Adaptive two-stage maximum likelihood estimation for cellular radio

Adaptive two-stage maximum likelihood estimation for cellular radio

A.B. Sesay

Indexing terms: Cellular radio, Maximum likelihood estimation, Nonselective fading channels

~~ ~~

Abstract: The problem of optimal detection over nonselective fading channels for cellular radio is treated from the viewpoint of adaptive two-stage maximum likelihood estimation (MLE). Data for transmission are divided into blocks, each preceded by one or more reference symbols. For selective fading, the receiver consists of an adaptive filter (AF) and a two-stage maximum likelihood estimator (TSMLE). For nonselective fading, TSMLE is preceded by a fixed filter. For quadrature modulation, the complex output of the filter contains two sets of unknowns, the quadrature data symbols (QDS) and the quadrature gains (QG) between transmitter and filter output. Stage I generates MLE estimates of the QG during the reference period and Stage I1 generates QDS estimates during the data intervals. AF coefficients are adjusted using a QR-decomposition least squares (QRD-LS) algorithm. A likelihood ratio test is performed to produce noise-free data estimates which Stage I in turn can use, in a decision aided manner, to update its gain estimates for the next interval. The quadrature gain estimates can be used to further correct reference carrier phase errors. Nonrecursive and recursive estimation procedures are developed. Bit error rate (BER) expressions are presented for 4-PSK and 44- DQPSK. Numerical results show that the scheme meets the North American Digital Cellular (NADC) system performance specifications (IS-54) with a margin. The BER floor is an order of magnitude lower than those of other schemes reported in the literature. For nonselective fading the BER floor is practically eliminated. This, and results reported elsewhere (Haeb and Meyr, 1989), lead us to conclude that schemes performing quadrature gain estimation can be used to lower error probabilities, attractively.

1 Introduction

1.1 Nonselective fading Nonselective fading is characterised by multiplicative random amplitude (gain) and phase variations of the carrier. These are the effects of a highly nonstationary channel. Linear modulation and their coherent reception

0 IEE, 1994 Paper 98051 (E8). first received 21st January and in revised form 28th July 1993 The author is with TRLabs/Electrical & Computer Engineering Dept., University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4

IEE Proc.-Commun., Vol. 141, No. I , February 1994

are the preferred schemes but require perfect gain and phase estimates to compensate the distortions at the receiver. This is a difficult task because phase synchron- isation methods for fading channels suffer instability and tracking problems. In an attempt to overcome these problems, several schemes have been proposed [l-71. These works are based upon quadrature channel gain estimation. References 4 to 6 propose partially coherent reception while References 1 to 3 propose coherent reception schemes. They all propose Kalman filtering, in one form or the other, to generate an MMSE estimate of the channel gain. They also assume that the filter order is sufficient to provide accurate gain estimates; another practical problem. The compensation procedures, however, differ. In References 4 and 5, the gain estimates are used to construct a likelihood ratio test for 2-PSK. In Reference 1, gain estimation is followed by a MAP detector. in References 2, 3 and 6, multiplicative distortion is compensated by dividing the received signal samples by the gain estimate followed by a MAP detector for M-PSK. Assuming perfect estimates, the coherent schemes exhibit superior performance. However, this is at the expense of additional receiver complexity. Moreover, the compensation procedure is, practically, not straight- forward for M-PSK and the appropriate filter order must be determined.

Here, we follow the general approach of these works but employ two-stage ML estimation theory which we first proposed [SI, as an alternative to maximal ratio- combining (MRC). MRC is a scheme that first weights the individual diversity signals according to their SNR, then aligns their phases and sums them. The phase alignment requires fast and stable phase tracking loops for the cellular radio channel. For the two-stage procedure all channel amplitude and phase variations are lumped into the quadrature gains, which are Gaussian processes, and use ML methods for their estimation. As pointed out [l, 41, since quadrature gain estimation is equivalent to phase estimation, the nonlinear loop requirement of phase tracking is eliminated. Also, two-stage ML estimation avoids such problems as filter order selection, and stability and tracking requirements.

To facilitate gain estimation at the TSMLE receiver, it is proposed that data for transmission be divided into blocks each preceded by at least one known symbol; redundancy is not greater than 10%. At the receiver, two estimation stages are used, Stage 1 generates ML gain estimates and stage I1 utilises them to generate ML estimates of data symbols in the next interval. A likelihood ratio test is performed to provide clean data estimates which Stage I, in turn, uses to update its gain estimates for the next interval. The procedure is repeated until the entire block is decoded. Gain estimates are also used to correct reference phase errors. The complete

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scheme is relatively simpler to implement than other reported schemes with comparable performance [ 1-73. In Reference 3, known symbols are time multiplexed with data symbols, however, in their case these are passed through a Kalman smoother and an interpolator to provide gain estimates. The signal samples are then divided by the estimates to compensate multiplicative distortion. This is different from the two-stage MLE scheme. The fundamental difference with the other schemes is that TSMLE does not use any filtering at all for gain estimation. Furthermore, TSMLE applies likelihood ratio test on an ML estimate of the data while the other schemes apply MAP detection or likelihood ratio tests to gain-weighted versions of the received samples. In this work nonrecursive and recursive two-stage MLE procedures are developed. The resulting error probability is analysed and a closed form expression is presented. It is shown that, for 2-PSK, the error probability performance is the same as that required by Kam [4, eqn. 161, under identical conditions.

1.2 Selective fading In selective fading, transmitted data is further affected by severe channel dispersive effects. These impairments cause a severe degradation in the BER performance and must be mitigated. The traditional approach has been the use of decision feedback equalisation (DFE) and maximum likelihood sequence estimation (MLSE) techniques. These methods do not include gain estimation in their schemes. A wide range of schemes have been studied for NADC [9-211 in the last few years. The DFE schemes can be classified into the following categories:

(a) DFE with explicit channel estimation [9, lo]. This is the conventional approach where the DFE parameters are computed from the channel estimates.

