EXTENDED MLSE DIVERSITY RECEIVER FOR THE TIME- AND FREQUENCY-SELECTIVE CHANNEL Brian Hart ([email protected]), D.P.Taylor ([email protected]) Department of Electrical and Electronic Engineering, University of Canterbury, New Zealand Abstract This paper develops an MLSE diversity receiver for the time- and frequency-selective channel corrupted by additive Gaussian noise, when linear constellations are employed. The paper extends Ungerboeck’s derivation of the Extended MLSE receiver for the purely frequency-selective channel to the more general channel with diversity. The new receiver structure and metric assume a known channel. The major contributions of this paper are as follows: (i) the derivation of a finite-complexity diversity receiver that is Maximum Likelihood (ML) for all linear channel models and sources of diversity; (ii) a benchmark, in that the new receiver’s performance is a lower bound on the performance of practical systems; (iii) insight into matched filtering and ML diversity receiver processing for the time- and frequency-selective channel; (iv) bounds on the new receiver’s BER for ideal CSI, in a fast Rayleigh fading channel with multiple independently faded paths. I. INTRODUCTION Shannon’s channel capacity theorem[1] proves that coding schemes exist to transmit information with an arbitrary low probability of error, assuming the information rate of the source does not exceed the channel capacity. However, the coding delay is not constrained and there is no guarantee that coding is actually helpful in a system when the transmission delay is fixed. For instance, even the simplest two-state interleaved trellis code in the frequency-flat, Rayleigh fading channel requires an interleaving delay of at least ∆s for optimal performance, according to [2] ∆ > 153 . fD (1) where f D = vf c c is the one-sided Doppler spread; v is the receiver speed; fc is the carrier frequency; and c is the speed of light. For a receiver operating at 900MHz, travelling at walking pace (4km/h), the Doppler spread is 3.3Hz and the interleaving delay is 0.46s. Furthermore, the code’s performance is significantly impaired by reducing the interleaving. In channels with significant frequency-selectivity, there is implicit delay diversity, the error rate is less dominated by long, deep fades, and so codes can enhance the system performance. The codes still require the interleaving of (1) for optimal performance. However, a robust communication system cannot rely on Doppler or delay spread, so we argue that coding will be insufficient for future systems that offer ATM-like services: i.e. a common digital transmission system providing high bandwidth, a low error rate, and a low delay. Therefore, this paper analyses receiver structures for frequency diversity and receiver space diversity. Maximal Ratio Combining (MRC)[3] is optimal for slowly varying, frequencyselective channels. However, rapidly varying channels introduce additional pulse distortion which must be explicitly removed to avoid an ISI-induced error floor[4]. By following Ungerboeck’s Extended MLSE receiver for the frequency-selective channel[5], this paper derives an MLSE diversity receiver, assuming the channel is known (ideal channel state information, CSI). The new receiver extends the idea of MRC to rapidly varying channels, and is known as the “Extended MLSE diversity receiver for the time- and frequency-selective channel,” or the EMLSE-tf diversity receiver. II. RECEIVER DERIVATION The transmitter maps a binary source to a sequence of M-ary complex phasors, {ai}. The transmission interval is assumed to extend sufficiently far into the past and future that the end points can be assumed to be ±∞ without penalty. In complex baseband, the transmitter sends a(t ) = ∑ α i h(t − iT ) (2) i where h(t) is the transmitter pulse shape; and T is the symbol [ ] = 1. period. The phasors are normalised so that E α i 2 The diversity receiver receives D signals (threads). They may arise from different antennae or different carrier frequencies. The dth thread (d = 1 ... D) is the transmitted signal distorted in time and frequency by the channel, modelled generally by a complex, time-varying, filter, zd(t,x), and complex Gaussian noise, nd(t), is added, resulting in the D threads, y(t) = [y1(t) ... yD(t)]T, where each thread is described by y d (t ) = a(ξ)∗ z d (t , ξ) t =ξ + nd (t ) = ∞ d d ∫ a(t − ξ)z (t , ξ)dξ + n (t ) (3) −∞ For space diversity, the noise comes from different antennae and front-end electronics. For frequency diversity, the noise occupies different bandwidths. Therefore we assume independent noise in distinct threads. The channel, zd(t,x), may be correlated between threads, but we shall see that this has no influence on the receiver structure for ideal CSI. Substituting (2) into (3), we have y d (t ) = ∑ α i c d (t − iT , iT ) + n d (t ) (4) i where cd(t-iT,iT) is the effective pulse shape, combining the effects of the transmitter pulse shape and the time-varying distortion of the dth channel, as expressed by c d (t − iT , iT ) = ∞ d ∫ h(t − ξ − iT )z (t , ξ)dξ −∞ (5) Equation (4) is the familar notation for linear modulations, but now there is an extra parameter, iT, since the effective pulse shape is time varying. The MLSE receiver searches all allowed symbol sequences in the transmission interval and chooses the one with maximum likelihood. Conditioned on the symbol sequence and the channel process (i.e. ideal CSI), the log-likelihood of y(t) matches the log-likelihood of the complex, zero-mean, Gaussian noise vector, n(t) = [n1(t) ... nD(t)]T, ( ) l y α ,z y α , z = ln (n) = ∑ ln (n d (t )) = ∑ l y α ,z ( y d (t ) α, z d ) (6) D D d =1 d =1 which is just the sum of each thread’s noise log-likelihood, ( ) ∞ ∞ y (t1 ) − ∑ αi c (t1 − iT , iT ) i −∞ −∞ ly α ,z y d (t ) α, z d ~ − ∫ × ( −1 Rnn t1 ∫ (7) − t2 ) y(t2 ) − ∑ α k c(t2 − kT , iT ) dt1dt2 k since the noise is independent between threads. The overbar denotes complex conjugation; the noise autocovariance is defined by Rnn (τ) = ( ) convolution. Following [5], the log-likelihood of each thread can be transformed as ) { } l y α ,z y d α, z d ~ ∑ 2ℜ α i mid − ∑ ∑ α l sld,k α k where d l ,k d i m and s mid = i l (8) k d −1 d ∫ ∫ y (t 2 )Rnn (t1 − t 2 )c (t1 − iT , iT )dt1dt 2 −1 = y d (t1 )∗ Rnn (t1 )∗ c d (−t1 , iT ) (9) t1 =iT ∞ ∞ ∫ d d −1 ∫ c (t1 − lT , lT )Rnn (t1 − t 2 )c (t 2 − kT , kT )dt1dt 2 −∞ −∞ In slow fading, the presence of c d (.) in the mid equation is equivalent to MRC, since it removes the channel’s phase distortion, weights the signal according to the depth of the fade, and then matched filters the signal. As with Ungerboeck’s EMLSE-f receiver[5], there is no need for a noise-whitening filter. The sld,k term cancels or equalises the ISI introduced by the channel’s time- and frequency-selectivity. In the time-invariant channel, the ISI is fixed, so Ungerboeck’s EMLSE-f receiver is able to precompute sl,k. This is not possible in time-varying channels. Substituting (8) into (6) leads to l y α ,z (y α , z) ~ d =1 ∑ 2ℜ{α i mi } −∑ ∑ α l sl ,k α k i l where mi and sl,k are the sum of all the threads’ sld,k terms respectively, (10) k mid and D ∑ sld,k (11) d =1 sl ,k = sk ,l , can exploited by properly grouping the double summation of (10) to obtain the iterative metric[5], ∞ i −1 ∞ ) ∑ 2ℜαi mi − ∑ si ,k α k − αi 2 si,i = ∑ li ( ly α , z y α , z ~ i =−∞ k =−∞ i =−∞ (12) which can be calculated more efficiently. As we shall see, this additive metric leads to a straightforward implementation using the Viterbi algorithm. Having assumed ideal CSI, the statistics of the channel are irrelevant to the metric. The analysis thus applies to all noiselimited, linear channels, including dispersive, L-path[7], Rayleigh, Ricean, log-normal and AWGN. When ideal CSI is not available, the performance of this receiver still leads to a lower bound on a practical receiver’s performance. III. WHITE NOISE For the important case of white noise with a two-sided passband power spectral density of N0/2, we have in complex baseband Rnn (τ) = N 0 δ(τ) . Since the metric of (12) is used for