Symbol Error Prob With Channel Estimate Errors

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A New Method for Calculating Symbol Error Probabilities

of Two-Dimensional Signalings in Rayleigh Fading with

Channel Estimation Errors 1

Xiaodai Dong, Member, IEEE,

Norman C. Beaulieu, Fellow, IEEE

Department of Electrical and Computer Engineering,

University of Alberta, Edmonton, Alberta, Canada T6G 2V4

Email:

xdong, beaulieu

@ee.ualberta.ca

Tel: 780 492-6989 Fax: 780 492-0194

1This work was presented in part at the 2002 IEEE Symposium on Advances in Wireless Communications, Victoria, Canada, September

2002, and the 2003 IEEE International Conference on Communications (ICC2003), Anchorage, USA, May 2003.


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Abstract

A general analytical framework for evaluating the performance of coherent two-dimensional signaling in fre-

quency flat Rayleigh fading with channel estimation is proposed in this paper. A new and simple analytical ex-

pression for the symbol error rate of an arbitrary polygonal two-dimensional constellation in Rayleigh fading in

the presence of channel estimation errors is presented. This framework is applicable to many current channel es-

timation methods such as pilot symbol assisted modulation and minimum mean square error estimation where thefading estimate is a complex Gaussian random variable correlated with the channel fading. The sensitivity of var-

ious high-level two-dimensional signaling formats to static and dynamic channel amplitude and phase estimation

errors in Rayleigh fading can be easily studied using the derived formula. The new exact expression makes it pos-

sible to optimize constellation parameters and various parameters associated with channel estimation schemes. It

also provides insights into choosing an appropriate signaling format for a fading environment with practical channel

estimation methods used at the receiver.

Index Terms

Channel estimation errors, fading channels, high order modulation, imperfect channel estimation, minimum

mean square error (MMSE), M-ary phase shift keying (MPSK), pilot symbol assisted modulation (PSAM), quadra-

ture amplitude modulation (QAM), symbol error probability, two-dimensional (2-D) modulation formats.

I. INTRODUCTION

The perfect coherent detection of arbitrary polygonal two-dimensional (2-D) signaling in frequency flat

fading channels has been well studied. Tractable analytical results for determining the performances of

2-D signaling schemes are available for various fading models [1]-[2]. These results assume perfect fad-

ing channel state information. In reality, channel state information is only available through estimation

algorithms with inherent channel estimation errors. Therefore, it is of practical interest and importance to

study the performances of coherent signaling formats in the presence of channel estimation errors. Com-

mon channel estimation methods studied in the literature are pilot symbol assisted modulation (PSAM)

and minimum mean square error (MMSE) estimation. A classical paper on pilot symbol assisted mod-

ulation for Rayleigh fading was presented by Cavers in 1991 [3] where he derived the optimum Wiener


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interpolation filter to minimize the variance of the estimation error and studied the system performance for

binary phase shift keying (BPSK), quaternary phase shift keying (QPSK) and 16-ary quadrature amplitude

modulation (QAM). The optimum Wiener filter requires a priori information of the channel and is compu-

tationally complex. Therefore, several suboptimal interpolation filters have been proposed among which

the sinc interpolator is most widely used due to its simple implementation and close-to-optimum perfor-

mance [4]. The bit error rate (BER) of pilot symbol assisted M-ary QAM (MQAM) in Rayleigh fading

using the sinc interpolator has been analyzed in [5]. An upper bound to the symbol error rate (SER) for

pilot symbol assisted QAM in Rayleigh fading was presented in [6], and the optimum MMSE interpolator

using an infinite number of pilot symbols was examined. The performance of MMSE channel estimation

has been investigated for M-ary PSK (MPSK) and MQAM in Rayleigh fading in [7] and for Ricean fading

in [8]. The analyses in these two papers employed the orthogonality between the channel estimate and the

channel estimation error. Their work has been further extended by Wilson and Cioffi [9] to eliminate the

orthogonality requirement. The probability density functions (PDFs) of the receiver decision variables

for Rayleigh fading channels with maximal ratio diversity combining, and for single branch Ricean fading

channels in the presence of imperfect channel estimation were derived. The PDFs were employed to cal-

culate the SER and BER of 16-QAM with channel estimation errors. However, the analysis in [9] is only

applicable to MQAM. In summary, results are only available for MPSK and MQAM with small Mvalues;

results for arbitrary 2-D signaling are not available. Furthermore, there has been no analysis that is able to

unify these previously published results and to account for more general case of arbitrary 2-D signalings.

We present in this paper a general method of analysis for determining symbol error probabilities of ar-

bitrary 2-D signalings in Rayleigh fading with static or dynamic channel estimation errors. It is applicable

to any fading estimate which is jointly Gaussian with the actual fading. This includes, but is not lim-

ited to, MMSE and pilot symbol assisted modulation schemes. Our analytical method is new and applies

very generally to modulation formats not previously studied as well as to modulation formats previously

studied using other methods which are not applicable to the general modulations considered here. The

paper is organized as follows. Section II derives the error probability expression for coherent 2-D signal-


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ing in Rayleigh fading with channel estimation errors. Section III applies the newly derived formula to

the analysis of dynamic estimation errors when PSAM and MMSE estimation are employed as well as to

the analysis of static estimation errors. Section IV discusses the numerical SER results for various 8-ary

and 16-ary constellations that are of practical interest or are commonly studied in the literature. Finally,

conclusions are given in Section V.