(b) DFE with channel interpolation [16-181. Kalman filtering is used to obtain channel impulse response estimates with a fewer number of samples and interpolation is used to obtain estimates of intermediate samples. This reduces the excessive demand on channel tracking. This scheme requires removal of phase ambiguity if K/~-DQPSK is used.

(c) DFE with phase-locked-loop (PLL). PLL is first used to perform phase and frequency offset compensation before applying DFE [12].

(d) Bidirectional DFE [lS, 18, 191 uses pre- and post- qualisation to reduce error propagation.

The MLSE based schemes studied for NADC can be classified into the following categories :

(a) Conventional MLSE with a least mean square (LMS) algorithm. The LMS algorithm is used to estimate the channel impulse response [18,21] followed by MLSE of the data.

(b) MLSE with bidirectional equalisation [18]. This is similar to DFE with bidirectional equalisation except that MLSE is used in place of DFE.

(c) MLSE with direct update Viterbi equaliser [18] uses 4-state Viterbi algorithm (which is lower in complexity than the conventional) to estimate the received signal. The reference states, used for the metric computation, are updated without having to track the channel impulse response variations.

(d) MLSE with channel impulse response interpolation [18]. This is similar to DFE with channel interpolation except that it uses phase-alignment techniques to remove phase ambiguity when used for n/4-DQPSK, and MLSE in place of DFE.

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(e) MLSE with phase-locked loop (PLL) [20]. This is the same as DFE with PLL but uses sequence estimation in place of DFE.

These DFE and MLSE based schemes are preceded by a fixed filter which is matched to a transmit filter and none of them perform gain estimation for compensating amplitude and phase variations. In this work we propose an alternate scheme which uses a single adaptive receiver filter to perform the function of the fixed filter as well as to compensate dispersion effects. The output of the adaptive filter is a nonselective fading signal with only amplitude and phase variations. TSMLE is then used to compensate amplitude and phase variations. A likelihood ratio test is then used to decode the data. The new scheme performs neither DFE nor MLSE of the data. Expressions for predicting lower and upper bounds to the resulting BER performances are provided for 4-PSK and rrI4-DQPSK. Simulations and theoretical BER performance results are also presented to validate the theoretical developments.

2 System model

Fig. 1 shows the general system model in an equivalent low-pass form. The baseband modulated data ( I k , Q,) are

Nyquist

n(t) Fig. 1 Baseband model for communication over selective fading channel

passed through a pulse shaping filter denoted by At). The result is a bandlimited waveform $t). As s(t) propagates through the channel, with the impulse response h(t, T), it undergoes random amplitude and phase variations as well as dispersion in time and frequency. The waveform is further corrupted by additive Gaussian noise n(t). The corrupted signal is first ideal low-pass filtered (ILPF) and sampled at the Nyquist rate or higher. The resulting dis- crete samples are-pryssed by a digital processor to extract estimates ( I k , Qk) of the transmitted symbols with minimum probability of error. This paper is concerned with the development and performance of the digital processor which provides estimates of the transmitted data with minimum probability of error.

The complex envelope of the transmitted signal is

where At) is a square-root raised cosine or a raised- cosine pulse shape. I, and Q, are the encoded data symbols given by I, = a cos d,, Q, = a sin d k . a = J(2E,) and 4, = Adk for 4-PSK, while a = J(E, ) and dk = d k - , + Adk for 4 4 DQPSK where Ad, is a mapping of Gray-encoded binary bits X,, into differential phase.

For convenience we isolate the desired components ( I k , ,Q,) of the kth signalling interval and express the received waveforms as

U,(?) = I k 4.(t. t - k T ) - Q k qs(t, t - kT) + wc(t)

us(?) = I k qs(t, t - kT) -k Qk qc(t. t - k T ) -k W,(t) (2) where q,(t, 7 ) and qJt, 7 ) are the convolutions of the real and imaginary parts of a complex channel impulse


response with the transmit pulse shape, respectively. w,(t) and w,(t) (see Section 8.1 for details) represent inter- symbol interference (ISI), cochannel interference (CCI), cross-talk interference (CTI), and noise. for nonselective fading there is no ISI. Hence, the receiver performs a fixed matched filtering followed by a sampler operating at the symbol rate 1/T samples/second. For selective fading the received signals are passed through an ideal low-pass filter (ILPF) followed by a sample operating at the Nyquist rate or higher. For the proposed scheme, the sampled signals are then adaptively filtered. Suppose u,(t) and uJt) are sampled at the Nyquist rate (or higher), with appropriate adjustments so that the factor rn = 2f0 T is an integer, where p(t) is bandlimited to fo. Then two vectors of sampled received sequences of length L, corresponding to each symbol pair (I,, Qk), are obtained, that is,

8, = a,(k)q, aJkk, + w,

U, = a,(kk, + a,(kk, + w, (3) where a,(k) and a,(k) consist of data symbols interleaved with (rn - 1) zeros. We shall use the valid assumption that, during transmission, the channel is essentially unchanged for the duration of K symbols, that is, q,(k, 4 = qAk, n + k ) = qAn) and q,&, 4 = qXk, n + k ) = qs(4 for k = 0, 1, . .. , K - 1. In general, K can be selected as described [SI.

3 Receiver for nonselective fading

For clarity of exposition, let { u l k r u z k , g1,,, g2.,} denote the sampled filter outputs. Then the quadrature components are given by

u l k = I k g 1 . k - Q k S 2 . k + w l k

u2k = I k g 2 . k f Q k 9 l . k + W 2 k (4) Here, wlk and w2, are independent, identically distributed Gaussian processes with zero mean and diagonal covariance matrix. gl. , and g2. , are link gains up to the filter outputs. Generally, only the data symbols I, and Q, are of interest. However, to detect these symbols with a smaller probability of error, it is desirable to know the link gains as well. In general, the gains can take on posi- tive and negative values randomly. There are now four unknowns and we propose two stages for their estimation.

