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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 10, OCTOBER 2006 1775

Performance of Convolutional CodesWith Finite-Depth Interleaving and

Noisy Channel EstimatesJittra Jootar, Student Member, IEEE, James R. Zeidler, Fellow, IEEE, and John G. Proakis, Life Fellow, IEEE

AbstractIn this paper, we derive the Chernoff bound of thepairwise error probability (PEP) and the exact PEP of convolu-tional codes in a time-varying Rician fading channel. With the as-sumptions that the channel estimator is a finite impulse responsefilter and the interleaving depth is finite, we are able to investigatethe estimation-diversity tradeoff resulting from the effects of theDoppler spread on the system performance via the channel-esti-mation accuracy and the channel diversity. In addition, we verifythat, in the special case when the pilot signal-to-noise ratio is infin-itely large and the channel estimator is well-designed, our analysis

leads to the same result as the existing perfect channel-state infor-mation analysis. Finally, the analytical results are compared withresults from Monte Carlo simulation, and the comparison showsthat the analytical results match well with the simulation results.

Index TermsChannel estimation, convolutional codes, diver-sity, estimation-diversity tradeoff, interleaving.

I. INTRODUCTION

PREVIOUS analytical studies on the performance of con-

volutional codes in a time-varying fading channel have fo-

cused on either imperfect channel-state information (CSI) or

finite-depth interleaving, while assuming the other to be per-

fect [1][4]. Since fading coefficients can be estimated withbetter accuracy in a slowly fading channel, a perfectly inter-

leaved system with noisy CSI in a slowly fading channel out-

performs the system in a fast-fading channel. However, when

the CSI is perfect but the interleaving is imperfect due to finite

interleaving depth, the performance is reversed, i.e., the system

in a fast-fading channel outperforms the system in a slowly

fading channel [2][4]. This is because the number of indepen-

dent fading realizations available for a codeword, the number

referred to as the channel diversity [5], of a fast-fading channel

is greater than that of a slowly fading channel.

In a practical system, where both CSI and interleaving are

not perfect and the imperfections contribute to the performance

Paper approved by C. Schlegel, the Editor for Coding Theory and Techniquesof the IEEE Communications Society. Manuscript received June 27, 2005; re-vised February 20, 2006. This work was supported in part by Ericsson underCore Grant 02-10109 and in part by the U.S. Army Research Office under theMultiuniversity Research Initiative (MURI) Grant W911NF-04-1-0224. Thispaper was presented in part at the IEEE 61st Vehicular Technology Conference,Stockholm, Sweden, May/June 2005.

J. Jootar was with the Department of Electrical and Computer Engineering,University of California at San Diego, La Jolla, CA 92093-0407 USA. She isnow with Qualcomm Inc., San Diego, CA 92121 USA (e-mail: [email protected]).

J. R. Zeidler and J. G. Proakis are with the Department of Electrical andComputer Engineering, University of California at San Diego, La Jolla, CA92093-0407 USA (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TCOMM.2006.881363

degradation of the system, the performance analysis has to take

into account both imperfections. Since increasing the Doppler

spread improves the system performance by increasing the

channel diversity, but degrades the performance by worsening

the channel-estimation accuracy, we expect to observe the

estimation-diversity tradeoff as a function of the Doppler

spread when both imperfect CSI and imperfect interleaving are

considered [5][8].

In order to address system performance in realistic operatingenvironment, there has recently been growing interest in the per-

formance analysis of coded systems with imperfect CSI and im-

perfect interleaving. However, earlier analyses did not model the

CSI accuracy as a function of the Doppler spread [9], or used

simple assumptions, such as noninterleaved codes [10], or dis-

cussed the tradeoff from simulation results without providing

any analytical analysis [8]. The analysis on the estimation-di-

versity tradeoff, we believe, was first presented in [7], where

the authors derived the optimal memory lengths and the error

exponent bounds for joint estimation and decoding, assuming

a block-fading channel. The block-fading assumption was also

used in later works [5], [6], where the pairwise error probability(PEP) for coded systems in a Rayleigh fading channel and a Ri-

cian fading channel was derived. Later, in [11], the block-fading

assumption was replaced with a more general channel model.

However, the authors used the assumption that the noise com-

ponents, after multiplying the received signals with the conju-

gate of noisy channel estimates, are Gaussian random variables.

This assumption caused the analytical results in [11] to be just

approximate, not the exact performance.