II. ERROR PROBABILITY OF 2-D SIGNALING

This section begins with the system modeling. We then present our symbol error probability analysis

of 2-D signalings in Rayleigh fading with channel estimation errors. Assume perfect symbol timing,

intersymbol interference (ISI) free transmission, and slow channel fading that is almost constant over one

symbol duration. The received signal z at any symbol interval given signal si

sIi

jsQi

i

1

M

transmitted can be expressed as

z

gsi

n (1)

where the zero-mean Gaussian random variable (R.V.) g

gI

jgQ represents the complex fading in-

troduced by the channel with variance Es in both of its real and imaginary parts, and n

nI

jnQ is a

zero-mean Gaussian noise R.V. with variance N0 in both of its real and imaginary parts. The fading g is

independent of the additive Gaussian noise n. Signal si is assumed to be normalized to yield unit aver-

age energy of the constellation. Various channel estimation methods, for instance, pilot symbol assisted

modulation and minimum mean square error estimation, generate a fading estimate g that is also Gaussian

and correlated with the true fading g. The specific expression for g depends on the particular estimation

method used. The most often used method for calculating the probability of error of a signaling format

is based on the conditional error probability given the channel fading g and the fading estimate g. Under

this conditioning, the problem is similar to the well known additive white Gaussian noise (AWGN) case.

The probability of error is then obtained by averaging the conditional error probability over the joint dis-

tribution of g and g. Previous publications [5]-[6] follow this approach. However, in the general case of

arbitrary 2-D constellations, this approach often leads to an intractable analysis. Averaging over the joint


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distribution of g

ej and g

ej usually involves a four-fold nested integration, and the conditional

probability P e

for arbitrary 2-D signaling usually has another level of integration. That is,

Pe

2

0

2

0

0

0P

e

f

dddd (2)

and P e

requires a single integration for general 2-D signaling as seen in [2], [1].

We propose in this paper a completely different approach to solving the problem and arrive at a surpris-

ingly simple yet general expression for the symbol error probability of arbitrary 2-D signaling in Rayleigh

fading with channel estimation errors. We begin with defining a decision variable as

D

z

g

g

gsi

n

g

si

t

(3)

And then,

t

z

g

si

D

si (4)

where t

rej can be considered as an equivalent noise term that is superimposed on the signal si and

includes the effects of both the fading estimation error and the AWGN. Assuming equally-likely transmis-

sion of signals, the maximum likelihood (ML) decision rule of a constellation in AWGN and in fading

with perfect channel sate information is equivalent to the minimum Euclidean distance criterion. In the

case of imperfect channel estimation, however, minimum Euclidean distance rule is no longer the opti-

mum. Nevertheless, in practice a receiver will likely use the estimated fading as if it were perfect channel

state information and draw up decision boundaries as in the AWGN case. This is because the receiver

may not have the necessary information, such as the maximum Doppler frequency, the signal-to-noise

ratio (SNR) and so on, needed in the PDF of the decision variable to determine the optimum decision

boundaries. Therefore, we consider the practical case where the minimum Euclidean distance rule is used

in the receiver. IfD falls into the decision region of signal si, a correct decision will be made; otherwise,

a symbol error will occur.

The decision variable D was also written as the sum of the signal and an equivalent noise term in [7] and

[8]. Furthermore, the probability density function of the noise term t was obtained from the conditional


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PDF as

ft r

2

0

2

0

0

0ft r

f

dddd (5)

which can be simplified for MMSE estimation as discussed in [7] and [8]. However, the four-fold nested

integration in (5) usually cannot be solved for other practical channel estimation schemes.

Here we present a new derivation of the PDF oft. Now, recognizing the fact that the decision variable Din (3) is the ratio of two complex-valued Gaussian random variables z and g, we first derive the distribution

of the ratio of two complex Gaussian R.V.s X

z and Y

g. Define variable D in a general form as

D

X

Y(6)

where X and Y are correlated zero-mean complex-valued Gaussian random variables. The joint PDF of

zero-mean complex Gaussian R.V.s X and Y was given by Wooding in a simple and compact form as [10]

fXY X

Y

2 L 1 exp

VHL

1V

(7)

where V

X

Y !

H with#

representing complex conjugation and AH the Hermitian transpose of matrix

A. Matrix L is the Hermitian covariance matrix given by

L $&

mxx mxy

mxy myy

'(

(8)

where mxx

E

X X !

, myy

E

YY !

and mxy

E

XY !

are the second moments of the zero-mean complex-

valued Gaussian R.V.s X and Y. Eqn. (7) can also be written as

fXY X

Y

2L

1 exp

L

1myy X

2

mxx Y2

m xyXY

mxyX Y ! (9)

and in polar coordinates X

rxejx and Y

ryejy ,

fXY rx

x

ry

y

rxry

2L

1 exp

L

1myyr

2x mxxr

2y

20

m xyrxryej

1x

y2 3

!

