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Impact of Measurement Errors on the ClosedLoop Power Control for CDMA Systems

Li-Chun Wang and Chih-Wen Chang

National Chiao Tung University, Taiwan

Email : lichun@cc.nctu.edu.tw

AbstractIn this paper we present a simple analytical modelto evaluate the impact of measurement errors on the closed-loop power control for code division multiple access (CDMA)systems. By introducing a new performance measurement, theprobability of false command in power control, PFC, a closedform formula for calculating PFC with consideration of mea-surement errors is presented. Furthermore, we drive a bit er-ror rate (BER) performance bound in terms of PFC for theCDMA system with the closed-loop power control (CLPC). Theproposed analytical approach can quantitatively evaluate theperformance of the CLPC taking into account of the effects ofmeasurement errors and Doppler frequency under a Rayleighfading channel. Through simulation and analysis, we show thatthe proposed analytical BER bound can accurately estimate theBER performance of the CLPC under the impact of measure-ment errors. Interestingly, we find that the CLPC is less sensi-tive to measurement errors due to the non-linear operation inthe one step up/down power control scheme compared with thevariable-step size power control.

Index TermsCDMA System, Power Control, MeasurementError.

I. INTRODUCTION

AC curate power control is one of key technologies to

achieve high capacity code division multiple access

(CDMA) systems. Power control errors may be resulted

from many factors [1], such as loop delay [2], quantiza-

tion errors [3], multi-path fading [4], [5], link-quality mea-

surement errors [6], and feedback errors [7]. Although the

closed-loop power control (CLPC) in the CDMA system has

been studied extensively in the literature [3], [6], [8], [9],

[10], fewer papers have analyzed the performance of power

control scheme subject to signal-to-interference ratio (SIR)

measurement errors.Previous works about the impact of measurement errors

on power control can be summarized as follows. The im-

pact of measurement errors on the open-loop power control

was studied in [6]. The authors in [1] discussed the filter-

ing effect in the measurement scheme for the WCDMA sys-

This work was supported jointly by the Lee and MIT center for network-ing research, and the National Science Council R.O.C under the contract90-2213-E-009-068, 89-E-FA06-2-4, and EX-91-E-FA06-4-4.

tems. In [10], [11], [12], the issue of joint minimization of

SIR measurement errors and power control errors are inves-

tigated in the form of a stochastic control problem, but the

SIR measurement errors in [10], [11], [12] are modeled as

white Gausssian noise. However, in [13], it has been found

that measurement errors tend to be log-normal distributed incellular channel with Rayleigh fading and shadowing.

The goal of this paper is to develop a simple and accurate

analytical model to evaluate the impact of log-normal dis-

tributed SIR measurement errors on the CLPC of CDMA

systems. The contributions of this work can be summa-

rized in two folds. First, to evaluate the impact of measure-

ment errors on the closed-loop power control, we introduce

a new performance measurement called the probability of

false power control command, PFC. The motivation of in-troducing the new parameter, PFC, is because the feedbackpower control command is the key to the accuracy of power

control scheme. We will present a close-form formula for

calculating PFC in terms of measurement errors. Second,we propose a simple BER bound in terms of PFC. Hence,by using the proposed analytical approach, the performance

for different SIR measurement schemes on the closed-loop

power control in the CDMA system can be easily obtained.

The rest of this paper is organized as follows. In Section

II, we briefly introduce power control and define the proba-

bility PFC. Section III derives the closed-form formula ofPFC in terms of measurement errors. In Section IV, we de-rive a simple BER bound for the CLPC in CDMA systems

with the parameter PFC. Section V shows some analyti-cal and simulation results. Section VI gives concluding re-

marks.

I I . PROBABILITY OF FALSE POWER CONTROL

COMMAND

A. Background

Generally speaking, we can categorize power control

models into two kinds. The first one is the open-loop power

control and the other one is the closed-loop power control. In

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the former scheme, a mobile terminal decides its own trans-

mission power by comparing the desired received SIR and

the target SIR to compensate path loss and shadowing. On

the other hand, the CLPC makes the base station determine

the up/down power control command by comparing the re-

ceived SIR with the target SIR. The block diagram of theCLPC scheme is shown in Fig.1. As shown in this figure,

the CLPC has two feedback loops where the inner loop is

for fast power adjustment, while the outer loop is for setting

the target Eb/No.

