covarianceof~e(n)E[~e(n)~eH(m)]=E"1 Nkn1 Nmk!H#=1 N(nkmk)(3.16)


Ifweletkk=l,thenEquation(3.16)canbeexpressedasfollowsE[~e(n)~eH(m)]=1 N(nkmk+ml)=1 N(nm)kei2 Nml| {z }A(nm)+1 N(nm)kei2 Nml| {z }B(nm)(3.17) NotethatN1Xk=0ei2 N(nm)k=0(3.18) andl1Xk=0ei2 N(nm)k=N1Xk=lei2 N(nm)k(3.19) Ifwetakethenormof(3.17),thenormofrstsummationterm,kA(nm)k,canbeexpressedasfollowskA(nm)k1 N(nm)kj| {z }jlj1


Also,kB(nm)kcanbeexpressedasfollowswith(3.19)kB(nm)k=1 N(nm)k=1 N(nm)k1 N(nm)kj1 Underthemildassumptionthatthecovariancesequencedecayssucientlyrapidly,whenN!1,(3.20)and(3.21)gotozero.Asaconsequence,theFouriertransformednoiseistemporallywhiteandhascovarianceR(n).


Whenthereisstronginterferenceinthereceivedsignals,itcanbebenecialtosuppresstheinterference,sothatcorrectsignaldecisionscanbemade.Oneapproachtoestimatethecovarianceoftheinterferenceistoinstantlyestimatethecovarianceofeachsubchannel,butnormallysuchinstantaneouscovarianceestimatesdonotworkwellifthereisonlyashorttrainingsequence.Oneapproachtocanceltheco-channelinterferenceissuggestedinLiandSollenberger[8],whichusesaminimummean-squareerror(MMSE)diversitycombiner(DC).Theyestimatethechannelsandthenoisecovarianceinthefrequencydomain,andusetheseestimatedvaluestosuppresstheinterference.Inthischapter,weintroducenovelmethodsforinterferencesuppressioninanOFDMsystem.Thesealgorithmsarebasedonalow-ordertimedomainmodeltoparameterizethematrix-valuedspectraldensityoftheinterference. 4.1 Channel Estimation Letusassumethatwereceivedmorethanonetrainingsequence(T1).AftertakingtheFouriertransformofthereceivedtrainingdata,theycanbeexpressedaszt(n)=LXl=0hlexpi2ln Nst(n)+1 Nkn(4.1) forn=0;:::;N1;t=1;:::;T,wherezt(n)standsforFourier-transformedknowndata.Wecanestimatethechannelfrom(4.1),assumingknowntrainingdata,usingvariousmethods. 16


Theparsimonyprinciple[9],explainsthatamongmodelsdescribingdatawell,themodelwiththesmallestnumberofparametersisabetterchoicethanothers.Forthisreason,itismoredesirabletoestimatetimedomainchannelsinsteadoffrequencydomainchannels,becausetheformerconsistofnr(L+1)parameters;howeverinthefrequencydomain,wewouldneedtoestimatenrNparameters.Channelestimationcanbedonebythefollowingprocedure. DenotethetransmittedandreceivedtrainingdatabeforeFouriertrans-formationbyxt(n)andyt(n).UsingToeplitzmatrices,thetrainingdataXt,channelH,noiseEtandreceiveddatasequenceYtcanbeexpressedbyYt=XtH+Et(4.2) whereXt=266666664xt(0)xt(1)xt(L)xt(1)xt(0)xt(1L).........xt(N1)xt(N2)xt(N1L)377777775 Thenegativeloglikelihoodfunctionfor(4.2)becomes,assumingthenoiseiswhiteminHTXt=1kYtXtHk2(4.5)


