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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 6, JUNE 20041019

Exact Error-Rate Analysis of Diversity 16-QAM

With Channel Estimation Error

Lingzhi Cao, Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE

Abstract—The bit-error rate (BER) performance of multilevel

quadratureamplitudemodulationwithpilot-symbol-assistedmod-

ulationchannelestimationinstaticandRayleighfadingchannelsis

derived, both for single branch reception and maximal ratio com-

biningdiversityreceiversystems.Theeffectsofnoiseandestimator

decorrelation on the received BER are examined. The high sen-

sitivity of diversity systems to channel estimation error is investi-

gated and quantified. The influence of the pilot-symbol interpola-

tion filter windowing is also considered.

Index Terms—Diversity, estimation error, fading channels,

maximal ratio combining (MRC), pilot-symbol-assisted modula-

tion (PSAM), quadrature amplitude modulation (QAM).

I. INTRODUCTION

T

have been studied by several authors. [1] gives the bit-error rate

(BER) of 16-QAM and 64-QAM over both the additive white

Gaussian noise (AWGN) channel and the Rayleigh fading

channel without diversity. A recursive algorithm for computing

the BER of M-QAM constellations over an AWGN channel

was given in [2]. Much work has focused on the derivation

of the symbol-error rate (SER) when diversity techniques are

used to compensate for the fading caused by the multipath

propagation. [3] presented the SER of M-QAM with maximal

ratio combining (MRC) diversity and selection combining (SC)

diversity in Rayleigh fading channels. In [4], precise analytical

expressions for the SER of MRC and equal gain combining

(EGC) in Nakagami fading channels were derived. [5] gave

a thorough study of the performances of two-dimensional

signaling schemes. The SER of 16-QAM on Rayleigh, Ricean,

and Nakagami fading channels with MRC, EGC, and SC was

derived. The SER for M-QAM with MRC reception was given

in terms of alternate forms of the

communication theory in [6].

References [1]–[6] assume that perfect channel state infor-

mation (CSI) is available to the receivers. Meanwhile, the per-

formances of coherent demodulation and coherent combining

can be severely degraded if good fading CSI is not available, as

shown in [7]. Study of a variable-rate, variable-power M-QAM

HE performances of multilevel quadrature amplitude

modulation (M-QAM) in different wireless environments

-functions, widely used in

Paper approved by P. Y. Kam, the Editor for Modulation and Detection for

Wireless Systems of the IEEE Communications Society. Manuscript received

March 12, 2003; revised October 2, 2003 and December 12, 2003.

The authors are with the Department of Electrical and Computer Engi-

neering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail:

lingzhi@ee.ualberta.ca; beaulieu@ee.ualberta.ca)

Digital Object Identifier 10.1109/TCOMM.2004.829516

in [8] also showed the sensitivity of the error rate of M-QAM to

estimation error.

Pilot-symbol-assisted modulation (PSAM) has proved to

be an effective method for channel estimation [9], [10]. Some

previous work has considered the performance of M-QAM

when using PSAM for channel estimation. [11] gives simple

upper bounds for the SER of M-QAM over a Rayleigh fading

channel in the presence of channel estimation error, and the

approximate BER of M-QAM with imperfect channel estima-

tion has been derived in [12]. None of [7], [8], [11], or [12]

considered diversity QAM. In this paper, we derive the BER

of 16-QAM using PSAM to provide estimates of the channel

amplitude and phase in the receiver demodulation process for

AWGN and Rayleigh fading channels, and for MRC diversity

systems, in which case, the PSAM is used to provide CSI for

both the demodulation process and the diversity combining.

Though we consider specifically 16-QAM, the analysis is

general and can be applied to general M-QAM with minor

modifications, but more cumbersome definitions, notations,

and development.

This paper is organized as follows. In Section II, the system

model as well as the 16-QAM modulation are described, and

someparametersusedinthelatersectionsarederived.TheBER

performance of 16-QAM in AWGN and in fading is derived in

Section III and the limitations of previous work are clarified. In

Section IV,the BERof 16-QAM with MRC diversityis derived.

Some examples calculated using our theoretical results are pre-

sented and discussed in Section V. Simulation results are also

presented for verification. Section VI concludes the paper.