(10)


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To obtain the PDF of D

rdejd

XY

, we define another complex variable F

rfejf

Y. Then, we have

the following relationships

rd rx

ry(11a)

d x

y (11b)

rf

ry (11c)

f y (11d)

The joint PDF ofD and F is obtained from (10) as

fDF rd d rf f

fXY rdrf d f rf f

J

rd

2 Lr3f exp 5

myyr

2d mxx

20

mxyrde

jd 3r2f

L6

(12)

where J is the Jacobian of the transformation (11) and it can be shown that J

1ry

1rf

. The PDF of

the complex variable D is then obtained by integrating the joint density (12) with respect to the complex

variable F,

fD rd d

0

2

0fD

F rd d rf f dfdrf

L

rd

myyr

2d mxx

20

mxyrde

jd 3

2 (13)

The next step is to derive the PDF of the combined noise term t

rej in (4). Rewriting the PDF of

D

Dx

jDy given by (13) in rectangular coordinates, we have

fD Dx

Dy

L

1

myy D2x D2

y mxx

20

mxy 3 Dx

20

mxy 3 Dy ! 2

(14)

It can be shown that (14) is equivalent to [9, eqn. (3)] but (14) is derived using a different method. More-

over, our later application of Craigs method to general 2-D signaling schemes will require the PDF of the


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combined noise term t in polar coordinate form. Since t

tx

jty

D

si, one has

tx

Dx

sIi (15a)

ty

Dy

sQi

(15b)

The Jacobian of transformation (15) is J

1, and thus the PDF oft can be written as

ft tx

ty

L

myy t2

x t2

y 2 myysIi

0

mxy 3 tx

2myysQi

0

mxy 3 ty

myy si2

mxx

2 0 mxy 3 sIi

2 0 mxy 3 sQi ! 2

(16)

Furthermore, the PDF of the complex-valued variable t expressed in polar coordinates can be obtained

from (16) by another change of variables r 7

t2x t2

y and tan 1

ty

9

tx as

ft r

L

r

cr2 b

r a 2

(17a)

where

a

myy si2

mxx

2 0 mxy 3 sIi

2 0 mxy 3 sQi (17b)

b

2myysIi cos

myysQi sin

0

mxy 3 cos

0

mxy 3 sin (17c)

c

myy

(17d)

Once the PDF (17) of the equivalent noise term taccounting for both the channel estimation error and the

additive white Gaussian noise is available, we apply Craigs method [11], [1], [2] to analyze the probability

of symbol error for arbitrary 2-D signaling. The main idea of Craigs method is to work directly with the

PDF of the noise term that is superimposed onto the signal si and divide erroneous decision regions into

smaller subregions. Each subregion can be conveniently represented by polar coordinates [11]. Since we

have expressed the decision variable D as the sum of si and t, we shift the origin of the coordinates to

signal si

sIi

jsQi to form new coordinates and work directly with t. Fig. 1 illustrates how to partition


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the erroneous decision region of signal si into subregions. Following analysis similar to that presented in

[2], the probability of error for the j-th erroneous subregion is given by

Pe j @@

@

@

2Aj

1Aj

R1

2

ft r

drd@

@

@

@

(18a)

R

xj sinj

sin

1 j j (18b)

where angles 1 j

2 j

j and distance xj are parameters specific to the geometry of the j-th erroneous

subregion and are illustrated in Fig. 1 for different erroneous subregions. It can be seen from (17) that

the PDF of the combined noise t is no longer Gaussian and does not have circular symmetry as in the

perfect channel state information case. The integration limits 1 j and 2 j must be angles related to the

j-th subregion in the new coordinates (si is the origin). In the open decision region case shown in Fig. 1(b),

1j is defined as the angle formed by the horizontal axis of the new coordinates and the line from signal si

to the intersection of two decision boundaries. The absolute value is needed in (18a) as 1 j might be larger

than 2 j, resulting in a negative value of the integral in (18a). Subregion 5 in Fig. 1(b) is one example

where 1 5 C 2 5. Substituting (17) into (18) and solving the inner integral with respect to r, we get

Pej

@

@

@

@

2Aj

1Aj

L

D

2a

b

R

a

b R

cR2

b

3 E 2

2b

3 E 2tan

1 F b 2cR G

H I

d@

@

@

@

(19a)

where [12, eqns. (2.175.2) and (2.172)] have been used and

4ac

b

2

4mxxmyy

mxy2

4

myy sIi sin

sQi cos

0

mxy 3 cos

0

mxy 3 sin !2

C

0 (19b)

L

mxxmyy

mxy2 (19c)

mxx

E

zz !

(19d)

myy E gg ! (19e)

mxy

E

zg !

(19f)


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The symbol error probability of any coherent two-dimensional constellation in Rayleigh fading with chan-

nel estimation errors is simply a weighted sum of terms given by eqn. (19a) [2],

Pe

N

j

Q1

wjPej (20)

where N is the total number of distinct erroneous subregions of the 2-D constellation and wj is the weight-ing coefficient [2]. The new single integral expression (19) is an exact solution to calculating the SER of

coherent 2-D signaling in Rayleigh fading with imperfect channel estimation. Note that only one level of

integration is required, in contrast to four levels of nested integration required by previous methods.