andtheMLestimateofchannel^Hisdeterminedby^H=TXt=1XHtXt1TXt=1XHtYt(4.6) Afterestimatingthechannel^H,thefrequencydomain^h(n)channelscanbeeasilycalculatedbyusingFouriertransformation^h(n)=LXl=0^hlei22 Nnl(4.7) 4.2 Estimation of with Piecewise Constant Model Theinterferencesignaline(n)consistsofthesamefrequenciesasdesiredsignal,andithaspropagatedoverachannelwithsimilarpropertiesasthechannelh(n).Hence,wecanexpectthatthepowerspectrumofe(n)issmoothandthischaracteristicscanbeusedtoestimateR(n).Followingthisreasoning,wecanexpectthatthecovarianceinaportionofthefrequencybandconsistofapproximatelythesamevalues.ThepiecewiseconstantmodelassumesthattheinstantcovarianceR(n)ofMadjacentsubchannelsconsistsofthesamevalueorapproximatelythesamevalue.Withthismodel,wepartitionthefrequencybandinN=Msegments,calculateR(n)foreachsubchannel,takeanaverageofR(n)valuesineachsegmentwhichconsistsofMadjacentsubchannelsandapplyeachaveragedvalue^R(n)toMsubchannelsineachsegmentinsteadofR(n).MoredetailedexplanationsareasfollowsUsinganestimatedchannel^Hfrom(4.6),wecalculateresidualsfromreceivedtrainingdata~Yt=(YtXt^H)T(4.8) where~Ytconsistsof~Yt=~yt(0)~yt(N1)(4.9)


Figure4{1:ExampleofthepiecewiseconstantmodelforaSISOsystem. WetaketheFouriertransformof~yt(n)asfollows~rt(n)=1 Nnk(4.10) WenextestimateR(n)foreachsubchannelby^R(n)=1 andthen,taketheaverageineachsegmenti=0;:::;N=M1,accordingtotheparameterM^RM(i)=1 Weuse^RM(i)forsegmentitosuppresstheinterferenceinsteadoftheinstan-taneousmomentbasedestimate^R(n).Figure4{1showsanexampleofthepiecewiseconstantmodelforaSISOsystem.Thedash-dottedlineisthetrue


covarianceoftheinterferenceandthethermalnoise,andthesolidlineisthees-timatedcovarianceusingourmethod.Hence,accordingto(3.12),themaximumlikelihooddetectionforinformationdataisexpressedasmins(Mi+m)2S^R1=2M(i)(z(Mi+m)^h(Mi+m)s(Mi+m))2(4.13) fori=0;:::;N=M1andm=0;:::;M1. 4.3 Automatic Selection of Model Order Togetthebestperformance,weneedtochoosethemodelorder,theparameterM,autonomouslyineachsequenceofdata.Asasub-optimalway,weuseamaximum-likelihood(ML)approachforselectingM.AfterestimatingthecovariancefordierentM,weinsertthosevaluesintothenegativelog-likelihoodfunctionwithtrainingsymbols,compareandselecttheminimumoneasthepropermodelorderfortheinformationdatasequence.Itisdesirabletochoosethemodel,thatexplainsthedata(interferenceandnoise)wellandhasaparametrizationassimpleaspossible.ButMLalwayschoosesthehighestmodelorder,whenthereisinterferenceinthetrainingsequence.Tocounterbalancethistendency,weaddapenaltyterminspiredbytheminimumdescriptionlength(MDL)method(moredetaileddiscussioncanbefoundinKay[10],Section6.8)totheMLruleMDL(M)=lnP(Yt;^RMj^H)+nM