B. Probability of False Power Control Command

Because the accuracy of power control commands will

strongly influence the performance of the closed loop power

control, it is important to investigate the impact of measure-

ment errors on the false power control command. Figure 2

illustrates how measurement errors will influence the power

control command. For example, as shown in Fig. 2, if theactual SIR value is below the target SIR value, the CLPC

will issue a power up command to increase the transmis-

sion power. For a small measurement error, the power con-

trol command will still be correct since the measured SIR is

below the the target SIR as shown in the left hand side of the

figure. However, if a large measurement error causes the re-

ceived SIR to become larger than the target SIR, then power

control command may change to be a wrong power down

command, as shown in the right hand side of the figure.

In order to evaluate the effect of measurement errors on

the closed-loop power control scheme, we define the proba-

bility of false command (PFC) in power control as follows.

PFC = Prob{ sgn (SI RT SI RM)= sgn (SI RT SI RA)} (1)

where sgn(x) is the operator to choose the sign ofx, SI RMis the measured SIR, SI RA is the actual SIR value, and theSI RT is the target SIR. From this definition, one can seethat the measurement error makes the sign of (1) change,

thereby issuing a false command in power control.

III. ANALYSIS

In this section, we will derive an analytical formula toevaluate the probability of false command in power control.

To begin with, we model the power control error (PCE) and

measurement errors as independent log-normal distributed

random variable as [9], [13]. Denote the SIR measurement

error by a random variable, Y, and the power control error by

a random variable, X. In the following, we will consider all

the variables in the dB domain. Denote SI RT and SI RA asthe target SIR and the actual SIR, respectively. Then SI RA

Received

Signal

ChannelMatch

Filter

Rake

Combiner

Channel

Decoder

CRC

Detection

Eb/NoMeasurement

Power Control

Step Size

Target Eb/No

Setting

+

T

Fig. 1. The block diagram of the closed-loop power control scheme.

can be expressed as the summation ofSI RT and power con-trol error, X, i.e.

SI RA = SI RT + X . (2)

The measured SIR, denoted as SI RM, can be expressed asthe summation ofSI RA and the measurement error, Y, i.e,

SI RM = SI RA + Y . (3)

From (2) and (3), we know

SI RM = SI RT + X+ Y . (4)

Substituting (2) and (4) into (1), we find that a false power

control command occurs when the following condition is

sustained:

sgn(SI RT (SI RT + X+ Y)) =sgn(SI RT (SI RT + X)) (5)

or

sgn(X+ Y) = sgn(X) . (6)Hence, the probability of false power control command

(PFC) can be written as

PFC = P(X+ Y < 0|X > 0)P(X > 0) +P(X+ Y > 0|X < 0)P(X < 0)

= P(X+ Y < 0, X > 0) +

P(X+ Y > 0, X < 0) . (7)

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Measurement

error Actual Eb/No

Target Eb/No

Measured Eb/No

No False Command

Actual Eb/No

Target Eb/No

Measured Eb/No

However, actual Eb/No < Target

Eb/No. We need to increase power.

Measured Eb/No < Target Eb/No

=>Issue a Power up command

Measured Eb/No > Target Eb/No

=> Issue a Power down command

PCEMeasurement error PCE

False Command occurs.

Fig. 2. Occurrence of false command in power control.

