Broadband Beamforming for Joint Interference Cancellation and Low-Complexity Turbo Equalization Mutlu Koca
Bernard C. Levy
Institut de Recherche en Informatique et Systemes Aleatoires (IRISA) 35042 Rennes, France Email:
[email protected]
Department of Electrical & Computer Engineering University of California Davis, CA 95616 USA Email:
[email protected]
Abstract— We propose a low-complexity turbo space-time equalizer for signals transmitted over a fixed broadband wireless channel. The transmitter employs trellis coded modulation (TCM) for its bandwidth and power efficiency, and the broadband transmission is severely impaired by both intersymbol interference (ISI) due to multipath dispersion of the wireless channel and co-channel interference (CCI) due to the presence of adjacent users. The proposed turbo equalizer consists of a broadband beamformer, which performs soft interference cancellation and space-time minimum mean-square error (MMSE) equalization, followed by a soft-in, soft-out (SISO) TCM decoder. A priori expected symbol values are used as soft inputs in the soft interference cancellation stage and then after beamforming, the elements of the output sequence are mapped onto extrinsic loglikelihoods or probabilities using a Gaussian assumption. Because these two operations make the beamformer a SISO processor, the front-end of the space-time receiver is suitable for iterative processing. The proposed structure is implemented with small antenna arrays and simulated for turbo equalization of 8-PSK TCM signals, and simulation results indicate a relatively close performance to “no interference” bounds after only a small number of turbo iterations.
I. I NTRODUCTION Trellis-based turbo equalization, introduced in [1] as a joint maximum a posteriori (MAP) equalization and decoding method of data protected by an error correcting code, has been shown to be very effective in overcoming the ISI effects of wireless channels. However, particularly because the frontend SISO channel decoder of the MAP turbo equalizer relies on trellis-search techniques, it is only applicable to situations where the number of states remains reasonable, which typically occurs for narrowband channels, i.e., channels with a short impulse response. For broadband channels, severe multipath dispersion causes the size of the corresponding ISI trellis to grow exponentially with the data rate, so that direct application of MAP turbo equalization becomes unfeasible. For this reason a number of suboptimal approaches has been proposed for the application of turbo equalization for broadband wireless systems. One such approach, proposed in [2,3], relies on first shortening the broadband channel to a length typical of narrowband channels and then employing a MAP turbo receiver for optimum equalization. The channel shortening is achieved by a space-time receiver consisting of an antenna array followed by
a broadband beamformer whose coefficients are chosen such that the effect of secondary paths is reduced. Note that because the beamformer output sequence is processed by the SISO channel decoder of the turbo equalizer, which assumes that the additive channel noise is white and Gaussian, the beamformer must be designed to ensure that the observation noise at its output is white. This requirement makes the beamformer design a constrained quadratic optimization problem for which the solution methods are discussed in detail in [2]. Then such beamformers are used in combination with a turbo equalizer in [3] for decoding received signals from a single user with 8-PSK and 16-QAM TCM constellations. Note that broadband beamformers are also effective in rejecting CCI which typically occurs due to the presence of interfering users in an adjacent cell whose signals usually occupy the same frequency band and employ the same modulation format. That is why, in [4] the design methodology of [2] is modified for broadband beamformers rejecting interfering users and shortening desired channel impulse response simultaneously so as to make MAP turbo equalization possible for channels suffering form both ISI and CCI. Another approach, introduced in [5] and described further in [6, 7] all for binary transmission, replaces the trellis-based SISO channel decoder of the turbo equalizer by a transversal filter capable of processing and generating soft information. This filter is used for soft interference cancellation and MMSE equalization and, despite its suboptimal performance in comparison to a MAP equalizer, it yields a feasible turbo receiver scheme for broadband systems because its complexity grows only linearly with the channel length. For this reason, in this paper we extend the low complexity approach of [5–7] to the problem of [4], namely joint CCI cancellation and space-time turbo equalization for broadband transmission protected by a TCM code and subject to both ISI and CCI. We propose a space-time turbo equalizer which consists of a linear antenna array followed by a broadband beamformer at its front-end and a SISO TCM decoder at its back-end. The proposed receiver performs soft interference cancellation prior to broadband beamforming using a priori expectations of the data symbols as soft inputs. Then the beamformer combines diversity observations after space-time filtering operations, and the elements of its decision sequence are mapped onto log-likelihood ratios or a posteriori probabil-
