EFFECTS OF ADDITIVE NOISE ON THE
THROUGHPUT OF CDMA DATA
COMMUNICATIONS
Penina Orenstein, David Goodman, Zory Marantz and Virgilio
Rodriguez
Department of Electrical and Computer Engineering1
Polytechnic University
Brooklyn, NY, USA
porenste@poly.edu
Abstract—We analyze the optimum transmitter power levels and
the optimum number of active terminals sending data to a
CDMA base station. The objective is to maximize the aggregate
throughput of the base station. We find that in the presence of
additive noise, received power balancing is suboptimal
mathematically. We consider N terminals transmitting at the
same data rate, with the power of the most distant terminal
(terminal N), fixed at its maximum value, and the power of the
other N-1 terminals varying. We conclude that the aggregate
throughput at the base station is maximized when the receiver
powers for the N-1 terminals are equal and larger than the
receiver power of the N-th terminal. This finding reduces the
complexity of the analysis to a univariate optimization problem.
A numerical analysis indicates the extent to which additive noise
reduces the optimum number of active terminals and the
maximum base station throughput.
Keywords-power control; radio resources management; power
balancing
I.
INTRODUCTION
We analyze the throughput of a CDMA base station
receiving data from N transmitters, all operating at the same
constant bit rate. We consider two resource management
issues: transmitter power control and the number of terminals
that should be admitted to the system in order to maximize
base station throughput.
Early work on uplink CDMA power control focused
on telephone communications and determined that to
maximize the number of voice communications, all signals
should arrive at a base station with equal power [1]. Initial
studies of power control for data communications focused on
maximizing the utility of each terminal, with utility measured
as bits delivered per Joule of radiated energy [2,3]. By
contrast this paper considers maximizing the aggregate
throughput of a CDMA base station.
In earlier work on this problem, Ulukus and
Greenstein [9] adjust data rates and transmitter power levels in
order to maximize network throughput. Lee, Mazumdar, and
Shroff [10] adapt data rates and power allocation for the
downlink, and provide a sub-optimal algorithmic solution
based on pricing. Sung and Wong [11] assume that the
terminals’ data rates are different but fixed, and maximize a
capacity function.
Our recent work [4-7] adjusts the power and rate of
each terminal to maximize ΣiβiTi, the aggregate weighted
throughput of a base station. Ti b/s denotes the throughput of
terminal i and the weight βi admits various interpretations,
such as priority, utility per bit, or a monetary price paid by the
terminal. This research assumes that the number of active
terminals is fixed, their data rates are continuous variables to
be optimized, and that the system is interference limited (noise
is negligible).
In the present paper, we set all the βi=1 and take the
data rates to be identical and fixed, but we view the number of
active terminals as a key variable to be optimized. We also
consider additive noise to be non-negligible, which is essential
when out-of-cell interference is significant, and included in the
noise term. The research reported in [8] dealt specifically with
the case when noise and out-of-cell interference* are
negligible, and found that the transmitter power levels should
be controlled to achieve power balancing. With power
balancing, all signals arrive at the base station with equal
power.
By contrast, the results in the present paper pertain to
the case when noise and interference from other cells are not
negligible. We show that with additive noise power balancing
leads to sub-optimal performance and that when one terminal
has a maximum power constraint, the optimum set of
transmitter power levels depends on the maximum received
SNR of the constrained terminal. Furthermore, we
demonstrate that when one terminal has a maximum power
constraint, the other terminals should aim for the same
received power, which depends on the maximum SNR of the
constrained terminal.
1
Supported in part by NYSTAR through the Wireless Internet Center for
Advanced Technology (WICAT) at Polytechnic University and by the
National Science Foundation under Grant No. 0219822
1
*To be concise we refer to the combination of noise and inter-cell interference
simply as “noise”. The analysis does not distinguish the two impairments. It
considers only their combined power.