wherenMisthenumberofestimatedparametersinRM.Thus,thismethodndsthemodelorderMwhichminimizesminMTXt=1N=M1Xi=0M1Xm=0^R1=2M(i)(zt(Mi+m)^h(Mi+m)st(Mi+m))2+TMN=M1Xi=0log^RM(i)+TnM wherest(n)istheknownreceiveddata.NotethatRM(i)istheaveragevalueofthemomentbasedcovariancematricesofMadjacentsubchannelsandzt(Mi+m)^h(Mi+m)st(Mi+m)=~rt(Mi+m)(4.16) Notethatthenormsquaretermofequation(4.15)canbewrittenTXt=1N=M1Xi=0M1Xm=0^R1=2M(i)(zt(Mi+m)^h(Mi+m)st(Mi+m))2=TXt=1N=M1Xi=0M1Xm=0Trf^R1=2M(i)~rt(Mi+m)~rHt(Mi+m)^RH=2M(i)g=TXt=1N=M1Xi=0M1Xm=0Trf^RH=2M(i)^R1=2M(i)~rt(Mi+m)~rHt(Mi+m)g=N=M1Xi=0M1Xm=0Trf^R1M(i)TXt=1~rt(Mi+m)~rHt(Mi+m)g=TN=M1Xi=0M1Xm=0Trf^R1M(i)^R(Mi+m)g=TN=M1Xi=0Trf^R1M(n)(^R(Mk)+^R(Mk+1)++^R(Mk+M1))g=TN=M1Xi=0MTrf^R1M(i)^RM(i)g=TNTrfIg(4.17)


whereTrfgstandsforthetraceofamatrix.Followingthisresult,Equation(4.15)canberewritteninthemoresimpliedformminMMN=M1Xi=0log^RM(i)+nM WeusethissimpliedmethodwithtrainingsequencestoselectthepropervalueofMandapplythechosenmodeltosuppresstheinterferenceinthedatasequences. 4.4 Numerical Examples for the Piecewise Constant Model Inthissection,weshowsomesimulationexamplesforournewalgorithmforanOFDMsystem.Inourexperiment,N=64subcarriers,1trainingsequenceandnr=2receiveantennasareused.ThechannelconsistsofL+1=5independentRayleighfadingchanneltaps,andhasanexponentiallydecayingpowerdelayprole:E[khlk2]/1:5l;l=0;:::;L.Thetrainingsequenceisanimpulsetoestimatechannel,andtheconstellationsizeforoursimulationisQPSK. Figure4{2showstheestimatedSymbol-error-rate(SER)versustheCarrier-to-Noiseratio(C/N)whenthereceivedsignalsarenotaectedbyinterference,andwhenonlythermalnoiseispresent.WepresenttheresultforthecaseswhenM=2;4;8,and64,andwiththeautonomousselection.wealsocomparetheresulttocoherentdetectionandconventionaldetection(Inthiscase,weusenoisecovariance^R(n)=Iforalln.)Thenon-coherentdetectionandourmethodwithM=64showalmostsameresult.ThereasonforthisisthatsettingM=64impliesthatthenoisespectrumisindependentofn.WiththeothervaluesofM,weseeaperformancelossrelativetoconventionaldetection.Ourauto-selectionmethodchoosesM=64modelorderforeachsequenceasweexpected.


23 Figure4{2:Symbol-error-rate(SER)forthepiecewiseconstantmodelwhen thereisonlythermalnoisepresent. InFigure4{3,wepresentthecasewhenthereisonestronginterferer present.Theco-channeluserhasthesamechanneldelayproleandpowerdelay proleasthedesireduserhas,andthesignaltointerferenceratiois10 dB .This gureshowspoorperformancesofcoherentdetectionandconventionaldetection withoutestimationofinterference,butourmethodsuccessfullymitigatesthe co-channelinterference.ForhighSNR,the M =4modelorderoutperformsthe othermodels,butintheregionoflessthan20 dB ,the M =8modelordershows somewhatbetterperformancethan M =4.ForeachSNR,ourauto-selection


Figure4{3:Symbol-error-rate(SER)forthepiecewiseconstantmodelwhenthereareonestronginterferencesignal(C/I=10dB)andthermalnoisepresent. methoddoesnotalwayschoosethebestmodel,whichhasthelowestsymbol-error-rate.However,theauto-selectionmethodsignicantlyoutperformsthecoherentmethodwithoutestimatingthecovarianceofinterference. Figure4{4showshowourautonomousselectionmethodperformswhenthesignal-to-noiseratiois30dB.OurmodelalwayschoosesM=64whenonlythermalnoiseispresentasweexpect.Whenonestronginterfererandthermalnoisearepresent,ourmethodusuallyselectsM=8andM=4.


