The differential space-time block codes described in the previous section can be extended to nT ≥ 2 transmit antennas. In this section, we present differential schemes with real signal constellations for three and four transmit antennas based on the codes ofX3 andX4, respectively. The derivation of the schemes can be found in [8] and is not included here.
7.4.1 Differential Encoding
Let us recall the encoding operation for space-time block codes with a real signal constella- tion of 2mpoints. At each encoding operation,kminformation bits arrive at the encoder and selectkconstellation signals to be transmitted. The encoder takesk modulated signals and generates codeword sequences based on the transmission matrix of a space-time block code.
For the code with four transmit antennas, the number of the modulated signals that the encoder takes at each block isk=4. Let us denote the modulated signals in theνth encoding
Differential STBC with Real Signal Constellations for Three and Four Transmit Antennas 233 operation byxν
xν =(x1ν, xν2, xν3, xν4) (7.38) wherexνi,i=1, 2, 3 and 4, is theith modulated signal in theνth encoding operation. For the (ν+1)th encoding operation, a block of km = 4m input bits arrives to the encoder.
The message block is used to choose a unit-length real coefficient vector, denoted byR, and given by
R=(R1, R2, R3, R4) (7.39)
Then, based on the previous modulated signalsxνand the current coefficientsR, the encoder computes the modulated signals for the(ν+1)th block as
xν+1=
4
i=1
Ri ãVi(xν) (7.40)
whereVi(xν)are vectors defined as
V1(xν)=(xν1, xν2, xν3, xν4) V2(xν)=(xν2,−xν1, xν4,−xν3) V3(xν)=(xν3,−xν4,−xν1, xν2)
V4(xν)=(xν4, xν3,−xν2,−xν1) (7.41) The four-dimensional vectors form an orthonormal basis for a four-dimensional real signal space. If the both sides of (7.40) are multiplied by ViT(xν), i = 1, 2, 3, and 4, due to orthogonality of the vectors ofVi(xν), we can represent the coefficients by
Ri =xν+1ViT(xν) (7.42)
Let us denote by V the set of the unit-length vectors R. Given the signal vector xν for theνth block, there exist 24m coefficient vectors that generate valid signal vectorsxν+1for the (ν+1)th block, corresponding to the all possible 4minput bits. Therefore, there is a one-to-one mapping between the unit-length vectors inV and the input message blocks. The key issue in constructing a differential scheme is to compute the set of coefficient vectors V and map each block of 4minformation bits to the set.
For example, let us consider a BPSK constellation of two signal points−12 and+12. The signal amplitude is divided by 2 such that the total transmitted power of the baseband signals in a system with four transmit antennas is one. Let the four reference modulated signals at the initial transmission, denoted byx01, x02, x03, x04, for the differential scheme be 1/2, i.e.
x0=(x01, x20, x03, x04)=
1 2,12,12,12
(7.43) For four input bits at the first encoding blockc1=(c11, c21, c31, c41), we choose four BPSK signals, denoted byx11, x12, x13, x14, based on the following mapping
ci1→x1i = (−1)ci1
2 , i=1,2,3,4 (7.44)
The coefficients can be calculated from (7.42) as R1= 12
x11+x12+x13+x14 R2= 12
x11−x12+x13−x14 R3= 12
x11−x12−x13+x14 R4= 12
x11+x12−x13−x14
(7.45) Therefore, a one-to-one mapping from each block of four input bits to a coefficient vector is defined by (7.44) and (7.45). This mapping is based on the reference signals given by (7.43) and is used throughout the whole data sequence.
