Detection Principles for Fast Rayleigh Fading Channels ... - Chalmers

Thesis for the degree of Licentiate of Engineering

Detection Principles for Fast Rayleigh Fading Channels using an Antenna Array ANDERS HANSSON

Department of Computer Engineering CHALMERS UNIVERSITY OF TECHNOLOGY Goteborg, Sweden 2000

Detection Principles for Fast Rayleigh Fading Channels using an Antenna Array

ANDERS HANSSON Technical Report No. 343L Department of Computer Engineering Chalmers University of Technology SE{412 96 Goteborg, Sweden Phone: +46 (0)31{772 1000

Contact information: Anders Hansson Department of Computer Engineering Chalmers University of Technology Horsalsvagen 11 SE{412 96 Goteborg, Sweden Phone: +46 (0)31{772 8442 Fax: +46 (0)31{772 3663 E-mail: [email protected] URL: http://www.ce.chalmers.se/staff/ahansson

Printed in Sweden Chalmers Reproservice Goteborg, Sweden 2000

Detection Principles for Fast Rayleigh Fading Channels using an Antenna Array ANDERS HANSSON Department of Computer Engineering, Chalmers University of Technology Thesis for the degree of Licentiate of Engineering. Abstract

Frequency- at, fast Rayleigh fading may be considered the most critical disturbance in wireless communication systems. In its most general form, it can be modeled as a multiplicative time continuous random (zero-mean complex Gaussian) distortion of the transmitted signal. As higher regions of the frequency spectrum are exploited in order to enable increased capacity in the wireless networks, fundamental knowledge of the fast Rayleigh channel will become of paramount importance. A common technique for improving the detection performance over fading channels is to employ a diversity strategy. This thesis mainly treats multiple antenna receivers, which is a way of achieving space diversity. First, single symbol detection is analyzed without coding and interleaving. An earlier derivation of the exact error probability expression is generalized to include diversity reception. The structure of the front-end processor is seen to be crucial and adaptive antenna arrays are unable to exploit the implicit diversity of space-time-selective Rayleigh channels. Secondly, sequence detection is treated over the time continuous fading channel. Now, the statistical coupling between consecutive symbols is taken into account. The phase ambiguity caused by the fading is circumvented by means of minimum-shift keying. Space diversity is concluded to be the most rewarding strategy for improving the detection performance. Finally, the information bit stream is coded and interleaved before modulation and transmission. An iterative decoding algorithm is derived for fast fading channels. This system yields superior performance compared with previously proposed designs.

Keywords: Fading channels, Rayleigh channels, correlation, diversity recep-

tion, multiple antennas, adaptive antenna arrays, sequence detection, serially concatenated coding, APP algorithm, iterative decoding. i

ii

Contents Acknowledgements 1 Introduction

1.1 Background and Objectives . . . . . . . . . . . . . . . . . . . 1.2 Related Work and Thesis Outline . . . . . . . . . . . . . . . . 1.3 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . .

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5 7 8

2 Characterization of Narrowband Fast Fading Channels for Array Reception 11 2.1 Channel Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 On the Slow Fading Assumption . . . . . . . . . . . . . . . . . 18 2.3 Generation of D Correlated Fading Processes . . . . . . . . . 19

3 Single Symbol Detection 3.1 3.2 3.3 3.4 3.5

On Optimum Diversity Reception . Suboptimum Diversity Reception . A Weighted Sum Approach . . . . Performance Analysis . . . . . . . . Calculated Results and Discussion . 3.5.1 A Note on Robustness . . .

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4 Sequence Detection

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5 Iterative Detection

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4.1 Receiver Front-End Processing . . . . . . . . . . . . . . . . . . 48 4.2 Metric Derivation and Trellis Description . . . . . . . . . . . . 50 4.3 Numerical Results and Discussion . . . . . . . . . . . . . . . . 55

5.1 Iterative Decoding of Serially Concatenated Codes . 5.1.1 The APP Algorithm for Fading Channels . . 5.1.2 The Log Domain APP Algorithm . . . . . . 5.2 Iterative Detection for the Slow Fading Assumption iii

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62 64 65 67

5.3 Numerical Results and Discussion . . . . . . . . . . . . . . . . 68

6 Conclusions and Suggestions for Future Work Appendix A \Barrett's Method" Revisited

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73 77

Acknowledgements Many people have contributed in dierent ways to this thesis. I owe very special thanks to Professor Tor Aulin for supervising my research. He is gratefully acknowledged for the ability to share his profound knowledge with me and for all the enlightening discussions in general. I further appreciate the patience of Dr. Lars Rasmussen, who has always been willing to answer any questions. Thanks are also due to my roommate at Chalmers, Andreas Cedergren, simply for putting up with all my discussions. Finally, I am grateful to the Swedish Foundation for Strategic Research for funding my graduate studies by their Personal Computing and Communication Grant No. 9706-01.

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Notation In order to keep the notation plain and simple, we will not distinguish between deterministic and random variables; this should not cause any confusion. Further, vectors and matrices are consistently written in boldface. Mathematical Symbols

I

diag(x1 ; : : : ; xN )

xT xH kxk h x; y i

jXj (X)k;` x Jk (x) Æ (x)  (x) (x) Æmn Pr(X ) E(x)

_

The identity matrix. An N by N diagonal matrix with the elements x1 ; : : : ; xN on the main diagonal. Transpose of a vector x. Hermitian transpose of a vector x. The l2 normpof a column vector x, i.e., kxk , xH x. The inner product of two column vectors x and y, i.e., hx; yi , xH y. The Kronecker matrix product. The determinant of a matrix X. The element on the kth row and `th column of a matrix X. Complex conjugate of a variable x. The kth order Bessel function of the rst kind. Dirac's delta function. Heaviside's step function. Euler's Gamma function. The Kronecker delta symbol. The probability of an event X . The expectation of a random variable x. Disjunction (logic or).

1

Fundamental System Parameters

Eb N0 =2 Tb %k   !c !m fm 0
(k ; 'k ) U D N L W Ko

The bit energy. Double-sided power spectral density of the baseband thermal noise representation. The time duration of one bit interval. The Rayleigh distributed envelope of the fading process during symbol interval k, assuming slow fading. The wavelength of the carrier wave. The circular wave number;  = 2=. The angular frequency of the carrier wave. The maximum Doppler angular frequency shift. The maximum Doppler frequency shift normalized to the symbol duration, i.e., fm , Tb !m =(2 ). The mean direction of arrival (DOA) of the received signal. Half the angle spread of the DOA, i.e., the DOA is assumed to be uniformly distributed over [0
; 0 +
]. The polar coordinate for the kth receiver antenna element. The oversampling factor of a sampling discretizer. The rate of the lter used for generating fading samples. The number of receiver antenna elements. The number of observables obtained per symbol interval. The receiver truncation length of the channel memory. The cuto frequency of the ideal lowpass pre- lter. The block length of the outer code.

