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The equalizers that are utilized to compensate for the IS1 can be classified according to their structure, the optimising criterion and the algorithms used to adapt the equalizer coef- f

Introduction To Equalizers

In most digital data transmission systems the dispersive linear channel encountered exhibits amplitude and phase distortion As a result, the received signal is contaminated by Intersym- bo1 Interference (ISI) In a system, which transmits a sequence of pulse-shaped information symbols, the time domain full response signalling pulses are smeared by the hostile dispersive channel, resulting in intersymbol interference At the receiver, the linearly distorted signal has to be equalized in order to recover the information

The equalizers that are utilized to compensate for the IS1 can be classified according to their structure, the optimising criterion and the algorithms used to adapt the equalizer coef- ficients, which are summarized in Figure 2.1 On the basis of their structures, the equalizers can be classified as linear or decision feedback equalizers Each of these structures will be discussed at a later stage with more emphasis on the Decision Feedback Equalizer (DFE) Equalizers can also be distinguished on the basis of the criterion used to optimise their coefficients The optimization is governed by the performance criteria used For example, when applying the mean square error criterion (MSE), the equalizer is optimised such that the mean squared error between the distorted signal and the actual transmitted signal is min- imized Various optimization criteria will be discussed in the following sections, with more emphasis on the MSE criterion

A range of adaptive algorithms can be invoked, in order to provide the equalizer the means

of adapting its coefficients to the time-varying dispersive channels We will not elaborate

on the fine details of these algorithms until the next chapter, where attention will be given

to the well known Least Squares (LS) algorithms, in particular to the Recursive Kalman Algorithm [86]

Before we proceed to highlight the different techniques of channel equalization, we will introduce the multilevel modulation schemes that are utilized throughout this treatise This

is necessary in order to study the performance of the equalizers in multilevel modulation environment, since the choice of the modulation mode will affect the performance of the equalizer

21

Adaptive Wireless Tranceivers

L Hanzo, C.H Wong, M.S Yee Copyright © 2002 John Wiley & Sons Ltd ISBNs: 0-470-84689-5 (Hardback); 0-470-84776-X (Electronic)

Equalizer Structures

l

Optimi

Sequence Estimator

Adaptive Algorithms

Variants

Figure 2.1: Classification of conventional equalizers on the basis oftheir structure, optimization criteria

and coefficient adaptation algorithms

2.1 COHERENT DEMODULATION OF SQUARE-OAM 23

Quadrature Amplitude Modulation

In this section, various multilevel modulation schemes are presented and their performances are analysed in a Gaussian environment The well known shaped Square Quadrature Ampli- tude Modulation (QAM) schemes are defined mainly by their phasor constellation arrange- ments Throughout this treatise, the modulation schemes used are BPSK, 4QAM, 16QAM and 64QAM, representing the square-shaped constellations associated with one, two, four and six bits per symbol The corresponding square-shaped phasor constellations are depicted

in Figure 2.2 with their corresponding assigned bit sequences

The phasor constellations shown in Figure 2.2 provide the location of each constella- tion point in terms of their in-phase (I) and quadrature-phase (Q) components, where each point is assigned a particular bit sequence Gray-coding is applied in order to assign the bit sequence to their respective constellation points, ensuring that the nearest neighbour constel- lation points have a Hamming distance of one Thereby the bit assignments are optimised in terms of minimising the Bit Error Rate (BER)

Gray-coding of the bit sequence gives rise to the formation of a number of different in- tegrity subchannels [4] These subchannels are formed in order to distinguish the different level of noise protection experienced by the bits in different subchannels In the QAM con- stellations mentioned above, each subchannel consists of two bits Thus for 16QAM, there are two different integrity subchannels, labelled as C l and C2, where the C l subchannel possesses a higher noise protection distance than the C2 subchannel The same applies to the 64QAM mode having three subchannels For a more in-depth understanding of QAM techniques, the reader is referred to Hanzo et al [4]

2.1.1 Performance of Quadrature Amplitude Modulation in

Gaussian Channels

The BER performance of QAM in a Gaussian environment is presented below, both the- oretically and using simulations The simulation results presented here assumed coherent

detection and perfect timing synchronisation between the transmitter and receiver

2.1.2 Bit Error Rate Performance in Gaussian Channels

The theoretical solutions for the BER performance of QAM in Gaussian channels have been quantified and published According to Proakis [87], the closed form theoretical solution to the BPSK and 4QAM BER performance, P,&,SK and P&AM, respectively, are

where 7 in the above equations represents the average channel signal to noise ratio (SNR) and the Q function is defined by Proakis as [87]