As a special case of 2-D signalings, the symbol error probability of MPSK in Rayleigh fading with

channel estimation errors can be obtained by assuming si

1 as

PeMPSK

M

L

D

2a1 sin2

9

M

b1 sin 9 M sin

9

M

1 a1 sin2

9

M

b1 sin 9

M

sin

9

M

c sin2

9

M

b1

3

E2

1

2b1

3

E2

1

tan

1 F b1 sin

9

M

2c sin

9

M

sin

9

M

G

1 H 6d

M

L

D

2a1 sin2

9

M

b1 sin 9 M sin

9

M

1 a1 sin2

9

M

b1 sin 9 M sin

9

M

c sin2

9

M

b1

3 E 2

1

2b1

3 E 2

1

tan

1 F b1 sin

9

M

2c sin

9

M

sin

9

M

G

1 H 6d (21a)

where

a1

myy

mxx

2 0 mxy 3 (21b)

b1

2

myy cos

0

mxy 3 cos

0

mxy 3 sin (21c)

c

myy (21d)

1

4

mxxmyy

mxy2

4

myy sin

0

mxy 3 cos

0

mxy 3 sin 2

(21e)

Note that the two integrals on the right side of (21a) are equal when the second moment mxy is real, i.e.,

0

mxy 3

0 and, hence, b1

b1 . Thus, when a channel estimation method has the property that


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the cross-correlation between the received signal z and the channel estimate g is real, the probability of

symbol error for MPSK is given by doubling the first integral. As will be shown in the next section, this

is the case for pilot symbol assisted modulation. It is worth pointing out that for MPSK, it is actually

easier to work with the decision variable D rather than t, because the erroneous decision region is more

conveniently described by the non-shifted original coordinates and the SER is given by

PeMPSK

M

0fD rd d drddd

M

0fD rd d drddd (22a)

M

fd dd

M

f

d dd (22b)

M

L

5

2

2

b2 d

3

E2

2

2b2 d

3

E2

2

tan

1 b2 dG

2 6dd

M

L

5

2

2

b2 d

3E

2

2

2b2 d

3E

2

2

tan

1 b2 dG

2 6dd (22c)

where fd is the marginal PDF ofd, the phase difference between the received signal and the channel

phase estimate, and

b2 d

2

0mxy 3 cosd 0 mxy 3 sind (22d)

2 4mxxmyy 4 0 mxy 3 cosd 0 mxy 3 sind2

(22e)

Another special case is the symbol error probability ofM-ary rectangular QAM, which is often referred

to as MQAM. This is readily obtained using (20) and (19), though the details are omitted for brevity. More

generally, we have written a computer program that calculates these geometric parameters automatically

for arbitrary 2-D constellations.

III. APPLICATIONS TO CHANNEL ESTIMATION ERROR ANALYSIS

This section applies the newly derived method to the performance analysis of 2-D signalings in the

presence of three different kinds of channel estimation errors.


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A. Static Channel Estimation Errors

Two parameters, amplitude error tolerance and phase error tolerance, were defined in [2] to describe

how sensitive a 2-D constellation is to carrier amplitude and phase tracking errors. These two parameters

provide some information about the robustness of a modulation format, but the effect of channel estimation

errors on the average SER of the modulation format is not known with accuracy. Here we investigate the

performance of 2-D signaling in Rayleigh fading with constant channel estimation errors caused by the

carrier amplitude and phase tracking components. Another scenario where static channel estimation error

is relevant is when amplifier nonlinearity is present in the system and the estimated amplifier nonlinearity

differs from the true amplifier nonlinearity by some constant error. In all these cases, the estimated fading

is related to the real fading by a constant factor. That is, g

gq exp

1

j2

where q is the amplitude estimation

error and is the phase estimation error. Parameters q

1 and

0 correspond to the perfect channel

estimation case. Therefore, the fading estimate g is also a zero-mean complex Gaussian R.V. correlated

to the true fading g. Letting X

z

gsi

n and Y

g

gq exp 1 j

2

, the second moments given signal si

transmitted are

mxx

2Es si2

2N0 (23a)

myy

2Es

q2(23b)

mxy

2Essi

qej

(23c)

The signal-to-noise ratio (SNR) per symbol is given by Es9

N0. The SER of arbitrary 2-D signaling in the

presence of static channel estimation errors can be readily obtained from (19) and (23).

The second moments are obtained from (23) with si

1 for MPSK. The probability of error for MPSK

in Rayleigh fading with static channel estimation errors is then given by (21) and (23).

B. Pilot Symbol Assisted Modulation

In a PSAM system, pilot signals are periodically inserted into the data stream for every L

1 data

symbols. Hence, a data frame of length L consists of a pilot symbol at its zero-th position and L

1 data


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symbols at positions from 1 to L

1. The estimated channel fading at the pilot position of the k-th frame

is obtained by gkp zk 0 9 pk where pk is the pilot symbol of the k-th frame and zk 0 is the received signal at

this pilot position. The received signal at the l-th data symbol position of the current frame is given by

zl glsl nl (24)

where gl is the zero-mean complex Gaussian R.V. representing the multiplicative channel fading, sl is the

l-th data symbol taking values from an M-ary signal set and nl is the additive Gaussian noise. Channel

fading gl is estimated from K pilot symbols, that is, the previous K1 R K

1

9

2T

pilot symbols, the

current pilot symbol and the subsequent K2 R K9 2 T pilot symbols as

gl

K2

k

Q

K1

hklg

kp

K2

k

Q

K1

hkl

V

gkp nkp

pkW

(25)

where R XT

is the floor function, gkp is the actual channel fading at the k-th pilot position, and hk l h

l

kL is the interpolator coefficient for gkp and is dependent on the current symbol position l. Proposed

interpolators include the optimum Wiener interpolator [3], a Gaussian interpolator [13] and the simple

sinc interpolator [4]. As an example, the sinc interpolator with a Hamming window is given by

hn

wn

X

sinc`

n

L a

K1L b n b K2L (26a)

where

w

n

0

54

0

46cosV

2n

KL

1

2R

KL2

T

KL

1W

(26b)

