Low Complexity Soft Interference Cancellation for MIMO-Systems
Low Complexity Soft Interference Cancellation for MIMO-Systems Steffen Bittner, Ernesto Zimmermann and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Technische Universit¨at Dresden, D-01062 Dresden, Germany Email: [email protected]
Abstract— Iterative equalization has emerged as an efficient means of achieving near-capacity detection performance in multiple-antenna (MIMO) systems. However, many proposed detection strategies still exhibit a very high complexity which may render them unsuited for practical implementation. In this paper, we show that the appropriate use of apriori knowledge during iterative equalization based on soft interference cancellation enables to drastically reduce detection complexity. More specifically, we propose to take into account only a subset of constellation points in the calculation of detector soft output, by considering the vicinities of the interference reduced received signal and the constellation points supported by the a-priori knowledge. Additionally, a threshold rule on symbol probabilities is used to reduce complexity in the calculation of soft symbols and residual noise during soft interference cancelling. Our results show that the computational effort required for detection can be lowered by as much as 50% for 16-QAM and 96% for higher order constellations (64-256-QAM), without any significant loss in performance.
I. I NTRODUCTION Spectrum space in the radio frequency band is a scarce and expensive resource. In order to satisfy the demands of modern data services, this resource has to be used as efficiently as possible. One way to achieve high spectral efficiency is to use multiple transmit and receive antennas to transmit several spatially multiplexed data streams in the same time-frequency bin. Recently, a lot of research has been spent on serially concatenated systems consisting of an outer channel code and (inner) MIMO channel. It has been shown in [1] and [5] that by appropriately exploiting the correlation across the received signals it is possible to approach the capacity limit of the system. However, these LISS or sphere detection based algorithms still require a considerable computational effort. In this work we take a different approach, which is based on methods from soft successive interference cancellation [2] and still delivers and accepts log-likelihood values (Lvalues) which are needed for an iterative detection process. We obtain the L-values by soft cancelling of previously detected data symbols and compute new soft symbols using channel state and a-priori information. In order to further improve the accuracy of the detector output, we use the total (residual) interference plus noise power which is continuously updated during the cancellation steps and proportional to the variance of the soft symbols. A main problem of this approach is its still considerable computational complexity. Especially for systems
using higher order constellations, the distance, soft symbol and variance computations are a challenging task. We present an algorithm which drastically reduces the number of distance computations, based on the available apriori information. We also propose an algorithm to efficiently approximate the soft symbol and variance computation with the help of a probability threshold. Both algorithms are evaluated by numerical simulations and allow a substantial reduction in complexity without any significant performance degradation. The remainder of this paper is organised as follows. In Section II we describe the system model and the principles of an iterative detection process using channel state information and a-priori knowledge. In Section III the implemented soft interference cancellation algorithm is described in more detail. In addition, the two proposed complexity reduction techniques are presented. In Section IV we evaluate the performance of the new algorithms by computer simulations before we finally draw conclusions in Section V. II. S YSTEM M ODEL A. MIMO-Model Binary Source
u
Outer Encoder
c
Rate R
s ... H ...
Interleaver
AWGN Hard Decision Binary Sink
Constellation Mapper
x
n y
SISO Decoder
LA,Dec
LE,Dec
-1
LE,Det
MIMO Detector
LA,Det
Fig. 1. Transmission model with outer PCC encoder, MIMO channel and iterative receiver (soft SIC MIMO detector and SISO decoder).
We consider a MIMO (multiple-input, multiple-output) system with T x transmit and Rx receive antennas as described in Figure 1. Let u be a vector of information bits which are encoded by the outer encoder and interleaved. The resulting code bit stream is separated into blocks x containing T x·N independent binary digits. Here, N represents the number of bits per symbol and therefore allows to separate between M = 2N different constellation points. Each block x = (x1 , · · · , xT x )T consists of T x binary vectors xt = (xt1 , · · · , xtN ) of N bits. As part of the transmission process, every single block
is mapped onto a T x × 1 complex vector of symbols s = (s1 , · · · , sT x )T whose components are taken from some complex constellation C (e.g. 16-quadrature amplitude modulation (QAM)). These components are obtained using the mapping function st = map(xt ), t = 1, · · · , T x (e.g., Gray mapping). The transmission energy is equal to E[ksk2 ] = Es , with each component obeying the energy constraint of Es /T x. Let y be a vector of received symbols according to y = Hs + n
(1)
where H is a dimension Rx × T x MIMO channel matrix perfectly known at the receiver. Each entry of the channel matrix is an independent realisation of a complex Gaussian process with zero mean and variance 1/2 per real dimension, while n describes a vector of zero mean independent complex Gaussian noise values with variance N0 /2 per real dimension. B. Iterative Detection and Decoding At the receiver we consider an iterative Detection-DecodingSystem consisting of a serially concatenated MIMO detector and an outer decoder. The latter is based for example on the original BCJR algorithm or its max-log-MAP approximation [6]. The MIMO detector provides information to the outer decoder based on the received signal, the channel state information, and the a-priori knowledge from the same outer decoder. The information between detector and decoder is exchanged in an iterative fashion using the so called loglikelihood values (L-values) as presented in [4]. Knowing the received signal y, the L-value of bit xtn is defined as: P [xtn = +1|y] . (2) L(xtn |y) := ln P [xtn = −1|y] Using Bayes’ theorem and under the assumption of statistically independent bits, the joint probabilities can be split into products. We can rewrite (2) as P x∈Xtn ,+1 p(y|x) · P [x] L(xtn |y) = ln P , (3) x∈Xtn ,−1 p(y|x) · P [x] where Xtn,±1 is the set of 2T x·N −1 bit blocks x with xtn equal to ±1. The second term in (3) represents the a-priori knowledge fed to the detector from the decoder. The MIMO channel introduces interference among the transmitted signals at the receiver. The conditioned probability density in (3) is therefore given by the complex Gaussian distribution: 1 1 2 exp − ky − Hsk . (4) p(y|x) = (πN0 )Rx N0 However, for the L-value computation only the second term is relevant – the constant scaling factor can thus be omitted. To evaluate the numerator and denominator of (3) it is useful to recursively apply the so called ”Jacobian Logarithm”: ln(ea1 + ea2 ) = max(a1 , a2 ) + ln(1 + e−|a1 −a2 | ).
