Capacity Estimation for a DS-CDMA System in Nakagami-m Fading
PANAGIOTIS VARZAKAS
Department of Informatics and Telecommunications
University of Thessaly
3o Km Old Road Lamia-Athens
GREECE
Abstract: - In this paper, a novel closed-form expression of the Shannon average channel capacity per user for
direct- sequence code-division multiple access (DS-CDMA) systems, operating in Nakagami-m fading, with
optimum RAKE reception, is obtained. Numerical results are also presented to illustrate the proposed
mathematical analysis and to point out the effect of the fading severity on the user’s average channel capacity.
Key-Words: DS-CDMA systems, Channel capacity, Nakagami-m fading.
Received: July 15, 2021. Revised: January 11, 2022. Accepted: February 10, 2022. Published: March 3, 2022.
1 Introduction
RAKE reception, with maximal ratio
combining (MRC), is an effective way to anticipate
multipath signal fading, due to its ability to resolve
additional multipaths, resulting in an increased
diversity gain, [1]. Moreover, channel capacity, in
the Shannon sense, is a significant criterion for the
design and the performance evaluation of digital
communication systems, [2]. Thus, an estimation of
the average channel capacity based on optimal rate
adaptation to channel fading and constant transmit
power could indeed provide the maximum
transmission rates, if channel side information were
available at the receiver.
In this paper, a closed-form analytical
expression for the Shannon channel capacity per
user in direct-sequence code-division multiple
access (DS-CDMA) systems with MRC RAKE
reception is obtained, extending the results of [3]
and [4] for the important Nakagami-m fading
channel model. Numerical results are also presented
to illustrate the proposed mathematical analysis. In
these results, it is pointed out the effect of the fading
severity on the user’s capacity and a comparison
with the capacity of the additive white Gaussian
noise (AWGN) channel is also given.
2 System and channel model
We consider a non-cooperative DS-CDMA
wireless system that consists of K simultaneous
users, each transmitting with the same average
power. Bandwidth spreading is accomplished at the
transmitter by multiplying the information data by a
broadband code sequence. Each user transmits a
signal of bandwidth Wss after spreading the actual
signal bandwidth W by the system's processing gain
ss
p
W
G
W
.
The channel capacity of each user of such a
system, i.e. a single user’s conditional channel
capacity in the Shannon sense, called hereafter
“channel capacity per user”, will clearly depend on
the level of cooperation among the K>1 users or,
equivalently, on the multiple-access interference
(MAI) power. This channel capacity per user will be
given by the Shannon-Hartley theorem when
arbitrarily complex coding and delay is applied, [5],
while the total MAI power, caused by even a small
number of interfering users, will tend to be Gaussian
distributed, [6].
Each user’s RAKE receiver has L taps
corresponding to L resolvable signal paths of the
multipath tapped delay line channel model, whose
fading amplitudes and phases are perfectly known.
1
m ss
L T W
, where Tm is the total multipath
delay spread of the Nakagami-m fading channel on
the condition that the transmitted signal bandwidth
Wss is much greater than the coherence bandwidth
Wcoh of the fading channel, with
x
be the
maximum integer less than or equal to x.
2.1 AWGN channel
For a single user transmitting signal, of
bandwidth W, in the AWGN channel, the received
signal-to-noise ratio (SNR) is
0
P
Z
N W
where N0
is the double-sided noise power spectral density and
P is the received power. When this user operates in
the considered DS-CDMA system, his signal will
clearly be affected by the Gaussian distributed MAI
of all the other (K-1) simultaneous users, the
power of its is
1
MAI
K
P
P
. The received
spread signal-to-interference-plus-noise power ratio
(SINR) (prior despreading) will then be
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DOI: 10.37394/23204.2022.21.6
Panagiotis Varzakas
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1
ss
p
Z
Z
G K Z
. Accordingly, the
channel capacity per user is [2, Eq. (15-2-8)]:
ss 2
C log (1 )
user ss
W Z
(1)
and the spread channel incorporated will appear
with a total channel capacity Ct, available to all
users, being equal to the sum of the corresponding
individual rates, i.e.
t user
C K C
.