We also find that in order to maximize base station
throughput with any power control algorithm, the number of
active transmitters, N, should be limited to N ≤ N*, where N*,
is a property of the frame success rate f(γ), the probability that
a terminal’s data packet is received successfully as a function
of γ, the received signal-to-interference-plus-noise ratio
(SINR). The specific form of f(γ) depends on the details of the
CDMA transmission system, including the packet size,
modem configuration, channel coding, antennas, and radio
propagation conditions. Our analysis applies to a wide class of
practical frame success functions, each characterized by a
smooth S-shaped curve [4].
The next section presents the CDMA transmission
system and a statement of the throughput optimization
problem. The analysis in Section III considers the effects of
non-negligible noise and out-of-cell interference.
II.
THE OPTIMIZATION PROBLEM
A data source generates packets of length L bits at each
terminal of a CDMA system. A forward error correction
encoder, if present and a cyclic redundancy check (CRC)
encoder together expand the packet size to M bits. The data
rate of the coded packets is Rs b/s. The digital modulator
spreads the signal to produce Rc chips/s. The CDMA
processing gain is G=W/Rs, where W Hz, the system
bandwidth, is proportional to Rc. Terminal i also contains a
radio modulator and a transmitter radiating Pi watts. The path
gain from transmitter i to the base station is hi and the signal
from terminal i arrives at the base station at a received power
level of Qi =Pi hi watts. The base station also receives noise and
out-of-cell interference with a total power of σ2=Wη0 watts,
where η0 is the one-sided power spectral density of white
noise. The base station has N receivers, each containing a
demodulator, a correlator for despreading the received signal,
and a cyclic redundancy check decoder. Each receiver also
contains a channel decoder if the transmitter includes forward
error correction.
In our analysis, the details of the transmission system are
embodied in a mathematical function f(γ), the probability that
a packet arrives without errors at the CRC decoder. The
dependent variable γ, is the received SINR. For terminal i,
γi = G
Pi hi
N
∑P h
j =1
j ≠i
j
j
+σ 2
=G
(1)
Qi
N
∑Q
j =1
j ≠i
j
+σ 2
Acknowledgment messages from the receiver inform the
transmitter of errors detected at the CRC decoder that have not
been corrected by the channel decoder. The transmitter
employs selective-repeat retransmission of packets received in
error.
Our earlier study [8] assumes that intra-cell
interference dominates the total distortion and study system
performance when σ2=0. When σ2>0, we define the signal-tonoise ratio of receiver i as si =Qi/σ2 and rewrite Equation (1) as
2
γi =G
si
.
N
∑s
j
(2)
+1
j =1, j ≠ i
In cases of practical interest f(γ) is a continuous, increasing Sshaped function of γ, with f(0)=2-M≈0 and f(∞)=1 [4].
If the probability of undetected errors at the CRC
decoder is negligible, the throughput of signal i, defined as the
number of information bits per second received without error,
is:
Ti =
L
Rs f (γ i )
M
b/s,
(3)
The aggregate throughput, Ttotal, is the sum of the N individual
throughput measures in Equation (3). Assuming that L, M, and
Rs are system constants, we analyze the normalized throughput
of N simultaneous transmitters as UN defined as
N
M
(4)
Ttotal = ∑ f (γ i ) .
LRs
i =1
UN is dimensionless and bounded by 0≤ UN≤ N.
The aim of our optimization study is to find the
transmitter power levels, Pi, that maximize UN.. We then
examine the maximum throughput as a function of N in order
to find the number of simultaneous transmitters that result in
the highest normalized throughput. To find the optimum
transmitter power levels, it is convenient mathematically to
maximize Equation (4) with respect to the received powers Q1,
Q2,…,QN. To do so, we differentiate Equation (4) with respect
to each of the received power levels Qi . We then examine the
N derivatives under the power balancing condition Qi =Q for
i=1,2,…,N. Under this condition, all of the derivatives are
equal. They have the following properties.