(5)
The correction term ln(·) can be implemented via a look-uptable or even omitted, to speed up computation at the expense
of some performance degradation [6]. The L-values from the detector can thus be approximated by a difference of two maximum operations:
L(xtn |y) ≈
max
x∈Xtn ,+1
max
x∈Xtn ,−1
Tx
Tx
YY 1
2 P [xtn ]
y − Hs + ln N0 t=1 n=1
−
YY 1
2 P [xtn ] .
y − Hs + ln N0 t=1 n=1 (6)
−
−
N
N
Equation (6) can be implemented by a soft interference cancellation algorithm, which will be described in more detail in the following section. III. S OFT I NTERFERENCE C ANCELLATION A. Detection Strategy The principle of soft interference cancellation is shown in Figure 2. The basic idea is to subtract the contributions by y˜2
y˜3
y˜1
σn2ˇ 3 = N0
L(x31 )
CU −R23 s¯3 −R13 s¯3
σn2ˇ 2 L(x21 )
CU −R12 s¯2 σn2ˇ 1
L(x11 ) CU... Computational Unit
CU
Fig. 2. Example of a soft SIC 3 × 3 MIMO detector. Soft symbols are denoted as si . The variance of these soft symbols contributes to the overall 2 . noise σn ˇi
previously detected signal from the received signal [3]. For this interference cancellation we do not use hard decided symbols, as this would lead to unjustifiable error propagation. We rather use soft symbols resulting from an expectation computation, as well as the variance of these symbols which directly relates to their reliability [2] (and thus the expected residual noise after interference cancellation). In order to perform the interference cancellation, we have to transform the channel matrix H such that it is causal with respect to the antenna index t, i.e., the currently detected symbol depends only on previously detected signals. It is obviously advantageous to detect the strongest antenna first, in order to minimize the effects of error propagation. For this purpose we use the MMSE based sorted QR decomposition introduced in [7]. To reduce the computational effort, we suggest not to use the post-sorting-algorithm since it does not significantly increase the performance in a coded system. Since Q is a unitary matrix, the distance computation in (6) can now be rewritten as ky − Hsk2 = k˜ y − Rsk2 , where y ˜ = QH y.
Exploiting the upper triangular structure of R, the L-value from the detector is given by: Tx
2 X 1
L(xtn |y) ≈ max − 2 y˜t − R(t,t) st − R(t,j) s¯j s ,x =+1 t
σnˇ t
tn
+
N X
j=t+1
ln P [xtn ] −
n=1
max
st ,xtn =+1
Qu
Qu 0000
0100
1100
1000
0000
0100
1100
1000
0001
0101
1101
1001
0001
0101
1101
1001
0011
0111
1111
1011
0010
0110
1110
1010
In
In
··· .
0011
0111
1111
1011
0010
0110
1110
1010
(7)
where the a-priori bit probabilities are defined as [4]: LA (xtn ) + Atn . (8) 2 The normalization term Atn can be omitted as it cancels out in the calculation of the detector L-values in (7). As stated before we use soft interference cancellation. The desired soft symbol is given by: ln P [xtn ] := xtn
s¯j
M X
:= E[sj ] = M X
=
m=1 N Y
sm ·
m=1
sm · P [sm ] P [xmn ]
(10)
(11)
Note that we use the L-value including channel state information and a-priori knowledge for computing the soft bit and thus attain a good estimation for the transmitted symbol. Since we cannot expect perfect cancellation we have to deal with residual interference after the cancellation step. This leads us to a new channel model where the Gaussian term is replaced by an interference-plus-noise term [2]: Tx X
R(t,t) st + n ˇ t = R(t,t) st +
Search areas for a 16-QAM constellation
a system using higher order constellation such as 64-QAM or even 256-QAM, the task of computing the expectation and variance Rx − 1 times during interference cancellation requires a tremendous effort – far too high for a practical implementation. We will show in the following section how this effort can be reduced substantially. B. Complexity Reduction
where sm represents one possible complex transmitted symbol from C. The required bit probabilities P [xmn ] are defined by: 1 + xtn · x ¯tn 2 with the help of the so called soft bit: L(xtn ) x ¯tn = E{xtn } = tanh . 2
Fig. 3.
