Performance Comparison of Layered Space Time Codes

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Performance Comparison of Layered Space Time Codes

Ka Leong Lo, Slavica Marinkovic, Zhuo Chen and Branka Vucetic

The School of Electrical and Information Engineering, The University of Sydney, NSW 2006, Australia

Abstract—Multiple antenna systems have a potential to provide a high capacity in wireless communication systems. The spectral efficiency of space time trellis coding (STTC) is limited by the en- coder structure. The layered space time (LST) architecture can overcome this problem. Three different LST schemes are pre- sented. An improved iterative parallel interference canceller (PIC) method is applied at the receiver. A significant performance im- provement is achieved compared to the standard PIC. Simulation results of three various layer structures are compared with low density parity check (LDPC) and convolutional codes as compo- nent codes.

I. INTRODUCTION

In [1], Foschini and Gans showed that a high bandwidth efficient communication can be achieved in a wireless fading environment with multi-transmit and receive antennas. Fur- thermore, the horizontally (HLST) and diagonally (DLST) layered space time structure were proposed to realize this con- cept [2]. In LST system, the received signal is a superposi- tion of the transmitted coded symbols scaled by the multipath fading coefficients and corrupted by additive white Gaussian noise (AWGN). Interference suppression and interference cancellation techniques are employed in the detector and each constituent code can be decoded individually [2], [3]. In addition, a great performance improvement can be realized by using iterative detection/decoding techniques [4], [5], [6].

Recent simulation results [8], [9] show that performance of LDPC is close to that of turbo codes. The important property of LDPC codes is that the minimum Hamming distance of the code increases linearly with the code length [7]. A simple and effective iterative probabilistic decoding algorithm is used to decode this class of codes. The decoding complexity is linearly proportional to the code length. Frame errors are easily detected by the parity check syndrome.

In this paper, the simulation results of the three proposed LST structures are compared. The LST structures with LDPC codes are compared to the respective structures with convolutional codes. The simulation results show that the architectures with LDPC codes outperform those with convolutional codes if there is no interference from other antennas. This can be

This project was supported by Norman I Price Fellowship and Nortel Net- works.

approached by a system with sufficient number of receive antennas. It shows that the performance of LST is dominated by minimum squared Euclidean distance

of the component code when the diversity gain is large. This result is consistent with the conclusion in [10] and [11]. Furthermore, an improved iterative PIC [12] is applied as the detector in LST structures and is shown to achieve the performance of a minimum mean square error (MMSE) iterative receiver, although the complexity of PIC is much lower than that of MMSE.

.

M

Encoder

Encoder 1

Encoder

Encoder 1 Source

S/P Source

S/P

(LST-b)

(LST-a)

Source

M M M

S/P

M

(LST-c) Encoder

Fig. 1. Layered space time transmit architectures

The remainder of the paper is organized as follows. Sec- tion II reviews the system model of LST structures. Section III describes three different proposed LST architectures. Sec- tion IV presents the iterative standard PIC, improved PIC and MMSE detections. The improved PIC detector with combining is shown to achieve MMSE performance with similar complexity of standard PIC. Section V presents the simulation results and the conclusion is given in section VI.

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II. SYSTEM MODEL

We consider a wireless system with transmit and ^!

receive antennas in a quasi-static flat Rayleigh fading fading channel. We assume that the fading coefficients remain the same over a frame and change independently from one frame to another. The information bits are encoded by the constituent code(s) to generate a matrix C of rows. Then each row of C is interleaved independently, modulated and transmitted by a separate antenna. The transmitted symbol of antenna^", at time

# , is denoted by^$

%. H represents the complex channel matrix with dimension! '&() and entries, denoted by^*+ , represent the fade coefficient from transmit antenna^" to receive antenna

,

. The signal at receive antenna^, at time^# is given by

- +

%/.

0

1

*+

$

%32

+

%547698

" 8

)

4:698

, 8

) ; (1)

where ⁺^% is a statistically independent sample of AWGN with identical variance at each of the receive antenna. The receiver is partitioned into a detector and a decoder for each layer. The iterative decoding principle is applied to pass the probability estimates between the detector and decoders.

