Space-Time Coding phần 5 potx

Figure 3.7 Bit error rate performance for STBC of 3 bits/s/Hz on Rayleigh fading channels withone receive antenna... Figure 3.8 Symbol error rate performance for STBC of 3 bits/s/Hz on R

108 Space-Time Block Codes

3 bits/s/Hz and a variable number of transmit antennas are shown in Figs 3.7 and 3.8,respectively The performance of an uncoded 8-PSK is plotted in the figures for comparison

Figure 3.7 Bit error rate performance for STBC of 3 bits/s/Hz on Rayleigh fading channels withone receive antenna

Figure 3.8 Symbol error rate performance for STBC of 3 bits/s/Hz on Rayleigh fading channelswith one receive antenna

For transmission with two transmit antennas, the rate one code Xc2 and 8-PSK modulation

are employed For three and four transmit antennas, 16-QAM and the rate 3/4 codes Xh3and Xh4 are used, respectively Therefore, the transmission rate is 3 bits/s/Hz in all cases.For Fig 3.7, we can see that at the BER of 10−5, the code Xh

4 is better by about 7dB and

2.5 dB than the code Xc2and the code Xh3, respectively

Figures 3.9 and 3.10 show BER and SER performance, respectively, for STBC of 2bits/s/Hz with two, three, and four transmit antennas and one receive antenna on Rayleigh

fading channels The STBC with two transmit antennas is the rate one code Xc2with QPSK

modulation The STBC with three and four transmit antennas are the rate 1/2 codes Xc

3and

Xc

4, respectively, with 16-QAM modulation It can be observed that at the BER of 10−5,

the code with four transmit antennas gains about 5 dB and 3 dB relative to the codes withtwo and three transmit antennas, respectively

The BER and SER performance for the codes with 1 bit/s/Hz, a variable number ofthe transmit antennas and a single receive antenna are illustrated in Figs 3.11 and 3.12,

respectively The STBC with two transmit antennas is the rate one code Xc2 with BPSK

modulation The STBC with three and four transmit antennas are the rate 1/2 codes Xc3and

Xc4, respectively, with QPSK modulation It can be observed that at the BER of 10−5, the

code with four transmit antennas is superior by about 8 dB and 2.5 dB to the codes withtwo and three transmit antennas, respectively

The simulation results show that increasing the number of transmit antennas can provide

a significant performance gain The increase in decoding complexity for STBC with a largenumber of transmit antennas is very little due to the fact that only linear processing isrequired for decoding In order to further improve the code performance, it is possible toconcatenate an outer code, such as trellis or turbo code, with an STBC as an inner code

110 Space-Time Block Codes

Figure 3.9 Bit error rate performance for STBC of 2 bits/s/Hz on Rayleigh fading channels withone receive antenna

Figure 3.10 Symbol error rate performance for STBC of 2 bits/s/Hz on Rayleigh fading channelswith one receive antenna

Figure 3.11 Bit error rate performance for STBC of 1 bits/s/Hz on Rayleigh fading channels withone receive antenna

Figure 3.12 Symbol error rate performance for STBC of 1 bits/s/Hz on Rayleigh fading channelswith one receive antenna

112 Space-Time Block Codes

3.8 Effect of Imperfect Channel Estimation

on Performance

In this section, the effect of imperfect channel state information on the code performance

is discussed We start with the description of the channel estimation method used in thesimulations [6] The channel fading coefficients are estimated by inserting pilot sequences

in the transmitted signals It is assumed that the channel is constant over the duration of

a frame and independent between the frames In general, with n T transmit antennas we

need to have n T different pilot sequences P1, P2, , P n T At the beginning of each frame

transmitted from antennas i, a pilot sequence P i consisting of k symbols

P i = (P i,1, P i,2, , P i,k ) (3.64)

is appended Since the signals at the receive antennas are linear superpositions of all

trans-mitted signals, the pilot sequences P1, P2, , P n T are designed to be orthogonal to eachother

During the channel estimation, the received signal at antenna j and time t can be

rj = (r j

1, r2j , , r k j )

nj = (n j

The receiver estimates the channel fading coefficients h j,iby using the observed sequences

rj Since the pilot sequences P1, P2, , Pn T are orthogonal, the minimum mean square

error (MMSE) estimate of h j,i is given by [6]

Since n j t is a zero-mean complex Gaussian random variable with single-sided power spectral

density N0, the estimation error e j,i has a zero mean and single-sided power spectral density

N0/ k [6]

Effect of Antenna Correlation on Performance 113

Figure 3.13 Performance of the STBC with 2 bits/s/Hz on correlated slow Rayleigh fading channelswith two transmit and two receive antennas

The performance of the STBC with imperfect channel state information at the receiver is

shown in Fig 3.13 In the simulation, QPSK modulation and the rate one code Xc2with twotransmit and two receive antennas are employed It is assumed that the channel is described

as a slow Rayleigh fading model with constant coefficients over a frame of 130 symbols.The pilot sequence inserted in each frame has a length of 10 symbols The simulation resultsshow that due to imperfect channel estimation, the code performance is degraded by about0.3 dB compared to the case of ideal channel state information Note that the degradation

in code performance also accounts for the loss of the signal energy by appending the pilotsequences