(b) Second search area based on the a-priori knowledge.
(9)
n=1
P [xtn = ±1] =
(a) First search area around interference reduced received signal.
R(t,j) (sj − s¯j ) + n ˜t.
To reduce the amount of required calculations, two algorithms were developed. The first one reduces the number of distance computations and basically employs a union of two search areas. The second one deals with the task to approximate the soft symbol and variance computations. In the first proposed complexity reduction technique, one search area covers the constellation point closest to the interference reduced received signal and the appropriate counterhypotheses for each bit based on the max-log approximation. Thus one has only to compute N + 1 distances in the first detection-decoding iteration, as shown in Figure 3(a) for a 16-QAM constellation. Here, “x” indicates the received signal point surrounded by the closest constellation point 0001 and the appropriate counter-hypotheses. However, if a-priori information is available, finding the (unbiased) maximum likelihood estimate does not necessarily help, since this point has the minimum distance but does not necessarily maximise the two terms in (7).
j=t+1
(12) The variance of this new noise term is equivalent to the interference-plus-noise power σn2ˇ t introduced at each antenna and is given by: σn2ˇ t
= N0 +
Tx X
Qu 0000
0100
1100
1000
0001
0101
1101
1001 In
2
|R(t,j) | · Var[sj ].
(13)
0011
0111
1111
1011
0010
0110
1110
1010
j=t+1
The variance of the soft symbols can be computed by M X
(14)
Fig. 4. Union of the first and the second search area corresponding to the Figures in 3.
where we can reuse the symbol probabilities required for soft symbol computation in (9). It is easily seen that for
We therefore extend the first search area by a fixed number of constellation points and introduce a second search area
Var[sj ] =
(sm − s¯j ) · P [sm ],
m=1
0
IV. S IMULATION R ESULTS The proposed algorithms were tested by simulating a 4 × 4 MIMO System, with simulation parameters equivalent to the setup in [5]. We assumed the receiver to know the channel perfectly and the channel gains to remain constant over the transmission of one vector QAM symbol but change statistically independently between the different transmitted signal vectors (i.e., ergodic fading). The detector works according to the soft interference cancellation algorithm with the proposed modifications. For the decoding process we used a rate R = 1/2 parallel concatenated G = [7R , 5] turbo code, where 7 indicates the memory two feedback polynomial G(D) = 1 + D + D2 and 5 the memory two feedforward polynomial G(D) = 1 + D2 . The decoder performed 8 internal turbo decoding iterations and the interleaver size is 9216 bits. Moreover we performed 4 detector-decoder-iterations. Figure 5 shows the performance of a 16-QAM coded system for different complexity schemes. The genie bound of such a system would be at 6.2 dB. The full complexity setup represents a loss of approximately 1.4 dB to this bound. In this setting we computed the distance to all 16 possible constellation points and for the expectation and variance computation we also used all 16 possible symbols. In the first complexity reduction setup, the probability threshold for the expectation and variance computation was set to 1% but we still computed all 16 Euclidean distances for the
10
−1
10
−2
Genie 16−QAM
BER →
10
−3
10
−4
10
Full Complexity 16 Distances, 1% Prob.−Threshold min. 5 Distances, 1% Prob.−Threshold 5
5.5
6
6.5
7
7.5
Eb/N0 [dB] →
8
8.5
Fig. 5. Simulation results of a 16-QAM SIC receiver after 4 Det.-Dec.Iterations. Transmission over a 4 × 4 MIMO channel, outer PCC code R = 1/2. The performance degradation due to introducing the proposed complexity reduction techniques is negligible.
metric/L-value computation. One can see that there is no performance degradation due to using this approximation. As for the second complexity reduction technique, we reduced the number of distance computations by considering only the union of our two search areas (the probability threshold was again set to 1%). Interestingly, we see that this results in a performance loss of only less than 0.1 dB, with respect to the full complexity setup – but only needs a fraction of the computation effort. In the setup in Figure 5, a minimum of 5 constellation points were used for the first search area. However, finding the maximum in equation (7) using only a subset of constellation points does not always correspond to the global maximum over all possible symbols. This rule applies especially for higher QAM constellations as shown in Figure 6 for a 256-QAM 0
10
−1
10
−2
10
BER →
based on the a-priori knowledge. The centre of this area is the constellation point corresponding to the sign decision of the a-priori L-values provided by the decoder. The other constellation points are found by bit flipping of the sign decision. These form the constellation points supported by the a-priori information. Let the the sign decision of the apriori knowledge be equal to 0101, as illustrated in Figure 3(b). The corresponding symbol represents the centre of the second search area, surrounded by the constellation points resulting from the proposed bit flipping computation. The second search search area will thus always cover N + 1 constellation points. Finally, the union of the two search areas is used to compute the Euclidean distances and derive the detector L-values, as shown in Figure 4. The second algorithm uses a probability threshold to exclude unlikely symbols from the computation of the soft symbols and their reliability. The basic idea is that all symbols for which the probability of occurrence lies below the threshold will contribute only slightly to the expectation and variance computation. Since we use a binary coded system, one unlikely bit value already excludes half of the possible complex symbols in C. As there is a direct relation between bit probability and symbol probability, one could also use a bit based instead of a symbol based probability threshold. In the reminder of this paper, however, we will stick to the symbol based probability threshold. Both approaches lead to a huge complexity reduction, without any significant loss in performance, as will be shown in section IV.