3.1 Nonrecursive ML estimation In the kth interval, 0 < k < K - 1, Stage I generates ML channel gain estimates (G1,,, G2.,), while Stage I1 generates ML data estimates {f,, Q,}, given the gain estimates of Stage 1. For Stage 1, we as_um_e that signal samples { u l i , u Z i } and data estimates { I i , Qi } , i = 0, 1, ..., k, are available and want to utilise some or all to generate channel gain estimates { G I , , , &,}. For Stage 11, we assume that signal samples { u l i , uZi} and gain estimates {i,. i - G2, i - l } r for i = 0, 1, . . . , k are available-and_ want to utilise some or all to obtain data estimates {I,, Q,} for the kth interval. To solve the two-stage estimation, we note that the problem can be represented by the following linear Gaussian model [22]:

(5 ) V, = H k X k + W, where V, is an m-by-1 vector of observable random variables of the sampled signals, H , an rn-by-p matrix of rank p of observable nonrandom variables and X , is a p-by-1

IEE Proc.-Commun., Vol. 141, No. I , February I994

vector of nonobservable constants. For Stage I, H , is composed of data estimates up to the kth interval and for Stage 11, gain estimates up to the (k - 1)th interval. X , comprises current channel gains for Stage I, and data symbols for Stage 11. W, is an m-by-1 vector of nonobservable random variables. The dimension rn depends upon the number of intervals we utilise while p depends upon modulation levels; p = 2 for 4-PSK. As an example, suppose we want to use the current and previous observation interval (i.e. k and k - 1) to estimate the channels gains for 4-PSK, then rn = 4, p = 2.

For Stage I, H, comprises known data and X k the unknown gains. For Stage 11, H, and X , comprise, respectively, known gain estimates and unknown data. All quantities for which k - 1 < 0 are identically zero. Also, data symbols are assumed known for k = 0. Ini- tially, this is achieved by transmitting known symbols at the start of each block. As a start-up procedure, one can transmit more than one known symbol so that all ele- ments of H , become nonzero before actual data is transmitted.

3.1.1 MLE Stage I : channel gain estimation: Consider the kth signalling interval and assume we have available, signal samples and data symbol estimates up to the kth (i.e. the components of the vectors V, and matrix H, are known). We want to generate an estimate of the current gain vector X , = g, = bl.k, g2, J'. The ML gain estimate in the kth observation interval can be obtained by using standard statistical methods. Simplification of the expression for the ML estimate shows that this operation strips the modulation from the signal samples. The expression for the gain estimate is given in Section 8.2.

3.1.2 MLE Stage I I : data symbol estimation: Consider the kth signalling interval and assume we have available, signal samples up to the kth and channel gain estimates up to the (k - 1)th. At this stage we want to generate an estimate of the current data vector xk = d, = [ I , , QJr. The data estimates in the kth interval are determined by standard statistical methods. The expression for the estimates are given in Section 8.2.

Except for a normalisation factor the result is similar to the detection statistics used [23]. Simplification of the expression for the ML estimate shows that the operation indeed compensates multiplicative and phase distortions. This is different from References 3 and 6 where compensation is assumed to be achieved by dividing the received signal samples by the gain estimates.

3.2 Recursive estimation

3 2 . 1 ML estimator stage I : estimation of channel gains: The procedure discussed so far is nonrecursive and requires the inversion of pby-p matrices at each stage. It is suitable for the situation where there is no phase variation due to variations in the speed of the mobile unit. If the mobile unit does not change speeds while data is being estimated, the matrix inversions involved in gain estimation need to be performed only during the reference period. However, when the mobile unit changes speed, the channel phase (and hence the gain) can change rapidly while data is being estimated. For the data to be estimated correctly the gain must therefore be updated throughout the data block. To reduce the amount of computation in that case, recursive estimation of the gain is proposed. Recursive estimation replaces matrix inversion with scalar division. Using

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standard methods [24] it can be shown that a recursive relation for updating the gain estimate in the data interval is given by

o k + 1 = 9 ^ k + P k + l u : + l e k + l p k = ( D : D k ) - ' (6) where e , + , = u k + l - u , + , i , is an a priori error between the vector U,+, and its estimate U,+&, when the gain estimate of the previous interval is used. The initial gain estimate is obtained from the reference interval using the nonrecursive method.

3.2.2 ML Stage II: recursive estimation of data: We need not develop a recursive relation for the data. We simply update the gain matrix with the current gain estimate and then compute the data estimate according to Section 8.2.

3.3 Detection and probability of error performance The data vector estimate from Stage I1 can be simplified as a, = A,d, + B, W, where B, W, is a vector of residuals. If the gain estimate is accurate (i.e. g, % &,) A, is the identity matrix (in general, A, is close to the identity matrix). This condition is met in the start-up interval during which known symbols are sent. In general, the condition is practically met as far as the majority of decisions are correct and gain estimates continue to be improved. To further improve the gain estimates, a phase estimate computed as 4 = tan-'(9^*.(,-,)/~,,(,-,)), can be used to correct the reference phase during demodulation. This phase correction, in turn, reduces cross-rail interference. As observed [4], there is also the probability of this estimate suddenly diverging from the true value. In our application, this can occur when the channel is in deep fade and the detector makes a sequence of errors resulting in a degradation of the estimates. This can be alleviated by reinitialising the data and gain matrices periodically. This fits quite well into the North American cellular standard [25] which inserts training symbols in each TDMA frame.