Figure 2.2: QAM phasor constellations

2.1 COHERENT DEMODULATION OF SOUARE-OAM 25

Following the approach outlined by Hanzo et al [4], the closed form solution of the 16QAM and 64QAM modulation mode in terms of its individual subchannel BER perfor- mance can be written as :

The notations PiXyQAM represent the BER of subchannel x of the y-point QAM, while

P:vyQAM denotes the average BER of y-point QAM, assuming that the probability of transmis- sion over any of the subchannels is equal The theoretical BERs from the expressions above were compared to the experimental BERs and the results are presented graphically in Figure

2.3 The term - experimental - will be used throughout our discourse in the context of results generated by computer simulation

In this section, the performance results of coherent QAM in a non-dispersive Gaussian

channel were presented In the next section, the performance of the QAM modems in a Rayleigh flat fading environment is evaluated by assuming perfect channel estimation and the employment of a matched filter at the receiver

2.1.3 Bit Error Rate Performance in a Rayleigh Flat Fading Environ-

ment

In this section the performance of the multilevel modulation schemes outlined in Section 2.1 is presented and analysed in a Rayleigh flat fading environment [88] Theoretical closed form solutions are presented and compared with the experimental results for each multi- level scheme utilized The theoretical solution for BPSK followed the approach outlined by

* 64QAM C l - Simulated

0 64QAM C2 - Simulated

0 64QAM C3 - Simulated 16QAM C 1 - Theoretical I6QAM C2 - Theoretical

_ _

X 16QAM C1 - Simulated

4 I6QAM C2 - Simulated QPSK - Theoretical

V QPSK - Simulated BPSK - Theoretical BPSK - Simulated

Figure 2.3: Theoretical and simulated BER performance of a BPSK, 4QAM, 16QAM and 64QAM

over a Gaussian channel

2.1 COHERENT DEMODULATION OF SQUARE-OAM 27

Proakis [87] , where perfect channel estimation and the employment of a matched filter [89] was assumed at the receiver Since the matched filter was matched to the channel variations for all symbols transmitted, its transfer function K M F ( z ) , can be written as:

where E represents the Rayleigh-distributed attenuation factor for the received signal and 0

denotes the random phase variation, which is distributed uniformly between 7r and - 7 r

For a fixed attenuation E , we can express the instantaneous SNR y, at the receiver as

y = e 2 y ~ , where is the channel SNR in a Gaussian environment Upon utilizing Equation 2.1, we can characterize the BER performance of BPSK, assuming a fixed attenuation E

as [87]:

For a random attenuation E , we can obtain the theoretical BER performance by integrating

P k P S K ( y ) over the probability density function (PDF) of y, p(y) for all possible values of

Now, by exploiting that Q(.) = 0.5(1 -erf( L)) Jz and employing Equation 2.16 and 2.14,

we rewrite Equation 2.15 as :

(2.18)

We can exploit the fact stated by Gradshteyn et al [90]:

The same derivation can be extended to the higher-order modulation schemes discussed

in Section 2.1 Therefore the theoretical BER performance of 4QAM, 16QAM and 64QAM over flat Rayleigh-fading channels can be written as :

Due to the linear, dispersive channel, the received linearly distorted instantaneous signal can

be visualised as the superposition of several information symbols in the past and in the future

The so-called post-cursor IS1 and pre-cursor IS1 can be explained by using a hypothetical

channel impulse response, as shown in Figure 2.5

The channel impulse response shown in Figure 2.5 can be viewed as constituted by three distinct parts The tap which possesses the highest relative amplitude hz, is termed the main

tap The taps that exist before the main tap h0 and h l , are called pre-cursors and the taps

that follow the main tap, namely h3 and h.4, are referred to as post-cursors The energy

of the wanted signal is conveyed mainly by the contribution of the main channel tap In addition to that, the received signal will also contain energy contributed by the convolution

of the pre-cursors with the future transmitted symbols (relative to the channel main tap) and the convolution of the post-cursors with the past transmitted symbols, which are termed pre-

cursor IS1 and post-cursor ISI, respectively Thus the received signal is distorted due to

the superposition of the wanted signal, pre-cursor IS1 and post-cursor ISI Having defined the

effects of a dispersive channel in terms of the post- and pre-cursor ISI, a range of equalization techniques used to combat the effects of this dispersive channel is introduced in the next section