−3
10
−4
10
256 Distances, 1% Prob−Threshold min. 9 Distances, 1% Prob−Threshold min. 15 Distances, 1% Prob−Threshold 13
13.5
14
14.5
Eb/N0 [dB] →
15
15.5
16
Fig. 6. Simulation results of a 256-QAM SIC receiver after 4 Det.-Dec.Iterations. Transmission over a 4 × 4 MIMO channel, outer PCC code R = 1/2. A higher number of constellation points (as opposed to N +1 points used in 16-QAM transmission) must be considered in metric computation to achieve good performance.
scenario. We therefore increased the number of constellation
points lying in the first search area in this example to 15 instead of 9. This wider coverage area leads to a significant performance improvement for the shown 256-QAM constellation. It is worth mentioning that the genie bound is around 11.5 dB so that there is still a 3.5 dB gap. 20 19
Distance Computations →
18 17 16 15 14 13 12 11 10 10
min. 9 Distances, 1% Prob.−Threshold min. 15 Distances, 1% Prob.−Threshold 11
12
13
14
15
16
Eb/N0 [dB] →
17
18
19
20
Fig. 7. Number of constellation points within the search area for 256-QAM transmission, corresponding to Figure 6.
Considering the complexity reduction, we computed less than 7 distances in the cliff region of interest around 7.6 dB, for a 16-QAM system. We thus achieved a complexity reduction of more than 50%. For higher modulation schemes, e.g. 256-QAM as shown in Figure 7, we even attained a reduction of more than 93% without any significant loss in performance. The influence of the probability threshold over the detectiondecoding iteration for a 256-QAM system is shown in Figure 8. The graphs show the number of symbols which passed the 25
No. of Symbols→
20
15
10
5
0 10
1st Iteration 2nd Iteration 3rd Iteration 4th Iteration 11
12
13
14
15
16
Eb/N0 [dB] →
17
18
19
20
Fig. 8. Impact of the 1% probability threshold on the number of symbols considered in soft symbol computation, as a function of the detector-decoderiterations and SNR. 256-QAM transmission is used, corresponding to the setup in Figure 6.
1% threshold and thus contributed to the soft symbol and variance calculation. Even in the first detector-decoder iteration, where no a-priori information is available, the achieved complexity reduction is more than 90% and increases dramatically
with the iteration index until only a single symbol passes the threshold. On average over the detector-decoder iterations we achieved a reduction in computational effort of more than 96%. V. C ONCLUSION In this paper we presented an iterative equalization system for multiple antenna systems. We used soft interference cancellation based detection, supported by an interference-plus-noise term to mitigate the effects of error propagation. Furthermore, we proposed two algorithms which substantially reduce the computational complexity. The first algorithm uses the provided a-priori knowledge to derive highly accurate L-values, considering only a minimum number of potentially relevant constellation symbols. The second algorithm uses a probability threshold in order to calculate the symbol expectation (soft symbols) and variance (residual noise) at very low complexity. Computer simulation results show that we can reduce the complexity drastically – by over 90%, for 256-QAM transmission – without any significant loss in performance. The proposed algorithms thus make it possible to implement even very high order modulation schemes on modern digital signal processors and thus attain high spectral efficiencies and data rates with MIMO systems. ACKNOWLEDGMENT The authors would like to thank Wolfgang Rave for inspiring ideas and discussions. R EFERENCES [1] S. B¨aro, J. Hagenauer and M. Witzke, Iterative Detection of MIMO Transmission Using a List-Sequential (LISS) Detector, Proceedings of the IEEE International Conference on Communications (ICC’03), vol. 4, pp. 2653-2657, May 2003. [2] W. J. Choi, K. W. Cheong and J. M. Cioffi, Iterative Soft Interference Cancellation for Multiple Antenna Systems, Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC’00), No. 1, pp. 304-309, 2000. [3] A. Gersho and T. L. Lim, Adaptive Cancellation of Intersymbol Interference for Data Transmission, Bell Systems Technical Journal, vol. 60, pp. 1997-2021, 1981. [4] J. Hagenauer, E. Offer and L. Papke, Iterative Decoding of Binary Block and Convolutional Codes, IEEE Transactions on Information Theory, vol. 42, pp. 429-445, Mar. 1996. [5] B. M. Hochwald and S. ten Brink Achieving Near-Capacity on a MultipleAntenna Channel, IEEE Transactions on Communications, vol. 51, No. 3, pp. 389-399, Mar. 2003. [6] P. Robertson, E. Villebrun and P. Hoeher, A comparison of optimal and suboptimal MAP decoding algorithms operating in the log domain, Proceedings of the IEEE International Conference on Communications (ICC’95), pp. 1009-1013, Jun. 1995. [7] D. W¨ubben, R. B¨ohnke and V. K¨uhn and K. D. Kammeyer, MMSE Extension of V-BLAST based on Sorted QR Decomposition, Proceedings of the IEEE Semiannual Vehicular Technology Conference (VTC Fall 03), Oct. 2003.