To decide upon the data symbol that was transmitted a likelihood ratio test is performed on the data estimate. Using standard methods, the average bit error probability is given by (see Section 8.3)

2E,B,_ J.: + U:- ,/U: 1'2

P(&IH1) = - - - ] (7) 2 ' 2 '[ 1 + 2EbB,-,/.: +.:-,/a: where B, is the variance of the quadrature gains for the kth interval and defined as B, = cl E[Bg. l]E[a~,l]. E[Bg, is the average power associated with the Ith path of the channel and E [ E ~ , ~ ] is a result of nonideal sampling (its value is unity for ideal sampling). U : - , and U:

are variances of the interference plus noise components in the (k - 1)th and kth intervals, respectively. In the limit of noise- and distortion-free gain estimates in the (k - 1)th interval and no interference in the kth data interval (that is, U:- I = 0 and U: = N0/2), the expression in eqn. 7 is identical with the result reported by Kam [4, eqn. 161, for 2-PSK using similar assumptions.

The overall scheme is shown in Fig. 2. The received waveform is demodulated using a reference carrier from a voltage controlled oscillator (VCO) with an initial phase estimate which may be set equal to zero. The demodulated signal is then sampled at the data symbol rate. During the reference or start-up period only known data symbols are present in the received signal. A replica of these known data symbols are also present at the receiver in the block denoted by d o . During the reference period,

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the contents of do and the received samples are used to generate ML estimates of the quadrature gains. These gains serve two purposes: for updating the phase esti-

received signal

likelihood ratio test

7 j k Fig. 2 nels

Adaptive two-stage M L E receiver Jor nonselective fading chan-

mate for the VCO, and for generating ML estimate of data symbols at the end of the reference period. A likelihood ratio test is then performed to decide on the actual symbol transmitted. If recursive gain estimation is desired after the reference period, this can be accomplished by using the decoded symbol in a decision feedback mode.

4 Receiver for selective fading

Receiver optimisation for selective fading can be divided into: (a) gain estimation at the start of each data frame, (b) data symbol estimation and (c) filter optimisation in the reference period. We shall denote the receive filter by the vector g, of length L, -= L o , where L, is the input sequence length. U, and U, shall denote the inphase and quadrature components of the filter output. The overall scheme for selective fading is the same as that of Fig. 2 except for an adaptive filter which is inserted after the demodulation.

4.1 MLE Stage I Starting with an initial filter setting, ML estimate of the gains at the sampling time n = n,, are then computed. Expressions for these estimates are given in Section 8.4.

4.2 MLE Stage II In the data intervals, the channel gains are assumed known and the data estimates are computed. Expressions for these estimates are given in Section 8.4.

4.3 Optimisation of receiver filter The ML estimates of Stages I and I1 are obtained assuming the receive filter is optimum. We shall now develop procedures for its optimisation. The filter optimisation must be accomplished during the reference period when the data are known.

4.3.1 Least squares solution: The exponentially weighted, prewindowed, least squares solution [26] is the weight vector g,(n) that satisfies the following system of equations: A,(n)gc = b, where A,(n) is an n-by-L, input data matrix whose ith row is a snapshot of the filter input

I E E Proc.-Commun., Vol. 141, No. I , February 1994

vector and b, a vector of some desired data vector, which is composed of the known symbols in the reference period.

A suitable method with superior numerical properties for solving the least square problem is the QR- decomposition using Givens rotations [26]. The desirable features of this method are:

(a) It can be implemented using pipelined and modular structures known as systolic arrays [27]. Such arrays allow parallel, real-time processing and suitable for VLSI.

(b) It produces optimal results independent of the condition of the data matrix [26].

Given the matrix A,(n), QR-decomposition solves the least squares problem in real-time by applying a sequence of elementary Givens rotation [26] to reduce the problem into the form

R,(nk,(n) =PI(") (8) where R,(n) = Q(n)A1'2(n)A,(n) is an upper triangular matrix, p,(n) = Q(n)A1l2(n)bc(n) and Q(n) is an n-by-n unitary matrix. Eqn. 8 can be solved by back-substitution or the weights extracted directly during the matrix reduction process [28]. QR-decomposition, its superior numerical properties with finite precision arithmetic and its systolic implementation have been treated exhaust- ively in the literature. We shall therefore not delve into them here.

4.4 Performance prediction Because of symmetry, we shall restrict attention to inphase branch. The output of the inphase branch of receiver can be expressed as

the ' the

f, = I, + CCI (n,) + CTI (nk) + ISI (nk) + AI^ (9) where the right-hand side components are, respectively, the desired signal, the CCI, the CTI, the IS1 and the noise components at the sampling instant n = n k .

4.4.1 4-PSK: Using standard methods, an upper bound on the bit error probability is given by (Appendix 8.5)

where E, = E,[1 - Ej+, I Rj1l2, R j is a correlation coeff- cient of delayed versions of the gains and 5: = u:B. A lower bound to the unconditional bit error probability is (Section 8.5)

PL(&) = - { 1 - [ 2 8 k + 1]1'2}

2 + 2 b k

4.4.2 xI4-DQPSK: The probability of bit error for 4 4 - DQPSK on a nondispersive channel with no interference is presented in Appendix 8.5. When interference is present, approximate upper and lower bounds to the probability of error can be obtained by using the worst case average SNR, 7. = pkEb/5: and the best case average SNR, 7' = bk E, B/u:, respectively, in place of the actual average SNR. The validity of these approx- imations is supported by simulations.

5 Sample timing estimation

In order not to degrade BER, accurate sampling points must be selected in each signalling interval. Techniques

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that address sample timing extraction for TDMA are reported in References 29 and 31.