ities (soft outputs) using a Gaussian assumption. These three operations at the front end form a SISO module making it suitable for iterative processing. This paper is organized as follows. In Section II, the transmission system and the signal model are presented. The design of space-time turbo equalizer is presented in Section III. Then, Section IV presents simulation results for the proposed receiver, which is implemented with small antenna arrays and employed for 8-PSK TCM signals. Finally, this paper is ended with some conclusive remarks in Section V. II. S IGNAL M ODEL
d
TCM Encoder
c
x
Interleaver
s(t)
Transmitter
for 1 i N . Note that a sampling rate above the Nyquist rate can also be employed for channel shortening in which case the expression in (2) corresponds to each of the subsequences obtained through oversampling. In this expression, hi;k represents the discrete-time channel impulse response with length L = L1 + L2 + 1 acting on the k -th transmitted sequence as seen by the i-th antenna element, and vni denotes the complex additive white Gaussian noise observed by the i-th antenna with zero mean and intensity 2 . Note that, the first summation term in (2) corresponds to the ISI effect whereas the second summation term represents the combined effect of CCI, both of which are usually characterized by long channel impulse responses. Thus the application of a broadband beamformer is necessary for joint rejection of interfering signals and equalization of the channel for the desired TCM signals. z1n
Multipath Channel v i (t) z1 (t) d
Turbo Space−Time Decoder
y
Array Receiver
zN (t)
. . .
...
+ .. .
v1 (t)
+
z in
We consider the transmission system in Fig. 1 where, a TCM encoder encodes and maps the elements dn of a data sequence d into points of a PSK signal constellation using the Ungerboeck set partitioning rules [8]. The elements cn of the resulting complex symbol sequence c are reordered by an interleaver, producing xn of the sequence x, hereafter referred to as the desired signal. We further assume that the transmission also suffers from CCI of the same type as the desired signal, where both the desired signal and K interfering signals x1 : : : xk : : : xK are independent, identically distributed (i.i.d.) complex, zero mean random sequences. The modulator uses a signalling waveform fT (t) with baud interval T , and the modulated signals admit the complex baseband representation
sk (t) =
m
fT (t , lT )xkl :
(1)
for 0 k K where the superscript 0 corresponds to the desired signal. The signals are transmitted through a frequency selective fixed multipath channel. We assume a quasistatic channel model, meaning the channel is time invariant during the transmission of a packet or block, and changes independently from one block to another. The corresponding multipath signal is received by an N -element evenly-spaced linear antenna array, where the first element is used as reference point for all observations. Then the resulting waveform is sampled at the baud rate and the sampled noisy observation sequence at the output of the i-th antenna element can be expressed as
zni =
L2 X
L2 K X X k i hi;l 0 x0n,l + hi;k l xn,l + vn l=,L1 k=1 l=,L1
W i (z) . . .
Transmission system
X
. . .
vN (t)
Modeled as a noisy convolutional encoder
Fig. 1.
1
W (z)
+
yn
. . .
zN n
WN(z)
Fig. 2.
Broadband beamformer
The structure of such a broadband beamformer is shown in Fig. 2, where a FIR filter W i (z ) with 1 i N is applied to each of the antenna array observation sequences zni and the resulting outputs are combined to generate a single observation sequence yn . Assuming that each branch filter W i (z ) has P coefficients, the beamformer is designed to minimize the effects of secondary paths and of interfering users. Note that defining T zn = zn1 : : : zn1 ,P : : : znN : : : znN,P T K (3) xn = x0n+L1 : : : x0n,P ,L2 : : : xK n+L1 : : : xn,P ,L2 1 T vn = vn : : : vn1 ,P : : : vnN : : : vnN,P
as the vectors collecting all channel values, data symbols and noise components processed by the beamformer at time-n, we can rewrite (2) as
where
zn = Hxn + vn
(4)
3 H1;0 H1;1 : : : H1;K 6 H2;0 H2;1 : : : H2;K 7 7 6
(5)
2
H = 6 .. . . . . .. 7 4 . . . . 5 N; 0 N; H H 1 : : :HN;K
(2) is the
NP
K (P + L) channel matrix regrouping all P
(P + L) channel coefficient matrices Hi;k defined as 2
hi;k : : : hi;k 0 , L 1 6 i;k 6 0 h,L1 : : : 6
. Hi;k = 6 6 ..