Because existing analyses are limited to specific assumptions,

such as block fading [5][7], which is not an accurate assump-

tion for several wireless systems, or Gaussian noise component

[11], the primary focus of this paper is to derive the PEP withthe assumptions that are more general and the model that incor-

porates implementation issues, such as the choice of the pilot

filter. Consequently, the system performance in a realistic sce-

nario can be calculated from the analysis without having to re-

sort to lengthly simulations, allowing optimization studies of

various design parameters, such as the pilot filter coefficients,

the interleaving depth, and the pilot-to-signal power ratio. We

would like to note that the material presented in this paper was

presented in part in [12], and also note that the analysis builds

mainly upon the work on imperfect CSI by Cavers [13] and the

work on noninterleaved codes with imperfect CSI by Nobelen

and Taylor [10]. In addition, because the mathematical model

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1776 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 10, OCTOBER 2006

Fig. 1. System block diagram.

of this system is similar to the mathematical model of the corre-

lated maximal ratio combining (MRC) system, the method used

here also resembles the ones used in [14][16].

The paper is organized as follows. Section II introduces thesystem model, which includes the transmitter at the base station,

the frequency-selective Rician fading channel, and the receiver

at the mobile unit. In Section III, we derive the Chernoff bound

of the PEP and the exact PEP of the system. In addition, we

also verify in Section III that for a special case when the CSI is

perfect, our analysis agrees with the existing perfect CSI anal-

ysis. Discussions of the results and conclusions are presented in

Sections IV and V, respectively.

II. SYSTEM MODEL

For the rest of this paper, the following notation will be used.A lowercase bold letter denotes a vector, and an uppercase bold

letter denotes a matrix. The element in the th row and the th

column of matrix is denoted by , and the element in

the th row (column) of a column (row) vector is denoted

by . The superscripts , , denote the complex conju-

gate, the matrix transpose, and the matrix Hermitian operation,

respectively. The determinant of a matrix is denoted by .

The length- column vector of ones, the square identity matrix,

and the square zero matrix of order are denoted by , ,

and , respectively.

The system considered is a downlink binary phase-shift

keying (BPSK) direct-sequence code-division multiple-access

(DS-CDMA) system. A complex baseband representation ofthe system is illustrated in Fig. 1, where the base stations

transmitter, the frequency-selective fading channel, and the

mobiles receiver are shown in the upper left corner, the upper

right corner, and the bottom section of the figure, respectively.

A. Transmitter

We assume that there are signal streams transmitted

from the base station. The streams consist of one pilot

stream (zeroth stream) and data streams assigned to users

( th stream for the th user). The pilot stream is

spreadwith the orthogonalcode , where denotes the chip

time index, and denotes the period or the spreading gain of

. Similarly, the th BPSK data stream , which

is the interleaved convolutionally coded BPSK signal, is spread

with the orthogonal code , which has the same period .

After spreading, the th signal stream is scaled by , and

the signals from all branches are combined and scrambled bythe base-station-dependent complex long code . The signal

after scrambling can be expressed as

(1)

The signal is then passed through an impulse modulator

and a pulse-shaping filter with frequency response , where

satisfies the Nyquist condition for zero inter-

symbol interference (ISI) [17], i.e.,

when

when and

is any nonzero integer

(2)


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JOOTAR et al.: PERFORMANCE OF CONVOLUTIONAL CODES WITH FINITE-DEPTH INTERLEAVING AND NOISY CHANNEL ESTIMATES 1777

where denotes the inverse Fourier transform of

, and denotes the chip period. Examples of functions

with these properties are functions in the family of square-root

raised cosine pulse-shaping filters [17].

B. Channel

The channel is assumed to be a time-varying frequency selec-

tive Rician fading channel with resolvable paths and impulse

response

(3)

where denotes the fading coefficient corresponding to the

th path, denotes the Dirac delta function, and denotes

the delay associated with the th path. We also assume that the

s are integers, i.e., the delays are multiples of the chip period,

and that .The fading coefficient is assumed to be a circu-

larly symmetric complex Gaussian random variable with

real-valued mean and autocovariance function

. Note that

represents the energy of the line-of-sight (LOS) component of

the fading channel, and represents the autocorrelation

function of the diffuse component of the fading channel. In addi-

tion, we assume that the fading coefficients from different paths

are independent; thus,

when . An analysis for the Rayleigh fading channel is

simply a special case when . Although we assume a

time-varying channel, we restrict ourselves to the case when

the channel changes slowly enough that the fading coefficients

appear to be constant over one symbol period .

Finally, the thermal noise is assumed to be additive

white Gaussian noise (AWGN) with variance .

