Encoder Structure for STTC

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

Let us consider an encoder of space-time trellis codedM-PSK modulation withnT transmit antennas as shown in Fig. 4.1. The input message stream, denoted byc, is given by

c=(c0,c1,c2, . . . ,ct, . . . ) (4.1) wherect is a group ofm=log2M information bits at timet and given by

ct =(c1t, c2t, . . . , cmt ) (4.2)

Space-Time Coding Branka Vucetic and Jinhong Yuan c2003 John Wiley & Sons, Ltd ISBN: 0-470-84757-3

cm c1

ããã

ãã

(g10,1, . . . , g10,n

T) (g11,1, . . . , g11,n

T) (gν1

1,1, . . . , g1ν1,nT)

(gm0,1, . . . , gm0,n

T) (gm1,1, . . . , gm1,n

T) (gνm

m,1, . . . , gmνm,nT)

x1, x2, . . . , xnT

Figure 4.1 Encoder for STTC

The encoder maps the input sequence into anM-PSK modulated signal sequence, which is given by

x=(x0,x1,x2, . . . ,xt, . . . ) (4.3) wherext is aspace-time symbol at time t and given by

xt =(xt1, xt2, . . . , xtnT)T (4.4) The modulated signals,xt1, xt2, . . . , xtnT, are transmitted simultaneously throughnT transmit antennas.

4.2.1 Generator Description

In the STTC encoder as shown in Fig. 4.1,mbinary input sequencesc1,c2, . . . ,cmare fed into the encoder, which consists ofmfeedforward shift registers. The k-th input sequence ck =(ck0, ck1, ck2, . . . , ctk, . . . ),k=1,2, . . . m, is passed to thek-th shift register and multiplied by an encoder coefficient set. The multiplier outputs from all shift registers are added modulo M, giving the encoder output x = (x1,x2, . . . ,xnT). The connections between the shift register elements and the moduloM adder can be described by the followingm multiplication coefficient set sequences

g1=[(g0,11 , g0,21 , . . . , g0,n1

T), (g1,11 , g1,21 , . . . , g1,n1

T), . . . , (gν1

1,1, gν1

1,2, . . . , g1ν

1,nT)]

g2=[(g0,12 , g0,22 , . . . , g0,n2

T), (g1,12 , g1,22 , . . . , g1,n2

T), . . . , (gν2

2,1, gν2

2,2, . . . , g2ν2,nT)]

...

gm=[(g0,1m , g0,2m , . . . , g0,nm

T), (g1,1m , g1,2m , . . . , g1,nm

T), . . . , (gνm

m,1, gνm

m,2, . . . , gmν

m,nT)]

Encoder Structure for STTC 119

where gkj,i, k = 1,2, . . . , m, j = 1,2, . . . , νk, i = 1,2, . . . , nT, is an element of the M-PSK constellation set, andνk is the memory order of the k-th shift register.

The encoder output at timet for transmit antennai, denoted byxti, can be computed as xti =

k=1 νk

j=0

gkj,ictk−j modM, i=1,2, . . . , nT (4.5) These outputs are elements of anM-PSK signal set. Modulated signals form the space-time symbol transmitted at timet

xt =(xt1, x2t, . . . , xntT)T. (4.6) The space-time trellis coded M-PSK can achieve a bandwidth efficiency of mbits/s/Hz.

The total memory order of the encoder, denoted byν, is given by ν=

k=1

νk (4.7)

whereνk,k=1,2, . . . , m, is the memory order for the k-th encoder branch. The value of νk for M-PSK constellations is determined by

νk =

ν+k−1 log2M

(4.8) The total number of states for the trellis encoder is 2ν. Themmultiplication coefficient set sequences are also called thegenerator sequences, since they can fully describe the encoder structure.

For example, let us consider a simple space-time trellis coded QPSK with two transmit antennas. The encoder consists of two feedforward shift registers. The encoder structure for the scheme with memory order ofν is shown in Fig. 4.2.