comparisons only, the scale factor of N0 can be neglected. Then δ(τ ) −1 using Rnn ∝ δ(τ) simplifies mi and si,k to ( τ) = N0 mi = ∞ ∞ sl , k = The conjugate symmetry of sl,k, are defined as −∞ −∞ sld,k = D ∑ mid −1 E n d (t )n d (t + τ) ; Rnn (τ) is its inverse 1 2 −1 kernel[6], and satisfies Rnn (τ)∗ Rnn (τ) = δ(τ) , where ∗ denotes ( mi = si ,k = D ∞ ∑ ∫ y d (t1 )c d (t1 − iT , iT )dt1 d =1 −∞ D ∞ (13) ∑ ∫ c (t1 − iT , iT )c (t1 − kT , kT )dt1 d d d =1 −∞ We note that mi can be considered as the output at t = iT (i.e. symbol-rate sampled) of the sum of matched filters for the timeand frequency-selective channel (MF-tf). The signal is filtered by a filter matched to the effective pulse shape in each thread. Modern digital receivers operate in discrete time. The receiver first applies a zonal anti-aliasing filter to the received signal, with transfer function, rect ( fTr ) , where Tr = T/r. The filtered signal is sampled at t = lTr, so the receiver takes r 1 samples per symbol period. Subscripts are used to denote time, so that yl = y(lTr), cl-ir,ir = c(lTr-irTr,irTr), nl = n(lTr). The integrals of (13) are replaced by summations, resulting in min {ir , kr } mi = D ir + Lr −1 ∑ ∑ yld cld−ir ,ir d =1 l = ir si ,k = D ∑ + Lr −1 ∑ cld−ir ,ir cld− kr ,kr (14) d =1 l = max {ir , kr } where the factor of Tr is neglected, since the metrics are used for comparison only. The absolute bandwidth of a time-limited pulse is infinite, so strictly speaking a sampling rate of 1/Tr is inadequate. However, we can normally use another bandwidth definition (such as the -40dB bandwidth), add the maximum Doppler spread, and sample accordingly. In this case, the discrete and continuous time receivers are equivalent. For the special case of effective pulses of less than one symbol in duration (i.e. a short transmitted pulse that is shorter than T even after delay-spread smearing), there can be no ISI, the receiver makes hard, symbol-by-symbol decisions, and the branch metric reduces to a Euclidean distance, li ~ mi − α i si ,i 2 . Exact BERs can be calculated for this case. Here, the EMLSE-tf diversity receiver’s decisions have the same geometric interpretation as the decisions of a receiver for the AWGN channel, except that the signal space is warped in a time-varying manner, according to si,i. V. PERFORMANCE EVALUATION We seek the discrete-time receiver’s BER in a fast Rayleigh fading channel with multiple independently faded paths in white noise. The analysis of the trellis-based receiver follows Forney[8]. The BER is upper bounded by summing the pairwise error probabilities between the transmitted sequence and some erroneous sequence, weighted by the number of bit errors that the error event introduces, and the probability that the transmitted sequence is sent. For the Rayleigh fading channel, the required random variables are Gaussian quadratic forms in IV. RECEIVER STRUCTURE cld−ir ,ir and nld , and their probability is evaluated by computing The MLSE receiver employs exhaustive comparison to find the maximum metric and thus the maximum likelihood sequence. This task is undertaken efficiently by a Viterbi processor, which has finite complexity provided the ISI extends over only a finite number of symbols. This condition is satisfied if the pulse shape has a finite duration, h(t) = 0, t < 0, HT ≤ t, and the delay spread is finite, zd(t,x) = 0, x < 0, t ≤ x, for some pulse length, H, and maximum delay, t. Define L = H + τ T . Then the ISI residues. The problem is a common one in Rayleigh fading channels, so we omit the details here, since the solution is well covered in the literature[5,8,9]. Suffice it that the autocorrelation term, si,k, in (13) is zero for k − i ≥ L , and the branch metric is 1 2 ( ) E clb−ir ,ir cmd− kr = 1 2 ( cld−ir ,ir , is given by ∞ ∞ ∫ ∫ h((l − ir )Tr − ξ1)h ((m − kr )Tr − ξ2 ) × −∞ −∞ (16) ) E z (lTr , ξ1 )z (mTr , ξ2 ) dξ1dξ2 b d A Wide Sense Stationary, Uncorrelated Scattering[10] channel model is assumed, with the conventional J 0 2πλx ( ) given by i −1 2 li ~ 2ℜα i mi − ∑ α i