II. SYSTEM AND CHANNEL MODELS

A. PSAM

The PSAM system structure considered here is identical to

that considered in [12], and the reader is referred to [12, Figs. 1,

3, and 7]. A block diagram of the PSAM system structure con-

sidered here is shown in [12, Fig. 1]. The transmitter period-

ically inserts the known pilot symbols into the data sequence

via a multiplexer. Let

denote the th symbol transmitted in

the th data frame. The symbols are formatted into frames of

symbols, with the first pilot symbol (

data symbols (

), as depicted in [12, Fig. 7]. At

the receiver side, the pilot symbols are extracted and input to an

estimation/interpolator filter. Thus, knowledge of the channel

amplitude and phase at the times of the pilot symbols is used to

estimate the channel amplitude and phase at the times of other

symbols.

) followed by ()

0090-6778/04$20.00 © 2004 IEEE

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1020 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 6, JUNE 2004

Denote the symbol interval as

timing recovery. The received signal sample at the th symbol

in the th frame,

, is given by

and assume perfect symbol

(1)

where

Rayleigh-distributed amplitude sample,

uniformly distributed on

16-QAM signal sample, and

variance

in both the real and imaginary parts. The channel

estimate for the th pilot symbol,

is the complex channel gain sample with

, and phase sample

is the transmitted

is the AWGN sample with

;

, is given by

(2)

where

symbol in the th frame, and

the th frame. The fading at the th symbol in the th frame is

estimated from

adjacent pilot symbols with

from previous frames, one from the current frame, and

symbols from subsequent frames, where

estimate is given by

is the complexchannel gain corresponding to the pilot

is the known pilot symbol in

pilot symbols

pilot

. The

(3)

where

are the interpolation coefficients of the estimation filter [10].

Since the channel fading and the noise are independent

complex Gaussian random variables, the channel estimate

, which is a weighted sum of zero-mean complex jointly

Gaussian random variables, is also a zero-mean complex

Gaussian random variable [13, Ch. 8], [14, Ch. 6]. Since

and are correlated complex Gaussian random variables, the

joint probability density function (jpdf) of their amplitudes,

and , and their phase difference,

following the discussion given in [13, Ch. 8]. The subscripts for

the complex channel gain and its estimate have been omitted

in the following discussion when ambiguity can not arise, for

notational simplicity. Similarly, the dependence on , the time

difference between two samples, has also been suppressed

for notational brevity. The jpdf,

(8–102)], is with the notation used here

,,

, can be derived

, given in [13, eq.

(4a)

where

(4b)

(4c)

(4d)

(4e)

(4f)

The second-order joint central moments,

rived in Section II-B, and

and , are de-

is derived in Section II-C.

B. Derivation of the Covariance Between

Thecomplexfadinggaincanbeexpressedintermsofquadra-

ture components as

and

(5)

In an omnidirectional scattering Rayleigh fading channel, the

autocorrelation and cross-correlation are given by [15]

(6)

and

(7)

respectively, where

the first kind [16] and

Using (2), (3), (6), and (7) in (4d) and (4e), the covariances,

and, are determined as

is the zeroth-order Bessel function of

is the maximum Doppler frequency.

(8a)

(8b)

C. Derivation of

The variance of each component of the complex Gaussian

fading estimate is obtained by using the definition (4c) with (2)

and (3). It is given by

(9)

From (6)

(10)

and assuming that the pilot-symbol energy

average data symbol energy

is equal to the

, one can rewrite (9) as

(11)

The results of (8a), (8b), and (11) are used in the jpdf (4a) of

the amplitudes and phase difference of the channel fading and

its estimate.