2

Miscellaneous

b`k (i) s`k (i) f (t) n(t) r(t) r`k C`k (i) Cs Z`k (i) F`k

A sequence of information bits transmitted during symbol intervals k up to ` on the ith hypothesis. A sequence of baseband modulator waveforms transmitted during symbol intervals k up to ` on the ith hypothesis. The multipath-fading vector random process. The thermal noise vector random process. The received continuous-time process. A column vector of observables obtained during symbol intervals k up to `. The covariance matrix for r`k conditioned on the message sequence s`k (i). The array spatial correlation matrix. A diagonal matrix comprising samples of the message sequence s`k (i). The covariance matrix for samples of the fading process obtained from time index k up to `.

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Chapter 1 Introduction \I nd all books too long."

{Voltaire

This chapter provides a background to the problems tackled in this thesis, and also formulates a number of research objectives motivated by the identi ed problems. The problems are further put in perspective to related work and the thesis exposition is outlined. Finally, the principal contributions are brie y summarized without entering into details. 1.1

Background and Objectives

Nowadays, there is a lot of focus on wireless communication, i.e. using electromagnetic waves for conveying information. As soon as it comes to wireless communication, it is essential that signals do not interfere with each other in order to extract information reliably. The frequency spectrum, which is a limited resource given by nature, must therefore be divided among dierent services. Since the spectrum is becoming a scarce resource as more and more communication systems are deployed, it is of major concern to use the spectrum as eÆciently as possible. Also, mobile users of tomorrow will demand the same level of service as if they were using a xed network, i.e., future networks must appear to be transparent to the users. To enable wireless multimedia applications, the wireless networks need to convey data at rates that are many orders of magnitude beyond what is possible with today's technology. In order to obtain these high-capacity links, higher regions of the spectrum must be exploited. However, due to the nature of radio wave propagation, the received signal power varies in space and time which is referred to as fading. If the 5

data rate is kept constant, the fading becomes faster at higher frequencies. This eect causes considerable impairment to reliable communication, unless counteractions are taken. It may thus be concluded that fundamental knowledge of how to deal with fast fading channels will become of paramount importance in the near future. The quest for conceptual understanding of communication over such fast fading channels is also the main driving force behind this thesis. A frequent proposal to meet the high-capacity demands is simply to increase the transmission rate [Mor97], which at the same time mitigates the fast fading. However, in the case of real-time information, bandwidth requirements should be dictated by the medium of discourse, i.e., to convey the message faster than it originates only costs more bandwidth. This idea has the blessings of one of the most prominent advocates of information technology, namely Professor Nicholas Negroponte who is Founding Director of MIT's Media Lab [Neg95]: \There really are some natural laws of bandwidth that suggest that squirting more bits at somebody is no more sensible or logical than turning up a radio's volume to get more information." In accordance with this philosophy, our overall goal is to develop spectrum eÆcient communication schemes. Moreover, in sharp contrast to the common belief that fast fading deteriorates detection performance more than slow fading, a recent study by Hansson and Aulin [HanU99] shows that the presence of fast fading actually improves the performance, provided the receiver is properly designed. This quite remarkable result motivates the use of detector structures that exploits fast fading, e.g., multicarrier schemes with multiple low-rate channels. Still, the system in [HanU99] has a performance that is clearly inferior to what is achievable over the additive white Gaussian noise (AWGN) channel. It has, however, been shown that the capacity of a fading channel is the same as the capacity of an AWGN channel [Gal68]. A well-known way to improve communication over fading channels is to employ a diversity strategy, i.e., to provide the receiver with several dierently fading replicas of the same information-bearing signal. Common techniques are frequency diversity by means of dierent carriers, time diversity by means of repeated transmissions, and space diversity by means of multiple antennas. Other ways to accomplish diversity exist, such as angle-of-arrival diversity, and polarization diversity. It should also be noted that the fading itself gives rise to a diversity eect called implicit diversity [Ken65]. This last eect explains the improvement achieved in [HanU99]. An initial aim of the thesis is to extend the results of [HanU99] to space diversity reception. As soon as such a starting-point has been established, the objective is to derive new low-complexity detection algorithms that further improves the performance. 6

An important principle for achieving better detection performance over fading channels is known as interleaving. The idea is to permute the bits of a coded word before transmission, and then reorder them before decoding. Hopefully, it becomes rather unlikely that a long stream of consecutive code bits fades out in a power dip. Interleaving may thus be regarded as an eÆcient way of achieving time diversity. After analyzing fundamental detection algorithms, a sophisticated system that combines interleaving and coded modulation will be employed in order to further close the gap to capacity. 1.2

Related Work and Thesis Outline

Before designing and analyzing eÆcient wireless communication systems, it is essential to understand the physical characteristics of radio channels. Chapter 2, then, will initially present an overview of the fundamentals of radio wave propagation over land-mobile radio channels. A classic work on this topic was edited by Jakes [Jak74], while two more recent publications are due to Parsons [Par92] and Saunders [Sau99]. Early models only accounted for time-selectivity of the radio channel, and later also frequency-selectivity was added. Now, with the advent of adaptive antenna arrays, the additional concept of space-selectivity is being incorporated in the channel modeling. A tutorial, which reviews some of the present emerging spatial propagation models, is due to Ertel et al. [Ert98]. In chapter 2, we will adopt and slightly modify a model proposed by Raleigh et al. [Ral94][Nag96]. In chapter 3, the single symbol detection principles in [HanU99] are extended to multi-antenna reception. The description is founded on theories derived already in the fties and the sixties when ionospheric scattering was developed for long-distance communication [Kad64][Ken65][Tur61][VTr68]. A weakness of the single symbol detector is that is does not take advantage of the memory induced by the fading, i.e. the statistical coupling between individual information bits is not utilized. Thus, chapter 4 treats sequence detection and adjacent data symbols are then taken into account in the detector. Sequence detection for Rayleigh fading channels was investigated in e.g. [Han97][Cas96][Lod90][Yu95], and later combined with uncorrelated diversity reception [Yu95] as well as correlated diversity reception [Har98]. The derivation in chapter 4 is founded on the traditional approach taken in [Han97][Cas96], instead of the somewhat unconventional approach in [Yu95][Har98]. Ever since the introduction of the so-called \turbo codes" [Ber93], much research has been pursued on the combination of interleaving and concatenated coding, as well as on iterative detection of these schemes. Even though 7

many results are available for the piecewise constant uncorrelated Rayleigh fading channel [Li99][Bru99][Pel99], few results have been obtained for the fast fading channel [Mar98]. Chapter 5 investigates iterative detection of interleaved coded modulation over fast fading channels. 1.3