It can be seen from (25) that gl is also a zero-mean complex Gaussian R.V. because it is a linear combi-

nation of g K1

p

gK2p that are jointly Gaussian R.V.s. Let X

zl and Y gl . To calculate the SER of


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PSAM 2-D signaling, the only quantities needed from (19) are the second moments

mxx

2Es sl2

2N0 (27a)

myy

HlCHTl

2N0

pk2Hl

2 (27b)

mxy

K2

kQ

K12Eshk l slJ0

2fD kL

l T

(27c)

Hl h

K1 l hK2 l ! (27d)

Ckn E gkpg

np ! 2EsJ0 2fD k

n T

(27e)

where Ckn is the element with index k n of the covariance matrix C for fadings at pilot symbols, fD is

the maximum Doppler frequency and T is the symbol duration. Pilot symbol pk is assumed to have unit

average energy, i.e., pk2

1 in (27). The signal-to-noise ratio per symbol is given by Es9

N0. Since the

second moments myy and mxy are dependent on the symbol position l where l

1

L

1, the overall

symbol error probability should be averaged over the L

1 positions within a frame.

In the case of pilot symbol assisted MPSK, the second moments are given by (27) with sl 1. Since

mxy in (27c) is real, the two integrals in (21a) are equal. The probability of error for pilot symbol assisted

MPSK in Rayleigh fading is thus given by two times the first integral in (21a).

C. Minimum Mean Square Error Estimation

It is well known that the estimate and the estimation error are uncorrelated in minimum mean square

error estimation and that the average estimation error power is the difference of the power of the variable

to be estimated and the power of the estimate. Since the fading and its estimate are complex Gaussian

R.V.s, MMSE estimate g and the estimation error e

g

g are independent. Denote the mean power of

the estimation error by 22e and the fading power by 2Es. The estimation error power is a measure of the

quality of the estimation algorithm. Let X

z and Y

g. The second moments required by (19) in the


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case of MMSE channel estimation are given by

mxx 2Es si2

2N0 (28a)

mxy

E

gsi

ng

!

E

e

gsig !

myysi (28b)

myy

2Es

22e (28c)

This leads to

a

mxx

myy si2

C

0 (29a)

b

0 (29b)

c

myyC

0

(29c)

Substituting (29) into (19), we have for MMSE estimation

PMMSEej

@

@

@

@

@

2A

j

1Aj

L

sin2

1 j j

2c

a sin2

1j

j

cx2j sin2!

d @@

@

@

@

(30)

Eqn. (30) can be further simplified to a closed-form expression given by

PMMS Ee

j

2

j

1

j

2

G

cuj

27

cu2j a

c de

tan

1$&

1

a

cu2jtan

2 j

1 j j

'(

tan

1$&

1

a

cu2jtanj

'( fh

(31)

where uj

xj sinjC

0 and [12, eqn. (2.562.1)] has been used. For MMSE MPSK, the symbol error

probability can be written in closed-form as

PMMS EeMPSK

M

1

M

G

cu

G

cu2

a D

tan

1F i 1

a

cu2tan

MH I

(32)

where u

sin M, and a and c can be obtained from (29) and (28) with si 1.

Both (31) and (32) are new results. Substituting (29) into (17), we obtain an equivalent PDF expression

in polar coordinates for the combined noise term to that given by [7, eqn. (22)]. Polar coordinates facilitate


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the present analysis and lead to simplified and generalized results for arbitrary 2-D signaling. While [7,

eqn. (22)] is valid only for MMSE channel estimation, (17) applies to any channel estimations that are

jointly Gaussian with the actual channel fading. Note that our new closed-form SER expression (31)

applies to arbitrary polygonal 2-D constellations with MMSE estimation in Rayleigh fading, compared to

the single integral results given in [7] for SERs of 4-PSK, 8-PSK, 8-AMPM, 16-PSK and 16-QAM.

IV. NUMERICAL EVALUATIONS

We present in this section some numerical results for various 8-ary and 16-ary constellations as appli-

cations of the theoretical analysis presented in Sections II and III. The 8-ary signalings considered are

8PSK, the 8-ary rectangular set, the 8-ary max-density, the 8-ary triangular set, (4,4) and (1,7). The 16-ary

signalings considered are the 16-ary hexagonal set, 16 rectangular QAM, the 16-ary triangular set, (4,12),

the 16-ary max-density, (5,11), 16 star-QAM, rotated (8,8) and (1,5,10). The signal space diagrams for

these constellations can be found in [2] and are not presented here due to space limitations. Figs. 2-4 show

the SERs of the 8-ary and 16-ary signal sets as functions of the average signal-to-noise ratio per bit,Eb

N0,

in the presence of static channel estimation errors. The SNR per bitEb

N0is related to Es

N0by a factor of

log2M

1. The derived single integral (19) with finite integration limits is easy to evaluate numerically

and generates results with high accuracy.

