Part IV MULTIPLE ACCESS AND ADVANCED TRANSCEIVER SCHEMES 363
12.2 Error Probability in Flat-Fading Channels
12.2.2 Computation of Average Error Probability – Alternative Method
In the late 1990s, a new method for computation of the average BER was proposed, and shown to be very efficient. It is based on an alternative representation of the Q-function, and allows easier averaging over different fading distributions ([Annamalai et al. 2000], [Simon and Alouini 2004]).
Demodulation 235
Alternative Representation of the Q-Function
When evaluating the classical definition of the Q-function Q(x)= 1
√2π ∞
x
exp(−t2/2) dt (12.59)
there is the problem that the argument of the Q-function is in the integration limit, not in its integrand. This makes evaluation of the integrals of the Q-function much more difficult, especially because we cannot use the beloved trick of exchanging the sequence of integration in multiple integrals. This problem is particularly relevant in BER computations. The problem can be solved by using an alternative formulation of the Q-function:
Q(x)= 1 π
π/2 0
exp
− x2 2 sin2θ
dθ forx >0 (12.60) This representation now has the argument in the integrand (in a Gaussian form, exp(−x2)), and also has finite integration limits. We will see below that this greatly simplifies evaluation of the error probabilities.
It turns out that Marcum’s Q-function, as defined in Eq. (12.40), also has an alternative repre- sentation:
QM(a, b)=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩ 1 2π
π
−π
b2+absinθ
b2+2absinθ+a2exp(−12(b2+2absinθ+a2))dθ forb > a≥0 1+ 1
2π π
−π
b2+absinθ
b2+2absinθ+a2exp(−12(b2+2absinθ+a2))dθ fora > b≥0 (12.61) Error Probability in Additive White Gaussian -Noise Channels
For computation of the BER in AWGN channels, we find that the BER for BPSK can be written as (compare also Eq. 12.30):
BER=Q(
2γS) (12.62)
= 1 π
π/2 0
exp
− γS
sin2θ
dθ (12.63)
For QPSK, the SER can be computed as (compare Eq. 12.35):
SER=2Q(√
γS)−Q2(√
γS) (12.64)
= 1 π
3π/4
0 exp
− γS
sin2θ sin2(π/4)
dθ (12.65)
and quite generally forM-ary PSK:
SER= 1 π
(M−1)π/M
0 exp
− γS
sin2θsin2(π/M)
dθ (12.66)
For binary orthogonal FSK, we finally find that BER=Q(√
γS) (12.67)
= 1 π
π/2 0
exp
− γS
2 sin2θ
dθ (12.68)
Example 12.5 Compare the SER for 8-PSK as computed from Eq. (12.66) with the value obtained from the union bound forγS=3 and 10 dB.
First using Eq. (12.66) withM=8 andγS=3 dB we have SER= 1
π
(M−1)π/M 0
exp
− γS
sin2θsin2(π/M)
dθ
=0.442 (12.69)
and forγS=10 dB we getSER=0.087, where the integral has been evaluated numerically.
The 8-PSK constellation has a minimum distance of dmin=2√
ESsin(π/8). The nearest neighbor union bound on the SER is then given by
SERunion-bound=2ãQ dmin
√2N0
=2ãQ 2√
ESsin(π/8)
√2N0
=2ãQ
2γSsin(π/8)
(12.70) Thus, with γS=3 dB we get SERunion-bound=0.445, and with γS=10 dB we get SERunion-bound=0.087. In this example the union bound gives a very good approximation, even at the rather low SNR of 3 dB.
Error Probability in Fading Channels
For AWGN channels, the advantages of the alternative representation of the Q-function are rather limited. They allow a simpler formulation for higher order modulation formats, but do not exhibit significant advantages for the modulation formats that are mostly used in practice. The real advan- tage emerges when we apply this description method as the basis for computations of the BER in fading channels. We find that we have to average over the pdf of the SNRpdfγ(γ ), as described in Eq. (12.50). We have now seen that the alternative representation of the Q-function allows us to write the SER (for a given SNR) in the generic form:
SER(γ )= θ2
θ1
f1(θ )exp(−γf2(θ ))dθ (12.71) Thus, the average SER becomes
SER= ∞
0
pdfγ(γ )SER(γ )dγ (12.72)
= ∞
0
pdfγ(γ ) θ2
θ1
f1(θ )exp(−γf2(θ ))dθdγ (12.73)
= θ2
θ1
f1(θ ) ∞
0
pdfγ(γ )exp(−γf2(θ ))dγdθ (12.74) Let us now have a closer look at the inner integral:
∞
0
pdfγ(γ )exp(−γf2(θ ))dγ (12.75)
Demodulation 237
We find that it is the moment-generating function of pdfγ(γ ), evaluated at the point −f2(θ ).
Remember (see also, e.g., Papoulis [1991]) that the moment-generating function is defined as the Laplace transform of the pdf ofγ:
Mγ(s)= ∞
0
pdfγ(γ )exp(γ s)dγ (12.76)
and the mean SNR is the first derivative, evaluated ats=0:
γ = dMγ(s) ds
s=0
(12.77) Summarizing, the average SER can be computed as
SER= θ2
θ1
f1(θ )Mγ(−f2(θ ))dθ (12.78) The next step is then finding the moment-generating function of the distribution of the SNR.
Without going into the details of the derivations, we find that for a Rayleigh distribution of the signal amplitude, the moment-generating function of the SNR distribution is
Mγ(s)= 1
1−sγ (12.79)
for a Rice distribution it is
Mγ(s)= 1+Kr
1+Kr−sγ exp
Krsγ 1+Kr−sγ
(12.80) and for a Nakagami distribution with parameterm:
Mγ(s)=
1−sγ m
−m
(12.81) Having now the general form of the SER (Eq. 12.78) and the form of the moment-generating func- tion, the computation of the error probabilities become straightforward (if sometimes a bit tedious).
Example 12.6 BER of BPSK in Rayleigh fading.
As one example, let us go through the computation of the average BER of BPSK in a Rayleigh- fading channel – a problem for which we already know the result from Eq. (12.52). Looking at Eq. (12.63), we find thatθ1=0, θ2=π/2:
f1(θ )= 1
π (12.82)
f2(θ )= 1
sin2(θ ) (12.83)
Since we consider Rayleigh fading:
Mγ(−f2(θ ))= 1 1+sinγ2(θ )
(12.84)
so that the total BER is, according to Eq. (12.78):
SER= 1 π
π/2 0
sin2(θ )
sin2(θ )+γ dθ (12.85)
which can be shown to be identical to the first (exact) expression in Eq. (12.52) – namely:
BER= 1 2
1− γ
1+γ
(12.86)
Many modulation formats and fading distributions can be treated in a similar way. An extensive list of solutions, dealing with coherent detection, partially coherent detection, and noncoherent detection in different types of channels, is given in the book by Simon and Alouini [2004].