D. Modulation and Demodulation of Square QAM

[1] describes the modulation and demodulation of square

M-QAM. The signaling constellation and the Gray code

mapping of the transmitted bits is shown in [12, Fig. 3]. The

transmitted bits are first split into in-phase (I) and quadrature

(Q) bit streams to modulate the I and Q carriers. The demodu-

lation of the received signal is implemented by extracting the

I and Q components separately, and deciding the transmitted

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CAO AND BEAULIEU: EXACT ERROR-RATE ANALYSIS OF DIVERSITY 16-QAM WITH CHANNEL ESTIMATION ERROR 1021

Fig. 1. 16-QAM I and Q bit demappings (after [12, Fig. 5]).

bits according to the decision boundaries. In the 16-QAM

constellation, every symbol is represented by four bits in the

sequence of

,,, , where

bits and

andare the quadrature bits. The I and Q bit

streams are Gray encoded by assigning the bits 01, 00, 10, 11

to the levels

, , , and

decision distance, shown in Fig. 1. Due to the symmetry of

the constellation, the demodulation scheme is same for the I

and Q components. Each symbol has two most significant bits

(MSBs)

and , and two least significant bits (LSBs)

. Following the analysis in [1], we calculate the MSB BER

, and the LSB BER , separately. Then, the BER

16-QAM is the average of

and

and are the in-phase

, respectively, whereis the

and

of

. That is

(12)

The BER with channel estimation error is derived in the next

section.

III. BER PERFORMANCE ANALYSIS

The BER of 16-QAM with channel estimation error is deter-

mined by averaging the conditional error rate, conditioned on

thechannelamplitude,channelamplitudeestimate,andchannel

phase estimate error, across the channel amplitude, channel am-

plitude estimate, and channel phase estimate error. The infor-

mation symbols are assumed to be transmitted with the same

probability. By symmetry, the BERs for the I and Q bit streams

are the same. We consider estimation that is both corrupted by

AWGN and by decorrelations between an information symbol

and the pilot symbols caused by the time-varying fading. We

also account for the cross-quadrature-carrier intersymbol inter-

ference (ISI) that occurs because of the phase error.

In the case of perfect channel estimation, the receiver can

construct the decision boundaries perfectly. However, in the

case of imperfect channel estimation, the decision boundaries

are set with the imperfect channel estimate. The situation is

shown in Fig. 2, where it is illustrated that the phase error

causes rotation of the decision boundaries. Further clarification

is provided in Fig. 3; the true phase is

phase is ; so, there is a phase error

a signal amplitude degradation, replacing, for example,

. The phase error also causes cross-quadrature ISI

with values

or

the symmetries and sign changes of

and , it suffices to include only the

values

in the BER averaging.

and the estimated

. The phase error causes

by

. Note further that due to

Fig.2.

are those relative to the rotated signal constellation.

Newdecisionboundariessetbytheimperfectestimate.Thecoordinates

Fig. 3.Cross-quadrature interferences due to imperfect estimation.

It is informative to compare these with the decision bound-

aries shown in Fig. 1 for the case of perfect channel estimation.

AsseeninFig.1,thedecisionboundaryfortheMSBbitsiszero

withperfectestimation.Thisdecisionboundaryisthesamewith

imperfectestimation,asthefadingamplitudedoesnotaffectthe

decision boundary in the MSB case, as also seen in Fig. 2. The

BER of the MSB conditioned on

for the ISI from the cross-quadrature components, is

, , and, and accounting

(13)

where

[18]; when

decision boundaries for the LSBs are located at

assumption that the channel information is perfectly known. In

is the complementary error function [17],

. As seen in Fig. 1, the,

under the

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1022 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 6, JUNE 2004

the presence of channel estimation error, the decision bound-

aries are located at

, as shown in Fig. 2. If a demodulated

quadrature sample amplitude is greater than

to “1”; if it is greater than

mapped to“0”. Thus, when a logical “1” is transmitted, thecon-

ditional BER can be expressed as

; when a logical “0” is transmitted,

the conditional BER is given by

. The BER of the LSB conditioned on

, , and and accounting for the cross-quadrature interfer-

ence, is then

, it is mapped

and less than, it is

(14a)

(14b)

(14c)

(14d)

(14e)

where

noise ratio (SNR) per bit.

Define the integral (15) as shown at the bottom of the page.