UN =
∂U N
∂Qi
= 0; σ 2 = 0;
Qi =Q
∂U N
∂Qi
> 0; σ 2 > 0 (5)
Qi = Q
These formulas indicate that when performance is
limited by intra-cell interference (σ2=0), it is possible that
maximum throughput occurs when all signals arrive at the
base station at the same power level. The optimization
problem is more complex when σ2>0.
Our prior work [8] shows that in the absence of
additive noise, UN is maximized when all signals have the
same SINR γ=G/(N-1) ≈γ*, where γ*, the preferred SINR, is a
property of the frame success function f(γ). Therefore, with
σ2=0, the optimum number of active terminals is an integer
near 1+G/γ*.
III. MAXIMUM THROUGHPUT WITH NOISE PRESENT
In this paper we take into account the effects of
additive noise and interference from other cells. The total
power in these impairments is σ2=Wη0 watts (and we refer to
them together as “noise” to be concise). The noise appears at
the receiver as an additional signal that does not contribute to
the overall throughput. The system has to use some of its
power and bandwidth resources to overcome the effects of the
noise.
The effects of noise depend on the power limits of
practical terminals. With unlimited power, we would increase
all the received powers Qi indefinitely until the effect of the
noise is negligible. To account for the power limits, let Pi,max
denote the power of the strongest possible signal transmitted
by terminal i
and Qi,max=Pi,maxhi, the power of the
corresponding received signal. The maximum signal-to-noise
ratio of terminal i is si,max=Qi,max/σ2. In our analysis, we order
the labels of the terminals such that Q1,max≥Q2,max≥…≥QN,,max.
In many situations this ordering implies that terminal 1 is
closest to the base station and terminal N is most distant.
A. Two terminals
With Q2,max≤Q1,max, it is reasonable to assume that
when terminal 2 is admitted to the system, it transmits with
maximum power to achieve Q2=Q2,max. In this case, Equation
(4) becomes
Q1
+
U2 = f G
2
Q
+
σ
2,max
Q 2 ,max
f G
Q +σ 2
1
(6)
If we introduce the normalized quantity z=s1/s2,max and for
conciseness we set ρ=1/s2,max, then we can rewrite equation (6)
as
Gz
G
+ f
,
si = Qi / σ 2
(7)
U 2 = f
1+ ρ
z+ρ
The first-order condition for optimality can be expressed as
Gz
G
∂U 2
G
G
−
=0
=
(8)
f '
f '
∂z
1 + ρ 1 + ρ ( z + ρ ) 2 z + ρ
If we evaluate equation (11) using the first-order condition in
(10), we obtain an expression (12), whose sign determines the
nature of the turning point. For a maximum, we require that
this expression is negative, i.e.
f ' ' (γ 1 )
f ' ' (γ 2 )
2
+
+
<0
(12)
2
2
γ 2 f ' (γ 2 )
f ' (γ 1 )
f ' (γ 2 )
Because f’(γ) >0 for all γ, a sufficient condition for a
maximum is that both
a) f ' ' (γ 1 ) < 0 and f ' ' (γ 2 ) < 0
b)
f ' ' (γ 1 )
f ' (γ 1 )
2
+
f ' ' (γ 2 )
f ' (γ 2 )
2
>
2
γ 2 f ' (γ 2 )
For the class of functions considered here, there is a quantity
γ~ for which f ' ' (γ ) < 0 when γ > γ~ . Therefore if γ 2 > γ~ and if
z>1, it follows that f ' ' (γ 1 ) < 0 as required.
For the numerical examples in this paper, we refer to
the frame success rate function for the non-coherent frequency
shift keying modem and the frame size M=80 considered in
previous work on power control for wireless data [2-7]. In this
case, f (γ ) = (1 − 0.5e −γ / 2 ) 80 .
