An APP Algorithm for Fading Channels using Forward-Only Prediction Anders Hansson†, ‡, Tor Aulin†, and Keith M. Chugg‡ † Department of Computer Engineering ‡ Communication Sciences Institute Chalmers University of Technology University of Southern California 412 96 Göteborg Los Angeles, CA 90089-2565, USA {ahansson, tor}@ce.chalmers.se {hansson,
[email protected]} ABSTRACT A multiple-antenna receiver is proposed for achieving a diversity effect that partly overcomes the severity of continuous-time frequency-flat Rayleigh fading channels, and fast fading channels in particular. The error performance is further improved by bit interleaving and channel coding, where the encoders/channel is viewed as a serially concatenated system: a convolutional code (CC) constitutes the outer code, whereas a differential encoder and the fading channel (having truncated memory) form a joint inner code. In order to obtain a feasible detector structure, it is desirable to perform iterative decoding, which implies the need for an APP (a posteriori probability) algorithm that can operate over fading channels. For this purpose, the well-known BCJR algorithm (due to Bahl, Cocke, Jelinek, Raviv [1]) is generalized to handle channels having memory. Numerical results indicate that iterative decoding becomes more powerful when the exploited channel memory depth is extended, but the chief performance gain is due to space diversity provided by the multiple antennas. INTRODUCTION Shortly after Berrou et al. announced that interleaved parallel concatenated convolutional codes (PCCC), socalled “turbo codes” [2], yield performance close to the theoretical Shannon capacity limits, serially concatenated codes (SCC) were shown to offer comparable (and in some cases even superior) performance [3]. The nearoptimal performance over additive white Gaussian noise (AWGN) channels is partly due to the interleaver embedded in the code structure, which forms an overall concatenated code with very large memory. The analysis in [3] revealed that the use of a recursive systematic convolutional encoder (RSC) as inner encoder guarantees an interleaver gain [3]. This fact further indicates that continuous phase modulation (CPM) [4] should be equally suitable as an inner encoder (CPM is a recursive trellis code), achieving bandwidth efficient modulation at the same time. Such a system, SCCPM, has recently proved to give performance comparable to turbo codes both over AWGN channels [5] as well as over piecewise constant memoryless Rayleigh fading channels [6]. To
simplify the treatment of the fading channel, the latter study assumes the detector has somehow attained perfect channel state information (CSI). Even without the ideal assumption of perfect CSI, excellent performance has been achieved for piecewise constant memoryless Rayleigh channels [7]. Such channel models have unfortunately a somewhat limited validity [8]. A few results have also been presented for PCCC over slow and fast Rayleigh fading channels [9]. The encoder proposed in [9] has the drawback of being highly complex (it consists of two RSCs, two interleavers, one deinterleaver, and one differential encoder), leading to excessive decoding complexity. This paper analyzes an SCCPM system over continuous-time fading channels, i.e., over channels where the rate of variation of the fading process cannot be neglected [10]. To somewhat reduce the receiver complexity, only the most elementary non-trivial CPM member, namely MSK (minimum shift keying) [11], [12] is treated. SYSTEM AND CHANNEL DESCRIPTION An all-zero tail is first appended to the information bits, before the bits are fed to an outer CC. This is done in order to provide perfect knowledge of the terminal state of the outer trellis. Next, the coded package of bits is bit interleaved, MSK modulated, and transmitted over the equivalent lowpass channel, which comprises D diversity links. In each link the transmitted sequence s1K (t ; i ) of K waveforms is perturbed by two time-varying random processes, one is the multiplicative fading vector f (t ) (dimension D by 1), while the other is the additive thermal noise vector n(t ) (dimension D by 1). The variable i in s1K (t ; i ) refers to the hypothesis label, i.e., i ∈ {1, , 2 K +1} [12]. The distorted signal is received by D identical, omnidirectional antenna elements, and can be expressed as follows by using a complex baseband notation [13]:
