THEORETICAL ANALYSIS OF A SIMO OFDM SYSTEM
2.5 HER Analysis
Consider the transmission of a digital modulated signal x (t) over flat slow Rayleigh fading channels using coherent demodulation with Lth order diversity. As shown in Fig. 3.16 through Fig. 3.21, the received signal component from the lth diversity channel is
rl(t)
=
al (t)expfjlW)}x(t) +wl (t), 1=1, 2, ,L (2.14)It is assumed that (a) the channel fading processes are mutually statistically independent, (b) the additive white Gaussian noise processes are mutually statistically independent, and (c) the
channel fading processes and additive noise processes are independent of each other. For a slow fading channel, the complex channel gain can be assumed to be a complex constant over each symbol interval. The demodulator in each channel is optimum for an A WGN channel (e.g., using filters matched to the orthonormal function
rpn(t), n=1, 2, ... , N, which define the signal space of the transmitted signal x (I),as discussed in section 2). Therefore, the output of the demodulator of the lth branch at the end of the Kth symbol interval isa •. exp (j {}. )i k
+ ii
Ik wherea,.
exp(;B,.)
is the complex channel gain of the Ith channel over the kth symbol interval,i.
is the vectorrepresentation of the transmitted signal over the k th symbol interval in the N-dimensional signal space and is also the demodulator output for an A WGN channel, and
iii, =
(n.,•.),n, •.2' _.,n,•.N) is the corresponding vector representation of the noise component at the demodulator output due to WI(t).As discussed in chapter 3, it can be easily shown thatiii.11
2=
E, (Es is the symbol energy), and that each component iniii,
is a Gaussian random variable with zero mean and variance No/2 that is independent of any other noise component in the same diversity channel or in a different channel.Consider maximal ratio combining, the decision variable for the kth transmitted symbol can be represented by
L
1',
=L
[a" exp( - j 0 " )][aIkJ exp( j 0" ) i, +n" 1
I=}
L
Where g.
= La
2"/=1
and -nj; =L~ a It exp -(-0J IA:)-nil<.
1=1
For the noise vector,
ii"
=(n.".), n".2' ...•. ,n'k.N)' each component isL
n,,k
= La"
exp( -jB,,)n".n,n=
1,2, ,N.[,=1
=
Ebi a
Ik 2 , ••••••••••••••••••••••••••••••••• (2.15) No 1=1Therefore, given the weighting gain
a
lkexp(- j()lk),1=
1,2, ,L, each noise component, nlc,n is a Gaussian random variable with zero mean and variance2
No ~
2CT k,n = -L..,a1k •
2
1=1For BPSK, N =1. The victorsrk,xkand
nkcan
be represented by the corresponding scalar variables rk,xk,and nk, respectively. The decision variable for the kth transmitted symbol is Where xk= IE:
for symbol "1" and xk= -IE:
for symbol "0" as in the SNR per bit at the output of the combiner for the kth symbol is then[tal/xkf
= --- -~-
L
NoLal/
/=1
Where Eb is the SNR value for the A WGN channel with ...
a
lk =1 and L=1. In a rayleigh Nofading environment, the
a
lk •s
are iid rayleigh random variables with parameteru
a2.Therefore, Yk follows a chi-square distribution with 2Ldegrees of freedom. Its pdf is given by
x ~ 0, (2.16)
Where Tc
=
2ua2Eb /N0 is the average SNR per bit in each diversity channel the mean SNR per bit after the combining isTb
=
E[y.]=
LTc,This increases linearly with L. Fig. 3.16 shows the pdf of the SNR per bit for maximal ratio combining with Tc =1. It can be observed that, asLincreases, the pdf curve shifts from left to right, meaning that the chance for a small instantaneous SNR is greatly reduced. Comparing the curves in Fig. 3.16 with those in Fig. 3.21, we see that maximal ratio combining indeed gives better performance as the chances for a small instantaneous SNR are further reduced from those with selective diversity, especially with a large L. This is because maximal ratio combining makes use of the signal components in all the diversity channels and the mean SNR per bit increases linearly with L .However, in selective diversity, at any time, only the signal from the best channel is used for detection. On the other hand, if we compare the two curves with L=2, the difference between them is not significant. From this, we can conclude that, with a low order of diversity reception, selective diversity, equal-gain diversity, and
maximal ratio combining diversity achieve similar transmission performance, with the best performance exhibited by maximal ratio combining diversity and the worst performance exhibited by selective diversity.