Figure 1 is a numerical example with ρ=1 and
G=16.2, the processing gain that produces U2=1 at z=1. The
figure shows throughput as a function of z, and graphs for the
two sides of Equation (10). It shows that the “power-balanced”
solution, z=1, is sub-optimal and that equation (10) has
solutions at z=1.41 and z=0.65. Throughput is maximum
when z=1.41 and a minimum when z=0.65. Note that if
terminal 1 transmits at a power greater than 1.41P2,max, then
the total throughput will still be better than having just one
terminal transmit; however, it is no longer optimal.
Using the substitutions
γ1 =
Gz
1+ ρ
γ2 =
,
G
z+ρ
,
(9)
we obtain the first-order condition for optimality:
γ 1 f ' (γ 1 )
z
=
γ 2 f ' (γ 2 ) z + ρ
(10)
Equation (5) implies that with σ 2 > 0 , power balancing is
suboptimal and that there are transmitter powers with z>1 that
produce higher throughput than the throughput obtained when
z=1.
To establish a necessary condition for a maximum we
∂ 2U 2
need to evaluate the sign of
at the critical point. In
∂z 2
general,
∂ 2U 2
∂z 2
=
γ 1 2 f ' ' (γ 1 )
z2
+
1
∂
( z + ρ ) 2 ∂γ 2
(γ
2
2
f ' (γ 2 )
)
(11)
3
Figure 1: The aggregate throughput U2 is maximum at power
ratio z=1.41
B. Arbitrary Number of Terminals
As with the previous analysis we begin by writing N-1 the
first-order conditions that need to be satisfied simultaneously:
We now move the analysis beyond the case of two
terminals. We would like to determine the optimum receivepower vector when there are N terminals, where the power of
the weakest terminal (N) is fixed. We begin with a brief
analysis for N=3 and find that the optimum solution depends
on a single variable, which greatly simplifies the problem. We
then extend the results to the more general case.
∂U N γ i
f ' (γ i ) −
=
zi
∂z i
B.1
j
j ≠i
i =1
i = 1,.., N − 1
(17)
j
This system of equations can be reduced to N-2 equations of
the form
z
γ i f ' (γ i )(G + γ i )
i = 1,.., N − 2
= i
(18)
γ i+1 f ' (γ i +1 )(G + γ i +1 ) z i+1
N=3 Terminals
As with N=2, we assume that the transmitter power
of the weakest terminal is fixed at P3=P3,max, so that s3=s3,max
andwe introduce the normalized quantities zi =si /s3,max, and
ρ=1/s3,max. The throughput equation can then be expressed as
Gz 2
Gz1
G
. (13)
+ f
+ f
U 3 = f
z1 + z 2 + ρ
z1 + 1 + ρ
z2 + 1 + ρ
We first evaluate the partial derivatives for terminals 1 and 2
and obtain
∂U 3 γ 1
γ 2
γ 2
=
f ' (γ 1 ) − 2 f ' (γ 2 ) − 3 f ' (γ 3 )
z1
Gz 2
G
∂z1
(14)
2
2
∂U 3
γ1
γ2
γ3
=−
f ' (γ 1 ) +
f ' (γ 2 ) −
f ' (γ 3 )
∂z 2
Gz1
z2
G
For a critical point we require that both partial derivatives
equal zero, from which we obtain the first-order condition
γ 1 f ' (γ 1 )(G + γ 1 ) z1
=
(15)
γ 2 f ' (γ 2 )(G + γ 2 ) z 2
Condition (15) can be met if γ 1 = γ 2 = γ~ → z1 = z 2 which
implies that a solution exists if the receive power of the two
“non-fixed” terminals is equal, yet not necessarily the same as
the receive power of the fixed terminal. This solution may not
be unique. However, it is useful both for mathematical
analysis and for practical implementation.