..
6 4 0
.
::: :::
0
: : : hi;k L2 hi;k : :: 0
0 ::: ::: 0 hi;k L2 0 : : : 0
. . . . . . . . . .. . .. . . . . . i;k i;k 0 h,L1 : : : h0 : : : hi;k 0 i;k : L: :2 hi;k : : : 0 hi;k : : : h 0 ,L1 L2 ..
3
and e is a N (P + L) all-zero column vector except for its L + 1-th entry.
7 7 7 7: 7 7 5
The vector output of soft interference canceller is then applied to the beamformer where the coefficient vector of (8) is chosen so as to minimize the MSE between the beamformer output and data symbol x0n
(6)
JMSE(w) = E [jx0n , yn j2 ] = E [jx0n , wH ~zn j2 ]
With beamformer coefficient vector defined as
wn = [w01 : : : wP1 : : : w0i : : : wPi : : : w0N : : : wPN ]T ;
which follows from the well-known projection theorem as (7)
the output sequence can be expressed as
yn = wH zn
(8)
and the optimum wn minimizing the MSE between the beamformer output sequence and the desired signal s0 (nT ) can be computed without difficulty. However, because the broadband beamformer alone is not a SISO processor, it cannot be used as the front-end processor of a turbo equalizer. In order to form a turbo equalizer with the broadband beamformer at the front-end, modifications must be made to use soft inputs and to produce soft outputs, as described in the next section. III. L OW C OMPLEXITY T URBO S PACE -T IME E QUALIZATION In this section we describe the low-complexity space-time turbo equalizer employing the serially concatenated M-ary decoding iteration shown in Fig. 3. z SIC / Broadband
y
Beamformer
APP Mapper
Deinterleaver
La (c ) SISO TCM Decoder
APP Demapper
Interleaver
Fig. 3.
L e (c )
Space-time turbo equalizer
The proposed receiver employs a preliminary stage of eliminating excess interference before broadband beamforming in a process called soft interference cancellation. This operation uses the a priori symbol expectation which is defined as
x = E [x] = for a symbol
X
Xj 2S
Xj P (x = Xj )
(9)
x taking values from an M-ary alphabet S =
fX0 ; X1; XM ,1 g
to compute the expected interference according to the signal model (4). The resulting “interference free” signal model is written as ,
~zn =zn , H x n , x0n e , =H xn , x n + x0n e + vn where
,1 wn = H Rn HH + j x0n j2 hhH + 2 I h
(10)
T K x n = x0n+L1 : : : x0n,P ,L2 : : : xK (11) n+L1 : : : xn,P ,L2
(13)
where h = He, and Rn is the diagonal covariance matrix
Rn = Cov[xn , x n ; xn , x n ]:
(14)
Once the beamformer coefficient vector is computed the corresponding output sequence yn can easily be obtained from (8) as , yn = wnH zn , Hxn + x0n h (15) which is extrinsic by construction due to fact that the softinterference cancellation step removes any a priori information. The main motivation in the SISO broadband beamformer design is to employ the soft interference canceller/broadband beamformer in a turbo-type equalizer/decoder structure. This means extracting extrinsic information from the beamformer output sequence in the form of either log-likelihoods or Mary a posteriori probability vectors that can be used by a SISO decoder at the back-end. That information is produced using the assumption that the beamformer output sequence y is the output of an AWGN channel having x0 as its input, i.e.,
yn = n xn + vn
La (x)= 0 x
(12)
(16)
where xn = x0n , n is the equivalent amplitude of the signal at the channel output and vn is a complex white Gaussian noise with zero mean and variance n2 . This is equivalent to saying that the estimates are complex Gaussian distributed, i.e., yn N (n xn ; n2 ). The parameters n and n2 are calculated at each time instant as a function of the equalizer structure. With this model, extrinsic symbol probabilities can be computed as
p(x0 = X j y ) p(yn j Xj ) pe (x0n = Xj j y) n i j n = P P (xn = Xj ) Xj 2S p(yn j Xj ) yn ,n Xj , e 2n2 =P (17) , yn ,2n2n Xr : Xr 2S e Finally, because computation of extrinsic probabilities in (16) requires the symbol statistics, n and n are updated at each time instant with the beamformer coefficient vector as , n =wnH E[~zn j x0n = x] , Hxn + x0n h =wnH h n =wnH cov[~z(n); ~z(n) j x0n = x] wnH =wnH h(1 , hH wn ):
(18) (19)
L(c) = [L(c1 ); L(c2 ); : : : ; L(cM )] :
(20)