III. LAYERED SPACE TIME ARCHITECTURES

The transmitters and receivers of three LST structures are shown in Figs. 1 and 2, respectively. In the structure denoted by LST-a [5], a single channel code is employed. After serial to parallel conversion, the encoded bit streams are fed into

) independent interleavers (

4

;<;

4

). Each interleaved sequence is mapped by a symbol mapper M and transmitted through a permanently assigned antenna. The LST-a receiver shown in Fig. 2 first detects the transmitted streams, de- interleaves each stream and converts parallel streams into one serial stream. A channel decoder operates on the output of the serial to parallel conversion and generates soft decoded sequence. Each of the decoded sequence is interleaved and fed back to the detector.

In the structure denoted by LST-b [2], [3], [4] [5], the information stream is first demultiplexed into parallel sequences, each of which is independently encoded, interleaved, symbol mapped and transmitted through a permanently assigned antenna. At the receiver, the data streams are detected and independently de-interleaved and decoded. This scheme is equivalent to the HLST scheme in [2], [3] and [5].

The structure, denoted by LST-c [4] is obtained from LST- b by introducing a spatial interleaver prior to time interleavers

4

;<;

4

as shown in Fig . 1. Hence the codeword symbols of each encoder are transmitted over different antennas. Let us consider a system with ^.>= . The operation of can be

expressed as

?@

A!B

B C B D B

EGFHFHF

B C B CC B CD B CE

FHFHF

B D B DC B DD B DE

FHFHF

B E B EC B ED B EE

FHFHF IKJ

L

;MON

?@@@@@@A B B EC B DD B CE . ..

B C B C B ED B DE . ..

B D B CC B

D B EE . ..

B E B DC B CD B

E . ..

I JJJJJJ

L

(2)

in which an element of the codeword matrix, denoted by ^P

%, represents the encoded bit of layer^" at time^#. Similarly, the

decoupled data streams are de-interleaved and decoded at the receiver. This structure is similar to the DLST architecture.

However, unlike the DLST in [2], [3] and [5], the left bottom of the encoded matrix C is not empty and thus this scheme is more spectrally efficient than the DLST.

QSRTU

RTU

V

(LST-c) (LST-b) (LST-a)

S/P P/S

Decoder

Decoder 1

Decoder 1 Decoder

RTU

V

W

X

W X

WS

R V

R U

Y R V

R U

QR

W X R TU

U

Y

WZ

[

Y

Decoder

RTU

V

Det

R TU

U

R TU

U R V

R U

Det

Fig. 2. Layered space time receiver architectures

We assume that the component codes in layers are identical for both LST-b and LST-c structures. In all schemes, the receiver uses an iterative joint detection and decoding.

For example, the LST-b detector (Det) shown in Fig. 2 re- ceives sequences,^W ⁴ ^;;<;⁴ ^W

4

;;<;

4W

SX

, where^W

is given by^W

.

\- 4 - 4

;;<;

4 -

]_^, representing the received sequence for ^"th receive antenna, of length^`. The decoder estimates the transmitted sequences,^a ⁴ ^;;<;⁴ ^a

4

;;<;

4 a

, where^a

is the estimated sequence transmitted by antenna^". After demodulation and dein- terleaving, the transmitted sequence estimates form the input to the channel decoder. The soft output of the decoder is interleaved and fed-back to the detector. In the next iteration, the interference caused by symbols from all other antennas is can- celled by subtracting the soft decoder output on these symbols from the received signal.

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IV. ITERATIVEDETECTION

We consider a standard parallel interference canceller (PIC- STD) [12], [13], its improved version referred as, a parallel interference canceller with decision statistics combining (PIC- CMB) [12] and an MMSE detector [14], [15], [16], as the signal detectors. These detectors are chosen because they offer a good performance-complexity trade-off, particularly when the number of transmit antennas is high and the optimal joint detection and decoding becomes impractical.

In the first iteration, the PIC detectors are equivalent to a bank of matched filters, matched to a vector of channel gains. In the second and later iterations, the estimated mean of the transmitted symbols, from the decoder output is used for interference cancellation.