Note that in the dB domain both X and Y are Gaussian ran-dom variables with zero means and standard deviations of

X and Y , respectively. Because

P(X+ Y < 0, X > 0) = P(X+ Y > 0, X < 0), (8)

we can simplify PFC as

PFC = 2P(X+ Y < 0, X > 0)

= 2P(Y < X,X > 0)

= 2

0

x

fXY (x, y)dydx , (9)

where fXY(x, y) is the joint probability density function ofrandom variables Xand Y. Because measurement errors aremajorly influenced by the measurement scheme rather than

power control method, we assume the measurement error Xis independent of power control error Y. Then, we have

PFC = 20

x

12XY

exp(x2

22X)

exp(y2

22Y)dydx

= 2

0

x

1

2XYexp(

x222X

)

exp(y2

22Y)dydx . (10)

Let x

= x/X and y

= y/Y . Then we can express PFCas

PFC = 2

0

X

Yx

1

2exp(

12

x2)

exp(12

y

2)dy

dx

. (11)

Next, we change the coordinate axis to polar coordinate axis

by letting x

= r cos and y

= r sin . Then PFC can bewritten as

PFC = 2

/2tan1

X

Y

0

1

2e1

2r2rdrd

=1

2 tan

1 XY

. (12)

Now we want to analyze the impact of PFC on the BERperformance. From [14], we know that for the received bit

energy to noise density ratio b, the bit error rate of BPSKmodulated signals in Rayleigh fading channel is

Pb1 =1

2

1

b

1 + b

. (13)

In the AWGN channel, the BER of BPSK modulated signals

is

Pb2 = Q(

2b) (14)

where

Q(x) =1

2x

e

t2

2 dx . (15)

Since the purpose of power control is aimed to remove

Rayleigh fading, we expect that the BER performance of the

CLPC will be closed to (14) if the power control perfectly

removes the impact of Ralyeigh fading; and on the other

hand, the BER performance of the CLPC will be close to

(13) if there is no power control. Consequently, we propose

an upper bound on the BER performance for the closed-loop

power control subject to measurement errors as follows.

Pb,FC 2PFCPb1 + (1 2PFC)Pb2

PFC1 b1 + b+(1 2PFC)Q(

2b) , (16)

where Pb1 and Pb2 are defined in (13) and (14). Note thatbecause the maximum of PFC is 0.5, we use 2PFC to ap-proximate the case without power control. We will evaluate

the accuracy of the proposed upper bound (16) via simula-

tion in the next section.

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TABLE I

SIMULATION PARAMETERS.

Spreading Factor 4

Doppler Frequency 5-30Hz

Power Control Period 0.667mSec

Power Control Step Size 1 dBTarget Eb/No 5 dB

Modulation Scheme BPSK

30 20 10 0 10 20 30 400

0.02

0.04

0.06

0.08

0.1

0.12

PCE in dB ( The measurement error is modeled by Log N(0,52) )

Pro

bility

PCE (Simulation)LogNormal Fitting

Theoretical PFC

=0.301

Simulative PFC

=0.309

Fig. 3. The power control error statistics with a measurement error ofstandard deviation equal to 5 dB, i.e. Y =5 dB, with fd = 20 Hz, andtarget Eb/No = 5 dB.

IV. NUMERICAL RESULTS

In this section, we simulate the uplink performance for a

single user under a flat Rayleigh fading channel. We assume

the measurement errors are log-normal distributed random

variables [13]. Other parameters used in the simulation are

listed in Table I.

Through simulation, Figure 3 shows the probability den-

sity function of power control errors (PCE) subject to mea-

surement errors with standard deviation of 5 dB. As shown

in the figure, the PCE (expressed in dB domain) exhibits

the similar characteristics of normal distribution. This re-

sult confirms our assumption that the PCE is lognormal dis-

tributed random variable even with the influence of measure-

ment errors.Figure 4 shows the impact of measurement errors for dif-

ferent Doppler frequencies on the the probability of false

power control command, PFC. One can note that thehigher the Doppler frequency, the lower the PFC. This phe-nomenon can be explained as follows. Because the radio

channel will change faster at a higher Doppler frequency, the

power control mechanism usually cannot follow fast chan-

nel variations. Thus, in this situation a larger power con-

1 2 3 4 5 6 7 8 9 100.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Y

(STD. of measurement error in dB)

PFC

fd=5 Hz

fd=15 Hz

fd=30 Hz

Theory

Simulation

Fig. 4. The probability of false power control command ,PFC, subject tomeasurement errors for different Doppler frequencies, where fd = 30, 15,and 5 Hz, and target Eb/No = 5 dB.

trol error will occur. Due to a larger gap between the tar-

get SIR and the actual SIR, it is less likely to change the

power control command from power up to power down

or vice versa. In other words, a larger PCE due to higher

Doppler frequency can tolerate large measurement errors,

thereby having a smaller PFC as shown in the figure. In Fig-ure 4, we also verify the accuracy of the analytical results of

PFC derived from (12) by comparing with simulation. Asshown, simulation results ofPFC are close to the analyticalresults derived from (12).