Thesis Contributions

Even though the geometry of the reviewed channel model is rede ned, this contribution is of secondary importance. More important is the fact that the presented model is used without making any simplifying assumptions, unlike the author of the model who sets the angle spread of the received signal to zero [Nag96] as soon as it comes to designing a detector|possibly for simplifying matters. Next, a method for computer simulation of a time continuous fading process [Ver93] is extended to include the case of multiple correlated fading processes. Also, the generation principle is shown to have an illustrative geometrical interpretation that contributes to a better understanding of the underlying problem. This work was presented at European Wireless '99 [HanA99]. Chapter 3 extends the results of [HanU99] to include multiple antenna receivers. The subsequent study of so-called adaptive antenna arrays is quite unique, and concludes that adaptive arrays are incapable of gaining much from the implicit diversity eect. Thus, the front-end processing is obviously highly crucial. One of the prime analytical contributions in chapter 3 concerns the calculation of an exact error probability expression, which e.g. involves nding the eigenvalues of a matrix-valued Fredholm equation. The main results presented in chapter 3 will appear in the April 2000 issue of IEEE Transactions on Communications [Han00a] and similar material will also be presented at 2000 IEEE International Symposium on Information Theory [Han00b]. Chapter 4 extends the the results of [Han97] to antenna arrays and continuous phase modulation (CPM) [Aul79b][And86]. Even though a related problem was treated in [Har98], that authors of [Har98] neither included the underlying geometry nor the spatial dispersion of the received signal in their problem formulation. Chapter 4 further gives a joint trellis description of the combination of modulation and fading. The description has the advantage of being general in the sense that it could easily be modi ed to deal with other types of channels having memory. The system proposed in chapter 5 is novel in itself, i.e., iterative decoding of coded interleaved CPM over fading channels [Han97]. Comparable 8

performance has, however, been demonstrated in [Mar98], but only for a single-antenna system of excessive complexity. Chapter 5 also generalizes the well-known BCJR algorithm [Bah74] into a neat form that copes with channels having memory. The trick partly lies in the way the joint trellis is de ned in chapter 4. At the time of writing, parts of the material in chapter 4 and chapter 5 have been submitted for publication.

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Chapter 2 Characterization of Narrowband Fast Fading Channels for Array Reception \The question is complex and life is short."

{Protagoras

A transmitted radio wave interacts with various physical objects within the wireless channel, such as buildings, hills, trees, and moving vehicles. The number and locations of these scattering objects describe a certain propagation environment. This structure varies with time, and is also dependent on the heights of the antennas, particularly regarding the local environment. The interaction of the wave with the surrounding objects is a very complex process that involves diraction, refraction, and re ection. Moreover, waves of dierent frequencies interact dierently (in the millimeter wave band scattering by precipitation, and at some frequencies absorption by water vapor and atmospheric gases like oxygen, has to be taken into account). The described interaction manifests itself as multipath propagation, i.e., the transmitted wave reaches the receiver through many dierent propagation paths. To each path we associate an attenuation factor, a time delay, a Doppler shift, and a direction of arrival (DOA). The Doppler shift is an apparent shift in frequency, and is caused by motion of the transmitter, the receiver, and/or any object in the channel. The principle of superposition leads to constructive and destructive interference of the waves, which is observed as a variation in the received signal power, referred to as multipath fading. Locations in the spatial interference pattern where the signal power is extremely low are called deep fades. These locations are spatially separated by distances typically comparable to the wavelength of the carrier, and 11

thus, this type of fading is also known as short-term fading or fast fading. The deep fades constitute a major problem since they make the average error probability decrease only as the inverse of the average signal-to-noise ratio (SNR). Multipath propagation results in the dispersion of the signal in dierent domains, including Doppler (or frequency) spread, delay (or temporal) spread, and angle (or spatial) spread. The Doppler spread (or frequency dispersion) is inversely proportional to the coherence time, which is a measure of the time-selectivity of the channel. Thus, two signals that in the time domain are separated more than the coherence time will be aected dierently. Clearly, a slowly varying channel has a large coherence time, or equivalently, a small Doppler spread. The reciprocal of the delay spread (or multipath spread) is an approximation of the coherence bandwidth of the channel, which is the maximum frequency separation for which two signals are considered to be correlated (aected similarly by the channel). The delay spread can also be interpreted as a measure of the range of time delay over which the average power output of the channel is essentially non-zero. When the signaling interval (symbol duration) is of the same order of magnitude as the delay spread, successive symbols are smeared together, an eect referred to as intersymbol interference (ISI). An equivalent statement is that ISI does not occur when the transmitted signal pulse has a bandwidth that is much smaller than the coherence bandwidth.1 In such a case, the channel is said to be frequency-nonselective (or frequency- at), and consequently there is no time dispersion. Angle spread refers to the spread of angles of the propagation paths. Angle spread causes space-selective fading and is characterized by the coherence distance, which is de ned as the maximum spatial separation for which two signals are correlated. The larger the angle spread, the shorter the coherence distance. In addition to the multipath fading, topographical changes in the channel give rise to long-term fading. This eect is due to shadowing by objects, such as buildings and hills, and is usually modeled by means of a log-normal distribution [Yac93]. For this reason long-term fading is also known as shadowing or log-normal fading. A common way to counteract these diÆculties is to introduce more transmitters or to adjust the transmitted power. Many text books use the term \slow fading" as a synonym of shadowing. However, also slowly (with respect to the transmission rate) varying multipath fading is often referred to as slow fading. In order to avoid any confusion due to 1 If

we assume that the bandwidth of the signal pulse is approximately equal to the inverse of the signal duration.

12

such a dual terminology, \slow fading" will solely be used when it comes to slow multipath fading. Another consequence of the environment is the path loss, i.e. the loss in received signal strength when the distance to the transmitter is increased. Generally, this eect is much more severe in an urban environment as well as for a high frequency, compared with a rural area and a lower frequency, respectively. 2.1

Channel Modeling

Models can roughly be divided into two categories: site-speci c models based on measurement data and general statistical models. Models in the rst category require measurement data as input and can be expected to yield a reasonable multipath channel description. Liberti and Rappaport, however, give two strong reasons for using a model from the second category [Lib96]. The most obvious advantage is that the statistical approach provides mathematically tractable models that are useful for general system performance analysis. Secondly, all of the assumptions underlying the model are clearly stated and may be compared with the environment for which the model is being applied. In this report a channel model proposed by Raleigh et al. will be used [Ert98]. In its original version [Ral94] some parameters should be chosen to match empirical data. However, after a few simpli cations the model is solely based on geometrical and statistical assumptions [Nag96]. Let us explore these assumptions, as well as the properties of the model. Since shadowing and path loss are slowly varying phenomena compared with the transmission rate, they may be neglected when it comes to our intention to design and analyze detection strategies [Ste87]. Hence, these long-term variations are not included in the model. The most severe multipath fading arises when the received signal has no specular component (p. 580 [Woz65]). This Rayleigh-fading case often occurs in practice, and is described by the model. For simplicity, and without loss of generality, the propagation geometry is restricted to the horizontal plane (i.e. only azimuth angles are considered), but the results could easily be extended to include the eects of an elevation angle [Aul79a]. The treated con guration consists of a single antenna transmitter and a multiple antenna receiver. All receiver elements are considered to be identical and omnidirectional, i.e., each receiver element is assumed to have an essentially nondirectional antenna pattern in azimuth. The propagation environment under consideration is a scattering struc13