In TDMA reception, it is necessary to synchronise to a frame/slot. Assuming this has been established, the sample timing location can be expressed as nj = + z j where nsync is the sample location corresponding to the frame/slot sync. Sample timing estimation is the selection of an optimal correction factor ' t j during the preamble time of the TDMA frame. Because of phase variations in the channel, z j is usually selected through a 2D search (over time and phase). However, because two-stage MLE compensates phase variations, only a 1D search is necessary. The selection of z j can therefore be done by solving the minimisation problems:

NP N, min [Adi - A$i(rj)]' min 1 [Zi - fd'tj)I2 (12)

i = 2 i = 2

for n/4-DQPSK and 4-PSK, respectively. N , is the number of preamble symbols. For n/4-DQPSK, A+i is the true differential phase angle of the ith and ( i - 1)th symbol, while A+i(zj) is the estimated differential phase angle prior to decoding. For 4-PSK, Ii is the true symbol while Ii(' t j) is the estimated symbol of the ith interval. The initial value 'tl must be specified.

6 Numerical results and discussions

For simulations and theoretical BER performance stud- ies, a square root raised cosine filter with a roll-off factor of 0.35 and truncated to ten symbols is used as the transmit filter. The receive filter is initially matched to the transmit filter and then adaptively adjusted during transmission. A two-path channel model is used per IS-54 [25, 321 to evaluate the tolerance of delay spread. The mean-square value of the channel impulse response is given by h2(z) = C6(z) + D ~ ( T - 'tI) where z1 is the arrival time of the delayed path relative to the main path. C and D are the average main path and delayed powers, respectively (D = 0 implies nonselective fading). C/D denotes average main-path-to-delayed path power ratio. The data rate is 48.6 kbps with a slot time of 6.67 ms and carrier frequency of 850 MHz.

Furthermore, the received signal is normalised with respect to the mean square value S , of the interfering signal. In that case. So for the desired signal is replaced by (see Section 8.6)

L S - "(Q - l)y y = 3.5 (13)

L" 0 -

where Q = D/R is a cochannel interference reduction factor that specifies the worst case S / I , see Appendix 8.6 for details. This allows us to obtain results that are independent of cell radius R. Also, we assume that the desired and interfering signals undergo equal loss by setting Lo/L,, equal to 1.0. To isolate the influence of each individual distortion factor, we assume the absence of the others.

6.1 Nonselective fading performance Fig. 3 illustrates theoretical and simulation BER versus &/No under perfect gain estimation conditions. The close agreement of theory and simulations supports the assumptions made in deriving the theoretical BER performance prediction. The simulation results were obtained by averaging over lo6 frames of 200 independent runs and mobile speeds of up to 120 km/h. The insen- sitivity to Doppler effects is due mainly to the schemes ability to strip phase distortions. The performance of the

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two-stage MLE is superior to other gain estimations schemes, by about 3dB to 7dB. The closest in performance to the two-stage MLE are the scheme reported

1 0 - 7 7 0 10 20 30 40 50 60 70

Eb/No,dB Fig. 3 & = 0 2 5 LI theory (two-stage MLE) b simulation (two-stage MLE) c Aghamohammadi & Meyr [6] d Kam 141 e Haeb & Meyr [I ]

Bit error ratefor perject gain estimation

[4, 61. For perfect timing and no interference, error floors are practically eliminated. The absence of error floors confirms the potential of the two-stage MLE to compensate multiplicative and phase distortions.

For minimal interference, cochannel cell separations must be carefully predetermined. Cochannel cell separations are preselected based on the tolerable worst case S / I . Fig. 4 illustrates BER comparison of the two-stage

100 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

0 10 20 30 40 50 60 70 Et."O.dE

Fig. 4 b, f, j : 2-stage MLE a, e, i: Aghamohammadi & Meyr c, g. k : Kam d , h. I : Haeb & Meyr

Bit error rate comparisonfor SI1 = SO dB, 30 dB, 20 dB

MLE and other gain estimation schemes for SI1 = 50 dB, 30 dB and 20 dB assuming perfect gain estimation and the absence of other inferences. Two-stage MLE still exhibits superior BER performance but the scheme of Aghamohammadi and Meyr exhibits slightly lower error floors. However, the lower error floors are insignificant compared to the two-stage MLE performance advantage before the error floors. The comparison here is also valid for the effect of delay spread and timing error on the BER. Timing error and delay spread cause BER floors

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similar to those caused by cochannel interference. For a BER level of 0.03 and SNR = 20 dB, the scheme can tolerate timing error of up to 20% and 30% delay spread. For SI1 = 20 dB (or Q = 4.95), the effect of cochannel interference is equivalent to 20% timing error and 15% delay spread.

6.2 Selective fading performance

6.2.7 4-PSK: Fig. 5 illustrates theoretical bounds and simulation results of BER versus Eb/No for values of delay spreads of up to the symbol time T and CID = 3 dB. As predicted, the simulation results are well within the theoretical bounds. The upper bound is rea- sonably tight especially for TIT < 1/3.

10-7L , . . . . , . . . , . . , . . . . , . . . 0 10 20 30 40 50 60 70

q N o . d B Bit error rate uersus E J N , with fractioml delay spread as Fig. 5

parameter CID = 3 dB LI simulations r /T = 0 to 1

Fig. 6 illustrates theoretical BER versus TIT when Eb/No = 60 dB and SI1 is infinite. For TIT < 1.2, the increase in BER is less than an order of magnitude and

10-7t.. . . . . . . . . . . . , . I 0 05 1 15 2

TIT Bit error rate versus fractional delay spread with CID as Fig. 6

paramerer E J N , = b0 dB

insignificant for T/T < 0.6 for CID values of up to 3 dB. These results show dramatic improvement over the fixed filtering. The system with adaptive filtering can tolerate timing errors much better than the fixed filtering. The improvement is dramatic, in excess of two orders of magnitude.

There is no noticeable improvement from the use of adaptive filters with respect to cochannel interference. It

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http://Et."O.dE

is therefore important that the interference reduction factor (i.e. the ratio of cochannel reuse distance to cell radius) be selected sufficiently large.

