frequency-domain sc/mmse iterative equalizer with mf ...

FREQUENCY-DOMAIN SC/MMSE ITERATIVE EQUALIZER WITH MF APPROXIMATION IN LDPC-CODED MIMO TRANSMISSIONS † Graduate

Toshiaki Koike† , Hidekazu Murata‡ , and Susumu Yoshida† School of Informatics, Kyoto University, Yoshida-hommachi, Sakyo-ku, Kyoto, 606-8501 Japan ‡ Graduate School of Science and Engineering, Tokyo Institute of Technology Ookayama 2–12–1, Meguro-ku, Tokyo, 152-8552 Japan † {koike, yoshida}@hanase.kuee.kyoto-u.ac.jp, ‡ [email protected]

Abstract— Multiple-input multiple-output (MIMO) transmission systems, exploiting multiple antennas at both Tx and Rx sides, have potential to dramatically increase the spectral efficiency in richscattering wireless links. At the receiver, the so-called space-time equalizer can adaptively deal with dispersive channels and jointly decode multiplexed signals. Recently, low-density parity-check (LDPC) codes have drawn lots of attention for achieving excellent error-correction performance by means of the low-complexity belief propagation (BP) algorithm. This paper explores an iterative spacetime equalizer employing soft cancellation followed by frequencydomain linear filtering, and evaluates the error-rate performance in LDPC-coded MIMO transmissions over time-varying frequencyselective fading channels.

I. I NTRODUCTION Digital wireless communications exploiting diversity of space, time and frequency have been extensively investigated in order to achieve high capacity. For wideband transmissions, frequency-selective fading channels cause severe intersymbol interference (ISI) impairments. Equalization techniques such as decision feedback equalizer [1, 2] and maximum likelihood sequence estimation (MLSE) equalizer can deal with ISI channels and yield path diversity gains. This paper describes a low-complexity frequency-domain (FD) iterative equalizer employing minimum mean-square error (MMSE) linear filter and matched filter (MF). More recently, multiple-input multiple-output (MIMO) transmission systems using multiple antennas have received much attention for achieving highly spectral efficient communications. Although this scheme provides good performance in rich-scattering environments, performance degradation occurs under correlated Nakagami-Rice fading channels due to reduction of space diversity gains. For improving diversity order, several attractive schemes have been studied, such as coded MIMO transmissions making use of time diversity of trellis-coded modulations or turbo-coded modulations. In this coded MIMO transmission system, the space-time (ST) equalizer can exploit inherent diversity effects; path diversity by ISI equalization, space diversity by multiple Rx antennas, and time diversity by decoding. In particular, the MLSE equalizer has excellent performance through the use of the vector Viterbi algorithm extended to super-trellis structure. However, in general, the computational complexity of the MLSE receiver is considerably high, although several complexity reduction schemes such as M-algorithm and sphere decoding [3, 4] are available. To realize a lowcomplexity ST receiver, the separate and iterative approach

of equalization and decoding has been proposed in [5–8], which is referred to as soft cancellation followed by MMSE (SC/MMSE) iterative equalizer. This SC/MMSE equalizer has been investigated in turbocoded MIMO transmissions. However, turbo decoding based on maximum a posteriori (MAP) has high complexity in general. In order to further reduce the complexity, the SC/MMSE equalizer in low-density parity-check (LDPC)coded transmissions is described in this paper. LDPC codes have drawn lots of attention for achieving excellent errorcorrecting performance by using the low-complexity belief propagation (BP) algorithm [9–11]. Furthermore, we introduce an FD linear filter [12–15] and an MF approximation [7] to the SC/MMSE equalizer for decreasing the complexity in significantly dispersive channels. This paper reveals that the FD-SC/MMSE-MF iterative equalizer offers good performance even in time-varying frequency-selective fading channels and asymmetric MIMO transmissions. II. S YSTEM D ESCRIPTION Notations: Throughout this paper, matrices and vectors are described by capital and lower case letters in boldface. Let X and x be a complex matrix and a complex vector, respectively. X T denotes the transpose and X † is the conjugate transpose of X.
x
refers to the Euclidean vector norm of x. The field of complex numbers and the set of natural numbers are designated by C and N, respectively. A. LDPC-coded MIMO Transmission The transmission system using J antenna branches at the Tx unit and D antenna branches at the Rx unit is denoted as a J × D system. Fig. 1 illustrates the LDPC-coded MIMO transmission system, in the case of J = D = 5. The Tx unit simultaneously transmits multiple co-channel signals in equal Tx power from J distinct Tx antennas after applying LDPC codes. Let {uj (k)} be the complex valued LDPC-coded sequence. Under frequency-selective (P + 1)path fading channels, the received sequence rd (k) ∈ C at the d-th Rx antenna and the k-th symbol is represented as rd (k) =