25 Figure4{4:Autonomousselectionofmodelorder,SNR=30dB,forpiecewiseconstantmodel. Wesummarizethecomparisonbetweendetectionusingourmethodand coherentdetection(usingperfectknowledgeofthechannelandthenoise covariance)inFigure4{5.Inthecasewhenthereceivedsignalsareonlyaected bythermalnoise,Figure4{5indicatesthatconventionaldetectionandour newmethodhavethesameperformance.Inthecasewhenreceivedsignalsare aectedbystronginterference,ourmethodshowsasignicantimprovement comparedtoconventionaldetection.


26 Figure4{5:Comparisonbetweendetection(usingpiecewiseconstantmodeland coherentdetectionusingperfectknowledgeofthechannelandthe noise covariance). 4.5 Estimation of R ( n ) with Moving Average Process Inthissection,weintroduceanalternativenewmethodforinterference suppression.Thismethodusesalow-ordertime-domainmodeltoparameterize thematrix-valuedspectraldensityofthenoiseandinterference.Westilluse thechannelestimationtechniquedescribedinSection4.1.Then,wecalculate residualsfromthetrainingsequenceusingestimatesofthechannel, ^ H ,in(4.6)


asfollows~Yt=(YtXt^H)T(4.19) Fromtheseresiduals,weestimate^Rlusingthebiasedauto-covarianceestimatesin(see,e.g.,SoderstromandStoica[9]andStoicaandMoses[11])^Rl=1 forl=0;:::where~Y+listhematrix~Ywiththerstlcolumnsremoved,and~Ylistheequalto~Ywiththelastlcolumnsremoved.WenextestimateR(n)viaatruncatedweightedFouriersumasfollows^R(n)=MXm=M1jmj Nmn(4.21) Becausethecovariancefunctionisconjugatesymmetricwecanset^Rm=^RHm.Truncatingthesumamountstoassumingthatthepowerspectrumofe(n)issmooth,whichisprobablyareasonableassumptioninpractice.Touse^Rfordetectionin(3.12),itisnecessarythat^Rispositivedeniteforn=0;:::;N1.Fortunately,thisisthecase.Toseewhythisisso,noterstthat^R(n)=1 where(n)isanNNmatrixwhose(k;l)thelementisequaltok;l(n)=(1jklj=(M+1))ei2(kl)n=NwhenjkljMandzerootherwise.But(1jklj=(M+1))eiwnisthecovariancefunctionofamovingaverageprocesswithcoecientsf1;eiw;:::;ei(M1)wgandhence(n)ispositivesemi-denite.Becauseofthenoisein~Yt,^R(n)ispositivedenitewithprobabilityone.


4.6 Auto Selection of Model Order Alsoforthemethodusingamovingaverageprocess,itisnecessarytondthepropermodelorderM.Asforthepiecewiseconstantmodel,wecanapplypenalizedMLtoobtainanautonomousselectionmethod.But,unfortunately,thisdoesnotappeartoworkthatwellforthemovingaverageprocessmodel.Wedonotknowtheexactreasonforwhythisdoesnotworkforthemovingaverageprocessmodel(inoursimulation,thevalueoftheMLmetricismuchbiggerthanthatofthepenaltyterm):butitisprobablyrelatedtothefactthattheestimates^R(n)and^h(n)weusearenotmaximumlikelihood.Instead,wehaveusedthedetecteddatasequencetodeterminethemodelorder.Forexample,ifwetransmitonetrainingsequenceandvedatasequences,weestimatethechannelandcovarianceusingthetrainingsequenceandthendetectrstdatasequence.Then,weusethedetecteddatasequencetoselectmodelorder,andthechosenmodelorderisusedtodetecttheotherdatasequences.Thus,ourautonomousselectioncanbeexpressedasfollowsminMN1Xn=0nlogj^R(n)j+^R1=2(n)(z(n)^h(n)^s(n)2o+nM where^s(n)isthedetecteddata. 4.7 Numerical Examples for Moving Average Process Inthissection,weshowsomesimulationexamplesusingthenewalgorithmwiththemovingaverageprocess.ThisexperimentwasdoneunderthesameconditionsasdescribedinSection4.4. Figure4{6showstheestimatedSymbol-error-rate(SER)versustheCarrier-to-Noiseratio(C/N)whenonlythermalnoiseispresent.WedemonstratetheresultsforthemodelordersM=0;2;4;6;8andforautonomousselectionamongthesemodels(incontrasttothepiecewiseconstantmethod,increasingthemodel