ture local to the transmitter, and a number of dominant re ectors within the propagation path. The transmitter radiation pattern uniformly illuminates all local re ectors within a few hundred wavelengths, and then reaches the receiver either directly or by the dominant re ectors. The wave fronts are further assumed to be plane at the receiver array, i.e., the subsequent analysis is restricted to the far- eld, or Fraunhofer region [Bal82]. Moreover, the channel is assumed to be frequency-nonselective. Let (` ; '` ) represent the polar coordinates for receiver element number `, and denote the DOA of the kth propagation path as k . The DOA, as well as '` , is measured counter-clockwise from the x-axis as shown in Figure 2.1. Further, let !m and !c denote the maximum Doppler angular frequency shift and the angular frequency of the carrier, respectively. In propagation path k, the phase of the transmitted signal will be transformed according to:    (2.1) !c
t ! (!c !m cos k )
t k + ` cosfk '` g ; c where k is the direction angle of the kth local scatterer with respect to the transmitter velocity vector, k is the time delay associated with path k relative to the origin of the array, and c is the speed of light. If the so-called narrowband assumption is invoked2 [Mon80], the equivalent lowpass [Pro95] received signal in element ` originating from a baseband modulator waveform si (t) in multipath k may be written as: rk;`(t) = Ak si (t)a` (k )e j (t) + n` (t); (2.2) k

where Ak is the attenuation factor associated with path k, n` (t) is a (complex) baseband representation of the white Gaussian thermal noise component, having zero mean and a double-sided power spectral density N0 =2, and

a` (k ) , exp fj` cos (k k (t)

'` )g ;

, !m (cos k ) t + !ck :

(2.3) (2.4)

In (2.3),  stands for the circular wave number, i.e.,  =2=, where  is the wavelength of the carrier. Propagation path number k gives rise to the following total received signal in the antenna array: rk (t) = [rk;1(t); : : : ; rk;D (t)]T = Ak si(t)a(k )e j (t) + n(t); (2.5) k

2 The

time delay associated with each propagation path is assumed to be negligible with respect to the data signal modulating the carrier, but signi cant with regard to the carrier phase. Even though this assumption is mostly satis ed by narrowband signals, it may also be satis ed by wideband signals.

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where D is the number of antenna elements, superscript T denotes the transpose, and

a(k ) , [a1 (k ); : : : ; aD (k )]T

(2.6)

is known as the array response vector, which is a function of the array geometry and the DOA. n(t) in (2.5) is the additive noise vector random process, or the noise vector process.

Figure 2.1: The propagation geometry. If we initially assume a nite number, ktot , of resolvable propagation paths, then the total received signal equals:

r(t)

= [r1 (t); : : : ; rD (t)]T = si (t) = si (t)f (t) + n(t);

ktot X k=1

Ak a(k )e

!

j

k

(t) + n(t)

(2.7)

where f (t) will be referred to as the multipath-fading vector random process, or simply the fading vector process. Note that both f (t) and n(t) are complex-valued. The statistics of these two vector random processes will be described below. First, assume that the noise vector process and the modulator waveform are statistically independent, and also that the noise and fading vector processes are statistically independent. Let us further suppose for simplicity that the noise processes in the dierent antenna elements are equally strong, and that they are independent and stationary. Hence, the noise covariance-function matrix may be written as: 



E n(t) nH (u) = N0 I Æ (t u); 15

(2.8)

Figure 2.2: The array trigonometry. where superscript H denotes the Hermitian (complex conjugate) transpose, I is an identity matrix, and Æ(
) is Dirac's delta function. For a Rayleigh-fading channel, the phases k (t) are modeled as independent identically distributed (i.i.d.) random variables according to a uniform distribution over [0; 2 ). In the sequel, we will assume a large number of paths, i.e. ktot ! 1, and f (t) will then approach a zero-mean complex Gaussian vector due to the central limit theorem [Woz65]. The next question is how to model the arrival angles k of the incident paths. Numerous distributions have been proposed, where perhaps the most appropriate is the Laplacian distribution [Ped97]. However, one of the simplest models is due to Salz and Winters [Sal94] who suggest a uniform distribution. The same modeling is used here, i.e., the DOA is assumed to be uniformly distributed over [0
; 0 +
), where 0 denotes the mean DOA and 2
the angle spread. It should be noted that the presented model could easily be extended to include the frequency selectivity that occurs when some incident energy is delayed of the same order of magnitude as the symbol duration [Nag96]. Let us now investigate the time-correlation properties of the fading vector. Following Naguib [Nag96] it can be shown that: 



E f (t) f H (u) = J0 (!m jt

uj) Cs;

(2.9)

where J0 (
) is the zero-order Bessel function of the rst kind, and Cs is the array spatial correlation matrix. This last factor is de ned as:

Cs ,

1 2


Z 0 +
0




16

a()aH ()d:

(2.10)

Besides making use of the well-known series representations: cosfx sin y g = J0 (x) + 2 sinfx sin y g = 2

1 X k=0

1 X

J2k (x) cosf2ky g

(2.11)

J2k+1 (x) cosf(2k + 1)y g

(2.12)

k=1

it is mostly a matter of trigonometry (cf. Figure 2.2) to show that element (m; n) of Cs exists in the following closed form: Z 1 0 +
(cosfmn sin 'mn ()g + j sinfmn sin 'mn ()g) d (Cs)m;n = 2
0
  1 X 2k
= J0 (mn ) + 2 J2k (mn ) cosf2k'mn (0 )g sinc  k=1 1 X

where





(2k + 1)
(2.13) +j 2 J2k+1 (mn ) sinf(2k + 1)'mn (0 )g sinc  k=0

mn ,

p

2m n cos f'm 'n g; (2.14)  2   m + 2mn 2n 'm arccos + I; (2.15) 'mn () ,  2 2m mn  the wave front reaches element n before element m; I = 10 i otherwise, and (to clear any confusion due to multiple de nitions in the literature): sin(x) sinc(x) , : (2.16) x From (2.9) and (2.13), we observe that the autocorrelation for each diversity link (m = n) is the Bessel function. This result is completely consistent with the classic single-antenna fading model rst derived by Clarke [Cla68], also known as the Jakes model [Jak74]. From the very same equations, i.e. (2.9) and (2.13), it is further noted that the cross-correlation coeÆcient between two diversity links depends on the antenna separation, the mean DOA, and the angle spread. Figure 2.3 illustrates how the matrix-valued correlation function in (2.9) depends on the time lag, expressed in number of symbols Tb , as well as on the antenna separation, expressed in number of carrier wavelengths . The plot assumes broadside reception, an angle spread of 60Æ (2
= 60Æ ), and a maximum Doppler shift of one tenth of the transmission rate (!m = 2 0:1=Tb ).