P
J


hp,j,d (k) uj (k − p) + nd (k),

(1)

p=0 j=1

where hp,j,d (k) ∈ C denotes the discrete channel impulse response of the p-th delayed wave (0 ≤ p ≤ P ) between the j-th Tx and the d-th Rx, nd (k) ∈ C is additive white

r

Time Kt

LDPC

LDPC LDPC

Frequency-Selective MIMO Channel

Training Period Information Period

LDPC-coded MIMO transmission system. (5 × 5 system)

Fig. 2.

x

Gaussian noise with variance of σ 2 . Here, the received signal is also expressed as a vector form:

p=0

⎛

K
b −1

Fig. 3.

j

p=0



(3) ⎞

T r(k)ρ
y1 (l), . . . , yD (l) ∈ CD×1 ⎟ ⎜ y(l)
⎟ ⎜ k=0 ⎟ ⎜ K
⎟ ⎜ b −1   ⎟ ⎜ T ⎜ x(l)
u(k)ρkl
x1 (l), . . . , xJ (l) ∈ CJ×1 ⎟ ⎟ ⎜ ⎟ ⎜ k=0 ⎟ ⎜ K
−1 b   ⎟ ⎜ T kl D×1 ⎟ ⎜ v(l)
n(k)ρ
v1 (l), . . . , vD (l) ∈ C ⎟ ⎜ ⎟ ⎜ k=0 ⎟ ⎜ P
⎟ ⎜   ⎟ ⎜ G(l)
¯ p ρpl
g (l), . . . , g (l) ∈ CD×J H 1 J ⎟ ⎜ ⎟ ⎜ p=0 ⎟ ⎜ P ⎟ ⎜
  T ⎠ ⎝ pl D×1 ¯ p,j ρ
gj,1 (l), . . . , gj,D (l) ∈ C g (l)
h kl

y

W

Bit Node J

x

Message Passing

Check Node

u

FD-SC/MMSE

In general, the number of taps of linear equalizer must be increased for overcoming ISI as the delay spread gets large. Consequently, high computational complexity is needed for time-domain (TD) linear equalization depending upon the channel memory P . Under significantly dispersive channels, FD equalization is known to have lower complexity than TD equalization. As shown in Fig. 2, framing a training sequence as a postamble enables the FD equalization. Fig. 3 depicts the signal processing scheme. The Rx unit employs channel identification, channel equalization, signal demultiplexing, decoding and decision by SC/MMSE iterative equalizer. For simplicity, it is assumed that the channel impulse response is fluctuated slowly enough; satisfying  ¯p
h ¯ p,1 , . . . , h ¯ p,J . Through the use of H p (k) ≡ H discrete Fourier transform (DFT), equation (2) is transformed into a frequency-domain expression: ⎛

MMSE Filter

r

B. FD-MMSE Linear Equalizer

y(l) = G(l) x(l) + v(l),

D

S/P

⎞ T  r(k)
r1 (k), . . . , rD (k) ∈ CD×1  T ⎜ ⎟ ⎜ u(k)
u1 (k), . . . , uJ (k) ∈ CJ×1 ⎟  T ⎜ ⎟ ⎜ n(k)
n1 (k), . . . , nD (k) ∈ CD×1 ⎟   ⎜ ⎟ ⎝ H p (k)
hp,1 (k), . . . , hp,J (k) ∈ CD×J ⎠  T D×1 hp,j (k)
hp,j,1 (k), . . . , hp,j,D (k) ∈ C

Hard Decision

IDFT

(2)

u

FD-Soft Replica Generator

DFT

H p (k) u(k − p) + n(k).

Frame format.

DFT

P


y

S/P

r(k) =

Postamble

Kb

IDFT

Fig. 1.