The PIC-STD detector output in the ^b th iteration for the symbol transmitted at time ^#, is arranged in a vector ^!c^a ^%d.

\ a

Hec

%f4

;;<;

4 a

ec

% ^

and is given by

a

c

% .hg

ji

W

%k

g a

l

cm3

% n 4 (3)

where^W ^% ^.

\

-S

% 4

;;<;

4 -

SX

% ^

is a ^& ⁶ vector of the received

signals and^l^a ^cm)^% ^.

\ a

o

pecm3

% 4

;<;;

4 ao ecm)

% ^

is an ^& ⁶ vector with symbol estimates as elements except for the^"th element which is set to zero. The estimates of the transmitted BPSK symbols are calculated by finding the mean

a

o ec

%

.hq c \o %

.r6s

a

ec ^ k q c \o % . k

6Os

a

ec

^p4 (4)

where^a

ec

is a vector of detector outputs in the ^b th iteration for transmit antenna^" and^q ^c

\o

%(.t6s

a

ec ^ is the a posteriori probability calculated by the decoder in the^b th iteration.

An iterative parallel interference canceller with combining was proposed in [12] for iterative joint detection and decoding to a direct sequence code division multiple access system.

The signal at the detector output is modelled as a Gaussian ran- dom variable with conditional mean^u

ec and variance^v

ec . In

a high interference scenario which occurs in a system with a large number of transmit antennas, the detector output in the second iteration becomes significantly biased towards the decision boundary. That is, the conditional mean of the decoder input signal in the second and later iterations is reduced from its nominal value. The other important observation is that for a large number of layers, detector outputs in consecutive iterations are low correlated in early iterations. Under these condi- tions, a combining method resembling the maximum ratio combining, gives an improved signal-to-noise ratio (SNR).

The optimal weighting coefficients, which maximize SNR after combining, are difficult to estimate or calculate. A practi- cal combining algorithm is obtained by using coefficients which depend only on the variances of the detector outputs in consecutive iterations.

An improved PIC estimate is formed by recursively combining the PIC estimate ^$^a

ec

% in the current iteration and an im-

proved PIC estimate

\ a$ ecm)

% ^xw from the previous iteration

\ a

$ ec

% ^w . \v

ecm)

^xw

\v

ecm)

^ w 2 v ec a

$ ec

% 2 v ec

\v ecm)

^ w 2 v

ec \ a

$ ecym3

% ^ w

(5) where^v ec and

\v

ecym3

^ w are the variances of the PIC estimate in the^b ^# ^* and improved PIC estimate in the

\b k

6^

# * iterations, respectively. The complexity of both PIC-STD and PIC-CMB is linear with the number of transmit antennas.

An iterative MMSE receiver performs parallel interference cancellation and then residual interference suppression [4]. The detector coefficients of an iterative MMSE receiver are derived by minimizing the mean square error between the transmitted signal and a detector output. In the first iteration, transmitted symbols are assumed to be uniformly distributed and hence their a priori probabilities are constant. In later iterations, they are recalculated using the decoder soft outputs in the previous iteration. The updated probabilities are then used to generate the new set of filter coefficients. The MMSE detector output in^b th iteration and for the symbol transmitted at time^# can be expressed as ^z ^c^% ^.

\|{

c% ^ a

!c

% , where ^a^!c^% is the parallel interference canceller output given by (3) and

{ c% is the matrix of MMSE filter coefficients. The detector filter coefficients for the transmit antenna^" are given by

{ ec

% .~}g

!

% gd2

v

H

m33

(6) where

is a column vector of all zeros except the^"th element which is⁶ and

% is the covariance matrix between the transmitted and estimated symbols in different layers, given by

% .

"

\6 k \ao

Hec

% ^

46

k \ao ec

% ^ 4

;<;;

4K6

k \ a

o

m3pec

% ^ 4

6S4K6

k \ a

o

<

pec

% ^ 4

;<;<;

46

k \ao ec

% ^ ^ ; (7)

The direct implementation of the iterative MMSE receiver based on matrix inversion requires a complexity of polynomial order in the number of transmit antennas [4]. However, for quasi-static fading channels, it is possible to implement adap- tive MMSE receivers, the complexity of which is linear in the number of transmit antennas.