K
[r1 (t )
L
rD (t )]T = r (t ) = s1K (t ; i ) f (t ) + n(t ) ,
(1)
where superscript T denotes the transpose. Note that both f (t ) and n(t ) are complex-valued. The waveform
sequence s1K (t ; i ) , the fading vector f (t ) , and the noise vector n(t ) are assumed to be statistically independent. Let us for simplicity assume that the noise vector is composed of independent, stationary, zero-mean, white Gaussian processes. If we in addition assume that noise processes in different diversity links are equally strong, the noise covariance-function matrix can be written as:
{
}
E n(t ) n H (t + τ ) = N0 Iδ (τ ) ,
(2)
where H denotes the Hermitian (complex conjugate) transpose, N0 is the double-sided power spectral density, I is the identity matrix, and δ (⋅) denotes Dirac’s delta function. Let the propagation geometry be restricted to the horizontal plane, and consider the Fraunhofer region (or the far-field; i.e., assume plane wavefronts). These two assumptions are reasonable, provided the distance between transmitter and receiver is sufficiently large [14], [15]. If we let xl be the (Euclidean) degeometrical vector for receiver antenna l , and notes a unit vector normal to the wavefront impinging from an angle θ , the array response vector (or steering vector) can be written as [16]:
g L
a(θ ) = exp { jη [1
g
jη [ D } , T
(3)
where η is the circular wavenumber, and g denotes scalar product. Now, following [17], the frequency-flat fading vector is zero-mean and has the following covariance-function matrix:
{
}
E f (t )f (t + τ ) = A (θ 0 , ∆) ⋅ J 0 ( 2π f mτ / Ts ) 1 = 2∆
H
θ 0 +∆
∫θ −∆ a(θ )a 0
H
(θ )dθ ⋅ J 0 ( 2π f mτ / Ts ) ,
lowpass filtered antenna signals, where the cutoff frequency W of the (ideal) lowpass pre-filter is chosen such that the loss of information becomes negligible.1 The observables are then used for computing soft information that is fed to the inner SISO (soft-input softoutput) module [3]. In addition, interleaved extrinsic APPs generated from the outer of the two constituent codes are fed to the inner SISO and used as a priori probabilities. Only in the very first iteration, before APPs have been generated for the outer code, no a priori information is provided. In a similar manner, the inner SISO computes extrinsic APPs that are deinterleaved and used as a priori information in the outer SISO. Information obtained about the inner tail bits is superfluous and simply dropped before deinterleaving. Finally, after a certain number of iterations, the detector arrives at a decision for each information bit. AN APP ALGORITHM USING FORWARD-ONLY PREDICTION The generation of APPs for the outer trellis is a standard procedure. See [3] for any details. However, the original BCJR algorithm [1] was derived for a discrete memoryless channel (DMC), while the branch metric in our inner tree depends on previously derived observables. This implies the need for a different APP algorithm. The authors of [9] face a similar problem, but their proposal is less general. The APP for edge number l at time index k can be written as:
(
=C⋅
(4)
(
∑ P(b1K (i)) P ( r1K b1K (i) ) P ( ek ,
b1K ( i )
=C⋅
∑
b1K (i ) : ek ,l
where the angle of arrival (AOA) has been modeled according to a uniform distribution over [θ 0 − ∆,θ0 + ∆ ] . Further, J 0 (⋅) is the zero-order Bessel function of the first kind and f m denotes the maximum Doppler frequency shift normalized to the symbol duration Ts . Below, we will consider the two fading rates f m = 0.01 and f m = 0.1 , which corresponds to slow and fast fading, respectively. From (4) it is seen that the fading induces infinite memory, i.e., the correlation extends over all time separations. The receiver first extracts N observables in each symbol interval 1 up to K , i.e., the received continuous-time process is “represented” by a sequence of N ⋅ K random variables. The obtained observation vector, say r1K , could be built up either by sampling or by an orthogonal expansion [10], [18]. In this paper, the observables will be derived by Nyquist sampling the
)
)
APP(ek ,l ) @ P ek ,l r1K = C ⋅ P r1K ek ,l P (ek ,l )
(
)
P r1K ; b1K (i ) = C ⋅
∑
l
b1K (i )
)
(5)
PkFW(i ) PkBW(i ).