2m + 2n

17




m;n

E f (t) f H (u)

jt uj [Tb ]

mn []

Figure 2.3: The correlation, given by equation (2.9), between two fading processes m and n versus time lag and spatial separation. 2.2

On the Slow Fading Assumption

In subsequent chapters, results will mainly be obtained for the normalized Doppler shift fm =0:1, which is regarded as fast fading. These results will be compared with results obtained for the slow fading rate fm =0:01. Note that the presented model is time-continuous and completely general in the sense that it includes both fast and slow fading. However, to simplify the treatment of digital communication over slow fading channels, researchers often tend to approximate the fading to be constant over a complete symbol interval k, i.e. f (t)  fk ; t 2 [(k 1)Tb ; kTb ].3 Further, it is frequently argued that the detector has been able to obtain perfect synchronization and perfect channel state information (CSI). These facts mean that the phase shift induced by the fading can be set to zero, i.e. fk , %k ej 0 , and that the Rayleigh distributed envelope %k of the fading is perfectly known. Even though the simpli cations involved in assuming a piecewise constant known fading process are rather crude, we will make this assumption in section 5.2, but solely for the sake of comparison with existing research material. Let us therefore describe the 3 When

this assumption can be made is far from clear; cf. [Cav92] for some indications.

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implications of such an assumption for a single antenna receiver. Under the extraordinary conditions described above, it seems reasonable to employ detection principles that are optimal for the AWGN channel, i.e., for every symbol interval k, the time-continuous received process is projected onto a nite set of basefunctions f'`;k g that span the signal space, so-called matched ltering [Woz65]:

r`;k = =

Z kTb

(k 1)Tb Z Tb (k 1)T Z Tb

r (t)'`;k (t)dt = ffrom (2.7)g f  (t)si (t)'`;k (t)dt +

b

Z kTb

(k 1)T

Z Tb

n (t)'`;k (t)dt b

 %k si (t)'`;k (t)dt + n (t)'`;k (t)dt (k 1)T (k 1)T , %k s`;k (i) + n`;k ; (2.17) superscript  denotes the complex conjugate. As soon as a set of b

b

where observation variables has been obtained, a probability density function (pdf) can be de ned for each hypothesis i, and these functions are then used for deriving a decision rule|something that will be addresses in section 5.2 when interleaved serially concatenated coding is studied under the slow fading assumption. For moderate and fast fading, however, it is not realistic to assume perfect synchronization and CSI. Instead, the detector is supposed to know or perfectly estimate the rst and second order statistics of the channel. In general, estimation introduces errors in the estimates, and therefore, section 3.5.1 treats the matter of robustness. 2.3

Generation of

D Correlated

Fading Processes

In order to study the performance of antenna arrays for the presented channel model, Monte Carlo simulations will be used in chapters 4 and 5. In this section, then, a method for computer generation of a discrete-time version of the fading vector process f (t) = [f1 (t); : : : ; fD (t)]T is derived. The material given in this section has partly been presented in [HanA99], where the generation principle as such can be seen as a generalization of a method described by Verdin and Tozer [Ver93], who restricted their investigation to a single fading process (D = 1). Also, Beaulieu has independently proposed an alternative approach [Bea99]. 19

Let z(t) = [z1 (t); : : : ; zD (t)]T comprise D independent complex-valued white Gaussian processes with i.i.d. real and imaginary parts, having zero means and (double-sided) unit power spectral densities. The generation principle is simply to lter the white noise through a multi-input multi-output (MIMO) lter:

f (t) =

Z 1

1

h(u) z(t

u)du:

(2.18)

All that has to be done is to nd the D  D matrix-valued impulse response h(t) in (2.18), so that the desired correlation properties are ful lled, i.e., the following must hold: n

E f (t) f

o

H (t +  )

=2

Z 1Z 1

1

1

h(u)hH (v)Æ( + u

v )dudv = CsJ0 (!m  ): (2.19)

First, the delta function will reduce the double integral in (2.19) to a single integral. This simpli ed problem has multiple solutions, two of which will be described in the following; one straightforward solution suitable for implementation, and one leading to a geometrical interpretation that hopefully sheds some light on the underlying problem. In the rst (and straightforward) solution, we note that Cs corresponds to a covariance matrix4, thus always allowing a Cholesky decomposition:

Cs = KsKHs :

(2.20)

Then, by introducing the relation

h(t) = Ksh(t);

(2.21)

equation (2.19) simpli es to the following: Z 1

1 h(u)h (u +  )du = J0 (!m  ): 2 1 Next, if (2.22) is Fourier transformed we get:5

jH (!)j2 = (! + !pm)2 (!2 !m) ; !m

4

!

Cs is a positive de nite Hermitian matrix, cf. equation (2.10).

5 Here

we have assumed that the transfer function H(!) is even.

20

(2.22)

(2.23)

where  (
) is Heaviside's step function. The trivial solution  (! + !m )  (! !m ) p H (! ) = 4 !m2 ! 2

(2.24)

is then inverse transformed to get the following impulse response [Gra92]: r

2! (3=4) (2.25) h(t) = p J1=4 (!m t) 4 m ; 2  t where (
) is Euler's Gamma function, and J1=4 (
) is the one-fourth-order Bessel function of the rst kind. Let us now turn back to (2.19) and consider an alternative to the Cholesky decomposition approach. After exploiting the properties of the delta function, write each element of the matrix-valued integrand as a sum, and change the order of integration and summation to arrive at: D Z 1 X d=1

1

2 6 4

3

h1d (u)h1d (u +  )


h1d (u)hDd (u +  ) 1 7 .. .. ... 5du = Cs J0 (!m  ): . . 2 hDd (u)h1d (u +  )


hDd (u)hDd (u +  )] (2.26)

From (2.22) and (2.25), it follows that the lters could be chosen according to: r (3=4) 2! hk` (t) = ck` p J1=4 (!m t) 4 m ; k; ` 2 f1; : : : ; Dg ; (2.27) t 2  where ck` is a complex-valued constant such that: D X d=1

ckd c`d = (Cs )k;` ; k; ` 2 f1; : : : ; Dg :

(2.28)

However, since the cross-correlation functions show the same time-lag dependency as the autocorrelations, it follows that we could start by generating D independent processes gk (t) having correct autocorrelation functions. This could be done by ltering D statistically independent white noise processes zk (t) through D identical lters with impulse responses given by (2.25). Next, from above, it is clear that the desired cross-correlations are achieved if the processes gk (t) are combined in the following weighting network: 2 6 4

f1 (t) .. . fD (t)

3

2

7 5

=6 4

c11


c1D .. . . . .. . . cD1


cDD 21

32 76 54

g1 (t) .. . gD (t)

3

7 5;

(2.29)

where the weights ck` are found as the solution to the non-linear system of equations (2.28). At this point, the reader might question the purpose of this alternative approach. The reason is that (2.28) has an illustrative geometrical interpretation. To see this, the ck` are rst ordered in vectors:

ck =



ck1




ckD

T

:

(2.30)

The system of equations (2.28) is equivalent to a set of inner products of the vectors ck . To simplify the exposition, assume real-valued cross-correlation coeÆcients (C)k;` and consider D =3; 8 2 < hck ; ck i = kck k = 1 k; ` 2 f1; 2; 3g ; k 6= `; (2.31) : hck ; c`i = cos k` = (Cs)k;` where k` is the angle between the two vectors ck and c`. Thus, the angle between two vectors determine the mutual cross-correlation (C)k;` between the corresponding processes, which can be viewed as the projection of one vector against the other. All vectors shall also have unit length, and they could e.g. be assumed to start in the origin and end at the surface of a unit sphere as illustrated in Figure 2.4 below.