6.2.2 +l-DQfSK: Fig. 7 illustrates theoretical bounds and simulation results of BER versus E J N , for values of

lower bound>\ 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I

0 10 20 30 40 50 60 70 E,/No dB

Bit error rate versus EJN, with delay spread as parameter Fig. 7 C/O = 3 dB o simulations r /T = 0 io 2

delays of up to twice the symbol time T and C/D = 3 dB. Unlike 4-PSK, the upper bounds are not so tight. However, we note that the simulation results are well within the bounds. For up to about &/No = 30 dB, these bounds can provide fairly good performance predictions.

Fig. 8 illustrates the BER performance with respect to delay spread. Although less dramatic as the 4-PSK case,

1 00

~ 162 e F b I) 10-4

I

=

10-6 0 0 4 08 1 2 16 2

T I T Fig. 8 parameter a fixed filter b adaptive filter

Bit error rate versus fractional deluy spread with C/O as

there is a significant improvement over the fixed filter performance [33] for different CID and SNR. The effect of timing error is similar to that of 4-PSK. Again, the interference reduction factor will have to be relied upon to reduce the effect of cochannel interference.

Fig. 9 provides a performance comparison of some of the schemes previously reported in the literature. The schemes not included in the comparison exhibit higher BER levels. Although all these schemes meet IS-54 requirement for Eb/No = 17 dB (that is BER = 0.03), their BER floors are higher than that of two-stage MLE with adaptive filtering. Whereas the BER floors of other schemes start at around 20 dB, that of the TSMLE starts

1EE Proc.-Commun., Vol. 141, No. I , February 1994

at around 40dB and at least an order of magnitude lower.

TSMLE with adaptive filtering is comparable in complexity with the least complex conventional DFE or

loo

.. .; ._.. '._ 0 '.. .

l c r 4 ~ ~ ~ ~ I I I I I . I I 3 ~~~~ :. 0 10 20 30 40 50

Eb'NO. d B Fig. 9 a TSMLE with adaptive filter b bidirectional DFE c MLSE with modified LMS d MLSE with phase-locked loop e conventional MLSE

Bit error rate comparison

MLSE that utilise fast adaptive algorithms. The only problem with the TSMLE scheme is that adaptation must be established within one symbol period. This requires sampling the incoming waveform at much higher than the Nyquist rate. However, for the North American Cellular, the data rate is so low that sampling at such a high rate and adaptation over a symbol period may not be an issue. For high data rate applications, the need for fast A/D and fast signal processors will be essential. The QR-decomposition and its parallel processing systolic implementation will be quite useful for high data rate applications. Since technology is rapidly moving towards extremely fast A/D and signal processors, in general, there should not be a concern in the use of TSMLE with adaptive filtering.

7 References

1 HAEB, R., and MEYR, H.: 'A systematic approach to carrier recovery and detection on digitally phase modulated signals on fading channels., IEEE Trans., 1989, COM-37, pp.. 74S754

2 SAMPEI, S., and SUNAGA, T.: 'Rayleigh fadmg compensation method for 16QAM in digital land mobile radio channels'. Pro- ceedings of the 39th IEEE Vehicular Technical Conference, VTC '89, 1989

3 AGHAMOHAMMADI, A., MEYR, H., and ASHEID, G.: 'A new method for phase synchronization and automatic gain control and linearly modulated signals on frequency-flat fading', IEEE Trans., 1991, COM-39, pp. 25-29

4 KAM, P.Y.: 'Reception of PSK signals over fading channels via quadrature amplitude estimation', 1EEE Trans., 1983, COM-31, pp. 1024-1027

5 KAM, P.Y.: 'Adaptive receiver with memory for slowly fading channels', IEEE Trans., 1984, COM-32, pp. 654-659

6 AGHAMOHAMMADI, A., and MEYR, H.: 'On the error probability of linearly modulated signals on Rayleigh frequency flat fading channels', IEEE Trans.. 1990, COM-38, pp. 1966-1970

7 VARSHNEY, P.K., and HADDAD, A.H.: 'A receiver with memory for fading channels', IEEE Trans., 1978, COM-26, pp. 278-283

8 SESAY, A.: 'Two-stage maximum likelihood estimation for diversity combining in digital mobile radio', IEEE Trans., 1992, COM-40, pp. 676-679

9 PROAKIS, J.G.: 'Adaptive equalization for TDMA digital radio', IEEE Trans., 1991, VT-40, (2), pp. 333-341

4s

10 CHENNAKESHU, S., NARASIMHAN, A., and ANDERSON, J.B.: ‘Decision feedback equalization for digital cellular radio’. Pro- ceedings of the IEEE International Conference on Communications, 1990, pp. 1492-1496

11 NAKAI, T., ONO, S., SHIMAZAKI, Y., and KONDOH, N.: ‘Adaptive equalizer for digital cellular radio’. Proceedings of the IEEE Vehicular Technology Conference, 1991, pp. 13-16

12 SHIMAZAKI, Y., NAKAI, T., ONO, S., and KONDOH, N.: ‘A decision feedback equaliser with frequency offset compensating circuit for digital cellular radio’. IEEE Vehicular Technology Con- ference, 1992, pp. 596-599

13 DARIA, G., PIER-MARINI, R., and ZINGARELI, V.: ‘Fast adaptive equalizers for narrow-band TDMA mobile radio’, IEEE Trans., 1991, VT-40, (2), pp. 392-404

14 NAKAJIMA, M., and SAMPEI, S.: ‘Performance of a decision feedback equalizer under frequency selective fading in land mobile communications’, IEICE Trans., 1989, J7ZB-11, (lo), pp. 515-523