29 Figure4{6:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen thereisonlythermalnoisepresent. orderheremeansincreasingthenumberofparametersweestimate).Also,to compareourmethodwithaconventionaldetectionscheme,weshowresultsfor coherentdetectionandconventionaldetection.Thisgureshowsthatthere isnotmuchdierencebetweenconventionaldetectionandthemethodusing movingaverageprocess. Figure4{7showsthecasewhenthereisonestronginterfererpresent( C=I =10 dB ).Fromthisgure,weseethatourmethodsuccessfullymitigates co-channelinterference.Likethepiecewiseconstantmodel,thismovingaverage


30 Figure4{7:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen thereareonestronginterferer(C/I=10dB)andthermalnoisepresent. processmodelsuppressestheinterferencemoreeectivelybyincreasingthe numberofparameters.Autonomousselectiondoesnotalwayschoosethemodel orderwhichshowsthebestSER. Figure4{8displayshowourautonomousselectionmodelperformswhenthe signal-to-noiseratiois30 dB .Ourmethodchooses M =0mostofthetimewhen onlythermalnoiseispresent.Whenonestronginterfererandthermalnoiseare present,ourmodelselectsmostly M =6butalso M =8.


31 Figure4{8:Autonomousselectionofmodelorder,SNR=30dB,formovingaverageprocess. Figure4{9showsacomparisonbetweendetectionusingourmethodandcoherentdetectionusingperfectknowledgeofthechannelandthenoisecovariance. NotethesimilaritybetweenFigure4{9andFigure4{5.


32 Figure4{9:Thecomparisonbetweendetectionusingmovingaverageprocess andcoherentdetectionusingperfectknowledgeofthechanneland the noise covariance.


InChapter2,weintroducedreceivediversity,whichisapopularmethodtocombatfadinginwirelesscommunications.Byinstallingmultipleantennasatreceiverside(usuallyatthebasestation),wecanmitigatechannelfadingaswellasincreasethechannelcapacity.Inrecentyears,researchershavefoundthatitispossibletoobtainthesameadvantageasreceivediversitybyusingmultipleantennasatthetransmitterside(thisiscalledtransmitdiversity).Usingmultipleantennasatbothsides,wecangetmuchmorebenetsthanwithreceivediversityortransmitdiversityalone.However,toexploitthesebenets,weneedacodingscheme,whichiscalledSpace-TimeCoding.Inthischapter,webrieyintroduceaMIMOsystem,space-timeblockcoding(STBC)andspace-timeOFDM(ST-OFDM),whichcombinesspace-timecodingandOFDM. 5.1 MIMO Systems and Space-Time Block Coding ConsideraMIMOsystemwhichconsistsofnttransmitantennasandnrreceiveantennas.Weassumethatthepropagationchannelsarelinear,time-invariant,andfrequencyat.LetXandYbentNtransmitteddataandnrNreceiveddatamatrices,respectively.Then,thereceiveddatacanbeexpressedinalinearmodelasfollows whereHisannrntchannelmatrixinwhichthe(m;n)thelementcorrespondstothechannelgainbetweenthereceiveantennamandthetransmitantennan; 33