si ,k α k − α i si ,i k =i − L +1 of the effective pulse samples, (15) This is a function of L symbols, (α i − L +1K α i ) , known as the hypothesis vector. A partially-connected trellis is built up, with ML-1 states, labelled by the first L-1 symbols of the vector, autocorrelation function in space, x[11]. Diversity is achieved through a receiver antenna array, comprising D antennae in a row, equispaced at A wavelength intervals. For frequency-flat fading, the expectation in (16) reduces to 1 2 State 64 748 $ i − L +1Kα$ i −1 , α$ i ), where the last symbol labels the current (α branch. ( J0 2 π ( ) E zb (lTr , ξ1 )z d (mTr , ξ2 ) = × J0 2 π During the ith symbol period, the receiver computes one value of mi and L values of si,k according to (11) or (13). The ML branch metrics are calculated according to (15), and then applied to a standard Viterbi processor. The EMLSE-tf space diversity structure is shown in Figure 1. ((d − b) Acosθ + (m − l) f DTr )2 + ((d − b) Asinθ)2 (17) where θ is the angle between the antenna array and the receiver velocity. For time- and frequency-selective channels, we assume a delay profile comprising N taps with with equal mean power. Then 1 2 Figure 1: The EMLSE-tf space diversity receiver. The antenna icon encompasses reception, amplification, and translation to baseband. ) E zb (lTr , ξ1 )z d (mTr , ξ2 ) = δ(ξ1 )δ(ξ1 − ξ2 ) × 1 N N −1 τn ∑ δ ξ1 − N − 1δ(ξ1 − ξ2 ) n =0 ((d − b) Acosθ + (m − l ) f DTr )2 + ((d − b) Asinθ)2 (18) In practice, only the dominant short error events are considered, where up to E symbols may be in error. If the fading is fast and the SNR high, then the dominant error events are short, so the truncated bound can validly neglect long error events. However, with slow fading, most errors occur in the deep fades. If the fading is very slow then the deep fade lasts hundreds of symbols at reasonable SNRs, and so do the error events. Thus the upper bound is tight and easily calculated only for fast fading and high SNR. The union of E = 1 error events due to the nearest neighbours of the errored symbol is asymptotically correct at high SNRs. The average bit energy to noise spectral density is given by ∞ Eb = N0 1 2 E ∫ 2 ∞ ∫ h(t1 − ξ 1 )z(t1 , ξ1 )dξ 1 dt1 −∞ −∞ N 0 log 2 M ∞ = ∫ −∞ space diversity dwarves the implicit channel diversity, and it is reliably available for all channels. h(t1 ) dt1 2 N 0 log 2 M (19) VI. RESULTS For figures 2 - 4, we use a pulse with a 50% excess bandwidth square-root raised cosine spectrum, but truncated in the time domain to H = 5 symbol periods. For figures 2 - 4, r = 2 samples per symbol are taken; for figures 5 and 6, r = 3 samples per symbol are taken. Figure 2 compares the performance of MRC and EMLSE-tf diversity receivers. Higher diversity can significantly reduce the required SNR to achieve a specified BER. The MRC receiver is normally constructed with a channel estimator to track the timeselective channel, but the very limited ability of the LMS and RLS (with an exponential forget factor) algorithms to predict the channel makes them unsuitable for fast fading channels, so an error floor results. Thus the conventional MRC diversity receiver cannot easily tolerate fast fading without an error floor, so instead it is plotted for slow fading. This slow fading curve is a lower bound on the MRC’s performance in fast fading. We see that the EMLSE-tf diversity receiver is able to exploit the implicit Doppler diversity of the channel[12], but the gain that it provides diminishes as the number of antennae increases. Notwithstanding, there is no error floor for the EMLSE-tf diversity receiver. Three curves are plotted for the EMLSE-tf diversity receiver: a lower bound that considers only one, length E = 1, error event; a nearest neighbour curve that for QPSK considers two length E = 1, error events and is asymptotically correct; and a union bound that considers all error events of up to E = 3 symbols in length. At low SNR, the longer events dominate since the bound is very loose, and truncation is not valid. At high