If the number of transmit antennas is small, the performance degradation due to thechannel estimation error is small However, as the number of transmit antennas increases,the sensitivity of the system to channel estimation error increases [6]

3.9 Effect of Antenna Correlation on Performance

Figure 3.14 shows the performance of the STBC with 2 bits/s/Hz on correlated slowRayleigh fading channels with two transmit and two receive antennas We assume thatthe transmit antennas are not correlated but the receive antennas are correlated The receiveantenna correlation matrix is given by

where θ is the correlation factor between the receive antennas In the simulation, the

correla-tion factor is chosen to be 0.25, 0.5, 0.75 and 1 It can be observed that the code performance

114 Space-Time Block Codes

Figure 3.14 Performance of the STBC with 2 bits/s/Hz on correlated slow Rayleigh fading channelswith two transmit and two receive antennas

is slightly degraded when the correlation factor is 0.25 However, relative to the case withuncorrelated antennas, the code is getting worse by 0.7 dB and 1.6 dB at a FER of 10−2for

the correlation factors of 0.5 and 0.75, respectively When the channels are fully correlated,the penalty on the code performance is about 4.2 dB at the same FER

Bibliography

[1] S M Alamouti, “A simple transmit diversity technique for wireless communications”,

IEEE Journal Select Areas Commun., vol 16, no 8, pp 1451–1458, Oct 1998.

[2] A Wittneben, “A new bandwidth efficient transmit antenna modulation diversity

scheme for linear digital modulation”, in Proc IEEE ICC93, pp 1630–1634, 1993.

[3] V Tarokh, H Jafarkhani and A R Calderbank, “Space-time block codes from

orthog-onal designs”, IEEE Trans Inform Theory, vol 45, no 5, pp 1456–1467, July 1999.

[4] V Tarokh, H Jafarkhani and A R Calderbank, “Space-time block coding for wireless

communications: performance results”, IEEE J Select Areas Commun., vol 17, no 3,

pp 451–460, Mar 1999

[5] V Tarokh, A Naguib, N Seshadri and A R Calderbank, “Combined array processing

and space-time coding”, IEEE Trans Inform Theory, vol 45, no 4, pp 1121–1128,

Bibliography 115

[7] V Tarokh and H Jafarkhani, “A differential detection scheme for transmit diversity”,

IEEE J Select Areas Commun., vol 18, pp 1169–1174, July 2000.

[8] H Jafarkhani and V Tarokh, “Multiple transmit antenna differential detection from

generalized orthogonal designs”, IEEE Trans Inform Theory, vol 47, no 6, pp 2626–

2631, Sep 2001

[9] B L Hughes, “Differential space-time modulation”, IEEE Trans Inform Theory, vol.

46, no 7, pp 2567–2578, Nov 2000

[10] B M Hochwald and T L Marzetta, “Unitary space-time modulation for

multiple-antenna communications in Rayleigh flat fading”, IEEE Trans Inform Theory, vol.

46, no 2, pp 543–564, Mar 2000

[11] B M Hochwald and W Sweldens, “Differential unitary space-time modulation”, IEEE

Trans Communi., vol 48, no 12, Dec 2000.

[12] B Hochwald, T L Marzetta and C B Papadias, “A transmitter diversity scheme for

wideband CDMA systems based on space-time spreading”, IEEE Journal on Selected

Areas in Commun., vol 19, no 1, Jan 2001, pp 48–60.

[13] T S Rappaport, Wireless Communications: Principles and Practice, Prentice Hall,

1996

This page intentionally left blank

In this chapter, we introduce an encoder structure of space-time trellis codes By applying

the space-time code design criteria, optimum space-time trellis coded M-PSK schemes for

various numbers of transmit antennas and spectral efficiencies are constructed for slowand fast fading channels The code performance is evaluated by simulations and comparedagainst the capacity limit The effects of imperfect channel estimation and correlated antennaelements on the code performance are also presented

4.2 Encoder Structure for STTC

For space-time trellis codes, the encoder maps binary data to modulation symbols, wherethe mapping function is described by a trellis diagram

Let us consider an encoder of space-time trellis coded M-PSK modulation with n T

trans-mit antennas as shown in Fig 4.1 The input message stream, denoted by c, is given by

c= (c0, c1, c2, , c t , ) (4.1)

where ct is a group of m= log2M information bits at time t and given by

ct = (c1

Space-Time Coding Branka Vucetic and Jinhong Yuan

2003 John Wiley & Sons, Ltd ISBN: 0-470-84757-3

118 Space-Time Trellis Codes

T ) (g ν1

Figure 4.1 Encoder for STTC

The encoder maps the input sequence into an M-PSK modulated signal sequence, which is

The modulated signals, x t1, x t2, , x n T

t , are transmitted simultaneously through n T transmitantennas

4.2.1 Generator Description

In the STTC encoder as shown in Fig 4.1, m binary input sequences c1, c2, , c mare fed

into the encoder, which consists of m feedforward shift registers The k-th input sequence

ck = (c k

0, c k1, c k2, , c t k , ) , k = 1, 2, m, is passed to the k-th shift register and

multi-plied by an encoder coefficient set The multiplier outputs from all shift registers are added

modulo M, giving the encoder output x = (x1, x2, , x n T ) The connections between

the shift register elements and the modulo M adder can be described by the following m

multiplication coefficient set sequences

Encoder Structure for STTC 119

where g k j,i , k = 1, 2, , m, j = 1, 2, , ν k , i = 1, 2, , n T, is an element of the

M -PSK constellation set, and ν k is the memory order of the k-th shift register.

