Performance Analysis Of Predetection EGC Receiver In Weibull ...
Performance analysis of predetection EGC
In this Letter we propose to evaluate these expectations by using the
receiver in Weibull fading channel
infinite series representation of cos and sine functions, as follows.
P.R. Sahu and A.K. Chaturvedi
P
1
x2k
cosðxÞ ¼
ðÀ1Þk
k¼0
2k!
The predetection equal gain combining (EGC) receiver is generally
P
1
known to have a performance that is close to the maximal ratio
sinðxÞ ¼
ðÀ1Þk
x2kþ1
ð3Þ
ð2k þ 1Þ!
combining (MRC) receiver while having relatively less implementa-
k¼0
tion complexity. The bit error rate (BER) of an EGC receiver for
binary, coherent and noncoherent modulations has been analysed for
Interestingly, this is easily integrated as shown below, and the final
an independent Weibull fading channel. Numerical results have been
expression is in the form of a convergent infinite series containing
compared with the available results for selection combining (SC) and
Gamma function.
MRC diversity receivers.
ð1 P1
E½cosðnox
ðÀ1Þk ðnoxlÞ2k
Introduction: The Weibull fading model exhibits an excellent fit to
l Þ ¼
0
k¼0
2k!
experimental fading channel measurements, for indoor [1] as well as
m
Â
xmÀ1
=gdx
outdoor environments [2]. In some recent work [3, 4] performance
g l
eÀxml
l
based on statistical parameters of the output signal-to-noise ratio
ð
m P
1 ðÀ1Þk ðnoÞ2k 1
(SNR) of switch-and-stay combining (SSC) and selection combining
¼
x2kþmÀ1eÀxm=g
l
dx
l
l
(SC) diversity receivers in Weibull fading channels has been obtained.
g k¼0
2k!
0
In [5] the average symbol error probability for the SSC receiver for
P
1
¼
ðÀ1Þk ðnog1=mÞ2k
several binary and multilevel modulation schemes has been studied.
k¼0
2k!
The outage probability as well as BER performance of SC and MRC
ð1
xm
diversity receivers has been analysed in [6].
Â
t2k=meÀtdt; substituting t ¼ g
The predetection EGC receiver is generally known to have a
0
performance close to a MRC receiver while at the same time being
P
1 ðÀ1Þk ðnog1=mÞ2k
2k
¼
G
þ 1
ð4Þ
less complex from the implementation point of view. Hence, the
k¼0
2k!
m
performance analysis of this receiver is of interest.
In this Letter we analyse the performance of a predetection EGC
E[sin(noxl)] can also be derived similarly and the final expression
receiver in independent Weibull fading channels for binary, coherent
obtained is
PSK and FSK, differential coherent PSK, and noncoherent FSK
modulations.
P
1
2k þ 1
E½sinðnox
G
l Þ ¼
ðÀ1Þk ðnog1=mÞ2kþ1
þ 1
ð5Þ
k¼0
ð2k þ 1Þ!