Tx1 Tx2 Tx3 Tx4 Tx5

Space-Time Equalizer with BP Decoding

LDPC

Ki

P/S

H

P/S

u LDPC

BP Decoding

Iterative space-time equalizer with frequency-domain filter.

where ρ
exp(−j2π/Kb ) denotes a rotation operator of DFT, and Kb is the transmission block length in symbol. y(l), x(l), v(l), and G(l) stand for frequency-domain Rx signal, Tx signal, noise signal, and channel impulse response of the l-th frequency component (0 ≤ l < Kb ), respectively.  Here, frequency-domain linear filter W (l)
w1 (l),  . . . , w J (l) ∈ CD×J based on MMSE weights is given by W (l) = G(l) Q−1 (l) =R

−1

(4)

(l) G(l),

Q(l)
G† (l) G(l) + σ 2 I J ∈ CJ×J R(l)
G(l) G† (l) + σ 2 I D ∈ CD×D

(5) 

where I m ∈ Cm×m denotes the m-dimensional identity matrix (m the FD-MMSE filter’s output  ∈ N). Subsequently, T ˜ (l)
x x ˜1 (l), . . . , x ˜J (l) ∈ CJ×1 is calculated as ˜ (l) = W † (l) y(l). x

(6)

Although FD-MMSE weights generation is required for each frequency component, lower-complexity linear equalization can be implemented compared with TD-MMSE, whose computational complexity depends on the channel memory P , especially in dispersive channels with large delay spread. C. FD-SC/MMSE Iterative Equalizer For iteration of ST equalization, the FD-SC/MMSE is applied, which employs FD filtering after canceling the soft ˆ j (l)
g j (l) x replica y ˆj (l), where xˆj (l) is the FD signal corresponding to the TD signal u ˆj (k) from the BP decoder. In the FD-SC/MMSE of the co-channel interference (CCI)

cancellation, the weight for the j-th Tx signal is written as wj (l) =

J


βi g i (l)g †i (l) + σ 2 I D

−1

g j (l),

i=1

where βi means the residual error coefficient of the SC/Simplified-MMSE [8]. Hence, the filter’s output yields  
ˆ i (l) . x ˜j (l) = w†j (l) y(l) − (7) y

TABLE I C OMPUTER S IMULATION PARAMETERS Modulation scheme LDPC coding rate Tx/Rx filter shaping Block length Training length Information length Channel acquisition Channel model

QPSK, 8PSK, 16QAM 2/3 Square-root Nyquist (roll-off: 0.5) Kb = 512 symbols Kt = 130 symbols Ki = 382 symbols Least-squares normal equations Equal level 8-path Rayleigh fading (fD Tb = 512/50000)

i
=j

Since this FD-SC/MMSE does not consider ISI cancellation, the achievable diversity gain is not enough even when the CCI signals are ideally cancelled. Although the soft cancellation of both CCI and ISI signals can be applied even to the FD-SC/MMSE, the computational complexity for weights generation is significantly increased. In consequence, we introduce the above-mentioned FD-SC/MMSE cancelling only CCI, and apply the following FD-SC/MMSE with MF approximation after several iterations for both improving the performance and reducing the complexity. D. FD-SC/MMSE-MF Iterative Equalizer After several iterations, highly reliable replicas can be generated. Therefore, by assuming that ISI and CCI signals are suppressed, an MF approximation technique [7] is applicable even to the FD equalizer for improving the achievable path diversity gain as follows:

 1  † ˆ (l) + γj x x ˜j (l) = g j (l) y(l) − G(l) x ˆj (l) . αj ⎛ ⎞
g j (l)
2 2 α
γ + σ j j ⎜ ⎟ γj ⎜ ⎟ P ⎜ ⎟
 2 ⎝ ⎠ ¯  γj
hp,j p=0

Employing this MF approximation reduces considerably the computational complexity of the FD-SC/MMSE equalizer. In addition, since γj is total power combined with D-branch and (P + 1)-path waves corresponding to the j-th Tx signal, MF approximation can ideally achieve full diversity gains of space and path diversity without residual errors. E. LDPC Code and BP Decoding LDPC code is a type of linear block codes, whose parity-check matrix has low-density non-zero elements. In general, irregular LDPC codes having nonuniform non-zero distributions offer excellent error-correcting performance as well as turbo codes especially in using long code sequence. BP decoding for LDPC is known to be suitable for parallel computations and can be implemented by low-complexity iterative processing, compared with turbo decoding. As LDPC code works as an error-check code, in iterative ST equalizations, several methods for performance improvements and computational complexity reductions are available; generating hard-decision replica, setting residual error coefficient zero, and avoiding iteration for the signal with no errors.