The analysis of the probability of bit error over a Rayleigh fading channel can be extended to the case with diversity, so that
1; = f
p~,(x)f,,(xJix, (2.17)Where 1;,lr(x) is the conditional probability of bit error given that the received SNR per bit is
r
=x. For coherent BPSK, from Eq. (2.4) we havePe1r(x)
=
Q( ~). . (2.18)Substituting Eqs.(2.18) and (2.16) into Eq. (2.17), it can be derived that
P, =[0.5(1- p)1"
f
(L-I+')[0.5(1+p)]', [13] (2.19)1",0 I
For Tc »1, we have 0.5(1+,u) andO.5(1-,u) '" 1/4Tc• Furthermore,
f (L-l+') = (2L-').
1=0 / L
As a result, for Tc
»
1, the probability of error can be approximated byThe following observation can be made: (a) Diversity can dramatically improve the transmission performance; (b) The improvement is most significant when L increases from 1 to 2. AsL increases, the BER curve moves towards that for an A WON channel. In fact, it can be proved that when L approaches infinity, the BER curve with diversity will converge to the AWON curve; (c) At a 10wTb value, diversity with L=8 may perform worse than diversity with L =4. This is because, for the same Tb value, Tc for L =8 is 3dB less than Tc for L =4 with a low the effect of the much larger channel noise in the case ofL =8 on BER is stronger than the effect of diversity gain achieved by increasing L from 4 to 8.
Example: Differential BPSK (DBPSK) with Selective Diversity It is known that the BER of DBPSK (BPSK with differential encoding) in an A WON channel with zero mean and two- sided noise pds Ni2 is given by [13].
P
b(rb)
= -exp(-1rb~
(2.20)2
The pdf of the instantaneous SNR per bit with selective diversity over L iid Rayleigh fading channels is fr(x) = ~exJ _~)[l_exp(_~)]L-l dx, x 2: 0, (2.21)
2t'c ~ Tc Tc
Where
or
c =r bEla
/2J
is the mean SNR per bit of each diversity channel. With Eq.(2.20), substituting Eq.(2.21) into Eq.(2.17), we obtain the BER of interest asWithout diversity (L=l), we have 1
Pb= 2[1+7J'
With diversity, L=2, 3, ...., using the binomial expansion
(1- xY-
1= ~(_l)I(L-l)xl,
1=0 I
it can be easily derived that Pbis
L
L-I I(L-l)~ [( 1+1) r
Pb =-L(-l) .Jexp - l+-x 2rc 1=0 lOre
The following observation can be made: (a) Comparing the BER curve for an A WON channel with that for the fading channel without diversity ( L =1), the transmission performance is severely degraded by the Rayleigh fading ; (b) Selective diversity can greatly improve the BER performance; (c) The performance improvement is more significant when L is increased from 1 to 2 than it is when Lis further increased from 2 to 4, and to 8. Note that the superior performance of BPSK with maximal ratio combining is obtained under the assumption that the receiver can have accurate estimates of the diversity channel gains
a
lk expUB
lk) and can thenweight the received signals accordingly. If the channel gain (especially the phase distortion) estimates are not accurate, the performance of BPSK with maximal ratio combining will not be as good. On the other hand, in addition to the much simpler receiver structure (achieved by selective diversity), DBPSK does not require accurate carrier phase synchronization and is robust to carrier phase distortion introduced by the fading channel as long as the phase distortion does not change much over a time duration of two symbol intervals (similar to 7f /
4-DQPSK discussed in section 2.2.1). Therefore, in practice, the performance difference between DBPSK with selective diversity and coherent BPSK with maximal ratio combining may not be significant.