B.2
γ j2
∑ Gz f ' (γ )
N
Extension to N terminals
The throughput equation for N terminals transmitting
to the base station, assuming that zi =si/sN,max, for i=1,…,N-1,
and zN=sN/sN,,max =1, can be expressed as
N
N
Gz i
UN =
f (γ i ) =
f N
(16)
i =1
i =1
zj +ρ
j
i
≠
i =1
∑
∑
∑
Condition (18) can be met if γ i = γ i +1 = γ~ for i=1,…,N-2,. It
follows that zi =zi+1 for i=1,…,N-2 from which we conclude
that a solution to the first-order conditions is zi =z=s/sN,max for
i=1,…,N-1. This greatly simplifies our problem since the
optimal solution now depends on one power ratio z, rather
than N power levels. We can therefore reduce the throughput
equation given in equation (16) as follows.
For each si =s, (i=1,…,N-1), we write
G
Gz
γi = γ =
; and γ N =
( N − 1) z + ρ
( N − 2) z + 1 + ρ
Then, the throughput equation can be expressed as
U N ( z ) = ( N − 1) f (γ ) + f (γ N )
(19)
Using the same approach as before, we can differentiate with
respect to z and obtain
∂U N ((1 + ρ )γ 2 f ' (γ )) f ' (γ N )γ N 2
=
−
(20)
G
∂z
Gz 2
which for a critical point yields the first-order condition:
γ 2 f ' (γ )
z2
=
(21)
γ N 2 f ' (γ N ) 1 + ρ
If z = 1 , the first order condition cannot be satisfied. Hence, as
with N=2, we conclude that for a turning point, z > 1 . To
classify this point we need to evaluate the sign of the second
derivative:
f ' (γ N 2 )
∂ 2U N γ N 4
'
'
(
)
f
γ
(
1
)
'
'
(
)
N
f
γ
=
+
−
+
N
G 2
f ' (γ ) 2
∂z 2
(22)
3
2 f ' (γ N )γ N
( N − 1) − γ ( N − 2)
2
zγ N
G
Since γ N 4 , G 2 , f ' (γ N ) 2 > 0 it follows that the sign of
equation (22) is the same as the sign of
f ' ' (γ N )
f ' ' (γ )
2
γ ( N − 2)
( N − 1) −
+( N − 1)
+
2
2
zγ
γ f ' (γ N )
f ' (γ )
f ' (γ N )
1424
3 144244
3 1N444
44
424444N44
3
A
B
C
(23)
Note that when N=2, expressions (23) and (12) are identical.
A sufficient set of conditions for a maximum is
4
a)
γ ( N − 2)
,
γ N ( N − 1)
A+ B < 0,
1< z ≤
b)
(24)
c) A + B > C
One way to satisfy condition (24b) is if both A < 0 and B < 0 .
For the class of functions considered here, f’’(γi )<0
when γ i > γ~. If we assume that both γ N > γ~ and z>1 then it
follows that both f ' ' (γ N ) < 0, f ' ' (γ ) < 0 as required.
This analysis so far suggests that a local maximum of
UN with respect to z1, z2, …, zN-1 occurs at some value of z such
that z=zi >zN. However, we need to ascertain whether the
maximum is at z=∞. A maximum at z=∞ suggests that it
might be better to exclude terminal N. The answer depends on
ρ. Since we have established that U N ( z > 1) is a maximum,
we can say that a sufficient condition for an interior solution is
U N (∞) < U N (1) , which implies that
G
G
( N − 1) f
< Nf
2
N
1
ρ
−
+
−
N
(25)
It follows that the largest value of N which satisfies equation
(25), say N*, is the maximum number of active terminals
which ensure an interior solution. If N>N*, the aggregate
throughput may still be maximized (z>1), however, the power
of the “non-fixed” terminals will approach infinity.
and the maximum throughput occurs at z<∞. The largest
value, N*=10 is the maximum number of active terminals
which will ensure an interior solution. Using condition (21)
we can numerically calculate z as z=2.24 which is indeed an
interior solution. For N>N*, condition (25) is no longer met
and the maximum approaches z=∞. This suggests that in a
practical setting, it would advisable to admit no more than N*
terminals (let Pi =0, for i>N*).