Fig. 2 demonstrates the effect of constant channel estimation errors on the SER performance of 8-ary

signal sets in Rayleigh fading. The performance of the (4,4) constellation will depend on the ring ratio

employed. The ring ratio of (4,4) is set at 2.4544, the optimized value for SNR 20 dB. It is evident that

the relative performances of these constellations with channel estimation errors are distinct from those

with perfect channel state information as given in [2]. Given constant amplitude error q

2 dB and phase

error

10 r , (4,4) and (1,7) have better error performance than 8PSK and the 8-ary rectangular set in a

Rayleigh fading environment. The (1,7) set saves about 2.04 dB power over 8PSK at SER

1e

3 and

(4,4) saves around 1.96 dB over 8PSK as shown in Fig. 2.

The performances of some 16-ary constellations are presented in Figs. 3 and 4 for the ring ratios speci-


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fied in the figure captions. The performance differences among 16-ary signal sets in Figs. 3 and 4 are more

pronounced than those of 8-ary signal sets in the presence of the same static channel estimation errors, as

expected. Circular constellations with two rings such as 16 star-QAM, rotated (8,8), (5,11) and (1,5,10)

are clearly good choices that are robust to constant channel estimation errors as shown in Figs. 3 and 4.

The 16-ary hexagonal set, 16 rectangular-QAM, the 16-ary max-density and the 16-ary triangular set no

longer yield acceptable performance in the presence of the amplitude error q

2 dB and the phase error

10 r . Sixteen star-QAM and rotated (8,8) signaling have comparable error performances. Comparing

Fig. 3 with Fig. 4, it is seen that under-estimating the channel amplitude has less adverse effect on the SER

than over-estimating the channel amplitude with the same q dB. Some simulation results are plotted in

Figs. 2 and 4 to verify the analytical SER results.

Figs. 5-9 present SER results for pilot symbol assisted modulation schemes which introduce dynamic

channel estimation errors to receiver detection. As described in Section III-B, we use a sinc interpola-

tor with Hamming windowing in our numerical evaluations. Note that the general SER expression (19) is

applicable to any linear interpolator and our study shows that Hamming windowing produces better perfor-

mance than rectangular windowing. Fig. 5 depicts the SERs of pilot symbol assisted 16 rectangular-QAM

and 16 star-QAM as a function of SNR in Rayleigh fading. Sixteen rectangular-QAM slightly outper-

forms 16 star-QAM for the perfect channel state information case, and the benefit is greater at higher

SNR. However, it is clear from Fig. 5 that when dynamic channel estimation errors inherent in PSAM

schemes are present, the performance degradation of 16 rectangular-QAM is quite severe. The SNR dif-

ference by numerical evaluation for star-QAM with and without channel estimation error at SER=1e

3

is about 2.4 dB, while for rectangular-QAM, the SNR difference is about 3 dB. This shows that star-QAM

is more robust to channel estimation errors than rectangular-QAM. Pilot symbol assisted 16 star-QAM

has a slightly better performance than pilot symbol assisted 16 rectangular-QAM, and at very high SNR

both constellations exhibit error floors. The error floor of 16 star-QAM, however, is less than that of 16

rectangular-QAM. Simulation results agree well with our theoretical analysis. It is also observed in our

study that performance differences between the two constellations are more pronounced at higher fading


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rates when channel estimation errors increase.

The choice of two parameters, frame length L and interpolation order K, depends on a number of

factors. The larger the frame length L, the less power loss incurred from the pilots inserted in the data

stream. However, L is upper bounded by the Nyquist Sampling Theorem at L b 12fDT

. For example, L

should be less than 16 for fDT

0

03. Therefore we choose L

15. We found that the PSAM system

would not work with larger values of L, and smaller L does not lead to much performance gain. Once L is

determined, the SER performance of 16 rectangular-QAM as a function of interpolation order K is plotted

in Fig. 6. Parameter K determines the buffer size and the detection delay and should be minimized without

sacrificing the error performance. As can be seen from Fig. 6, the smaller the average SNR of the channel,

the smaller the value ofK that is sufficient. It is obvious that choosing K

30 works well for SNR values

less than 40 dB, but is not sufficient for SNR = 50 dB, where K

34 reduces error floors noticeably. It

is observed in our study that K depends on the channel fading rate, the SNR and the frame size L. It does

not seem to depend strongly on the signal constellation used.

The ring ratio of pilot symbol assisted 16 star-QAM requires optimization for minimum SER. Note that

in slow Rayleigh fading with perfect channel state information, the optimum ring ratios are a function of

SNR and approach an asymptotic value as SNR gets very large [2]. The asymptotic optimum ring ratio

is 1.952 in Rayleigh fading. Fig. 7 shows the optimum ring ratio of PSAM 16 star-QAM as a function of

SNR for fDT

0

03, L

15, K

30 and 34. Obviously the optimum ring ratios no longer approach an

asymptotic value and are quite different from those of the perfect channel knowledge case. The optimum

ring ratio of 16 star-QAM is chosen at 2.18 in Fig. 5.

Fig. 8 shows that pilot symbol assisted (4,4) achieves the best SER performance among 8-ary signal

sets with fDT

0

03, L

15, and K

30 in Rayleigh fading. The ring ratio used for (4,4) is 2.4065,

optimized for SNR

20 dB. There is about 1

6 dB power savings for PSAM (4,4) over PSAM 8PSK at

SER

1e

3, which is rather significant when it comes to system design for high speed data. Error floors

are observed for all signalings at high SNR.