m
BER performance analysis: Performance analysis of an EGC receiver
usually needs the PDF of the output SNR, which is a function of the
sum of the fading envelopes of the individual branches. However, a
Numerical results: The BER expression (1) has been evaluated
closed form expression for the PDF of this sum is not available for
numerically for different values of L and m and curves for average
Weibull distribution. An alternative approach could be to use the BER
branch SNR against BER have been plotted. In numerical evaluation
expression derived in [7], provided the relevant expectations for the
the value of T and the number of terms in the infinite series have been
fading distribution can be determined. Hence, we begin by consider-
chosen to ensure an accuracy of Æ10À7 in BER. In Fig. 1 curves for
ing the BER expression [7],
coherent and differential coherent PSK modulations have been given
2 X
1
for L ¼ 2 and L ¼ 5 each one for m ¼ 2 and m ¼ 4. It can be observed
Pe ¼
A
T
nBn cosðtn À anÞ
ð1Þ
that the curve for coherent L ¼ 2, m ¼ 2 intersects the curve for
n¼1
differential coherent L ¼ 2, m ¼ 4 at an SNR of approximately
n odd
Q
p
P
2.5 dB. This indicates that before 2.5 dB the coherent receiver for
where A
L
L
n ¼
l¼1(
(E2{cos(noxl)} þ E2{sin(noxl)}), tn ¼
l¼1tanÀ1
m ¼ 2 gives better performance than the differential coherent receiver
{((E[sin(noxl)])=(E[cos(noxl)]))}, E(Á) is the expectation operator,
for m ¼ 4 while after that it is inferior. Similar observations can be
o ¼ 2p=T, and T is the period of the square wave used in deriving the
made from the results shown in [7] (Fig. 2). In Fig. 2 curves have been
infinite series expression for the sum of random variables in [8]. The
shown for coherent and noncoherent binary FSK modulations. A
minimum value of T required depends on the SNR as well as the desired
comparison of the results for coherent FSK with that of the SC
accuracy of Pe. In the above, the random variable xl denotes the fading
receiver [6] shows that at a BER of 10À3 there is a gain of
envelope of the lth branch. We assume it to be Weibull distributed with
approximately 1.8 dB in SNR for L ¼ 2, m ¼ 4. For the same values
a density function given by
of L and m a comparison with the MRC results in [6] shows that to
m
achieve a BER of 10À3 the EGC receiver requires only 0.2 dB more
f ðx
=g;
l Þ ¼
xmÀ1
x
g l
eÀxml
l ! 0;
m > 0
ð2Þ
SNR than the MRC receiver.
where E[X2
2=m
l ] ¼ gl
G(1 þ 2/m). In (1) Bn and an are independent of the
fading distribution and have been given in [7] (equation (27)) for
coherent PSK and FSK modulations. For differential coherent PSK and
noncoherent FSK modulations Bn has been denoted by Dn and an by dn
in [7] (equation (33)).
To evaluate (1) it is required to determine An and tn, which in turn
require expressions for E[cos(noxl)] and [sin(noxl)]. One possible
approach is to evaluate the integrals after multiplying the cos or sine
term with the PDF of al. For Nakagami-m distribution these integrals
can be readily found from standard integration and Fourier transform
tables as a closed form expression containing hypergeometric functions.
However, for Weibull distribution the integration is not straightforward
and is also not available in standard tables. Another approach could be
to evaluate the characteristic function and use its real and imaginary
parts. But the characteristic function is also not available. Although [6]
has derived the moment generating function, it is of limited use since
the expression is in terms of Meijer’s G function.
Fig. 1 BER for coherent and differential coherent binary PSK
ELECTRONICS LETTERS
20th January 2005
Vol. 41
No. 2
References
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Beaulieu, N., and Abu-Dayya, A.A.: ‘Analysis of equal gain diversity on
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Nakagami fading channels’, IEEE Trans. Commun., 1991, 39, (2),
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8
Beaulieu, N.C.: ‘An infinite series for the computation of the
Conclusions: We have analysed the performance of a predetection
complementary
probability
distribution
function
of
a
sum
of
EGC receiver in an independent Weibull fading channel. BER
independent random variables and its application to the sum of
performance for coherent PSK and FSK, differential coherent PSK
Rayleigh random variables’, IEEE Trans. Commun., 1990, 38, (9),
and noncoherent FSK modulations has been given for a varying
pp. 1463–1474
number of branches and fading parameters. The results have been
compared with the available results for SC and MRC receivers.
# IEE 2005
4 October 2004
Electronics Letters online no: 20057262
doi: 10.1049/el:20057262
P.R. Sahu and A.K. Chaturvedi (Department of Electrical Engine-
ering, Indian Institute of Technology, Kanpur, India)
ELECTRONICS LETTERS
20th January 2005
Vol. 41
No. 2