b1K (i ): ek ,l
where C is a normalization constant, b1K (i ) is the bit sequence on hypothesis i , and b1K (i ) : ek ,l denotes all bit sequences consistent with edge ek ,l . This quantity can be computed by the following recursive expressions, where the channel memory has been truncated to include only the L + 1 most recent symbol intervals: PkFW (i ) @ P(r1k ; b1k (i ))
( (i )) P (b (i )) P ( r
= P (r1k −1 ; b1k −1 (i )) P (bk (i )) P rk r1k −1 ; b1k (i ) ≈ P (r1k −1 ; b1k −1
k
= PkFW −1 (i ) ⋅ γ k (i ), 1
MSK is not strictly bandlimited.
k
)
rkk−−L1 ; b kk − L (i )
)
(6)
where Es is the symbol energy, it is possible to factorize the conditional covariance matrix for observables derived during symbol intervals k − L up to k (the variance of each complex-valued noise sample is 2WN 0 [23]):
and similarly:
(
PkBW (i) @ P rkK+1 ; b Kk +1 (i ) r1k ; b1k (i )
(
= P rkK+ 2 ; b kK+ 2 (i ) r1k +1 ; b1k +1 (i )
(
)
⋅ P (bk +1 (i )) P rk +1 r1k ; b1k +1 (i ) ≈P
(
rkK+ 2 ; b kK+ 2 (i ) r1k +1 ; b1k +1 (i )
(
)
)
)
⋅ P (bk +1 (i )) P rk +1 rkk+1− L ; b kk ++11− L (i )
)
H
=
Es I, Ts
)
H
}
b kk − L (i )
(
)
Any realizable detector must truncate the (infinite) channel memory and is inevitably suboptimal. However, the correlation is fortunately a decreasing function of the time lag, which causes the approximation entailed in truncating the memory depth to become increasingly accurate as the truncation length is extended. As soon as the memory is truncated, the tree folds into a trellis. Note that the size of the trellis is determined by the truncation length, which is a design parameter. If the channel is truncated to include only the L + 1 most recent symbol intervals, the modulation and the fading can thus described by a joint (time-invariant) trellis having 2L states. A trellis state is now defined as the L most recent data bits b kk − L+1 (i ) fed to the MSK modulator, while the branches are labeled b kk − L (i ) . Further, it is possible to terminate the inner trellis by appending L modulated zero tail bits to the transmitted waveform sequence, socalled post-interleaver flushing. From (5), the (approximated) APP is finally obtained by summing over all possible sequences consistent with the considered edge. Since sequences that merge in the forward (or backward) direction will have identical branch metrics for all future (or past) trellis sections, the summation in (5) can most efficiently be computed by means of the well-known BCJR recursions [1]. This was also realized by Gertsman and Lodge [19]. We will now give an expression for the branch metric that proves to be equal to the classical MLSD (maximum-likelihood sequence detection) metric [20]. Alternatively, using a set of linear prediction filters yield an equivalent result [20], [21], [22] (an approach taken in [9]). Let Zkk − L (i ) be a diagonal matrix comprising samples of the unperturbed message sequence skk − L (t ; i ) , and let Fkk− L denote the covariance matrix for samples of the fading process f (t ) taken from time k − L up to k , i.e., let Fkk− L be a discrete-time representation of (4). Since MSK has constant envelope, i.e.,
(
(
T = Z kk − L (i ) Fkk− L + 2WN 0 s I Z kk − L (i ) E s
BW (i ) ⋅ γ k +1 (i ). = Pk+1
Zkk − L (i ) Zkk − L (i )
{
Ckk − L (i ) @ E rkk− L rkk− L
(7)
(8)
)
H
(9) .