Figure 2.4: Graphical representation of equation (2.31). From Figure 2.4, it is clear that any set of cross-correlation coeÆcients (Cs)k;` does not correspond to realizable fading processes. Assume, e.g., that process one and two should be completely correlated, as well as process one and three. Now, is it possible to generate these three random processes, so that process two and three become completely uncorrelated? We expect the 22

answer to be negative. Let us start by specifying (Cs )1;2 and (Cs )1;3 . Then, with some trigonometry, it is straightforward from (2.31) to show that the following constraints hold for the coeÆcient (Cs )2;3 : q

  (Cs)1;2 (Cs )1;3 1 (Cs )21;2 1 (Cs )21;3  (Cs)2;3  q    (Cs)1;2 (Cs )1;3 + 1 (Cs )21;2 1 (Cs )21;3 :

(2.32)

The constraints (2.32) are illustrated in Figure 2.5, assuming (Cs)1;2 (Cs )1;3 .



Figure 2.5: Illustration of the inequalities (2.32), assuming (Cs )1;2  (Cs )1;3 . We may proceed in a similar manner when D  4. Even though the calculations are somewhat simpli ed by introducing hyperspherical coordinates, it proves to be hard to derive a complete set of inequalities corresponding to (2.32). Also, recall from (2.13) that the cross-correlation coeÆcients in general are complex-valued. Due to the fact that a correlation matrix must be positive de nite, the restrictions on the cross-correlation coeÆcients are, however, well known. E.g., the combination (Cs )1;2  1, (Cs )1;3  1, and (Cs )2;3  0 does not correspond to solely positive eigenvalues, and hence it does not correspond to a correlation matrix. These \impossible" choices of cross-correlations are simply not physically interesting, i.e., they do not correspond to a realizable antenna geometry|no matter what the DOA and the angle spread might be. As long as the cross-correlation coeÆcients are 23

speci ed by (2.13) we are guaranteed that the fading processes are possible to generate. Finally, a few words on practicalities. A discrete-time representation suitable for computer simulations is found by sampling the derived impulse response in (2.25). Note that some windowing technique is necessary to obtain a nite impulse response (FIR) lter [Pro92]. Moreover, in order to reduce computation times, a multi-rate ltering technique is recommended [Ver93][Wes83]. 6

4

Fading Envelope [dB]

2

0

−2

−4

−6

−8

Process 1 Process 2 Process 3

−10

−12

0

10

20

30

40

50

60

70

80

90

100

Time [Number of Samples]

Figure 2.6: Example of the envelopes of three correlated fading processes, with !m = 2 0:05, (C)1;2 = 0:9, (C)1;3 = 0:8, and (C)2;3 = 0:6. Figure 2.6 shows the Rayleigh fading envelopes of three processes with real-valued cross-correlation coeÆcients (C)1;2 = 0:9, (C)1;3 = 0:8, (C)2;3 = 0:6, and a maximum Doppler angular frequency shift equal to !m = 2 0:05. A 1024-tap Hanning window was used in order to obtain a FIR lter. Further, the multi-rate ltering technique was employed, meaning that a 1024=U -tap convolution was performed on every iteration u = 1; : : : ; U ; thus yielding U samples of the fading process before updating the input sequence (consisting of white noise samples). This technique requires that !m < (=U )( =Tb ) [Ver93], where =Tb is the sampling rate. The auto- and cross-correlation functions were estimated and closely matched the theoretical ideal functions. See Figures 2.7 and 2.8. 24

Autocorr. Autocorr.

1 0.5 0 −0.5 1 0.5 0 −0.5 1 0.5 0 −0.5

1 0.5 0 −0.5 1 0.5 0 −0.5 1 0.5 0

0

0

0

0

0

10

10

10

10

10

10

20

20

20

20

20

20

30

30

30

40

40

40

50

50

50

60

60

60

70

70

70

Time [Number of Samples]

30

30

30

40

40

40

50

50

50

60

60

60

70

70

70

Time [Number of Samples]

25

90

Theoretical Process 1

80

90

Theoretical Process 2

80

90

Theoretical Process 3

80

90

Theoretical Process 1 and 2

80

90

Theoretical Process 1 and 3

80

90

Theoretical Process 2 and 3

80

100

100

100

100

100

100

Figure 2.8: Cross-correlations v. theoretical ideal ones for three processes with !m = 2 0:05, (C)1;2 = 0:9, (C)1;3 = 0:8, and (C)2;3 = 0:6.

−0.5

0

Figure 2.7: Autocorrelations v. theoretical ideal ones for three processes with !m = 2 0:05, (C)1;2 = 0:9, (C)1;3 = 0:8, and (C)2;3 = 0:6.

Autocorr. Cross−corr. Cross−corr. Cross−corr.

26

Chapter 3 Single Symbol Detection \Science is a collection of successful recipes."

{Paul Valery

For simplicity, and without loss of generality, assume that binary signaling is used under equal a priori probability, i.e., the transmitter communicates one of two dierent equally likely modulator waveforms si (t); i 2f0; 1g. Assume further (merely for simplifying the notation) that the modulator waveforms are real-valued. One of these two waveforms is then transmitted over the channel and received as:

r(t) = si(t)f (t) + n(t);

(3.1)

where the properties of the fading vector process f (t) and noise vector n(t) are given by (2.9) and (2.8), respectively. Equation (3.1) is illustrated in Figure 3.1.

Figure 3.1: Schematic channel model. In each diversity link k, the modulator waveform si (t) is perturbed by fading fk (t) and noise nk (t). 27

3.1

On Optimum Diversity Reception

The detection problem that arises, i.e. discrimination of two Gaussian signals in AWGN, was treated in the fties and the sixties [Kad64][Tur61]. In order to use the classic method of a likelihood ratio test (LRT) [VTr68], the continuous-time received process must rst be \represented" by a sequence of observables. We could either sample to obtain the representing sequence, or we could make an orthogonal expansion of the received process and use the expansion coeÆcients as observables [Kad64][Tur61]. Such a transition from continuous time to discrete time will henceforth be referred to as discretization. The crux of the matter is obvious from Figure 3.1: the product of modulation and fading, which can be seen as random modulation, spans an in nite dimensional signal space [Woz65]. This means that optimum1 detection requires an in nite number of observables. However, we can at least start by extracting a nite number of observables:

r = [r1; : : : ; rN ]T :

(3.2)

The LRT is then a quotient of the two conditional density functions [VTr68] [VdB95]: 1
1 p (rjs0(t))  jC(0)j exp rH (C(0)) r s0 (t) = 1
? 1; (3.3) H (C(1)) 1 r s1 (t) p (rjs1(t))  jC(1) exp r j N

N

where the conditional covariance matrix is de ned as follows: 



C(i) , E r rH jsi(t)

; i 2 f0; 1g :

(3.4)

It has been proved that an optimum detector structure can be found by manipulating (3.3) into a form so that we can let N ! 1 [VTr68]. In the absence of additive noise (i.e. when n(t)  0), r converges to r(t) in a meansquare sense when N ! 1. We are, however, interested in the received process when it is corrupted by additive noise, and the representation is then invalid. Fortunately, it can be shown that r (in the limit) does preserve all of the relevant information contained in the received process. Precisely stated, even in the presence of white Gaussian noise, a receiver operating only upon r (in the limit) can be made to perform as well as one operating upon all of r(t). This observation is a consequence of what is called the theorem of irrelevance [Woz65]. Initially there was some criticism on the lack of convergence proofs for the discrete-time representations, but [Kad64] 1 Throughout

this report, \optimum" refers to minimum error probability.