V. SIMULATIONRESULTS FORLST CODES

A. Comparison of LST Structures

In this section, we present simulation results of the three considered LST architectures in a multi-path quasi-static fading channel. A rate 1/2 regular LDPC code with information length^S and a block-length^S ⁼ is chosen as the constituent code. The parity check matrix ^g) is constructed as in [7]. Hence, the matrix ^g) has a fixed column weight

. and a fixed row weight ^. . The binary encoded bits are Gray mapped and QPSK modulated prior to transmission, hence there are 252 symbols per frame. PIC-CMB is employed

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1e-04 1e-03 1e-02 1e-01 1e+00

0 2 4 6 8 10 12 14

Frame error rate

Eb/No (dB)

LST-b LST-a LST-c

Fig. 3. Performance comparison of three different LST for (^O ) = (4, 4)

as the detector in the iterative receiver. Five iterations are used between the detector and the decoder. These system param- eters are assumed in all simulations unless otherwise stated.

The bandwidth efficiency is given by ^.^¡ ^. ^)¢

o .

)

(^o /sec/Hz), where^-y£ is the data rate,^¤ is the bandwidth,^¢ is the code rate and^o is the number of bits per symbol in the signal set.

Fig. 3 shows the frame error rate (FER) performance of the three LST structures with ( ⁴ ) = (4, 4). As shown in Figs.

3 the LST-c has a superior performance compared to LST-a and LST-b. This can be attributed to a higher diversity gain of LST- c than that of the LST-b and its lower sensitivity to the decoder errors than the LST-a. Higher diversity gain is achieved in LST- c because the encoded codeword is transmitted by antennas.

Similar to LST-b, received sequences are decoded by channel decoders to provide more reliable feedback in the iteration structure. LST-a obtains a higher diversity gain than LST-b because the encoded sequence is transmitted through different antennas. Therefore, LST-a performs better than LST-b. However, LST-a is more sensitive than LST-b and LST-c to the decoder errors [5]. For example, in LST-b if three of four decoded sequences converge to the correct codewords in the first few iterations, the incorrect layer is likely to converge in later iterations.

On the other hand, if a single decoded sequence does not converge in LST-a, the detector will receive an unreliable feedback sequence. The decoder input of the next iteration in this case may not be improved. The error floor of LST-a in Fig. 3 is caused by this effect.

B. Comparison of Different Detection Techniques

In this section, we compare the performance of PIC-STD, PIC-CMB and MMSE for LST-b structure in a system with (⁾ ⁴ ⁾ ) = (4, 4), BPSK modulation and rate 1/2 LDPC as in the previous section. Fig. 4 shows that a significant error floor is observed with PIC-STD at high^¥ ^{£H¦¨§ª©} . In Figs. 5 and 6, no error floor is present with either PIC-CMB or MMSE. It can also be observed that MMSE converges slightly faster than PIC-CMB, but both detectors achieve FER of⁶ ^m¬« at less than

1e-03 1e-02 1e-01 1e+00

0 2 4 6 8 10 12 14

Frame error rate

Eb/No (dB)

PIC-STD 1st itr PIC-STD 2nd itr PIC-STD 3rd itr PIC-STD 4th itr PIC-STD 5th itr

Fig. 4. FER simulation result of LST-b with LDPC and PIC-STD for (^! )

= (4, 4)

12 dB. The main advantage of the proposed PIC-CMB detector is its low complexity compared to MMSE. The complexity of the MMSE detector generally as^®

\ « ^ , although it can be reduced to ^®

\ ^ as shown in [15], while the complexity of both PIC-STD and PIC-CMD are linearly proportional to the number of transmit antennas. In summary, PIC detector with combining has a complexity of standard PIC and achieves the MMSE performance.