The extrinsic information is passed to the SISO TCM decoding module, where it constitutes the a priori channel measurement for the outputs c of decoder. Since all data symbols d are equally likely, the a priori log-likelihood are set to zero at all iterations.The symbol-by-symbol BJCR algorithm is used in SISO TCM decoding whose details are not presented as both concepts are very well known. The SISO decoder generates a posteriori probabilities or log-likelihoods for both the data symbols d and the code symbols c. The information on data symbols is ultimately used to reach final decoding decisions at the end of the turbo equalization/decoding iteration, but in the iteration itself, the extrinsic information on code symbols obtained by removing the priori information from the a-posteriori information is interleaved and sent to the statistics generator whose output is fed back to the SISO broadband beamformer to start the next equalization iteration.
8-PSK TCM signal is transmitted over an additive white Gaussian noise (AWGN) channel with the same noise variance as in the multipath case. At the receiver, a single tap beamformer combines the array signals into a scalar output to which a turbo equalizer is then applied. For the second experiment, we simulate the transmission of only the desired signal without any co-channel interference. For the third experiment we assume that, in addition to the desired signal, a single interferer is present whose line of sight (LOS) multipath component is separated by a 15o angle from the LOS component of the desired signal. In the final experiment, we introduce a second interfering signal with a LOS component arriving with a 25o angular separation on the other side of the desired signal. Both the desired and interfering signals are assumed to be modulated with 8-PSK TCM and transmitted in blocks of 500 symbols. The fixed wireless channel corresponding to each signal is assumed quasi-stationary, i.e., the channel is stationary during the transmission of one block but changes independently from one block to another, and it is implemented according to the physical model described in [3]. For the interfering signals it is assumed that the power levels of the interfering users are lower than that of the desired signal and change randomly after transmission of every 5 data blocks. At the receiver a sampling rate of twice the baud rate is employed, i.e., Ts = T=2. Because most of the ISI introduced by a channel is concentrated in the center coefficients of its impulse response, the tail coefficients of the originally infinite channel impulse response are truncated, yielding a FIR model with L = 9 taps for both the desired and interfering channels. The broadband beamforming filters are implemented with P = 16 coefficients for each diversity branch. Fig. 4 - Fig. 6 show the bit error rate (BER) performance of the turbo equalizer when applied to the scalar output sequences associated to the four beamforming scenarios (the first one is essentially trivial) described above. 0
10
3 users
−1
10
−2
10 BER
Note that the quasistatic channel assumption, which makes the channel matrix H time-invariant during the transmission of each block, is not a key factor in the above description. Because the beamformer coefficients are computed at each time instant, the algorithm can easily be applied to the timevariant channels by updating H (and h) in (13), (15), (18), and (19). We can now summarize the turbo equalization mechanism as follows. The broadband beamformer receives N antenna array observations z1n : : : zN n . The equalization iterations starts with no a priori information, so the means are set to 0 in the first iteration. In the subsequent iterations, the expected values for the desired sequence symbols are computed using the a priori probabilities received from the SISO TCM decoder as in (9). Note that because such a priori information is never available for the interfering signals, their means are always kept as 0 in all iterations. The broadband beamformer combines the extrinsic diversity observations after soft interference cancellation and space-time filtering operations, and this sequence is mapped onto a extrinsic probabilities either before or after the interleaving. In this work, interleaving is performed on the scalar sequences for computational complexity reduction. Note that unlike the binary case where all functions that need to be evaluated can be expressed as scalar log-likelihood ratios, here the log-likelihood functions are M-vectors, allocating a reliability measure to each of the possible symbol values, e.g.,