III. P ERFORMANCE E VALUATIONS A. Computer Simulation Parameters Table I lists simulation parameters used in this study. The modulation schemes are QPSK, 8PSK and 16QAM. The LDPC code with coding rate of about 2/3 is used, whose column and row weights are two and six, respectively. The coded sequence has equal length of information period; Ki = 382 symbols long. The training sequence is Kt = 130, and the transmission block is Kb = 512 symbols long. The equal level 8-path Rayleigh fading channel is assumed, in which the normalized maximum Doppler frequency fD Tb is 512/50000, where Tb
Kb Ts denotes block interval and Ts is a symbol period. In the training sequence, channel impulse response is estimated by the leastsquares (LS) normal equations. Furthermore, the average between the estimated channel matrices in the preamble and the postamble is exploited for MMSE weights generation in order to overcome time-varying fading. At the receiver, the FD-SC/MMSE-MF iterative equalizer is employed. B. BER versus Eb /N0 in Symmetric MIMO Transmission In Fig. 4, the bit error rate (BER) performance as a function of average Eb /N0 per Rx antenna is shown under frequency-selective fading 5 × 5 MIMO channels. The number of iterations regarding ST equalization and BP decoding are five and four, respectively. For comparison, the BER performance in single-input multiple-output (SIMO) transmission system of 1 × 5 system is also presented. Notice that the QPSK 5 × 5 system achieves a BER of 10−4 at about Eb /N0 = 0.2 dB, and the degradation from 1 × 5 system is within 0.5 dB at a BER of 10−4 . In the 8PSK 5 × 5 system, a BER of 10−4 is achieved at about Eb /N0 = 2.8 dB, and the degradation from SIMO system is about 0.7 dB. In these QPSK and 8PSK 5 × 5 systems, the FD-SC/MMSE-MF equalizer offers good performance; degradation against SIMO system is within 1 dB in spite of the fact that the spectral efficiency is significantly increased by five times. However, in the 16QAM 5 × 5 system, the degradation is about 3.4 dB when the FD-SC/MMSE-MF iterative equalizer with 5-iteration is used. C. BER versus Eb /N0 in Asymmetric MIMO Transmission Different from the MLSE equalizer, the ST equalizer based on MMSE generally requires more Rx antennas than the number of Tx signals for mitigating a noise enhancement.

0

0

10

10

QPSK (0.0dB) 8PSK (2.5dB) 16QAM (6.0dB)

-1

10

Average Bit Error Rate

Average Bit Error Rate

QPSK 8PSK 16QAM

10-2 5x5 1x5

-3

10

-4

10

10-1

-2

10

10-3

-4

-5

0 5 10 Average Eb/N0 per Rx Antenna (dB)

15

Fig. 4. Average BER performance as a function of average Eb /N0 per Rx antenna under frequency-selective 8-path Rayleigh fading channels. (5 × 5 system, iteration of ST equalization: 5, iteration of BP decoding: 4)

10

0

1

2

3 4 5 6 7 8 Iteration of Equalization

9

10

Fig. 6. Average BER performance as a function of iteration of ST equalization under frequency-selective 8-path Rayleigh fading channels. (5 × 5 system, iteration of BP decoding: 4)

Average Bit Error Rate

100

MIMO systems. Average Eb /N0 per Rx antenna is set to be 0 dB, 2.5 dB and 6 dB for QPSK, 8PSK and 16QAM, respectively. Note that, the BER performance of 1-iteration denotes the performance of BP decoding after just single FD-MMSE linear equalization. In each performance, the BER improves as the number of iterations increases. The performance improvement is sufficiently achieved after about 5-iteration in both of QPSK and 8PSK. On the other hand, in 16QAM 5 × 5 system, a larger number of iterations for ST equalization is required.

-1

10

6x5

10-2 1x5 10-3 QPSK 8PSK 16QAM 10-4

-5

0 5 10 Average Eb/N0 per Rx Antenna (dB)

15

Fig. 5. Average BER performance as a function of average Eb /N0 per Rx antenna under frequency-selective 8-path Rayleigh fading channels. (6 × 5 system, iteration of ST equalization: 5, iteration of BP decoding: 4)