1e-03 1e-02 1e-01 1e+00

0 2 4 6 8 10 12 14

Frame error rate

Eb/No (dB)

PIC-CMB 1st itr PIC-CMB 2nd itr PIC-CMB 3rd itr PIC-CMB 4th itr PIC-CMB 5th itr

Fig. 5. FER simulation result of LST-b with LDPC and PIC-CMB for

(^O)¯O ) = (4, 4)

C. Comparison of LST Structures with Different Constituent Codes

In this section, we compare the performance of three LST structures and decoding complexity of convolutional and LDPC codes. A rate 1/2 convolutional codes with memory order^° ^. is considered. We use ( ⁴ ^b ⁴ ^° ) to denote a rate^b ^¦ convolutional code with memory^° . The generator polynomial in oc- tal form of this code is (53, 75), and the free Hamming dis- tances^O±

H²x²

is 8. The of the LDPC code ( ^.r , ^.r ) is calculated as about 11 [7]. The ^O< and the of these codes are given in Table I. The maximum a posteriori (MAP) and sum-product algorithms are employed to decode convolutional and LDPC codes, respectively. The MAP algorithm

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1e-03 1e-02 1e-01 1e+00

0 2 4 6 8 10 12 14

Frame error rate

Eb/No (dB)

MMSE-1st itr MMSE-2nd itr MMSE-3rd itr MMSE-4th itr MMSE-5th itr

Fig. 6. FER simulation result of LST-b with LDPC and MMSE for (^!O )

= (4, 4)

TABLE I

COMPARISON OF DISTANCE PROPERTY OF CONVOLUTIONAL CODE WITH

³µ´·¶

AND(^¸ ^´'¹ ,^º ^´¼» ) LDPCCODE WITH CODEWORD LENGTH500

Conv.^° ^. LDPC

<

8 11

16 22

[17] is an optimal trellis based decoding method which mini- mizes the symbol error probability. A MAP algorithm is used in iterative detection and decoding process. An LDPC code is represented by a factor graph or a belief network [7], [18], [8]. The sum-product algorithm is a probabilistic suboptimal method for decoding graph based codes. This is a syndrome decoding method which finds the most probable vector to sat- isfy all syndrome constraints. Although the soft-output of this decoder is not the true a posteriori probability because the factor graph of an LDPC code has many cycles, a good error correcting performance is achieved [19]. Furthermore, the probabilistic soft-input of the sum-product algorithm is calculated based on the channel model, CSI and the noise variance. Hence, the decoding algorithm is sensitive to channel estimation errors.

The decoding complexity of MAP algorithm increases ex- ponentially with the memory order^° . On the other hand, the complexity of decoding LDPC code is linearly proportional to the number of entries in the parity check matrix^g ^½) . Ta- ble II shows the complexity of the two decoding algorithms [17]. It shows the numbers of operations in a single decoding process. The total numbers of operations for an information bit in the MAP decoder are listed in column one. The sum- product decoder is an iterative algorithm, so column two shows the numbers of operations for an information bit in each sum- product iteration. Table II shows that LDPC decoder applies exponential operations to initialize the probabilities of the input sequence while the number of exponential operations of the MAP decoder is proportional to^O¾ . In the simulations, we ob- serve that in most cases, an LDPC code converges within 10

TABLE II

COMPLEXITY OFMAPANDSUM-PRODUCT ALGORITHM

MAP(per info. bit)Sum-Product(per info. bit in each iteration)

add. ^/¿/¿ ^¾ ^2À ^Ã ^«KÁ^Â

pmÄÂZÅÇÆÇÈ

multpl. ^/¿y/¿ ^¾ ^2ÀÉ ^ÂÁ

Ã Â Æ È

Ã

pmÄÂZÅÇÆÇÈ

exp. /¿/¿¾

2ÀÉ

pmÄÂZÅÇÆ

(initialization)

iterations. The numbers of decoding operations of the codes with codeword length 500 are listed in Table III, along which the results of sum-product algorithm, calculated on the basis of 10 iterations. Fig. 7 shows the performance of three LST