LikefortheSISOsysteminSection2.1,wecanderivethedetectionerrorprobabilityfromEquation(5.2).ByasimilarcalculationasinSection2.1 Next,weassume,forsimplicity,thatallchannelsinHareindependentandcomplexGaussianwithzeromeanandvariance2;hence,thechannelisRayleighfading.TheprobabilitydensityfunctionfortheRayleighfadingchannelscanthenbeexpressedas UsingtheChernobound,wegettheaverageerrorprobabilityasfollows(amoredetailedderivationcanbefoundin[5,Section4.3])EH[P(X!XojH)]=ZHP(X!XojH)p(H)dH=2 TheaboveequationshowsthattheaverageerrorprobabilityforaMIMOsystembehavesasP(error)/SNRnrnt(5.6)


whichmeansthatreceivediversityofordernrandtransmitdiversityoforderntareobtained,aslongasXXohasfullrank.Hence,thelengthofthetransmissionNhastobelongerthanthenumberoftransmitantennas,nt. Itisnoteasytoachievetransmitdiversitywithoutlosingtransmissiondatarateandincreasingthedecodingcomplexity.Space-timecodingisawaytoachievehighdatarateandlowdecodingcomplexity.Oneofwellknownspace-timecodingschemesistheAlamouticode[12],whichisatransmissionschemefortwotransmitantennas.Anothercaseisthelinearorthogonalspace-timeblockcode(OSTBC),whichhasasimplestructure.Considerthetransmissionofnssymbolsfs1;;snsg.Then,encodedlinearSTBChasfollowingstructureX=nsXn=1(snAn+i~snBn)(5.7) whereAnandBnarexedcodematrices,andsnand~snaretherealandimaginarypartsofthesymbolsn,respectively.OSTBChasthespecialmeritthattheMIMOchanneldecouplesintonsindependentscalarchannels,becauseofthefollowingorthogonalitypropertyofOSTBCXXH=nsXn=1jsnj2I(5.8) AmoredetailedexplanationforOSTBCcanbefoundinLarssonandStoica[5],Chapter7,Tarokhetal.[13]andLarssonetal.[14]. 5.2 Space-Time OFDM ST-OFDMisacombinationofspace-timeblockcodingandOFDMtoobtaintransmitdiversityforanOFDMsystem.ThemainideaofST-OFDMisidenticaltothatofOFDM.BecauseaST-OFDMsystemusesmultipletransmitantennas,itsendsNoencodedntNmatricesinwhichnssymbolsareencoded.Hence,nsNosymbolsaretransmittedduringNNosymboltransmittimeintervals.The


transmissionanddetectionproceduresforST-OFDMareasfollows.First,fromnsNosymbolstobetransmitted,encodeeachgroupofnssymbolstoNocodematricesG(n)forn=0;;No1.Second,taketheinverseFouriertransformofthecodematricesasfollowsX(n)=1 Nnk;n=0;:::;No1(5.9) Then,addtheCPwhichistakenfromNprelastcodematricesofthetransmitteddatamatrices,andselecttherstcolumnsofallcodematricestotransmitthemsequentiallythroughchannels,assumingthateachchannelhasLdelaytaps.Next,selectthenextcolumnsforthetransmissionuntilallthecolumnsofthecodematriceshavebeensentout.Atthereceiver,discardtheCP.ThenthereceivedcodematricescanbeexpressedasfollowsY(n)=LXk=0HkX(nk)+E(n)=1 N(nk)m+E(n)(5.10) AftertakingFouriertransformof(5.10)andfollowingsimilarderivationasin(3.4),thereceiveddatabecomeZ(n)=1 Nln=H(n)G(n)+1 Nln(5.11) whereH(n)=LXk=0Hkei2 Nkn(5.12)


Hence,MLdetectionforST-OFDMisequivalenttominimizingthefollowingmetrickZ(n)H(n)G(n)k2(5.13) MoreaboutST-OFDMcanbefoundinLarssonandStoica[5],Chapter8andLietal.[15].