The SERs of pilot symbol assisted 16-ary signal sets in Rayleigh fading are plotted in Fig. 9 with


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fDT

0

03, L

15, and K

30. Fig. 9 shows that 16 star-QAM has very comparable performance to the

rotated (8,8) set and achieves the lowest SER among the five signal sets in the medium to high SNR range.

Although not plotted here, it is also found in our study that the constellation (5,11) curve is slightly above

that of the 16-ary max-density, and (1,5,10) almost overlaps with rotated (8,8). The ring ratios used for 16

star-QAM and rotated (8,8) in Fig. 9 are 2.1674 and 2.1039, respectively.

Fig. 10 plots the SER curves of 16 rectangular-QAM, 16 star-QAM and QPSK in Rayleigh fading when

MMSE estimation errors are present. Suppose the variance of the estimation error is 5 dB, 16 star-QAM

suffers 4.86 dB power loss while 16 rectangular-QAM loses 5.8 dB power for SER at 1e

3, compared

to their corresponding perfect coherent detection. In addition, for symbol error probabilities smaller than

1e

2, 16 star-QAM saves about 0.6 dB power over 16 rectangular-QAM with 2e 5 dB. The QPSK

curves obtained agree with the results in [8, Fig. 6], except that the horizontal axis in Fig. 10 is SNR per

bit whereas it is SNR per symbol in [8, Fig. 6].

V. CONCLUSION

The impact of channel estimation errors on the performance of coherent 2-D signaling has been in-

vestigated in this paper. A new, simple, general method of analysis has been developed for performance

evaluation of arbitrary polygonal 2-D signaling in Rayleigh fading with imperfect channel estimation. The

symbol error probability of an arbitrary polygonal 2-D constellation is given by a single integral with finite

integration interval and is well-suited to numerical evaluation. The practical effects of channel estimation

errors on signaling performance have been examined by considering static channel estimation errors, pilot

symbol assisted modulation and minimum mean square error estimation. The SER expression of arbitrary

2-D signaling with MMSE estimation was further simplified to a closed form. The symbol error proba-

bilities of six 8-ary signal sets and nine 16-ary signal sets have been numerically evaluated and plotted

under different kinds of channel estimation errors. Pilot symbol assisted (4,4) displays significant power

advantage over conventionally used PSAM 8PSK. Pilot symbol assisted 16 star-QAM has been shown

to be more immune to channel estimation errors than PSAM 16 rectangular-QAM and its performance


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is comparable to PSAM rotated (8,8). It has also been demonstrated that parameters crucial to channel

estimation schemes and ring ratios of circular constellations can be easily and accurately determined using

the analytical results presented.

ACKNOWLEDGEMENT

The first author wishes to thank Mr. Lei Xiao for his assistance in simulations.

REFERENCES

[1] X. Dong, N.C. Beaulieu and P.H. Wittke, Error Probabilities of Two-Dimensional M-ary Signaling in Fading, IEEE Trans. Com-

mun.,vol. 47, pp. 352-355, Mar. 1999.

[2] X. Dong, N.C. Beaulieu and P.H. Wittke, Signal constellations for fading channels, IEEE Trans. Commun., vol. 47, pp. 703-714, May

1999.

[3] J.K. Cavers, An Analysis of Pilot Symbol Assisted Modulation for Rayleigh Fading Channels, IEEE Trans. Veh. Technol., vol. 40, pp.

686-693, Nov. 1991.

[4] Y.S. Kim, C.J. Kim, G.Y. Jeong, Y.J. Bang, H.K. Park and S.S. Choi, New Rayleigh Fading Channel Estimator Based on PSAM

Channel Sounding Technique, Proc. IEEE Int. Conf. Communications (ICC97), pp. 1518-1520, June 1997.

[5] X. Tang, M.S. Alouini and A. Goldsmith, Effect of channel estimation error on M-QAM BER performance in Rayleigh fading, IEEE

Trans. Commun., vol. 47, pp. 1856-1864, Dec. 1999.

[6] K. Yu, J. Evans and I. Collings, Performance Analysis of Pilot Symbol Aided QAM for Rayleigh Fading Channels, Proc. IEEE Int.

Conf. Communications (ICC02), Apr. 2002.

[7] A. Aghamohammadi and H. Meyr, On the Error Probability of Linearly Modulated Signals on Rayleigh Frequency-Flat Fading Chan-

nels, IEEE Trans. Commun., vol. 38, pp. 1966-1970, Nov. 1990.

[8] M.G. Shayesteh and A. Aghamohammadi, On the Error Probability of Linearly Modulated Signals on Frequency-Flat Ricean, Rayleigh,

and AWGN Channels, IEEE Trans. Commun., vol. 43, pp. 1454-1466, Feb./Mar./Apr. 1995.

[9] S.K. Wilson and J.M. Cioffi, Probability Density Functions for Analyzing Multi-Amplitude Constellations in Rayleigh and Ricean

Channels, IEEE Trans. Commun., vol. 47, pp. 380-386, Mar. 1999.

[10] R.A. Wooding, The Multivariate Distribution of Complex Normal Variables, Biometrika, vol. 43, pp. 212-215, June 1956.

[11] J.W. Craig, A New Simple and Exact Result for Calculating the Probability of Error for Two-Dimensional Signal Constellations, Proc.

IEEE Milit. Commun. Conf. (MILCOM91), Boston, MA, pp. 571-575, 1991.