Two binary input streamsc1 = (c10, c11, . . . , c1t, . . . ) andc2 =(c20, c12, . . . , c2t, . . . )are fed into the upper and lower encoder registers. The memory orders of the upper and lower encoder registers areν1andν2, respectively, whereν=ν1+ν2. The two input streams are delayed and multiplied by the coefficient pairs

g1=[(g0,11 , g0,21 ), (g1,11 , g1,21 ), . . . , (gν1

1,1, gν1

1,2)]

g2=[(g0,12 , g0,22 ), (g1,12 , g1,22 ), . . . , (gν2

2,1, gν2

2,2)] (4.9)

respectively, wheregkj,i ∈ {0,1,2,3}, k=1,2;i =1,2;j =0,1, . . . , νk. The multiplier outputs are added modulo 4, giving the output

xti = 2

k=1 νk

j=0

gj,ik ckt−j mod 4, i=1,2 (4.10)

The adder outputsx1t andxt2 are points from a QPSK constellation. They are transmitted simultaneously through the first and second antenna, respectively.

c2t c1t

(g20,1, g0,22 )

(g21,1, g1,22 ) (g11,1, g1,21 )

(g2ν

2,1, gν2

2,2) (g1ν

1,1, gν1

1,2) (g10,1, g0,21 )

(x1t, xt2)

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 binary sequence to the upper shift register can be represented as

c1(D)=c10+c11D+c12D2+c13D3+ ã ã ã (4.11) Similarly, the binary input sequence to the lower shift register can be written as

c2(D)=c20+c21D+c22D2+c23D3+ ã ã ã (4.12) whereckj,j =0,1,2,3, . . .,k=1,2, are binary symbols 0,1. The feedforward generator polynomial for the upper encoder and transmit antennai, wherei=1,2, can be written as

G1i(D)=g0,i1 +g11,iD+ ã ã ã +gν1

1,iDν1 (4.13)

where gj,i1 , j = 0,1, . . . , ν1 are non-binary coefficients that can take values 0,1,2,3 for QPSK modulation and ν1 is the memory order of the upper encoder. Similarly, the feedforward generator polynomial for the lower encoder and transmit antenna i, where i=1,2,can be written as

G2i(D)=g0,i2 +g21,iD+ ã ã ã +gν2

2,iDν1 (4.14)

wheregj,i2 ,j =1,2, . . . , ν2, are non-binary coefficients that can take values 0,1,2,3 for QPSK modulation andν2 is the memory order of the lower encoder. The encoded symbol

Encoder Structure for STTC 121

sequence transmitted from antennai 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

xi(D)=

c1(D) c2(D) G1i(D) G2i(D)

mod 4 (4.16)

A systematic recursive STTC can be obtained by setting G1(D)= 2

which means that the output of the first antenna is obtained by directly mapping the input sequencesc1andc2into a QPSK sequence.

4.2.3 Example

Let us assume that the generator sequences of a 4-state space-time trellis coded QPSK scheme with 2 transmit antennas are

g1=[(02), (20)]

g2=[(01), (10)]

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 takesm=2 bits as its input at each time. There are 2m=4 branches leaving from each state corresponding to four different input patterns.

Each branch is labelled by c1t c2t/xt1 xt2, where ct1 and ct2 are a pair of encoder input bits, and xt1 and xt2 represent two coded QPSK symbols transmitted through antennas 1 and 2, respectively. The row listed next to a state node in Fig. 4.3 indicates the branch labels for transitions from that state corresponding to the encoder inputs 00, 01, 10, and 11, respectively.

00/00 01/01 10/02 11/03

00/10 01/11 10/12 11/13

00/20 01/21 10/22 11/23

00/30 01/31 10/32 11/33

Figure 4.3 Trellis structure for a 4-state space-time coded QPSK with 2 antennas

Assume that the input sequence is

c=(10,01,11,00,01, . . . )

The output sequence generated by the space-time trellis encoder is given by x=(02,21,13,30,01, . . . )

The transmitted signal sequences from the two transmit antennas are x1=(0,2,1,3,0, . . . )

x2=(2,1,3,0,1, . . . )

Note that this example is actually a delay diversity scheme since the signal sequence transmitted from the first antenna is a delayed version of the signal sequence from the second antenna.

For STTC, the decoder employs the Viterbi algorithm to perform maximum likelihood decoding. Assuming that perfect CSI is available at the receiver, for a branch labelled by (xt1, xt2, . . . , xtnT), the branch metric is computed as the squared Euclidean distance between the hypothesised received symbols and the actual received signals as

j=1

rtj−

i=1

htj,ixti

(4.17) The Viterbi algorithm selects the path with the minimum path metric as the decoded sequence.

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