Abstract— Iterative equalization has emerged as an efficient means of achieving near-capacity detection performance in multiple-antenna (MIMO) systems. However, many proposed detection strategies still exhibit a very high complexity which may render them unsuited for practical implementation. In this paper, we show that the appropriate use of apriori knowledge during iterative equalization based on soft interference cancellation enables to drastically reduce detection complexity. More specifically, we propose to take into account only a subset of constellation points in the calculation of detector soft output, by considering the vicinities of the interference reduced received signal and the constellation points supported by the a-priori knowledge. Additionally, a threshold rule on symbol probabilities is used to reduce complexity in the calculation of soft symbols and residual noise during soft interference cancelling. Our results show that the computational effort required for detection can be lowered by as much as 50% for 16-QAM and 96% for higher order constellations (64-256-QAM), without any significant loss in performance.
I. I NTRODUCTION Spectrum space in the radio frequency band is a scarce and expensive resource. In order to satisfy the demands of modern data services, this resource has to be used as efficiently as possible. One way to achieve high spectral efficiency is to use multiple transmit and receive antennas to transmit several spatially multiplexed data streams in the same time-frequency bin. Recently, a lot of research has been spent on serially concatenated systems consisting of an outer channel code and (inner) MIMO channel. It has been shown in [1] and [5] that by appropriately exploiting the correlation across the received signals it is possible to approach the capacity limit of the system. However, these LISS or sphere detection based algorithms still require a considerable computational effort. In this work we take a different approach, which is based on methods from soft successive interference cancellation [2] and still delivers and accepts log-likelihood values (Lvalues) which are needed for an iterative detection process. We obtain the L-values by soft cancelling of previously detected data symbols and compute new soft symbols using channel state and a-priori information. In order to further improve the accuracy of the detector output, we use the total (residual) interference plus noise power which is continuously updated during the cancellation steps and proportional to the variance of the soft symbols. A main problem of this approach is its still considerable computational complexity. Especially for systems
using higher order constellations, the distance, soft symbol and variance computations are a challenging task. We present an algorithm which drastically reduces the number of distance computations, based on the available apriori information. We also propose an algorithm to efficiently approximate the soft symbol and variance computation with the help of a probability threshold. Both algorithms are evaluated by numerical simulations and allow a substantial reduction in complexity without any significant performance degradation. The remainder of this paper is organised as follows. In Section II we describe the system model and the principles of an iterative detection process using channel state information and a-priori knowledge. In Section III the implemented soft interference cancellation algorithm is described in more detail. In addition, the two proposed complexity reduction techniques are presented. In Section IV we evaluate the performance of the new algorithms by computer simulations before we finally draw conclusions in Section V. II. S YSTEM M ODEL A. MIMO-Model Binary Source
u
Outer Encoder
c
Rate R
s ... H ...
Interleaver
AWGN Hard Decision Binary Sink
Constellation Mapper
x
n y
SISO Decoder
LA,Dec
LE,Dec
-1
LE,Det
MIMO Detector
LA,Det
Fig. 1. Transmission model with outer PCC encoder, MIMO channel and iterative receiver (soft SIC MIMO detector and SISO decoder).
We consider a MIMO (multiple-input, multiple-output) system with T x transmit and Rx receive antennas as described in Figure 1. Let u be a vector of information bits which are encoded by the outer encoder and interleaved. The resulting code bit stream is separated into blocks x containing T x·N independent binary digits. Here, N represents the number of bits per symbol and therefore allows to separate between M = 2N different constellation points. Each block x = (x1 , · · · , xT x )T consists of T x binary vectors xt = (xt1 , · · · , xtN ) of N bits. As part of the transmission process, every single block
is mapped onto a T x × 1 complex vector of symbols s = (s1 , · · · , sT x )T whose components are taken from some complex constellation C (e.g. 16-quadrature amplitude modulation (QAM)). These components are obtained using the mapping function st = map(xt ), t = 1, · · · , T x (e.g., Gray mapping). The transmission energy is equal to E[ksk2 ] = Es , with each component obeying the energy constraint of Es /T x. Let y be a vector of received symbols according to y = Hs + n
(1)
where H is a dimension Rx × T x MIMO channel matrix perfectly known at the receiver. Each entry of the channel matrix is an independent realisation of a complex Gaussian process with zero mean and variance 1/2 per real dimension, while n describes a vector of zero mean independent complex Gaussian noise values with variance N0 /2 per real dimension. B. Iterative Detection and Decoding At the receiver we consider an iterative Detection-DecodingSystem consisting of a serially concatenated MIMO detector and an outer decoder. The latter is based for example on the original BCJR algorithm or its max-log-MAP approximation [6]. The MIMO detector provides information to the outer decoder based on the received signal, the channel state information, and the a-priori knowledge from the same outer decoder. The information between detector and decoder is exchanged in an iterative fashion using the so called loglikelihood values (L-values) as presented in [4]. Knowing the received signal y, the L-value of bit xtn is defined as: P [xtn = +1|y] . (2) L(xtn |y) := ln P [xtn = −1|y] Using Bayes’ theorem and under the assumption of statistically independent bits, the joint probabilities can be split into products. We can rewrite (2) as P x∈Xtn ,+1 p(y|x) · P [x] L(xtn |y) = ln P , (3) x∈Xtn ,−1 p(y|x) · P [x] where Xtn,±1 is the set of 2T x·N −1 bit blocks x with xtn equal to ±1. The second term in (3) represents the a-priori knowledge fed to the detector from the decoder. The MIMO channel introduces interference among the transmitted signals at the receiver. The conditioned probability density in (3) is therefore given by the complex Gaussian distribution: 1 1 2 exp − ky − Hsk . (4) p(y|x) = (πN0 )Rx N0 However, for the L-value computation only the second term is relevant – the constant scaling factor can thus be omitted. To evaluate the numerator and denominator of (3) it is useful to recursively apply the so called ”Jacobian Logarithm”: ln(ea1 + ea2 ) = max(a1 , a2 ) + ln(1 + e−|a1 −a2 | ).