It is interesting to note that as sN→∞, the value of N*
approaches 13, which is the optimal number of terminals
(G=128) when there is no noise: As shown in [8], maximizing
the number of active terminals is equivalent to optimizing the
function f(γ)/γ which is maximum at γ*=10.75. Thus,
N * ≅ 1 + G / γ * =12.9.
CONCLUSIONS
This research considers how to maximize the throughput of a
CDMA base station receiving data from N transmitters, all
operating at the same constant bit rate. The principal
conclusions are that the aggregate throughput at the base
station is maximized if N-1 transmitters aim for a target SINR
that is greater than the maximum SINR of the weakest
terminal. Further, the number of active terminals should not
exceed N*, where N* is the maximum number of terminals
that can simultaneously transmit whilst ensuring an interior
solution of the first and second order optimality conditions.
REFERENCES
[1]
Figure 2: Throughput difference as a function of number of
terminals (N). The values of N for which the difference is
negative have maximum throughput at z<∞.
Figure 2 shows a numerical example for G=128. The
[ ]1∞ = U N z =∞ − U N z =1 . The condition for
vertical axis is U N
an interior point (25) is [U N ]1∞ < 0 . The graphs indicate the
values of N that have an interior solution (1<z*<∞). For
example, when sN=0.5 (ρ=2), N≤10 satisfies condition (25)
5
R.D. Yates, “A framework for uplink power control in cellular radio
systems'', IEEE Journal on Selected Areas in Communications, vol. 13,
no. 7, pp. 1341--1347, September 1995.
[2] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, “Efficient power
control via pricing in wireless data networks”, IEEE Transactions on
Communications, vol. 50, no. 2, pp. 291-303, Feb.2002.
[3] D. Goodman and N. Mandayam, “Network assisted power control for
wireless data”, Mobile Networks and Applications, vol. 6, no. 5, pp. 409418, Sept, 2001.
[4] V. Rodriguez, "Robust modeling and analysis for wireless data resource
management" IEEE WCNC, Vol. 2 , pp. 717 -22 , March 2003
[5] V Rodriguez and D.J. Goodman "Prioritized throughput maximization
via rate and power control for 3G CDMA: the 2 terminal scenario", Proc.
of 40th Allerton Conf. on Comm., Control and Comp., Oct. 2002
[6] V. Rodriguez, and D.J. Goodman, "Power and data rate assignment for
maximal weighted throughput in 3G CDMA" IEEE WCNC, vol. 1, pp.
525 -31, March 2003
[7] V Rodriguez, and D.J. Goodman, "Power and Data Rate Assignment for
Maximal Weighted Throughput: A Dual-Class 3G CDMA Scenario"
IEEE ICC, vol. 1 , pp. 397 -401, May 2003
[8] D.J. Goodman, Z. Marantz, P. Orenstein, and V. Rodriguez,
“Maximizing the throughput of CDMA data communications”, IEEE
VTC, Orlando FL, Oct 6-9, 2003 (to appear)
[9] S. Ulukus. and L.J. Greenstein "Throughput maximization in CDMA
uplinks using adaptive spreading and power control" IEEE ISSSTA, vol.
2, pp: 565 -569, 2000
[10] J.W. Lee, R.R. Mazumdar and N.B. Shroff, "Joint power and data rate
allocation for the downlink in multi-class CDMA wireless networks",
Proc. of 40th Allerton Conf. on Comm., Control and Comp., Oct. 2002.
[11] C. W. Sung and W. S. Wong, "Power control and rate management for
wireless multimedia CDMA systems", IEEE Trans. Commun., vol. 49,
no. 7, pp. 1215-26, July 2002.
6
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