In the previous code performance analysis and code design, we considered only the worst case pairwise error probability upper bound. In order to get the accurate performance eval- uation, one possible method is to compute the code distance spectrum and apply the union bound technique to calculate the average pairwise error probability. The obtained upper bound is asymptotically tight at high SNR’s for a small number of receive antennas but loose for other scenarios [41]. A more accurate performance evaluation can be obtained with exact evaluation of the pairwise error probability, rather than evaluating the bounds.
This can be done by using residue methods based on the characteristic function technique [42] [43] or on the moment generating function method [44] [45].
Recall that the pairwise error probability conditioned on the MIMO fading coefficients is given by
P (X,X|H)ˆ =Q
$%
%& Es 2N0
L t=1
Ht(xt− ˆxt)2
(2.110)
Let
= Es 2N0
L t=1
Ht(xt− ˆxt)2 (2.111)
By using Graig’s formula for the GaussianQfunction [48]
Q(x)= 1 π
π/2
0
exp
− x2 2 sin2θ
dθ (2.112)
Exact Evaluation of Code Performance 83
we can rewrite for the conditional pairwise error probability P (X,X|H)ˆ = 1
π π/2
0
exp
− 2 2 sin2θ
dθ (2.113)
In order to compute the average pairwise error probability, we average (2.113) with respect to the distribution of. The average pairwise error probability can be expressed in terms of the moment generating function (MGF) of, denoted byM(s), which is given by
M(s)= ∞
0
esP() d (2.114)
The average pairwise error probability can be represented as [44]
P (X,X)ˆ = 1 π
π/2
0
E
* exp
− 2 2 sin2θ
+ dθ
= 1 π
π/2
0
∞
0
exp
− 2 2 sin2θ
P()d dθ
= 1 π
π/2
0
M
− 1
2 sin2θ
dθ (2.115)
For fast Rayleigh fading MIMO channels, the MGF can be expressed in a closed form as [44]
M(s)= )L t=1
1− sEs
2N0 nT
i=1
|xit − ˆxti|2 −nR
(2.116) where
s= − 1
2 sin2θ. (2.117)
Substituting (2.116) into (2.115), we get for the average pairwise error probability P (X,X)ˆ = 1
π π/2
0
)L t=1
1+ Es 4N0sin2θ
nT
i=1
|xti− ˆxti|2 −nR
dθ (2.118)
For slow Rayleigh fading channels, the MGF can be represented as [44]
M(s)=
det
*
InT −s(X− ˆX)(X− ˆX)H Es 2N0
+−nR
. (2.119)
If the rows of matrix(X− ˆX)are orthogonal, this MGF can be simplified as M(s)=
nT
)
i=1
1− sEs
2N0
L t=1
|xti − ˆxti|2 −nR
. (2.120)
In this case, the average pairwise error probability is given by P (X,X)ˆ = 1
π π/2
0 nT
)
i=1
1+ Es 4N0sin2θ
L t=1
|xti− ˆxti|2 −nR
dθ. (2.121)
Example 2.5
Let us consider a 4-state QPSK space-time trellis code with two transmit antennas. The code trellis structure is shown in Fig. 2.13. For the pairwise error event of length 2, illustrated by the thick lines in Fig. 2.13, matricesXandXˆ are
X=
* 1 1
1 1
+
, Xˆ =
* 1 −1
−1 1
+
(2.122) The codeword distance matrix for the error event is given by
A(X,X)ˆ =(X− ˆX)(X− ˆX)H
=
* 4 0
0 4
+
(2.123) Substituting (2.123) into (2.119) gives
M(s)=
1−2sEs
N0
−2nR
(2.124) The pairwise error probability for slow Rayleigh fading channels can be expressed as
P (X,X)ˆ = 1 π
π/2 0
*
1+ Es N0sin2θ
+−2nR
dθ (2.125)
=1 2
1−
Es/N0 1+Es/N0
2nR−1 k=0
2k k
1 4(1+Es/N0)
k
(2.126)
The pairwise error probability (2.126) is plotted in Figs. 2.14 and 2.15 for nR = 1 and nR = 2, respectively. The pairwise error probability upper bounds in (2.61), (2.64), and (2.65) are also shown in these figures for comparison.
The exact evaluation of the pairwise error probability based on the transfer function technique gives a transfer function upper bound on the average frame error probability or
00 01 02 03
10 11 12 13
20 21 22 23
30 31 32 33
Figure 2.13 Trellis structure for a 4-state QPSK space-time code with two antennas
Exact Evaluation of Code Performance 85
−5 0 5 10 15 20
10−5 10−4 10−3 10−2 10−1 100
Es/No (dB)
Pairwise Error Probability
exact (2.126) bound (2.65) bound (2.64) bound (2.61)
Figure 2.14 Pairwise error probability of the 4-state QPSK space-time trellis code with two transmit and one receive antenna
−10 −8 −6 −4 −2 0 2 4 6 8 10
10−5 10−4 10−3 10−2 10−1 100
Es/No (dB)
Pairwise Error Probability
exact (2.126) bound (2.65) bound (2.64) bound (2.61)
Figure 2.15 Pairwise error probability of the 4-state QPSK space-time trellis code with two transmit and two receive antennas
4 6 8 10 12 14 16 18 20 22 24 10−5
10−4 10−3 10−2 10−1 100
SNR (dB)
Bit error Rate
Upper bound Simulation
Simulation, 2Tx 2Rx
upper bound, 2Tx 2Rx upper bound, 2Tx 1Rx
Simulation, 2Tx 1Rx
Figure 2.16 Average bit error rate of the 4-state QPSK space-time trellis code with two transmit antennas and one and two receive antennas
bit error probability for space-time codes, which is asymptotically tight at high SNR’s and tighter than a Chernoff bound [44].
Figure 2.16 illustrates the performance comparison between the analytical and simulated average bit error rate of the 4-state QPSK space-time trellis code with two transmit antennas and one and two receive antennas on slow Rayleigh fading channels. In the analytical evaluating the average bit error rate, only the error events of lengths two and three are considered for simplicity.
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3
Space-Time Block Codes