Then, (15) combined with (12), (13), and (14) gives

is the energy per bit andis the average signal-to-

(16)

TABLE I

COEFFICIENTS IN BER EXPRESSION (16) AND (37) FOR 16-QAM

where the coefficients

the integral

plified into a single integration by integrating over

shown in (17a) at the bottom of the page, where

,,, are listed in Table I. Define

. can be sim-

and , as

(17b)

(17c)

(17d)

(17e)

(17f)

When

tion of the integration. We define the integral

and make a change of variables. Let

, where

spondingJacobiantransformationis

of

is obtained by integrating over

, we introduce a different method for simplifica-

,

, ; the corre-

.Thesimplifiedform

(18a)

where, by definition

(18b)

(18c)

(18d)

(15)

(17a)

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CAO AND BEAULIEU: EXACT ERROR-RATE ANALYSIS OF DIVERSITY 16-QAM WITH CHANNEL ESTIMATION ERROR 1023

Combining (16) with (17) and (18), the BER for single-branch

16-QAM with decorrelation error can be expressed as

(19)

Amoredetailedderivationisavailablein[19].TheBERexpres-

sionisthenacombinationofdefinitedoubleandsingleintegrals

with finite limits. It is simpler than the results in [12], which are

in the form of triple integrals and which take much longer to

evaluate, particularly for large SNR values.

For future comparisons, we derive the BER of 16-QAM in

Rayleigh fading with perfect channel estimation. The average

BER can be obtained by averaging the conditional BER (con-

ditioned on the SNR per bit) with respect to the SNR per bit,

according to

(20)

where

conditioned on the SNR per bit,

fading signal’s SNR per bit. The pdf

(14.3–5)].Rewriting(13)and(14)undertheconditionofperfect

estimation,

andgives

is the average BER in slow fading,is the BER

is the pdf of the

is given in [17, eq.

, and

(21a)

(21b)

Thus, the BER of 16-QAM in Rayleigh fading with perfect

channelestimationcanbe obtainedbycombining(12)with(20)

and (21) as

(22a)

(22b)

where, by definition

(22c)

(22d)

(22e)

The derivation of (22) is very close to that in [1], except that our

final result is expressed in terms of the SNR per bit, whereas the

result in [1] is expressed in terms of symbol SNR.

Previously, the BER of single-branch reception 16-QAM in

flat fading with imperfect channel estimation was derived in

[12], but the results are approximations. As [12] neither identi-

fies nor justifies that the analysis is approximate, we will clarify

the analytical approach used there and contrast it with the an-

alytical approach used here. Generally, the average BER can

be calculated by averaging the conditional BER for the AWGN

channel over the fading represented by the variables

. Hence

, , and

(23)

First, note that it is well known that the amplitude and the phase

of the Rayleigh fading channel are independent random vari-

ables. Thus

(24)

and the first-order jpdf factors into the product of the amplitude

pdf and the phase pdf. However, and importantly, the envelope

and phase random processes are not independent random

processes. Conceptually, even though the single sample of the

random amplitude process,

, is independent of the single

sample of the random phase process,

time, ), multiple samples of the amplitude will not be indepen-

dent of multiple samples of the phase (taken at multiple times).

For example, we expect that amplitude “hits” (rapid changes

from one sample to the next sample) will be accompanied by

phase “hits.” Experimental data reported in [20] shows this

dependency. Thus, in the PSAM system, the channel estimate

is a weighted sum of the estimates from the pilot symbols’

channel fadings (3), and thejpdf of the amplitude of the channel

gain

, and the amplitude of the channel gain estimate

not independent of the jpdf of the phase sample of the channel

gain , and the phase sample of the channel gain estimate .

In [12], the analysis and results are based on assuming the

phase error

, the amplitude

, are independent. This treatment is equivalent to using the

following equation:

(both taken at the same

, are

, and the amplitude estimate

(25)

which is only an approximation. The exact form is

(26)

In[12],thejpdf

product

scrutiny of [12, eqs. (38) and (40)]. This replacement requires

that the amplitude samples

samples

. It is clear from the analysis in [13, Ch. 8] that

andare not independent.

There is a second reason for which the BER results in [12]

are only approximate. Even in the absence of noise and other

system errors (for example, carrier recovery error or symbol

timing error), the channel estimate is corrupted by an error that

is Gaussian distributed. This error has its origin in the fact that

the fading at an information symbol’s time is decorrelated from

the fading at a pilot symbol’s time. Note particularly, that the

BER depends on the location of the information symbol, since

haseffectivelybeenreplacedbythe

.Thisconclusioncanbedrawnbycareful

are independent of the phase