The encoder output at time t for transmit antenna i, denoted by x t i, can be computed as

where ν k , k = 1, 2, , m, is the memory order for the k-th encoder branch The value of

ν k for M-PSK constellations is determined by

the scheme with memory order of ν is shown in Fig 4.2.

Two binary input streams c1 = (c1

0, c11, , c1t , ) and c2 = (c2

0, c12, , c2t , )arefed into the upper and lower encoder registers The memory orders of the upper and lower

encoder registers are ν1and ν2, respectively, where ν = ν1+ ν2 The two input streams aredelayed and multiplied by the coefficient pairs

120 Space-Time Trellis Codes

Figure 4.2 STTC encoder for two transmit antennas

4.2.2 Generator Polynomial Description

The STTC encoder can also be described in generator polynomial format Let us consider

a space-time encoder with two transmit antennas as shown in Fig 4.2 The input binarysequence to the upper shift register can be represented as

Encoder Structure for STTC 121

sequence transmitted from antenna i is given by

xi (D)= c1(D)G1i (D)+ c2(D)G2i (D) mod 4 (4.15)The relationship in (4.15) can be written in the following form

sequences c1and c2into a QPSK sequence

The trellis structure for the code is shown in Fig 4.3 The trellis consists of 2ν = 4 states,

represented by state nodes The encoder takes m= 2 bits as its input at each time Thereare 2m= 4 branches leaving from each state corresponding to four different input patterns

Each branch is labelled by c1t c2t /x t1 x t2, where c t1 and c t2 are a pair of encoder input

bits, and x t1 and x t2 represent two coded QPSK symbols transmitted through antennas 1and 2, respectively The row listed next to a state node in Fig 4.3 indicates the branchlabels for transitions from that state corresponding to the encoder inputs 00, 01, 10, and 11,respectively

122 Space-Time Trellis Codes

Assume that the input sequence is

trans-For STTC, the decoder employs the Viterbi algorithm to perform maximum likelihooddecoding Assuming that perfect CSI is available at the receiver, for a branch labelled by

Optimum space-time trellis coded M-PSK schemes for a given number of transmit antennas

and memory order are designed by applying the design criteria introduced in Chapter 2.For a given encoder structure, a set of encoder coefficients is determined by minimizingthe error probability It is important to note that the STTC encoder structure cannot guaranteegeometrical uniformity of the code [19] Therefore, the search was conducted over allpossible pairs of paths in the code trellis

As discussed in Chapter 2, the code design depends on the code parameter r and the number of receive antennas n R in the system If rn R <4, the rank & determinant criteria

are applicable, while the trace criterion is used if rn R≥ 4

To maximize the minimum rank r for matrix A(X, ˆ X) means to make the matrix full rank

such as r = n T However, the full rank is not always achievable due to the restriction of

the trellis structure For a space-time trellis code with the memory order of ν, the length of

an error event, denoted by l, can be lower-bounded as [13]

As we know the rank of A(X, ˆ X) is the same as the rank of B(X, ˆ X) For an error event path

of length l in the trellis, B(X, ˆ X) is a matrix of size n × l, which results in the maximum

Design of Space-Time Trellis Codes on Slow Fading Channels 123

Table 4.1 Upper bound of the rank values for STTC

Figure 4.4 The boundary for applicability of the TSC and the trace criteria

achievable rank is min(n T , l) Consider the constraint of the error event length in (4.18) The

maximum achievable rank for the code is determined by the value of min(n T , ν/2 + 1).

The upper bound of the rank values for STTC with various numbers of transmit antennas andmemory orders is listed in Table 4.1 It is clear from the table that the full rank is achievableonly for STTC with two transmit antennas For STTC with three and four transmit antennas,

in order to achieve the full rank, the memory order of the encoder is at least four and six,respectively

In the code design the number of receive antennas is normally not considered as a designparameter Considering the relationship between the maximum achievable rank, the number

of the transmit antennas and the memory order of an STTC shown in Table 4.1, we canvisualize the cases in which each criteria set is applicable in the code design The boundarybetween the rank & determinant criteria and the trace criterion is illustrated in Fig 4.4 Thepoints in the rectangular blocks are the cases where the rank & determinant criteria are to

be employed The trace criterion can be used for all other cases The figure suggests thatthe rank & determinant criteria only apply to the systems with one receive antenna

4.3.1 Optimal STTC Based on the Rank & Determinant Criteria

QPSK Codes with Two, Three and Four Transmit Antennas

For space-time codes with two or three transmit antennas and one receive antenna, the

max-imum possible diversity order rn is always less than 4 Following the rank & determinant