TABLE III

COMPLEXITY COMPARISON BETWEEN CONVOLUTIONAL CODE WITH

³µ´'¶

AND(^¸ ^´'¹ ,^º ^´'» ) LDPCCODE WITH CODEWORD LENGTH500

MAP^° ^. Sum-Product

add. 33500 45000

multpl. 82000 210000

exp. 32000 500

structures with ( ⁴ ) = (4, 4) and (2, 1, 5) convolutional code. It is shown that LST-c outperforms LST-b considerably and LST-a slightly. We compare the performance of LDPC and

1e-03 1e-02 1e-01 1e+00

0 2 4 6 8 10 12

Frame error rate

Eb/No (dB)

Conv (2, 1, 5) LST-b Conv (2, 1, 5) LST-a Conv (2, 1, 5) LST-c

Fig. 7. Performance comparison of three different LST structures with the (2,1,5) convolutional code as a constituent code for ( ) = (4, 4)

convolutional codes. Table IV shows the required^¥ ^£ ^¦p§ ^© for FER of⁶ ^m¬« in three LST structures with ( ⁴ ) = (4, 4). In LST-b, the LDPC outperforms the convolutional code but has a worse performance compared to (2, 1, 5) convolutional code in both LST-a and LST-c structures. Although the LDPC code has a higher distance than the convolutional codes, it has a worse performance. In addition, we notice in Fig. 3 that there exist error floors for the LDPC code in LST structures with^Ê ^.Ë= . However no error floor occurs for convolutional code in Fig. 7.

The reason for that is that the sum-product algorithm is more sensitive to decoding errors than the MAP decoder used for

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convolutional codes. As mentioned before, it is a probabilistic suboptimal algorithm and it is very sensitive to the estimation decoder errors.

TABLE IV

PERFORMANCE COMPARISON BETWEEN CONVOLUTIONAL CODE WITH

³µ´'¶

ANDLDPCCODE INLSTSTRUCTURES FOR(^O! ) = (4, 4)

Conv.^° ^. LDPC

LST-a 8.0 9.2

LST-b 11.6 11.0

LST-c 7.6 8.8

LST-c (perfect decoding feedback) 8.2 4.9

1e-04 1e-03 1e-02 1e-01 1e+00

-2 0 2 4 6 8

Frame error rate

Eb/No (dB)

Conv. (2, 1, 5) nR=4 (prefect decoding feedback) LDPC nR=4 (prefect decoding feedback) Conv. (2, 1, 5) nR=8 LDPC nR=8

Fig. 8. Performance comparison of LST-c with convolutional and LDPC codes for (^O ) = (4, 4) with perfect decoding feedback and (^O! ) = (4, 8)

Fig. 8 shows the performance of the two codes in LST-c with no interference. The LDPC code outperforms the convolutional code significantly because the and ^Ì< of the LDPC code are larger than those of the convolutional code. The last row of Table IV shows the required^¥ ^£ ^¦p§ ^© (in dB) of both codes achieving FER of⁶ ^mÍ« in the (4, 4) LST-c system with perfect decoding feedback. It shows that the performance difference between perfect and non-perfect decoding feedback of convolutional and LDPC codes are about 0.4 and 3.9 dB, respectively.

This means that the iterative joint detection and MAP decoding algorithm approaches no interference performance. As the number of receive antennas increases, the detector can provide better estimates of the transmitted symbols to the channel decoder. In this situation, the distance of the code dominates the LST system performance. Fig. 8 shows that the LDPC code outperforms this convolutional code in a (4, 8) LST-c system.

We conclude that the LDPC code has a superior error correction capability, but the performance is limited by feedback decoding errors in LST-a and LST-c structures.

VI. CONCLUSION

We have compared and shown the simulation results of three different LST structures. LDPC and convolutional codes are

used as constituent codes. The LST-c is shown to achieve the best performance because it has a high diversity gain and low sensitivity to the decoder errors. Furthermore, we have shown that the PIC detector with decision statistics combining has a complexity of the standard PIC. We conclude that it achieves the MMSE performance with much lower complexity. The decoding complexity and performance comparisons of LDPC and convolutional codes are also given. LDPC codes with good distance properties and low decoding complexity outperform convolutional codes, but the performance is limited by the feedback decoding errors.

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