28

put the preceding work on a more rigorous basis. These calculations, then, lead to a detector structure that involves time-varying lters with impulse responses that require solutions to integral equations based on the fading channel statistics|solutions that in general cannot be found in closed form. In order to keep the complexity of the receiver at a more moderate level, the number of observables will not be increased beyond a nite limit (say some xed N ), and under this restriction (3.3) still constitutes an optimum decision rule. Let us for convenience rewrite the LRT in (3.3) in its familiar quadratic form: (C(1)) 1

r

H


s0 (t) (C(0)) 1 r ? ln s1 (t)



jC(0)j  : jC(1)j

(3.5)

From a purely intuitive point of view, the optimum set of observables should be uncorrelated, and this should hold for any N . Otherwise, the number of observables would be eectively less than N . Karhunen-Loeve expansion (KLE) is optimal in this sense, because it leads to uncorrelated observables [VTr68]. A natural representation of the KLE is to use vector-valued eigenfunctions and scalar eigenvalues [VTr68], i.e., compute the observables according to:

rn =

Z Tb

0

rT (t) n(t)dt; n = 1; : : : ; N;

(3.6)

where [0; Tb ] is the signaling interval, and the vector-valued (D components) eigenfunctions n (t) are chosen to satisfy a matrix-valued Fredholm equation of the second kind [VTr68]:

k k (t) =

Z Tb

0

si (t)si (u)J0 (!m jt uj) Cs k (u)du; 0  t  Tb :

(3.7)

This discretization scheme will be referred to as VKLE, where the \V " is a reminder of the fact that a vector random process is expanded, and that the eigenfunctions are vector-valued. This scheme is illustrated in Figure 3.2. A quandary comes up here, since the kernel of the Fredholm equation, i.e. si (t)si (u)J0 (!m jt uj)Cs , involves the transmitted waveform, which of course is unknown to the receiver. Kadota, however, has shown that a simultaneously orthogonal expansion of two signals is possible [Kad66]. The problem is that his method seems hard to generalize; it is only exempli ed under very special conditions on the fading covariance-function. An alternative idea, which will be used in this chapter, is treated in [Ken65] and [HanU99], where the two random processes are forced to occupy 29

Figure 3.2: A detector based on the VKLE discretization scheme. disjoint subspaces of the total signal space. The two subspaces should be orthogonal to each other, i.e., the basefunctions of one subspace should be orthogonal to all the basefunctions of the other subspace. Such a partitioning of the signal space is easily created by using time-orthogonal modulator waveforms. For simplicity, let us use two modulator waveforms si (t) that are constant over the rst and the second halves of the signaling interval, respectively: 

t  Tb =2 ; si (t)  1 i i ifif 0T  b =2 < t  Tb :

(3.8)

The waveforms are thus orthogonal (in time). Without loss of generality, assume that N is an even number. Compute half of the observables (i.e. N=2) by projecting against N=2 basefunctions that occupy one of the two subspaces, and then, compute the rest of the observables by using N=2 basefunctions from the other subspace. For obvious reasons, it is important to choose the N=2 basefunctions in each subspace that correspond to most energy, i.e. the basefunctions that correspond to the largest eigenvalues. In accordance with the above discussion, choose basefunctions that satisfy the following two criteria [VTr68]: 1:

Z Tf Ti

Hk (t)`(t)dt = Æk`;

2: k k (t) =

Z Tf Ti

(3.9)

J0 (!m jt uj) Cs k (u)du; Ti  t  Tf ; (3.10)

where Æk` in (3.9) denotes the Kronecker delta symbol, and the limits of the integrals [Ti ; Tf ] equal [0; Tb =2] for the rst N=2 observables, and [Tb =2; Tb] for the last N=2 observables. Now, by applying Mercer's theorem [VTr68]

J0 (!m jt uj) Cs =

1 X k=1

30

k k (t)Hk (u)

(3.11)

it is straightforward to show that the conditional covariance matrices in (3.3) and (3.4) reduce to:

N=2 0 0 0



C(1) = 00 0N=2



C(0) =





+ N0 I; (3.12)

+ N0 I;

where

n = diag f1 ; : : : ; ng :

(3.13)

Note that the eigenvalues in (3.12) are real-valued due to the Hermitian kernel in (3.10)|as expected. If the determinants are equal under the two hypotheses, the right member of (3.5) reduces to a zero threshold. That this is the case for the VKLE scheme is trivial from (3.13). 3.2

Suboptimum Diversity Reception

VKLE is based on the solution of (3.10), but since this is ill-suited for implementation in a digital receiver, it is important to nd suboptimum schemes with comparable performance. Compute N=D observables in each antenna element k = 1; : : : ; D: 2

rk = 64

rk;1 .. . rk;(N=D)

3

3

2

k;1(t) Z T 7 7 6 . .. rk (t)k (t)dt; rk (t)4 5dt = 5= 0 0 k;(N=D) (t) (3.14) Z Tb

b

Naturally, these N random variables are stored as follows: 

r = rT1


rTD

T

:

(3.15)

Consider a submatrix (k; l) of the total conditional covariance matrix (i f0; 1g): n

C(i)k;` , E r r j

H k l si (t)

o

=

Z Tb Z Tb

0

0

n

2

o

k (t) E rk (t)r`(u)jsi(t)

H` (u)dtdu: (3.16)

31

An element (p; q ) of the submatrix (k; `) simpli es by using the same set of real-valued basefunctions in each antenna, i.e. by letting k;p(t)  p (t) for all k; (C(i)k;`)p;q = (Cs )k;`

Z Tb Z Tb

+ Æk`N0

0Z

Tb

0

0

p (t)si (t)si (u)J0 (!m jt uj)q (u)dtdu

p (t)q (t)dt; i 2 f0; 1g ;

(3.17)

If the basefunctions are chosen to satisfy a scalar-valued version of (3.10),2 i.e., if we perform a KLE in each antenna element, observables obtained in antenna k will be uncorrelated between themselves (and this is true for all k = 1; : : : ; D). By invoking Mercer's theorem (3.11) the following expression for the covariance matrices is found:

C(i) = Cs Li + N0 I; i 2 f0; 1g ;

(3.18)

where denotes the Kronecker matrix product [Hor91], and

L0 = L1 =



N=(2D) 0 0 0





0 0 0 N=(2D)



; (3.19)

:

Figure 3.3: A detector based on the Sub-KLE discretization scheme. The described discretization will be called Sub-KLE and is illustrated in Figure 3.3. Note that the threshold in (3.5) still equals zero, since the covariance matrix on one hypothesis is easily transformed to the covariance matrix on the other hypothesis by an even number of row and column permutations.3 kernel is now J0 (!m jt uj). exchange of two rows, or two columns, only changes the sign of the determinant.