Next, we evaluate the performance of the FD-SC/MMSEMF equalizer in asymmetric MIMO transmission systems, where more Tx antennas than the number of Rx antennas are utilized. Fig. 5 shows the BER performance under frequencyselective 6 × 5 MIMO channels. Note that, in the QPSK 6 × 5 system, the FD-SC/MMSEMF equalizer yields good BER performance; the degradation against the 1 × 5 system is within 1 dB at a BER of 10−4 . In the 8PSK 6 × 5 system, the degradation from the SIMO system is within 2 dB. However, considerable degradation is observed in the 16QAM 6 × 5 system. D. BER versus Iteration of ST Equalization Fig. 6 shows the BER performance as a function of the number of iterations regarding ST equalization in 5 × 5

E. BLER versus Iteration of BP Decoding Considering that BP decoding is expected to be implemented with lower complexity than ST equalization, we set 4-iteration times regarding BP decoding per ST equalization. Fig. 7 plots the block error rate (BLER) performance by changing a parameter of the number of iterations for BP decoding as a function of average Eb /N0 per Rx antenna over frequency-selective fading 5 × 5 MIMO channels. In each MIMO system, the BLER performance is improved as the iteration of BP decoding increases. The performance for 4-iteration is superior to that for 1-iteration by about 1.5 dB, and almost comparable to that for 8-iteration. F. BLER versus fD Tb Finally, we evaluate the performance against the channel fluctuation. In general, since FD-MMSE equalizer performs block-wise filtering, the performance is severely degraded in fast fading channels. In order to alleviate the effect of the channel variation, we introduce the zero-order interpolation of the channel matrix, which averages the LS estimated channel matrix in the preamble and that in the postamble of the transmission block. Fig. 8 shows the BLER performance for the parameter of fD Tb = 1/100, 1/50 and 1/25 as a function of average Eb /N0 per Rx antenna under frequency-selective fading 5×5

0

Average Block Error Rate

10

This result suggests that the middle point interpolation of the channel matrices is a simple but effective approach to improving FD-MMSE equalizer in fast fading channels.

1-BP 2-BP 4-BP 8-BP -1

10

16QAM 8PSK -2

QPSK

10

-3

10

-5

0 5 10 Average Eb/N0 per Rx Antenna (dB)

15

Fig. 7. Average BLER performance as a function of average Eb /N0 per Rx antenna on iteration times of BP decoding in frequency-selective 8-path Rayleigh fading channels. (5 × 5 system, iteration of ST equalization: 5) 100

Average Block Error Rate

1/100 1/50 1/25 10-1

8PSK

16QAM

-2

10

w/o Int.

-5

0 5 10 Average Eb/N0 per Rx Antenna (dB)

ACKNOWLEDGMENT This work is partially supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan, Grant-in-Aid for the 21th Century COE Program (no. 14213201), Grant-in-Aid for Scientific Research (A) (no. 16206040) of JSPS, and JSPS Research Fellowships for Young Scientists.

R EFERENCES

QPSK

10-3

IV. C ONCLUSION This paper has evaluated FD-SC/MMSE iterative equalizer with MF approximation in LDPC-coded MIMO transmissions. Through computer simulations, it has been confirmed that FD-SC/MMSE-MF with about 5-iteration provides substantial diversity gains in frequency-selective MIMO channels. In spite of MF approximation, the good performance can be offered in 5×5 and 6×5 MIMO systems under timevarying 8-path fading channels. In the 8PSK 6 × 5 system, the 12 bps/Hz spectral efficiency at 84 Mbps transmission rate is achievable at about Eb /N0 = 4 dB, provided that the carrier is 5 GHz band, the maximum delay is 1 µs and the mobile speed is around 60 km/h.

15

Fig. 8. Average BLER performance as a function of average Eb /N0 per Rx antenna for the parameter of fD Tb = 1/100, 1/50, 1/25 under frequency-selective 8-path Rayleigh fading channels. (5×5 system, iteration of ST equalization: 5, iteration of BP decoding: 4)

MIMO channels. For comparison, in the case of fD Tb = 1/25, the performance without zero-order interpolation of the channel matrix is also presented, which is designated by “w/o Int.” It is confirmed that the BLER performance for fD Tb = 1/50 has almost comparable performance to that for fD Tb = 1/100 in each transmission system. However, the BLER performance for fD Tb = 1/25 is degraded compared with them especially in high-level constellation; about 0.4 dB degradation in 8PSK 5 × 5 system, whereas about 1.1 dB degradation in 16QAM 5 × 5 system at a BLER of 10−3 . Moreover, the performance without interpolation is severely degraded; about 1.1 dB, 2.1 dB and 6.5 dB degradations at a BLER of 10−3 in QPSK, 8PSK and 16QAM, respectively.

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