SNR, the shortest error event dominates, and the bound is both valid and effective. Figure 3 repeats Figure 2, but now with both time- and frequency-selectivity. Again the MRC receiver is plotted for slow fading. Both receivers can exploit the channel’s implicit delay diversity, but the EMLSE-tf diversity receiver outperforms the MRC receiver by exploiting the implicit Doppler diversity. Figure 4 examines the effect of antenna separation on the BER. Most of the achievable diversity gain is achieved for separations above 0.15λ. Maximum diversity is achieved at 0.383λ, at the first zero of the autocorrelation function. We see that space diversity can be achieved without an undue antennae volume for 900MHz and higher carrier frequencies. Figures 5 and 6 show the BER behaviour over a range of fading rates and delay spreads. Figure 5 covers frequency-flat, fast fading channels. Figure 6 considers time-invariant, frequency-selective channels. Implicit diversity gains of 15dB and 12dB at a BER of 10-4 are observed for the extreme Doppler and delay spreads, but the BER improvement for low Doppler and delay spreads is minimal. By comparison, the gain due to VII. CONCLUSIONS An MLSE diversity receiver for the time- and frequencyselective channel has been derived, assuming ideal CSI and linear signal constellations. A method of bounding the receiver’s BER has been derived, and the bounds are tight for the frequency-selective, fast Rayleigh fading channel under consideration. The BER analysis shows that the implicit Doppler and delay diversity of the channel mildly improve the receiver’s performance, but that the explicit (space) diversity actually provides most of the benefits. The performance enhancement over conventional Maximal Ratio Combining is negligible at normal fading rates, but its error floor is removed. VIII. REFERENCES [1] C.E.Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J., vol 27, July 1948, pp 379-423 [2] R.van Nobelen and D.P.Taylor, “Analysis of the Pairwise Error Probability of Non-Interleaved Codes on the Rayleigh Fading Channel,” in Globecom Communications Theory MiniConference, San Francisco, 1994, pp181-185 [3] D.G.Brennan, “Linear Diversity Combining Techniques,” vol. 47, Proc. IRE, pp1075-1102, June 1959 [4] J.K.Cavers, “On the Validity of the Slow and Moderate Fading Models for Matched Filter Detection of Rayleigh Fading Signals,” Can. J. of Elect. & Comp. 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Commun. Sys., vol. 11, pp360-393, Dec. 1962 [11] R.H.Clarke, “A Statistical Theory of Mobile-Radio Reception,” Bell Syst. Tech. J., vol 47, July-Aug. 1968, pp9571000 [12] W.C.Dam, “An Adaptive Maximum Likelihood Receiver for Rayleigh Fading Channels,” M.Eng. thesis, Hamilton Ont. Canada: McMaster University, 1990 Figure 2: Upper (union) bound, nearest neighbour estimate and lower bound on the BER for EMLSE-tf diversity receiver with fDT = 0.1, and nearest neighbour BER estimate for MRC diversity receiver with fDT = 0 (An MRC receiver in fast fading has a BER floor). Error events up to E = 3 symbols are considered, t/T = 0, A = 0.383, θ = 0, ideal CSI and QPSK. Figure 3: Upper (union) bound, nearest neighbour estimate and lower bound on the BER for EMLSE-tf diversity receiver with fDT = 0.1, and and nearest neighbour BER estimate for MRC diversity receiver with fDT = 0 (An MRC receiver in fast fading has a BER floor). Error events up to E = 3 symbols are considered, t/T = 0.1, N = 2, A = 0.383, θ = 0, ideal CSI and QPSK. Figure 4: Nearest neighbour BER estimates for EMLSE-tf diversity receiver, with fDT = 0,t/T = 0, D = 2, ideal CSI and QPSK. Figure 5: Nearest neighbour BER estimates for EMLSE-tf diversity receiver, with fDT = 0, 0.01, 0.03, 0.1, 0.3; t/T = 0; D = 1, 2, 3; A = 0.383; θ = 0; H = 5; r = 3; ideal CSI and QPSK. Figure 6: Nearest neighbour BER estimates for EMLSE-tf diversity receiver, with t/T = 0, 0.01, 0.03, 0.1, 0.3; fDT = 0; D = 1, 2, 3; A = 0.383; θ = 0; N = 4; r = 3; H = 5; ideal CSI and QPSK.
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