This section provides a rudimentary introduction to channel equalizers A few equalizer struc- tures are presented along with the criteria used to optimise their coefficients Subsequently, the theoretical impact of these structures and their optimising criteria are highlighted Before

we proceed to discuss the structure and operation of the equalizers, their evolution history is briefly presented here

In the context of linear equalizers, the pioneering work was achieved mainly byTufts [91], where the design of the transmitter and receiver was jointly optimised The optimization was

Figure 2.4: Theoretical and simulated BER performance of a single tap optimum DFE for BPSK,

estimation

2.3 BASIC EQUALIZER THEORY 31

Figure 2.5: Channel Impulse Response (CIR) having pre-cursors, main tap and post-cursors

based on the minimisation of the MSE between the transmitted signal and the equalized sig- nal This was achieved under the Zero Forcing (ZF) condition, where the IS1 was completely mitigated at the sampling instances Subsequently, Smith [92] introduced a similar optimiza- tion criterion with and without applying the ZF condition Similar works as a result of these pioneering contributions were achieved by amongst others Hansler [93], Ericson [94] and Forney [95]

The development of the DFE was initiated by the idea of using previous detected symbols

to compensate for the IS1 in a dispersive channel, which was first proposed by Austin [96] This idea was adopted by Monsen [97], who managed to optimise the DFE based on minimiz- ing the MSE between the equalized symbol and the transmitted symbol The optimization of the DFE based on joint minimization of both the noise and IS1 was undertaken by Salz [98],

which was subsequently extended to QAM systems by Falconer and Foschini [99] At about the same time, Price [ 1001 optimised the DFE by utilizing the so-called ZF criterion, where all the IS1 was compensated by the DFE The pioneering work achieved so far assumed perfect decision feedback and that the number of taps of the DFE was infinite A more compre- hensive history of the linear equalizer and the DFE can be found in the classic papers by Lucky [ 1011 or by Belfiore [ 1021 and a more recent survey was produced by Qureshi [ 1031

In recent years, there has not been much development on the structure of the linear and decision feedback equalizers However considerable effort has been given to the investigation

of adaptive algorithms that are used to adapt the equalizers according to the prevalent CIR These contributions will be elaborated in the next chapter Nevertheless, some interesting

work on merging the MLSE detectors with the DFE has been achieved by Cheung et al

[104, 1051, Wu et al [106, 1071 and Gu et al [108] In these contributions, the structure

of the MLSE and DFE was merged in order to yield an improved BER performance, when compared to the DFE, albeit at the cost of increased complexity However, the complexity

Figure 2.6: Schematic of the transmission system

incurred was less, when compared to that of the MLSE

In the context of error propagation in the DFE, which will be explained in Section 7.1, this phenomenon has been reported and researched in the past by Duttweiler et al [ 1091 and more recently by Smee et al [ 1101 and Altekar et al [ 11 l] In this respect some solutions have

been proposed by amongst others, Tomlinson [112], Harashima [ l 131, Russell et al [ l 141 and Chiani [ 1 151, in reducing the impact of error propagation

In our subsequent discussion, the linear equalizer and the DFE are investigated using the Zero Forcing (ZF) and Minimum Mean Square Error (MMSE) criterion, with more emphasis

on the DFE structure In order to highlight the difference between the MMSE and ZF criteria, the linear equalizers based on these criteria are defined next

2.3.1 Zero Forcing Equalizer

In Figure 2.6, the basic schematic of our system is shown It simply consists of a pulse transmitter, the channel, an equalizer and a detector The transmitted bits are labelled as

s(lc), n ( k ) represents the Additive White Gaussian Noise (AWGN) samples, while the output

of the equalizer is denoted as :(IC) The channel and equalizer transfer functions are denoted

by H ( f ) and C ( f ) , respectively

The so-called zero forcing equalizer was devised and optimised by using the criterion This essentially implied forcing all the impulse response contributions of the concatenated transmitter, channel and equalizer to zero at the signalling instants nT for rL # 0, where T

was the signalling interval duration According to the zero IS1 constraint, in the frequency

domain the ZF criterion ensured the following relationship as stated by Lee et al [89] when assuming zero delay:

yielding the equalizer transfer function as the inverse of the channel transfer function :

(2.29)

Consequently, the equalizer was reduced to a Finite Impulse Response (FIR) filter having

an impulse response, which mimicked the inverse of the channel impulse response A good

insight into the behaviour of the equalizer can be obtained by studying the mean square error