15 LIU, Y.-J.: ‘Bidirectional equalization technique for TDMA communication systems over land mobile radio channels’. Proceedings IEEE,Globecom, 1991,pp.41.1.1-41.1.5

16 LO, N.W.K., FALCONER, D.D., and SHEIKH, A.U.H.: ‘Channel interpolation for digital moble radio communications’. Proceedings of the IEEE International Conference on Communications, 1991, pp. 25.3.1-25.3.5

17 SAMPEI, S.: ‘Computation reduction of decision feedback equalizer using interpolation for land mobile communications’. Proceedings IEEE,Globecom, 1991, pp. 16.1.1-16.1.5

18 KOILPILLAI, R.D., CHENNAKESHU, S., and TOY, R.L.: ‘Low complexity equalizers for US digital cellular system’. Proceedings IEEE, Vehicular Technology Conference, 1992, pp. 744-747

19 HIGASHI, A., and SUZUKI, H.: ‘Dual-mode equalization for digital mobile radio’, IEICE Trans., 1991, J74-Ell, (3), pp. 91-100

20 SHIINO, H., YAMAGUCHI, N., and SHOJI, Y.: ‘Performance of an adaptive maximum-likelihood receiver for fast fading multipath channel’. Proceedings of the IEEE Vehicular Technology Con- ference, 1992, pp. 380-383

21 LARSSON, G., GUDMUNDSON, B., and RAITH, K.: ‘Receiver performance for the North American digital cellular system’. Pro- ceedings of the IEEE Vehicular Technology Conference, 1991, pp. 1-6

22 GRAYBILL, F.A.: ‘Theory and application of the linear model’ (Duxbury Press, Massachusetts, 1976)

23 GLANCE, B., and GREENSTEIN, L.J.: ‘Frequency-selective fading effects in digital mobile radio with diversity combining’, IEEE Trans., 1983, COM-31, pp. 1085-1094

24 LEWIS, F.A.: ‘Optimal estimation with an introduction to stochas- tic control theory’ (John Wiley and Sons, New York, 1986)

25 IS-54: ‘Dual-mode mobile station-base station compatibility standard’. EIAFIA PN2398,1991

26 HAYKIN, S.: ‘Adaptive filter theory’ (Prentice-Hall, Englewood Cliffs, New Jersey, 1986)

27 KUNG, H.T.: ’Why systolic architecture?, Computer, 1982, 15, pp. 37-46

28 SESAY, A., and PA’ITON, M.: ‘QR-decomposition decision feedback equalisation and finite-precision results’, IEE Proc. F, 1993, 140, (2), pp. 89-97

29 CHENNAKESHU, S., and SAULNIER, G.J.: ‘Differential detection of +shifted-DQF’SK for digital cellular radio’. Proceedings of the IEEE Vehicular Technology Conference, 1991, pp. 186-191

30 ONO, S., KONDOH, N., and SHIMAZAKI, Y.: ‘Digital cellular system with linear modulation’. Proceedings of the IEEE Vehicular Technology Conference, San Francisco, 1989, pp. 4-49

31 SOLLENBERGER, N.R., and CHUANG, J.-C.-I.: ‘Low-overhead symbol timing and camer recovery for TDMA portable radio systems’, IEEE Trans., 1990, C-38, (10). pp. 1886-1892

32 IS-55: ‘Recommended minimum performance standards for 800 MHz dual-mode mobile station’. ElA/TlA PN2216, 1990

33 LIU, C.L., and FEHER, K.: ‘Bit error rate performance of n/4- DQPSK in a frequency-selective fast Rayleigh fading channel’, IEEE Trans., 1991, VT-40, (3), pp. 558-568

34 PAHLAVAN, K., and MATHEWS, J.W.: ‘Performance of adaptive matched filter receivers over fading multipath channels’, IEEE Trans., 1990, C-38, (12), pp. 2106-2113

8 Appendixes

8.1 Description of signals and interference This Appendix provides detailed description of signals and interference. In general, the propagation medium is a band limited, fading dispersive channel with the complex envelope of the composite impulse response being time-

46

invariant and defined by h(t, T ) = h,(t, T ) + jh,(t, T ) . The function h(t, T ) represents the complex path gain associated with delay 5 at time t. The components of the received signal in a noise-free environment, can be expressed [SI as

&(t) = I k qC(t, t - kT) - 1 Q k q,(t, t - kT)

‘&(t) = 1 Ikq,(t, t - kT) + 1 Qkq,(t, t - k T ) (14)

where q,(t, T ) and qdt, T ) are the inphase and quadrature components of the signalling pulse at the channel output. That is, q,(t, T ) = h,(t, r)*p(t), 4,(t, T ) = hdt, r)*p(t) where * denotes convolution. The functions q,(t, 5) and q.(t, T ) represent the inphase and quadrature gains, respectively, associated with delay T and time t, up to the receiver input.

In cellular systems with frequency reuse, another source of interference is a cochannel cell. In a system with M significant interfering cochannels, their contributions can be expressed as

1

M T

+ 1 QP’qp’(t, t - k T ) ] (15)

where ( I t ) , QP’) and [qp’(t, T ) , q?)(t, r)] are data symbols and signalling pulses at the output of the channel, respectively, for the nth interferer.

In a noise environment, white, complex Gaussian noise, n(t) = n,(t) + jns(t), of two-sided spectral height N, /2 , is added to form the composite received waveform.

wAt) = 1 CIj4,(t, t - j T ) - QjqXt, t - jT )1 j # k

+ iAt) + n,(t)

w,(t) = E CIiqdt, t - j T ) + Qjq,( t , t - j T ) l j # k

+ idt) + n,(t) (16)

82 ML gain and data estimates for nonselective fading

Using data estimates of the kth interval, the gain estimate is given by

Using the (k - 1)th gain estimates, the data estimate is given by

1 G l , k - l u l k + 9 ^ 2 , k - l U 2 k dk = - @ 2 . k - l U 1 k - 9 ^ 1 . k - l V 2 k

z k - l = s “ : , k - l + s ” : , k - l (I8)

8.3 Condition error probability for nonselective fading

Owing to symmetry, we consider the inphase branch. The data bit estimate can be expressed as I , = p k I k + rlk where p k I k is that part of the first row of Akdk that does not contain Qk while r lk consists of the first row of B, Wk


plus that part of the first row of A,d, that contains Q, as a result of cross-rail interference.