(5)
The correction term ln(·) can be implemented via a look-uptable or even omitted, to speed up computation at the expense
of some performance degradation [6]. The L-values from the detector can thus be approximated by a difference of two maximum operations:
L(xtn |y) ≈
max
x∈Xtn ,+1
max
x∈Xtn ,−1
Tx
Tx
YY 1
2 P [xtn ]
y − Hs + ln N0 t=1 n=1
−
YY 1
2 P [xtn ] .
y − Hs + ln N0 t=1 n=1 (6)
−
−
N
N
Equation (6) can be implemented by a soft interference cancellation algorithm, which will be described in more detail in the following section. III. S OFT I NTERFERENCE C ANCELLATION A. Detection Strategy The principle of soft interference cancellation is shown in Figure 2. The basic idea is to subtract the contributions by y˜2
y˜3
y˜1
σn2ˇ 3 = N0
L(x31 )
CU −R23 s¯3 −R13 s¯3
σn2ˇ 2 L(x21 )
CU −R12 s¯2 σn2ˇ 1
L(x11 ) CU... Computational Unit
CU
Fig. 2. Example of a soft SIC 3 × 3 MIMO detector. Soft symbols are denoted as si . The variance of these soft symbols contributes to the overall 2 . noise σn ˇi
previously detected signal from the received signal [3]. For this interference cancellation we do not use hard decided symbols, as this would lead to unjustifiable error propagation. We rather use soft symbols resulting from an expectation computation, as well as the variance of these symbols which directly relates to their reliability [2] (and thus the expected residual noise after interference cancellation). In order to perform the interference cancellation, we have to transform the channel matrix H such that it is causal with respect to the antenna index t, i.e., the currently detected symbol depends only on previously detected signals. It is obviously advantageous to detect the strongest antenna first, in order to minimize the effects of error propagation. For this purpose we use the MMSE based sorted QR decomposition introduced in [7]. To reduce the computational effort, we suggest not to use the post-sorting-algorithm since it does not significantly increase the performance in a coded system. Since Q is a unitary matrix, the distance computation in (6) can now be rewritten as ky − Hsk2 = k˜ y − Rsk2 , where y ˜ = QH y.
Exploiting the upper triangular structure of R, the L-value from the detector is given by: Tx
2 X 1
L(xtn |y) ≈ max − 2 y˜t − R(t,t) st − R(t,j) s¯j s ,x =+1 t
σnˇ t
tn
+
N X
j=t+1
ln P [xtn ] −
n=1
max
st ,xtn =+1
Qu
Qu 0000
0100
1100
1000
0000
0100
1100
1000
0001
0101
1101
1001
0001
0101
1101
1001
0011
0111
1111
1011
0010
0110
1110
1010
In
In
··· .
0011
0111
1111
1011
0010
0110
1110
1010
(7)
where the a-priori bit probabilities are defined as [4]: LA (xtn ) + Atn . (8) 2 The normalization term Atn can be omitted as it cancels out in the calculation of the detector L-values in (7). As stated before we use soft interference cancellation. The desired soft symbol is given by: ln P [xtn ] := xtn
s¯j
M X
:= E[sj ] = M X
=
m=1 N Y
sm ·
m=1
sm · P [sm ] P [xmn ]
(10)
(11)
Note that we use the L-value including channel state information and a-priori knowledge for computing the soft bit and thus attain a good estimation for the transmitted symbol. Since we cannot expect perfect cancellation we have to deal with residual interference after the cancellation step. This leads us to a new channel model where the Gaussian term is replaced by an interference-plus-noise term [2]: Tx X
R(t,t) st + n ˇ t = R(t,t) st +
Search areas for a 16-QAM constellation
a system using higher order constellation such as 64-QAM or even 256-QAM, the task of computing the expectation and variance Rx − 1 times during interference cancellation requires a tremendous effort – far too high for a practical implementation. We will show in the following section how this effort can be reduced substantially. B. Complexity Reduction
where sm represents one possible complex transmitted symbol from C. The required bit probabilities P [xmn ] are defined by: 1 + xtn · x ¯tn 2 with the help of the so called soft bit: L(xtn ) x ¯tn = E{xtn } = tanh . 2
Fig. 3.