[12] I.S. Gradshteyn and I.M. Ryzhik, A. Jeffrey (editor), Table of Integrals, Series, and Products. Fifth Edition, San Diego, CA: Academic

Press, 1994.

[13] S. Sampei and T. Sunaga, Rayleigh fading compensation for QAM in land mobile radio communications, IEEE Trans. Veh. Technol.,

vol. 42, pp. 137-147, May 1993.


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1

2,1

11,1

i

x = |OA|1

7

1,52,5

2,75

7

5

1,7

F

x = |OF|

x = |OG|

O

G

i

5

7

A

B

C

D

O

E

R

R = |OE|

2

1

3

4

(a)

0

x

S

x

x

5

7

6

0

S

(b)

Fig. 1. Decision regions and geometric parameters of a (a) closed region and (b) open region.


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0 5 10 15 20 25 30 35 40 45 5010

5

104

103

102

101

100

Average Eb/N

0 (dB)

SymbolEr

rorProbability

q=2 dB, =10o

8PSK8PSK simulation8 Rectangular8 Maxdensity8 Triangular(4,4)(1,7)

Fig. 2. Average SERs of 8-ary signal sets in Rayleigh fading with amplitude error q Q 2 dB and phase error Q 10 s .


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0 5 10 15 20 25 30 35 40 45 5010

5

104

103

102

101

100

Average Eb/N

0(dB)

SymbolErrorProbability

q=2 dB, =10o

16 rect.QAMTriangularHexagonal(4,12)Maxdensity(5,11)16 starQAMRot. (8,8)(1,5,10)

Fig. 3. Average SERs of 16-ary signal sets in Rayleigh fading with amplitude error qQ

2 dB and phase error Q

10 s . The ring ratios used

for (1,5,10), rotated (8,8), 16 star-QAM, (5,11) and (4,12) are 2.5933, 2.2893, 2.2947, 2.8487 and 3.1764, respectively.


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0 5 10 15 20 25 30 35 40 45 5010

5

104

103

102

101

100

Average Eb/N

0(dB)

SymbolE

rrorProbability

q=2 dB, =10o

16 Hexagonal16 rect.QAM16 rect.QAM simulation16 Maxdensity16 Triangular(4,12)Rot. (8,8)16 starQAM16 starQAM simulation(5,11)(1,5,10)

Fig. 4. Average SERs of 16-ary signal sets in Rayleigh fading with amplitude error qQ

2 dB and phase error Q

10 s . The ring ratios

used for (1,5,10), rotated (8,8), 16 star-QAM, (5,11) and (4,12) are 2.8939, 2.6987, 2.6846, 3.2253 and 3.497, respectively.


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0 10 20 30 40 50 6010

5

104

103

102

101

100

Average Eb

/N0

(dB)


L=15, K=30, fD

T=0.03

16 rect.QAM, simulation16 starQAM, simulation16 rect.QAM, theory16 starQAM, theory16 rect.QAM, perfect16 starQAM, perfect

Fig. 5. Average SERs of PSAM 16 rectangular-QAM and 16 star-QAM in Rayleigh fading with fDT Q 0 t 03 L Q 15 K Q 30 t


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5 10 15 20 25 30 35 4010

5

104

103

102

101

100

K


SNR=10 dB

SNR=20 dB

SNR=30 dB

SNR=40 dB

SNR=50 dB

16 rect.QAMfD

T=0.03

L=15

Fig. 6. Average SERs of PSAM 16 rectangular-QAM as a function ofK in Rayleigh fading with fDT Q 0 t 03 L Q 15 t


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5 10 15 20 25 30 35 402.1

2.15

2.2

2.25

2.3

2.35

2.4

2.45

2.5

Average Eb/N

0 (dB)

Optimum

RingRatio

K=30K=34

16 starQAML=15fDT=0.03

Fig. 7. Optimum ring ratios of PSAM 16 star-QAM in Rayleigh fading with fDT Q 0 t 03 L Q 15 K Q 30 and 34 t


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0 10 20 30 40 50 6010

5

104

103

102

101

100

Average Eb

/N0

(dB)


L=15, K=30, fD

T=0.03

8PSK8PSK simulation8 Rectangular(1,7)8 Maxdensity8 Triangular(4,4)

Fig. 8. Average SERs of PSAM 8-ary signal sets in Rayleigh fading with fDT Q 0 t 03 L Q 15 K Q 30 t


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0 10 20 30 40 50 6010

5

104

103

102

101

100

Average Eb

/N0

(dB)


L=15, K=30, fD

T=0.03

Hexagonal16 rect.QAMMaxdensityRot. (8,8)16 starQAM

Fig. 9. Average SERs of PSAM 16-ary signal sets in Rayleigh fading with fDT Q 0 t 03 L Q 15 K Q 30 t


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0 5 10 15 20 25 30 35 40 45 5010

5

104

103

102

101

100

Average Eb/N

0(dB)

SymbolE

rrorProbability

16 rect.QAM, MSE=5 dB16 starQAM, MSE=5 dB16 starQAM, perfect16 rect.QAM, perfectQPSK, MSE=1/3 [8]QPSK, perfect

Fig. 10. Average SERs of 16 star-QAM and 16 rectangular-QAM in Rayleigh fading with MMSE estimation and 2e Q 5 dB, and the SER

of QPSK with 2e Q 1 E 3.

Symbol Error Prob With Channel Estimate Errors

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