According to previous discussions, the data bit estimate, when conditioned on previous gain estimates through the matrix G,- , , current gain g, and current data bit I,, is Gaussian with mean and covariance given, respectively, by

ECi&I G&-l? 9x3 ' k l = pkdk

where a, is a function of G,_ and g,, and a, U; is the first element of the diagonal covariance matrix of d, .

For 4-PSK, the detection problem can be represented by the hypotheses testing model:

The likelihood ratio is L, = fJ(ak U:) and the test which minimises the error probability is: choose HI if L, i 0, otherwise choose H o . Since a, and U: are independent of hypotheses, the test can be reduced to checking the sign of 1,. In that case, it is not necessary to generate the estimate of U:. The average probability of bit error is given by

.f(G&- l r gk) d C k - 1 dgk (21)

where f(G,-,, g k ) is the joint density function of Gk-l and g,, and E is the error event. The conditional error probability in eqn. 29 can be shown to be given by the following expression :

P(E I HI, Gk- 9,) = - 2 1 erfc ([ 51 lI2) (22)

Because of the dependence of the conditional error probability on G k - l and g, through a,, a closed form expression for the average error probability is difficult to obtain directly. However, we can obtain a closed form if we use some justifiable assumptions that are utilised in the literature [l , 4-61, We shall then evaluate the double integral in eqn. 29 through simulations and compare the results with that of the closed form. Consider the case where the gain estimate (albeit noisy) is fairly accurate, that is, @ l , t - l rz gl, ,. Then p, rz 1, a, = l /zk-l and the conditional error probability is dependent only on previous gain estimates Gk-l through zkcl.

8.4 Gain and data estimates for selective fading The ML estimate for the inphase gain is

Li(n,) = [ I r uAnJ + Q. u,(n,)l/A (23)

where A = (If + Q:), r = 0, R - 1 and R is the number of reference symbols. I, and Q, are known reference symbols. The ML estimate of the quadrature gain at time n = n, is given by

SCnJ = [ I , u,(n,) - Q, uAn,)llA (24)

The estimates ci(n,) and Rn,) are Gaussian with zero mean and equal variance given by Var [&)I = Var [6(n,)] , = p, + u;/A, where uf = Var (gTw,). The ML data estim-


8.5 Conditional error probabilities for selective fading Using standard methods, the bit error probability conditioned on q n k ) and 6(n,.j is given by

J(E3 + C I j R , P[E I ci(nk), 6(n,)l = - erfc 2 [ J(G; ] (26)

where the summation term is the nonGaussian com- ponent of the IS1 contribution and R, is a cross- correlation coefficient between 44) and 4nk - k N ) [23]. U: is the sum of the mean-square values of the Gaussian components. Because of the difficulty in obtaining a closed form expression for the unconditional error probability, we examine two bounds: an upper bound and a lower bound.

8.5.1 Upper bound: It can be shown that an upper bound to the right-hand side of eqn. 26 is given by

The corresponding upper bound to the unconditional error probability is given by

P . ( E ) = ' ~ ~ " 2 2 n y k 0 rerfc[/($B)]

where y k = fl, + u f / A .

8.5.2 Lower bound: This corresponds to an interference-free environment. In that case the lower bound on the conditional error probability is given by

where uli,, = uf/B. Based upon analysis [29, 301, it can be shown that the

probability of bit error for nI4-DQPSK on a nondispersive Rayleigh channel with no interference can be expressed as

P(E) = I:n [m1 I 4 h 0 2 I4J d y l dy, (30)

where a = Y2 + A 4 + 4 4 , b = Y 2 + AI#J + 544 , Y i , i = 1, 2, are the received and true phase angles, respectively, of two successive transmitted symbols. I#Ji = na/8, n = 1, . . . , 8, under the constraint that A 4 = I#J1 - 42 = 4 4 . The conditional PDF mi I I # J i ) , i = 1,2, is given by

1 Pwil4i) =

where Oi = 4i - Y i , 44) = 4 4 sinz(Oi) + 271, dei) = tan- [ - 2 cos (Oi) /a(Oi) ] and 7 denotes the average SNR.

41

8.6 Worst case SI1 This Appendix presents an expression for the worst case SI1 and defines the interference reduction factor Q. The S/I at the receiving cell site with M interferers can be expressed as

where S , is the mean-square value of the nth signal aver- aged over channel variations. A seven-cell cluster system is illustrated by Fig. 10. The same set of frequency channels assigned to the centre cell A are also assigned to the cells denoted by B at multiple distances of D units. Due to fading there are only six significant interferers to A. With the centre cell site receiving and the desired mobile transmitting from the cell boundary, the separation is R. With the interfering mobiles transmitting from the cell boundaries closest to the receiving cell site, their distance is approximately D-R. Since received power is pro- portional to shadow loss L, and the inverse yth power of distance, the SI2 can be expressed as

n = l

Dividing numerator and denominator by R - Y and delin- ing interference reduction factor as Q = D/R, we obtain

. . _ _ . . . .. Fig. 10 A desired (antre all) B ialerferingah 0 desidmobilclocption x sienificant interfering mobile location

Signi$cant interfering cells in seven-cluster cellular system

48 IEE Proc.-Commun.. Vol. 141, No. I , February 1994

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Adaptive two-stage maximum likelihood estimation for cellular radio

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