(b) Second search area based on the a-priori knowledge.
(9)
n=1
P [xtn = ±1] =
(a) First search area around interference reduced received signal.
R(t,j) (sj − s¯j ) + n ˜t.
To reduce the amount of required calculations, two algorithms were developed. The first one reduces the number of distance computations and basically employs a union of two search areas. The second one deals with the task to approximate the soft symbol and variance computations. In the first proposed complexity reduction technique, one search area covers the constellation point closest to the interference reduced received signal and the appropriate counterhypotheses for each bit based on the max-log approximation. Thus one has only to compute N + 1 distances in the first detection-decoding iteration, as shown in Figure 3(a) for a 16-QAM constellation. Here, “x” indicates the received signal point surrounded by the closest constellation point 0001 and the appropriate counter-hypotheses. However, if a-priori information is available, finding the (unbiased) maximum likelihood estimate does not necessarily help, since this point has the minimum distance but does not necessarily maximise the two terms in (7).
j=t+1
(12) The variance of this new noise term is equivalent to the interference-plus-noise power σn2ˇ t introduced at each antenna and is given by: σn2ˇ t
= N0 +
Tx X
Qu 0000
0100
1100
1000
0001
0101
1101
1001 In
2
|R(t,j) | · Var[sj ].
(13)
0011
0111
1111
1011
0010
0110
1110
1010
j=t+1
The variance of the soft symbols can be computed by M X
(14)
Fig. 4. Union of the first and the second search area corresponding to the Figures in 3.
where we can reuse the symbol probabilities required for soft symbol computation in (9). It is easily seen that for
We therefore extend the first search area by a fixed number of constellation points and introduce a second search area
Var[sj ] =
(sm − s¯j ) · P [sm ],
m=1
0
IV. S IMULATION R ESULTS The proposed algorithms were tested by simulating a 4 × 4 MIMO System, with simulation parameters equivalent to the setup in [5]. We assumed the receiver to know the channel perfectly and the channel gains to remain constant over the transmission of one vector QAM symbol but change statistically independently between the different transmitted signal vectors (i.e., ergodic fading). The detector works according to the soft interference cancellation algorithm with the proposed modifications. For the decoding process we used a rate R = 1/2 parallel concatenated G = [7R , 5] turbo code, where 7 indicates the memory two feedback polynomial G(D) = 1 + D + D2 and 5 the memory two feedforward polynomial G(D) = 1 + D2 . The decoder performed 8 internal turbo decoding iterations and the interleaver size is 9216 bits. Moreover we performed 4 detector-decoder-iterations. Figure 5 shows the performance of a 16-QAM coded system for different complexity schemes. The genie bound of such a system would be at 6.2 dB. The full complexity setup represents a loss of approximately 1.4 dB to this bound. In this setting we computed the distance to all 16 possible constellation points and for the expectation and variance computation we also used all 16 possible symbols. In the first complexity reduction setup, the probability threshold for the expectation and variance computation was set to 1% but we still computed all 16 Euclidean distances for the
10
−1
10
−2
Genie 16−QAM
BER →
10
−3
10
−4
10
Full Complexity 16 Distances, 1% Prob.−Threshold min. 5 Distances, 1% Prob.−Threshold 5
5.5
6
6.5
7
7.5
Eb/N0 [dB] →
8
8.5
Fig. 5. Simulation results of a 16-QAM SIC receiver after 4 Det.-Dec.Iterations. Transmission over a 4 × 4 MIMO channel, outer PCC code R = 1/2. The performance degradation due to introducing the proposed complexity reduction techniques is negligible.
metric/L-value computation. One can see that there is no performance degradation due to using this approximation. As for the second complexity reduction technique, we reduced the number of distance computations by considering only the union of our two search areas (the probability threshold was again set to 1%). Interestingly, we see that this results in a performance loss of only less than 0.1 dB, with respect to the full complexity setup – but only needs a fraction of the computation effort. In the setup in Figure 5, a minimum of 5 constellation points were used for the first search area. However, finding the maximum in equation (7) using only a subset of constellation points does not always correspond to the global maximum over all possible symbols. This rule applies especially for higher QAM constellations as shown in Figure 6 for a 256-QAM 0
10
−1
10
−2
10
BER →
based on the a-priori knowledge. The centre of this area is the constellation point corresponding to the sign decision of the a-priori L-values provided by the decoder. The other constellation points are found by bit flipping of the sign decision. These form the constellation points supported by the a-priori information. Let the the sign decision of the apriori knowledge be equal to 0101, as illustrated in Figure 3(b). The corresponding symbol represents the centre of the second search area, surrounded by the constellation points resulting from the proposed bit flipping computation. The second search search area will thus always cover N + 1 constellation points. Finally, the union of the two search areas is used to compute the Euclidean distances and derive the detector L-values, as shown in Figure 4. The second algorithm uses a probability threshold to exclude unlikely symbols from the computation of the soft symbols and their reliability. The basic idea is that all symbols for which the probability of occurrence lies below the threshold will contribute only slightly to the expectation and variance computation. Since we use a binary coded system, one unlikely bit value already excludes half of the possible complex symbols in C. As there is a direct relation between bit probability and symbol probability, one could also use a bit based instead of a symbol based probability threshold. In the reminder of this paper, however, we will stick to the symbol based probability threshold. Both approaches lead to a huge complexity reduction, without any significant loss in performance, as will be shown in section IV.
