3.2 Alamouti Space-Time Code
3.2.4 Performance of the Alamouti Scheme
Now we show that due to the orthogonality between the sequences coming from the two transmit antennas, the Alamouti scheme can achieve the full transmit diversity ofnT =2.
Let us consider any two distinct code sequencesXandXˆ generated by the inputs(x1, x2) and (xˆ1,xˆ2), respectively, where (x1, x2) = (xˆ1,xˆ2). The codeword difference matrix is given by
B(X,X)ˆ =
x1− ˆx1 −x2∗+ ˆx2∗ x2− ˆx2 x∗1− ˆx1∗
(3.20) Since the rows of the code matrix are orthogonal, the rows of the codeword difference matrix are orthogonal as well. The codeword distance matrix is given by
A(X,X)ˆ =B(X,X)Bˆ H(X,X)ˆ
=
|x1− ˆx1|2+ |x2− ˆx2|2 0
0 |x1− ˆx1|2+ |x2− ˆx2|2
(3.21)
Since(x1, x2)=(xˆ1,xˆ2), it is clear that the distance matrices of any two distinct codewords have a full rank of two. In other words, the Alamouti scheme can achieve a full transmit diversity ofnT =2. The determinant of matrixA(X,X)ˆ is given by
det(A(X,X))ˆ =(|x1− ˆx1|2+ |x2− ˆx2|2)2 (3.22) It is obvious from (3.21) that for the Alamouti scheme, the codeword distance matrix has two identical eigenvalues. The minimum eigenvalue is equal to the minimum squared Euclidean distance in the signal constellation. This means for the Alamouti scheme, the minimum distance between any two transmitted code sequences remains the same as in the uncoded system. Therefore, the Alamouti scheme does not provide any coding gain relative to the uncoded modulation scheme, i.e.
Gc =(λ1λ2)1/2
du2 =1 (3.23)
For a pair of codewordsXandX, let us define the squared Euclidean distance betweenˆ the codewords, denoted bydE2(X,X), asˆ
dE2(X,X)ˆ = |x1− ˆx1|2+ |x2− ˆx2|2 (3.24) To obtain the average pairwise error probability, we compute the moment generating function (MGF) for the Alamouti scheme on slow Rayleigh fading channels based on (2.120) as
M(s)=
1−sdE2(X,X)ˆ Es 2N0
−2nR
(3.25) where
s= − 1 2 sin2θ The pairwise error probability in this case can be given by
P (X,X)ˆ = 1 π
π/2
0
1+dE2(X,X)ˆ Es
4N0sin2θ −2nR
dθ (3.26)
=1 2
1−
dE2(X,X)Eˆ s/4N0
1+dE2(X,X)Eˆ s/4N0
ã
2nR−1 k=0
2k k
1
4(1+dE2(X,X)Eˆ s/4N0) k
(3.27)
The performance of the Alamouti transmit diversity scheme on slow Rayleigh fading channels is evaluated by simulation. In the simulations, it is assumed that fading from each transmit antenna to each receive antenna is mutually independent and that the receiver has the perfect knowledge of the channel coefficients. Figure 3.3 shows the bit error rate (BER)
Alamouti Space-Time Code 97
0 5 10 15 20 25 30 35 40 45
10−5 10−4 10−3 10−2 10−1 100
SNR (dB)
Bit Error Rate
no diversity (1Tx 1Rx) Alamouti (2Tx 1 Rx) MRC (1Tx 2 Rx) Alamouti (2Tx 2Rx) MRC (1Tx 4Rx
Figure 3.3 The BER performance of the BPSK Alamouti scheme with one and two receive antennas on slow Rayleigh fading channels
performance of the Alamouti scheme with coherent BPSK modulation against the signal- to-noise ratio (SNR) per receive antenna. The BER performance of two and four-branch receive diversity schemes with single transmit antenna and maximal ratio combining (MRC) is also shown in the figure for comparison. Furthermore, we assume that the total transmit power from two antennas for the Alamouti scheme is the same as the transmit power from the single transmit antenna for the MRC receiver diversity scheme and that it is normalized to one.
The simulation results show that the Alamouti scheme with two transmit antennas and a single receive antenna achieves the same diversity order as a two-branch MRC receive diversity scheme as the slopes of the two curves are the same. However, the performance of the Alamouti scheme is 3 dB worse. The 3-dB performance penalty is due to the fact that the energy radiated from each transmit antenna in the Alamouti scheme is half of that radiated from the single antenna in the MRC receive diversity scheme in order that the two schemes have the same total transmitted power. If each transmit antenna in the Alamouti scheme was to radiate the same energy as the single transmit antenna in the MRC receive diversity scheme, the Alamouti scheme would be equivalent to the MRC receive diversity scheme. Similarly, from the figure, we can see that the Alamouti scheme with two receive antennas achieves the same diversity as a four-branch MRC receive diversity scheme but its performance is 3 dB worse. In general, the Alamouti scheme with two transmit andnR receive antennas has the same diversity gain as an MRC receive diversity scheme with one transmit and 2nR receive antennas.
Figures 3.4 and 3.5 show the frame error rate (FER) performance of the Alamouti scheme with coherent BPSK and QPSK, respectively, on slow Rayleigh fading channels. The frame size is 130 symbols. One and two receive antennas are employed in the simulations.
5 10 15 20 25 10−3
10−2 10−1 100
SNR (dB)
Frame Error Probability
Alamouti BPSK (2Tx 1Rx) Alamouti BPSK (2Tx 2Rx)
Figure 3.4 The FER performance of the BPSK Alamouti scheme with one and two receive antennas on slow Rayleigh fading channels
0 5 10 15 20 25
10−3 10−2 10−1 100
SNR (dB)
Frame Error Rate
Alamouti QPSK (2Tx 1Rx) Alamouti QPSK (2Tx 2Rx)
0 5 10 15 20 25
10−3 10−2 10−1 100
SNR (dB)
Frame Error Rate
Alamouti QPSK (2Tx 1Rx) Alamouti QPSK (2Tx 2Rx)
Figure 3.5 The FER performance of the QPSK Alamouti scheme with one and two receive antennas on slow Rayleigh fading channels
Space-Time Block Codes (STBC) 99