2 The 3 An

32

Figure 3.4: A detector based on the Sub-ON discretization scheme. A motivation for Sub-KLE is that it \only" requires the solution of a scalar-valued Fredholm equation|not the solution of the much more complex matrix-valued version as VKLE requires. It is, however, disappointing that the basefunctions still could not be analytically derived from (3.10) (even though the kernel is scalar-valued). Numerical methods may be applied, but note that we have to resolve for the basefunctions whenever the kernel changes, i.e., whenever the Doppler frequency shift changes. Even in this respect the VKLE scheme is less attractive, since the kernel also depends on the DOA and angle spread, in addition to the Doppler shift. It is thus desirable to nd another suboptimum way to compute the observables|a scheme that does not require the solution of (3.10). For the Jakes model (the single antenna case), various simple sets of orthonormal (ON) basefunctions have been proved to give almost the same performance as basefunctions that satisfy the (scalar-valued) Fredholm equation [HanU99]. An ON set that will be taken into considerationpis speci ed as follows: let the kth basefunction have constant amplitude N=Tb over [(k 1)Tb =N; kTb =N ], and let it be equal to zero elsewhere. When such a set is used in each antenna element, the discretization scheme will be referred to as Sub-ON. This structure is illustrated in Figure 3.4. 3.3

A Weighted Sum Approach

Nowadays, with the abundant progress in the signal processing area, adaptive antenna arrays are becoming increasingly popular [God97][Koh98]. Adaptive antenna arrays are frequently proposed for nulling interfering users, but the lion's share of all published work is based on communication systems free from perturbations due to fading and noise [God97].4 In real systems these ideal assumptions are hardly met, and it would thus be relevant to investigate 4 The

detection problem is rarely treated.

33

adaptive arrays in a more realistic environment|especially over our fast fading channel. This section, then, will lay the foundations for a subsequent detection performance analysis also for adaptive antenna arrays. The ad hoc starting-point for all adaptive antenna arrays is to form a weighted sum of the antenna signals:5

rsum (t) =

D X k=1

wk rk (t) =

D X k=1

wk (si (t)fk (t) + nk (t)) :

(3.20)

Next, represent the signal rsum (t) in discrete time:

rn =

Z Tb

rsum (t)n (t)dt; n = 1; : : : ; N: (3.21) 0 By considering real-valued basefunctions, element (p; q ) of the conditional covariance matrix becomes: (C(i))p;q =



D X D X

wk w`

k=1 `=1  Z Tb Z Tb ( s )k;` p (t)si (t)si (u)J0 (!m 0 0  Z Tb

C

jt uj)q (u)dtdu

+ Æk`N0

p (t)q (t)dt : (3.22) 0 Under the assumption that we receive the continuous-time sum of antenna signals in (3.20), the optimum projection in (3.21) is the Karhunen-Loeve expansion|optimal in the sense that each new extracted observable is uncorrelated with all the previously extracted observables, which means that each new observable brings completely unknown information. As stated before, the error rate is minimized by using those basefunctions that correspond to maximum symbol energy. Thus, according to Mercer's theorem (3.11):

C(0) = C(1) = where

W = (N=2) 5 Equation



W 0 0 0





0 0 0 W



D X D X k=1 `=1

+ V; + V;

wk w` (Cs )k;` ;

(3.20) is typical for narrowband signals [VVe88].

34

(3.23)

(3.24)

V = N0 I

D X k=1

jwk j2 :

(3.25)

As mentioned before, much research on deriving optimum antenna weights has been pursued. One of the most common algorithms is the least mean square (LMS). No details of this algorithm will be recapitulated within the scope of this report; instead the reader is directed to the literature [Com88]. The LMS algorithm assumes that a replica of the desired signal is available as a reference signal|something that of course is impossible in a practical system, but will be assumed here for simplifying reasons (cf. [Com88]). Simply stated, the LMS algorithm operates by aligning the phase of the signal from each antenna element with that of the reference signal, before summing the signals. Note that the array response vector in (2.6) contains the phase shifts for the D antenna elements (with respect to the origin in Figure 2.1), associated with a propagation path incident from an angle . Thus, if we suppose that there is no spread of the arrival angles of the propagation paths (i.e.   0 ), the LMS weights are directly found from (2.3): wk = exp f jk cos (0 'k )g ; k = 1; : : : ; D: (3.26) The erroneous assumption of zero angle spread could partly be viewed as a limited resolution of the receiving array. When the weights are chosen according to (3.26), irrespective of the angle spread (2
), the discretization is termed LMS-KLE. See Figure 3.5.

Figure 3.5: A detector based on the LMS/ML-KLE discretization scheme. The ideal weights, however, would be those weights that minimize the error probability, irrespective of the angle spread. This discretization technique will be referred to as ML-KLE (maximum-likelihood KLE). Once a 35

closed-form expression for the error probability has been derived, the question of optimum weights reduces to a pure optimization problem. In the next section it will become clear that this optimization problem is a nonlinear one, and hence a numerical method will be employed to nd the optimum weights. Also, there is the question of whether the optimum weights correspond to a global extreme value or not. In general, this may be diÆcult to prove, and therefore we have focused on a system with only two antennas where this can easily be veri ed. One might ask how such an ML-KLE detector could be implemented, but this is of no great importance, since this discretization scheme is only used as a reference system.

3.4

Performance Analysis

Let the set fn gNn=1 consist of the eigenvalues of the matrix C(0)(C(1)) 1 In appendix A it is shown that the error probability may be written as: Pr(e) =

N X Y n

1 : 1  = m n > > > >
m =n + n =m n=1 > > n= 6 m> > n6=k > :

N Y

1 : (A.9) 2 `=1 1 ` =m > > > > `= 6 m > > `6=k ; `6=n

brevity, we restrict ourselves to a zero threshold, i.e. T

79

9 > > > > > > =

 0.

80

References [Ana98]

A. Anastasopoulos and K. M. Chugg, \Iterative equalization/decoding of TCM for frequency-selective fading channels", Proc. 31 Asilomar Conf. on Signals, Systems and Computers, pp. 177{181, Paci c Grove, CA, Nov. 1997.

[And86]

J. B. Anderson, T. Aulin, and C.-E. Sundberg, Digital Phase Modulation, Plenum Press, 1986.

[Aul79a]

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