7.4.2 Differential Decoding
Now we consider differential decoding for the code X4. Note that xν and xν+1 are the modulated signal vectors for theνth and(ν+1)th block message, respectively. According to the transmission matrixX4, for each block of data, there are four signals received subse- quently, assuming that one receive antenna is used. We denote the received signals for the νth block message byr1ν, r2ν, r3ν, r4ν, and the received signals for the(ν+1)th block message byr1ν+1, r2ν+1, r3ν+1, r4ν+1. The received signals can be written in the following vector form
r1ν =
r1ν, r2ν, r3ν, r4ν
=V1(xν)+
nν1, nν2, nν3, nν4
(7.46) r2ν =
−r2ν, r1ν, r4ν,−r3ν
=V2(xν)+
−nν2, nν1, nν4,−nν3
(7.47) r3ν =
−r3ν,−r4ν, r1ν, r2ν
=V3(xν)+
−nν3,−nν4, nν1, nν2
(7.48) r4ν =
−r4ν, r3ν,−r2ν, r1ν
=V4(xν)+
−nν4, nν3,−nν2, nν1
(7.49) rν+1=
r1ν+1, r2ν+1, r3ν+1, r4ν+1
=V1(xν+1)+
nν1+1, nν2+1, nν3+1, nν4+1
(7.50) whereis a 4×4 matrix defined by
=
h1 h2 h3 h4 h2 −h1 −h4 h3 h3 h4 −h1 −h2 h4 −h3 h2 −h1
(7.51)
and hi is the fading coefficient for the channel from transmit antenna i to the receive antenna,nνi is the noise sample at theith symbol period in theνth block. Let us define the
Differential STBC with Real Signal Constellations for Three and Four Transmit Antennas 235 noise vectors in (7.46)–(7.50) as Nν1,Nν2,Nν3,Nν4, andNν+1, respectively. Thus, we can construct the decision statistics signals R˜i, i =1,2,3,4, as the inner product of vectors rν+1 andriν. The decision statistics signal can be computed as [8]
R˜i =rν+1ãriν
=
4
j=1
|hj|2ãV1(xν+1)ViT(xν)+ ˜Ni
=
4
j=1
|hj|2ãxν+1ViT(xν)+ ˜Ni
=
4
j=1
|hj|2ãRi+ ˜Ni (7.52)
whereN˜i is a noise term given by
N˜i =xν+1(Nνi)H+Nν+1HViT(xν)+Nν+1(Nνi)H. (7.53) Let us write the decision statistics signals in a vector form
(R˜1,R˜2,R˜3,R˜4)=
4
j=1
|hj|2
ã(R1, R2, R3, R4)+(N˜1,N˜2,N˜3,N˜4) (7.54) Clearly, the decision statistics signal vector is only a function of the differential coefficient vector. Since all the coefficient vectors in the set V have equal length, the receiver now chooses the closest coefficient vector fromV to the decision statistics signal vector as the differential detection output. Then the inverse mapping from the coefficient vector to the block of information bits is applied to decode the transmitted signal [8]. From (7.54), we can see that the scheme provides a four-level transmit diversity with four transmit antennas.
Thus far, we have described differential encoding and decoding algorithms for space-time block codes with four transmit antennasX4. For codes with three transmit antennasX3, the code rate is the same as forX4and the transmission matrixX3is identical to the first three rows ofX4. Therefore, the same differential encoding and decoding algorithms for X4 can be applied directly for the codeX3. In this case, the received signal can also be represented by (7.46)–(7.50), whereh4 in matrixis set to zero.
=
h1 h2 h3 0
h2 −h1 0 h3
h3 0 −h1 −h2
0 −h3 h2 −h1
(7.55)
The decision statistics signal vector at the receiver is given by (R˜1,R˜2,R˜3,R˜4)=
3
j=1
|hj|2
ã(R1, R2, R3, R4)+(N˜1,N˜2,N˜3,N˜4) (7.56) Therefore, this scheme can achieve three-level diversity with three transmit antennas and one receive antenna.
6 8 10 12 14 16 18 20 22 24 26 28 10−6
10−5 10−4 10−3 10−2 10−1
SNR (dB)
Bit Error Probability
D−STBC STBC
Figure 7.6 Performance comparison of the coherent and differential BPSK STBC with three transmit and one receive antenna on slow Rayleigh fading channels
0 5 10 15 20 25
10−6 10−5 10−4 10−3 10−2 10−1 100
SNR (dB)
Bit Error Probability
D−STBC STBC
Figure 7.7 Performance comparison of the coherent and differential BPSK STBC with four transmit and one receive antenna on slow Rayleigh fading channels
Differential STBC with Complex Signal Constellations for Three and Four Transmit Antennas 237
7.4.3 Performance Simulation
Figures 7.6 and 7.7 depict the FER performance of the BPSK differential STBC with three and four transmit antennas, respectively, on slow Rayleigh fading channels. The frame was assumed to be 130 symbols and one receive antenna was used in the simulations. The performance curves of the corresponding STBC with coherent detection are also plotted in the figures for comparison. For both cases, the differential scheme is worse by about 3 dB relative to the coherent scheme.