Note that the determinant of Ckk − L (i ) does not depend on the hypothesis i (due to the constant envelope), which leads to the following form of the (conditional) loglikelihood function:
{(
log p rk rkk−−L1 ; b kk − L (i )
(
∝ rkk−−L1
) (C H
k −1 k − L (i )
)
−1
)} (
rkk−−L1 − rkk− L
) (C H
k k − L (i )
)
−1
rkk− L .
(10) It is interesting to note that the forward and backward recursions use the same branch metric. Recently, socalled bi-directional algorithms have proved to achieve a performance gain for channels where the parametric uncertainty is modeled as a Gauss-Markov process [24]. NUMERICAL RESULTS Error performance in terms of bit error rate (BER) versus signal-to-noise ratio (SNR) has been obtained from Monte-Carlo simulations with a minimum bit error limit of 1000 errors. Samples of the fading vector f (t ) were generated by means of a method similar to the one described in [25] (the correlation coefficients were derived by Cholesky decomposition). Moreover, we investigated systems with a (7,5) convolutional encoder and a 1024 bit interleaver, assuming the mean AOA θ 0 = 45o and the azimuth spread 2∆ = 90o . Figure 1 shows the five first iterations for a twoelement array with antenna separation 0.5 carrier wavelengths, operating over a fading channel with bandwidth f m = 0.1 . Clearly, the receiver gains from increasing the exploited channel memory. Specifically, the iteration gain becomes more significant for large truncation lengths L , but recall from (10) that this performance improvement is achieved at the cost of considerably higher detection complexity. Figure 2 shows the three first iterations and assumes f m = 0.1 . Three different antenna constellation were investigated: a single antenna, a two-element array with antenna separation 0.5 carrier wavelengths, and a fourelement array with the elements equidistantly placed at a
0
10
D=1, f =0.1 m D=1, fm=0.01 D=2, fm=0.1 D=2, f =0.01 m
−1
10
−2
10 BER
circle with diameter 0.5 carrier wavelengths. Even though there is little to gain from running more than two iterations, there is obviously much to gain from spacediversity; the error performance of our proposed array receiver is superior to previously published results obtained for fast fading channels [20], [9]. An explanation to the moderate iteration gain is that only hard decisions are circulating between the inner and outer SISO modules after the second iteration. This might be improved by replacing MSK be a CPM scheme with larger memory, thus obtaining a stronger inner code. In figure 3, the two fading bandwidths f m = 0.01 and f m = 0.1 have been investigated for a single-antenna receiver, as well as for a two-element array with antenna separation 0.5 wavelengths. When the fading rate is slow compared with the transmission rate, i.e., when f m = 0.01 , iterating more than two times is seen to become somewhat more powerful.
−3
10
−4
10
−5
10
0
2
4
6 SNR [dB]
8
10
12
Figure 3: Error performance for various fading bandwidths and antennas. CONCLUSIONS
−1
10
BER
Truncation Length L=2 L=4 L=6
−2
10
−3
10
4
4.5
5
5.5
6
6.5
7
7.5
This paper has generalized the BCJR algorithm to be valid for fading channels. Moreover, numerical results indicated that iterative decoding offers a limited performance gain for fast fading channels, because most of the improvement is achieved in the second iteration. With a large truncation length L , the performance is significantly improved—but the cost is unfortunately high decoding complexity. However, a comparable or even superior gain is easily achieved by using additional receiver antenna elements. Stimulated by the good results in [24], the authors are currently working on generalizing the bi-directional algorithms to handle the more general channel model that was used in this paper.
SNR [dB]
Figure 1: Error performance for various truncation lengths. 0
10
D=1 (Single Antenna) D=2 (2−Element Array) D=4 (4−Element Array) −1
10
−2
BER
10
−3
10
−4
10
−5
10
0
2
4
6 SNR [dB]
8
10
12
Figure 2: Error performance for various antenna constellations.
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