−3
10
IV. S IMULATION R ESULTS In this section we present simulation results for the proposed space-time receiver for turbo equalization of 8-PSK TCM signals transmitted over fixed broadband wireless channels. The proposed receiver is implemented with 2, 3 and 4 antenna broadband beamformers. For each antenna configuration, we consider four transmission scenarios. The first scenario is the interference free broadband transmission case which provides a bound for the turbo equalizer performance. In this case, the
−4
10
0th turbo it. 4th turbo it No inteference 1 user
2 users
−5
10
0
2
4
6
8
10 SNR (dB)
12
14
16
18
20
Fig. 4. Turbo equalizer performance for 8-PSK TCM with a 2 antenna beamformer
0
10
0th turbo it. 4th turbo it No inteference −1
10
−2
BER
10
−3
10
−4
10
1 user
2 users
−5
10
0
2
4
6
8
10
12
3 users 14
16
18
20
22
SNR (dB)
Fig. 5. Turbo equalizer performance for 8-PSK TCM with a 3 antenna beamformer
V. C ONCLUSION In this paper, receiver diversity and spatio-temporal processing are employed to simultaneously reject interfering signals impinging on the array from other directions and perform turbo equalization for channel effects on the desired signal. When the broadband beamformer employs an antenna array with more elements than the number of transmitted signals, the proposed receiver is shown to be highly effective in rejecting interfering signals, and the significant performance gains achievable through the combination of antenna beamforming and turbo equalization are illustrated by simulations for 8-PSK TCM signals.
0
10
0th turbo it. 4th turbo it No inteference −1
10
−2
BER
10
−3
10
R EFERENCES
−4
10
1 user
−5
10
can be obtained only at a relatively high SNR. However the introduction of only one additional antenna to the beamformer yields a diversity gain of 5:5 dB for the same case as seen in Fig. 5. At the same 10,4 BER level, the diversity gain of a 3-antenna beamformer in the absence of CCI is 2:8 dB. Moreover the 3-antenna beamformer cancels the CCI from two interferers to such a level that turbo equalization becomes possible although a large SNR is again required to achieve a BER of 10,4 . In this case, a significant gain of 7:5 dB is obtained if we employ a beamformer with 4 antennas. At a 10,4 BER level, this beamformer yields a diversity gain beyond the 3-antenna configuration of 2:5 dB for the case of a single interferer, and 1:5 dB for the case with no CCI. More importantly, at a 10,4 BER level, the gap between the trelliscoded AWGN bound and the turbo equalizer performance after 4 iterations is 0:5 dB for a single user, 1:1 dB for two users and 3:5 dB for three users.
0
2
4
6 SNR (dB)
8
2 users
3 users 10
12
Fig. 6. Turbo equalizer performance for 8-PSK TCM with a 4 antenna beamformer
The turbo equalizer applies four decoding iterations after the first pass (zeroth iteration), and in cases where a BER of 10,4 can be achieved, it offers a gain of up to 2 dBs. In the absence of co-channel signals, for a BER of 10,4 , the performance of the space-time receiver is usually within a dB of the bound provided by the performance of a trellis-coded AWGN channel. Regarding the diversity gains and interference cancellation/equalization efficiency of the proposed receiver, note from Fig. 4 that a two antenna beamformer is only effective for cases where there is no CCI. When there are two interfering signals (3 users), the beamformer is unable to isolate the desired user, so that the turbo equalizer does not achieve low BERs at a reasonable SNR range. Even for a single interfering user case, the CCI effects of can be cancelled and a BER of 10,4 or less
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