Inthischapter,weextendourinterferencesuppressionalgorithmtotheST-OFDMcase. 6.1 Channel Estimation BecauseaST-OFDMsystemusesmultipletransmitantennasandtransmitsanntNinverseFouriertransformedcodematrixX,thenumberoftrainingsequences,whichconsistofNosymbolseach,isatleastN.LetusassumethatwehavereceivedNTtrainingsequences(T>1)andthateachchannelhasLdelaytaps.DenotetransmittedandreceivedcodematricesbyXt(n)andYt(n)forn=0;;No1;t=1;;T.Then,thereceivedtrainingdatacanbeexpressedasfollowsYt(n)=LXl=0HlXt(nl)+E(n);n=0;:::;No1(6.1) whereHlstandsforlthmatrixvalueddelaychanneltap.Byusingablock-Toeplitzmatrix,Equation(6.1)canberewrittenasYt(0)Yt(No1)=H0HLXt+Et(0)Et(No1)(6.2) 38


whereXt=266666664Xt(0)Xt(1)Xt(NoL1)Xt(1)Xt(0)Xt(NoL).........Xt(L)Xt(L1)Xt(0)Xt(No1)377777775(6.3) Notethatvec(ABC)=(CTA)vec(B)(6.4) wheredenotestheKroneckerproductandvec()isthevectorizationofamatrix.Byvectorizingbothsidesof(6.3),wegetyt=(XTtInr)h+et(6.5) whereyt=266664vec(Yt(0))...vec(Yt(No1))377775;h=266664vec(H0)...vec(HL)377775;et=266664vec(Et(0))...vec(Et(No1))377775(6.6) Maximumlikelihoodestimateofhin(6.5)isequivalenttominimizingthefollowingmetricminhTXt=1yt(XTtInr)h2(6.7) andtheMLestimateofthechannel^hisdeterminedby^h=vec(^Ho)Tvec(^HL)TT=1


Also,notethatafterestimatingthechannel^handreconstructing^H,^H(n)fortheatfadingsubchannelscanbecalculatedbytakingtheFouriertransformation ^H(n)=LXl=0^Hlei22 Nnl(6.9) 6.2 Estimation of with Moving Average Process First,notethatthereceivedcodematrix,Yt(n),consistsofNnumberofnr1columnvectors.VectorizationofYt(n)givesvec(Yt(n))=yt(n)=266664y1t(n)...yNt(n)377775(6.10) Next,wecalculatetheresidualsfromthevectorizationofthereceiveddatausingtheestimatedchannel^handtheblock-Toeplitzmatrix(6.3)asfollows~yt=yt(XTtInr)^h(6.11) Following(6.10),wecanexpress~ytas~yt=~yTt(0)~yTt(No1)T=~y1t(0)T~yNt(0)T~y1t(No1)T~yNt(No1)TT(6.12) Then,wereconstructthematrix~Yt(n)asfollows~Yt(n)=~ynt(0)~ynt(No1);n=0;:::;N1(6.13)


Followingthesameproceduresas(4.20),weestimate^Rlusingthebiasedauto-covarianceestimates^Rl=1 forl=0;:::where~Y+l(k)isthematrix~Y(k)withtherstlcolumnsremoved,and~Yl(k)istheequalto~Y(k)withthelastlcolumnsremoved.WenextestimateR(n)viaatruncatedweightedFouriersumasfollows^R(n)=MXm=M1jmj Nmn(6.15) Using^R(n),themaximumlikelihooddetectionrulefortheinformationsymbolsisexpressedasargminG^R(n)(Z(n)^H(n)G(n))2(6.16) Tochooseapropermodelorder,theautonomousselectionmethodcanbeexpressedasfollowsminMN1Xn=0nlogj^R(n)j+^R1=2(n)Z(n)^H(n)^G(n)2o+nM where^G(n)isthedetectedcodematrix. 6.3 Numerical Examples for the Moving Average Process WeshowsimulationresultsforouralgorithmforanST-OFDMsystemwithN=64subcarriers,nt=2transmitandnr=2receiveantennas.TheAlamouticodematrixisusedinourexperiment,andthechannelhasthesamedelaytapsandthesameaveragepowerdelayproleasinSections4.4and4.7.Twotrainingsequences,whicheachsequencebeinganimpulse,areused,andtheconstellationisQPSK.