Figure 5 shows the corresponding BER performance of

Fig. 4 including the impact ofPFC. Although in Fig. 4 itis demonstrated that a higher Doppler frequency leads to asmaller PFC, the BER performance at a higher Doppler fre-quency is still larger than that at a lower Doppler frequency

as shown in Fig. 5. It is implied that even with a larger

probability of false power control command, the closed-loop

power control scheme under a slower fading channel still

performs better than that under a fast fading channel.

Figure 6 shows the impact of very large measurement er-

rors on the probability of false command PFC in the closed-loop power control. One can see that as a measurement error

increases up to 30 dB, the probability of false power control

command tends to be saturated at 0.45. Note that in (16), the

maximum ofPFC approaches to 0.5 for the infinite measure-ment error.

Figure 7 illustrates the BER performance of the closed-

loop power control subject to very large measurement errors.

As shown in the figure, the theoretical values according to

formula (16) provide a reasonable bound on the BER per-

formance of the closed-loop power control subject to mea-

surement errors. Because PFC will be bounded to be 0.5even with large measurement errors, one can find that the

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1 2 3 4 5 6 7 8 9 1010

3

102

101

Y

(STD. of measurement error in dB)

BER

fd=5Hzfd=15Hzfd=30Hz

Fig. 5. The BER performance comparison of the the closed-loop powercontrol subject to measurement errors for different Doppler frequencies,where fd = 30, 15, and 5 Hz, and target Eb/No = 5 dB.

BER of the up-down closed-loop power control will also be

bounded under a large measurement error. In the figure, with

the target Eb/No equal to 10 dB, the BER is limited to below0.02 even with a measurement error with standard deviation

of 30 dB. Thus, it is implied that the up/down power control

scheme is robust to measurement errors.

Figure 8 compares the up/down closed-loop power con-

trol with the variable-step size closed-loop power control

from the standpoint of the sensitivity of power control er-

rors with respect to measurement errors. In our simula-

tion, the variable-step size power control is a power con-

trol mechanism with higher resolution in power adaptationsteps. As shown, unlike the variable-step power control has

a strong correlation between measurement errors and power

control errors, the impact of a larger measurement error on

the up/down closed-loop power control is less sensitive.

V. CONCLUSIONS

In this paper we have presented a simple and accurate

analytical expression for the probability of false command

(PFC) in the closed-loop power control (CLPC) subject tomeasurement errors. Moreover, we also propose a BER per-

formance bound in terms ofPFC. It is found that the larger

the Doppler frequency, the smaller the PFC, while for theBER performance, a larger Doppler frequency results in a

poor BER performance even with a smaller PFC. Interest-ingly, we find that compared with the variable-step power

control, the up/down CLPC is more robust to measurement

errors due to the non-linear operation in the up/down power

control scheme. The proposed analytical approach can be

easily applied to analyze the performance of the closed-loop

power control with different SIR measurement schemes.

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Y

(STD. of Measurement error in dB)

PFC

SimulationTheory

Fig. 6. Probability of false power control command, PFC, with a largestandard deviation of measurement error, Y, fd = 30 Hz, and targetEb/No = 5 dB.

5 10 15 20 25 30

103

102

101

Y

(STD. of measurement error in dB)

BER

Target Eb/N

0=5 dB

Target Eb/N

0=10 dB

Upper bound

Simulation

Fig. 7. BER performance of the closed-loop power control against mea-surement errors with stand-deviation,Y, and fd = 30 Hz.

Thus, an interesting future research extended from this work

is to apply the proposed analytical framework to incorporate

different SIR measurement schemes to evaluate the perfor-

mance of the closed-loop power control in CDMA systems.

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