−3
10
−4
10
256 Distances, 1% Prob−Threshold min. 9 Distances, 1% Prob−Threshold min. 15 Distances, 1% Prob−Threshold 13
13.5
14
14.5
Eb/N0 [dB] →
15
15.5
16
Fig. 6. Simulation results of a 256-QAM SIC receiver after 4 Det.-Dec.Iterations. Transmission over a 4 × 4 MIMO channel, outer PCC code R = 1/2. A higher number of constellation points (as opposed to N +1 points used in 16-QAM transmission) must be considered in metric computation to achieve good performance.
scenario. We therefore increased the number of constellation
points lying in the first search area in this example to 15 instead of 9. This wider coverage area leads to a significant performance improvement for the shown 256-QAM constellation. It is worth mentioning that the genie bound is around 11.5 dB so that there is still a 3.5 dB gap. 20 19
Distance Computations →
18 17 16 15 14 13 12 11 10 10
min. 9 Distances, 1% Prob.−Threshold min. 15 Distances, 1% Prob.−Threshold 11
12
13
14
15
16
Eb/N0 [dB] →
17
18
19
20
Fig. 7. Number of constellation points within the search area for 256-QAM transmission, corresponding to Figure 6.
Considering the complexity reduction, we computed less than 7 distances in the cliff region of interest around 7.6 dB, for a 16-QAM system. We thus achieved a complexity reduction of more than 50%. For higher modulation schemes, e.g. 256-QAM as shown in Figure 7, we even attained a reduction of more than 93% without any significant loss in performance. The influence of the probability threshold over the detectiondecoding iteration for a 256-QAM system is shown in Figure 8. The graphs show the number of symbols which passed the 25
No. of Symbols→
20
15
10
5
0 10
1st Iteration 2nd Iteration 3rd Iteration 4th Iteration 11
12
13
14
15
16
Eb/N0 [dB] →
17
18
19
20
Fig. 8. Impact of the 1% probability threshold on the number of symbols considered in soft symbol computation, as a function of the detector-decoderiterations and SNR. 256-QAM transmission is used, corresponding to the setup in Figure 6.
1% threshold and thus contributed to the soft symbol and variance calculation. Even in the first detector-decoder iteration, where no a-priori information is available, the achieved complexity reduction is more than 90% and increases dramatically
with the iteration index until only a single symbol passes the threshold. On average over the detector-decoder iterations we achieved a reduction in computational effort of more than 96%. V. C ONCLUSION In this paper we presented an iterative equalization system for multiple antenna systems. We used soft interference cancellation based detection, supported by an interference-plus-noise term to mitigate the effects of error propagation. Furthermore, we proposed two algorithms which substantially reduce the computational complexity. The first algorithm uses the provided a-priori knowledge to derive highly accurate L-values, considering only a minimum number of potentially relevant constellation symbols. The second algorithm uses a probability threshold in order to calculate the symbol expectation (soft symbols) and variance (residual noise) at very low complexity. Computer simulation results show that we can reduce the complexity drastically – by over 90%, for 256-QAM transmission – without any significant loss in performance. The proposed algorithms thus make it possible to implement even very high order modulation schemes on modern digital signal processors and thus attain high spectral efficiencies and data rates with MIMO systems. ACKNOWLEDGMENT The authors would like to thank Wolfgang Rave for inspiring ideas and discussions. R EFERENCES [1] S. B¨aro, J. Hagenauer and M. Witzke, Iterative Detection of MIMO Transmission Using a List-Sequential (LISS) Detector, Proceedings of the IEEE International Conference on Communications (ICC’03), vol. 4, pp. 2653-2657, May 2003. [2] W. J. Choi, K. W. Cheong and J. M. Cioffi, Iterative Soft Interference Cancellation for Multiple Antenna Systems, Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC’00), No. 1, pp. 304-309, 2000. [3] A. Gersho and T. L. Lim, Adaptive Cancellation of Intersymbol Interference for Data Transmission, Bell Systems Technical Journal, vol. 60, pp. 1997-2021, 1981. [4] J. Hagenauer, E. Offer and L. Papke, Iterative Decoding of Binary Block and Convolutional Codes, IEEE Transactions on Information Theory, vol. 42, pp. 429-445, Mar. 1996. [5] B. M. Hochwald and S. ten Brink Achieving Near-Capacity on a MultipleAntenna Channel, IEEE Transactions on Communications, vol. 51, No. 3, pp. 389-399, Mar. 2003. [6] P. Robertson, E. Villebrun and P. Hoeher, A comparison of optimal and suboptimal MAP decoding algorithms operating in the log domain, Proceedings of the IEEE International Conference on Communications (ICC’95), pp. 1009-1013, Jun. 1995. [7] D. W¨ubben, R. B¨ohnke and V. K¨uhn and K. D. Kammeyer, MMSE Extension of V-BLAST based on Sorted QR Decomposition, Proceedings of the IEEE Semiannual Vehicular Technology Conference (VTC Fall 03), Oct. 2003.