2.2 Nakagami-m fading channel
Assuming independent and identical
distributed (i.i.d.) input paths, the probability
density function (pdf) of the instantaneous received
signal-to-interference-plus-noise ratio (SINR),
,
i ss
,
in the lth, l 1,...,L ,branch of the ith, MRC RAKE
receiver is a Gamma distribution with two degrees
of freedom:
, , , ,
, , , ,
, ,
l i ss l i ss
mm
l i ss l i ss
l i ss
m
p e
m
(2)
where
, ,
l i ss
is the corresponding average received
SINR, resulting from the signals of all transmitting
users,
x
is the gamma function defined as
1
0
( ) t x
e t d
x
t
and m is the Nakagami-m
fading parameter
0.5
m. When m=1 the fading
envelopes are Rayleigh distributed and (2) reduces
to the exponential distribution. The statistics of
each interfering signal in (2) need not considered
separately since, either the total interference power
at the RAKE receiver output, or the MAI from the
(K-1) other users prior despreading, even for a
small number of users, tends to be Gaussian
distributed [6], and thus, it can directly be
incorporated in the Shannon formula regardless the
interference statistics.
When the K users are simultaneously
transmitting over a Nakagami-m fading channel,
the pdf of the combined SNR,
mrc
, assuming equal
path strengths, , ,
l i ss ss
, at the MRC RAKE
receiver output, will be given by [1]:
1mL
mrc
ss
ss
mL m
mrc
mrc
m
p e
mL
(3)
where the corresponding average received SINR,
mrc
, will be:
mrc
ss
L
(4)
The average total channel capacity,
t
C
, will
be given by averaging Ct over the pdf of the
combined SNR,
mrc
, at the RAKE receiver’s output
[7], i.e.
2
0
log 1
t ss mrc mrc mrc
C W p d
(5)
By replacing (3) and (4) in the above equation
and using [8, Eq. (32)], the average channel capacity
per user, normalized over the signal bandwidth, will
be given by a closed-form expression as:
1
1
1
mL
p
p
t
p
mL
mL G K Z
G
C
W K mL Z
mL G K Z
IZ
(6)
where
n
I
is defined as
1
1 ! ,
n
k
n
k
I n e n k
and
,
x
is the incomplete gamma function,
1
( , ) t
x
e t d
x
t
. It is interesting to note that,
for K=1, m=1 and Gp=1
(L=1), (6) directly leads to the expression of the
average channel capacity of a single user
transmitting, without spreading the signal bandwidth
W as it first appeared in [7]. In addition, considering
a DS-CDMA system, with a transmitted signal
bandwidth Wss that tends to infinity, i.e. with
p
G
, the average channel capacity per user
user
C will tend to the capacity of a AWGN
channel bandlimited to W, i.e.
2
log 1
user
C W Z
, since in this case, the
pdf of
mrc
in (3) will be given by the delta-function
[7].
3 Numerical results
By evaluating (6), numerical results are
presented in Figs. 1 and 2. In both figures a typical
urban area with Tm=1.5 µs is considered, with K =10
simultaneously transmitting active users, Gp=100
and W=20KHz. In Fig. 1,
/
user
C W
is plotted as
function of Z for RAKE reception and m=0.5, 1, 2
and 4. In the same figure, using (1), the
corresponding curve for the AWGN channel is also
plotted for comparison reasons. As it is clear
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DOI: 10.37394/23204.2022.21.6
Panagiotis Varzakas
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32
Volume 21, 2022
/
user
C W
increases with Z and a floor is observed
due to MAI. For a fixed Z , /
user
C W increases as
the fading severity decreases (i.e. m increases) and it
gets very close to the corresponding curve of the
AWGN for m=4 .
Fig. 1 /
user
C W versus Z for typical urban area
with Tm=1.5 µs, K=10 users, Gp=100, W=20KHz
and several values of m.
In Fig. 2, /
user
C W is plotted as function of
m for RAKE reception and Z=20, 25 and 30dB. As
m increases, /
user
C W increases, as it was also
observed in Fig. 1.
Fig. 2
/
user
C W
versus m for typical urban area
with Tm=1.5 µs, K=10 users, Gp=100, W=20KHz
and several values of Z.
4 Conclusions
In this paper, an analytical closed-form
expression for the average channel capacity per
user, in the Shannon sense, of a non-cooperative
DS-CDMA system over a Nakagami-m fading
channel under MRC RAKE reception has been
derived. Numerical results have pointed the effect of
the fading severity and MAI on the system’s
channel capacity.
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DOI: 10.37394/23204.2022.21.6
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