42 Figure6{1:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen thereisonlythermalnoiseispresent. Figure6{1showsthattheSymbol-error-rateperformanceversustheCarrierto-Noiseratio(C/N)foranST-OFDMsystemwhenonlythermalnoiseis present.Thisgureshowsthattransmitdiversityeectivelycombatsthechannel fadingcomparedtoFigure4{6. Figure6{2showsresultsforthecasewhenonestronginterferercomes fromasingleantennauser.CCIhasthesamechanneltapsandaveragepower delayproleasthedesireduserhas,andthesignaltointerferenceratiois10 dB Theinterferencesuppressingalgorithmoutperformstheonethatdoesnot


43 Figure6{2:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen stronginterferercomesfromasingleantennauser. takeinterferenceintoaccount.Also,thealgorithmsuppressestheinterference moreeectivelybyincreasingthenumberofparametersweestimate.However, Figure6{3showsthatwhenstronginterfererscomesfrommultipletransmit antennas,ourmethodcannotmitigateinterferencewell.Also,inthiscase, increasingthemodelorderdoesnothaveasignicanteectontheinterference suppression.


44 Figure6{3:Symbol-error-rate(SER)forthemovingaverageprocessmodelwhen stronginterfererscomesfromco-channeluser'smultipleantennas.


Co-channelinterferenceisamajorlimitingfactorforcellularcommunicationsystems,anditisexpectedtobecomeamajorbottleneckalsoforwirelesslocalareanetworks.Inthisthesis,wehavepresentedaconceptuallyverysimpleandcomputationallycheapmethodforsuppressionofco-channelinterferenceinsystemsthatuseOFDMmodulationandwhichhavemultiplereceiveantennas.Ourmethodcomesintwoversions:onethatparameterizestheinterferenceandnoisespectrumviaapiecewise-constantmodel,andonethatusesalow-ordermoving-averagemodel.Ineithercase,atechniqueinspiredbyMDLisusedtoautomaticallyselectthemodelorder,henceavoidinganyuserparameterstobeselectedpriortotheapplicationofthealgorithm.Simulationresultsshowthatourmethodcansubstantiallyimprovetheperformancecomparedtoconventionaltraining-baseddetection. Severalproblemsremainwhichcouldbethetopicoffuturestudies.First,themodelorderselectionstrategyusedisbynomeansoptimal(optimalsuchselectionmethodsingeneraldonotexist,seeLanterman[16])anditispossiblethatanalternativemethodsforchoosingthemodelordermayoutperformours.Second,theestimatesofthechannelandnoisecovariancethatweusearenotmaximum-likelihood,butneverthelesstheyarecomputationallysimpletoobtain.Itislikelythatusingmaximum-likelihoodestimatesinlieuofourproposedestimatesmayleadtoamethodthatworksbetterthanours.Themaximum-likelihoodestimatesgenerallydonotexistinclosedform,andobtainingthem 45


numericallyiscomputationallyburdensome.Yet,somelow-complexityapprox-imationstomaximum-likelihoodfortheproblemunderstudycanbefoundinJeremicetal.[17];usingtheestimatesproposedthereininsteadofoursubopti-malestimatesmightimproveperformance.Finally,astheinterferencesignalsareGaussianonlytowithinacertaindegreeofapproximation,usingamodelthatallowsfornon-Gaussianinterferencemayalsoimprovetheerrorrate. Wealsostudiedtheapplicationofouralgorithminasystemthatusesspace-timeOFDM.Thisscenarioismoredicult,becauseinterferencesignalsthatoriginatefromauserwithmultipletransmitantennashavearichercorrelationstructurethansignaloriginatingfromasingle-antennatransmitter.Forspace-timeOFDM,ouralgorithmwasabletosuppressco-channelinterferenceonlytoasmalldegree.Itispossiblethattheuseofamoreaccuratemodel,perhapsalongwithalargernumberofreceiveantennas,mayimprovethesituation.Thistopicisalsoleftforfuturework.


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