C. Receiver

At the mobiles receiver of user 1, the received signal can

be expressed as

(4)

After despreading, the pilot signal and the data signal at the

output of the accumulator corresponding to the th branch of

the RAKE receiver can be approximated as [18]

(5)

(6)

where , , and and

denote the summation of the self-noise and the thermal noise

components of the pilot and the data signals, respectively.

Conditioned on the transmitted data, the self-noise and the

thermal noise components can be approximated as zero-meanGaussian random variables with variances and

, respectively, where and

[18]. Thus, the variances of both and

are equal to . For simplicity,

this symbol-rate model will be used for the rest of the paper.

The channel estimator for the th branch of the RAKE

receiver [17] is assumed to be a -tap finite-im-

pulse response (FIR) filter with the filter coefficient vector. As a result, the channel

estimate can be written as

(7)

where .

III. ANALYSIS

Without loss of generality, we assume that the transmittedcodeword is an all-zero codeword which is mapped to an all-one

BPSK sequence . Due to interleaving, the PEP, which is the

probability that the decoder chooses the coded sequence

when was transmitted, is a function of and the structure

of the interleaver. Finding the PEP for each error pattern with

respect to a specific interleaver is tedious, and adds little insight

into the overall system performance [4]. Therefore, we will use

the approximation that an interleaving depth of a block inter-

leaver creates the same effect as separating consecutive symbol

errors by symbols [2]. As a result, the PEP can be simplified,

such that it depends only on the Hamming weight of the error

codeword, but not the structure of the interleaver nor the error

codeword itself. We also would like to note that this approxi-

mation is used here mainly to simplify the analysis, and an ex-

tension to the analysis without this approximation can be done

straightforwardly. The PEP, when a RAKE receiver is used with

a mismatched maximum-likelihood (ML) (Viterbi) decoder (see

[19] and references therein for the description of the mismatch

decoder), can be expressed as

(8)

where , , is the

interleaving depth, is the Hamming weight of the error code-

word, denotes the real part of the complex number , and

is the probability density function of .

A. Characteristic Function

Following the approach used in [13], can be written in a

quadratic form of complex Gaussian random variables , i.e.,

, where and

(9)


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The characteristic function of the quadratic form was derived by

Turin to be [20]

(10)

where and are the mean vector and the covariance matrixof , respectively. Due to the assumption that the fading coef-

ficients from different paths are independent, the characteristic

function of is simply the product of the characteristic func-

tions of for . Thus, the characteristic function of

becomes

(11)

Using the symbol-rate model from Section II-C, the mean

vector and the covariance matrix can be found, after some

math, to be

(12)

where

(13)

(14)

(15)

and is a square matrix of size with ones on the th

diagonal and zero elsewhere, is a square matrix of size

with , and is

the th column of .

B. The Chernoff Bound

The Chernoff bound of the PEP is an upper bound which is

often used in analytical studies due to its simplicity. For our

system, the Chernoff bound can be expressed as

(16)

where is a parameter to be optimized [17]. Notice that the

expectation can be written as a function of , i.e.,

(17)

Since is a convex function with respect to

(the second-order derivative of with respect tois equal to , and is always positive), we can

conclude that is equal to . Using

the characteristic function specified in (11), the set of can

be simplified, as shown in Appendix I, to be

(18)

where for are the eigenvalues of . Also

note that when the channel is a Rayleigh fading channel, which

is the special case of the Rician fading channel when ,

(18) reduces to

(19)

Substituting in (11), we can get , which

is the Chernoff bound of .

C. The PEP

In addition to the Chernoff bound, can also be used to

find directly. This is done by substituting the expression of

as a function of

(20)

into (8). Using Mellins inversion, we get

at (21)

where lies between the left half-plane poles and the imaginary

axis, denotes the number of negative poles of ,

denotes the th negative pole of , and at

denotes the residue of at the pole . The residue can be

calculated by

at (22)

where , denotes the th derivative

of , and is the order of the pole at .

Although the residue theorem leads to a desirable

closed-form expression of , it turns out to be cumbersome

for systems with large , which are the systems normally seen

in practice. For example, the third-generation UMTS-WCDMA

standard uses convolutional codes, rate 1/2 and rate 1/3, with

, respectively. In order to find for systemswith large , we resort to a numerical approximation called


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the GaussChebyshev approximation (suggested in [21] with a

correction in [22]), which can be expressed as

(23)

where denotes the imaginary part of , ,

and, in general, between 16 and

32 is sufficient [21]. In addition, is the same as defined for

(21).

D. Verification for Perfect CSI

Although this paper focuses on systems with noisy CSI, the

analysis can also be used to find the performance of systems

with perfect CSI, which is a special case of our analysis when

the pilot signal-to-noise ratio (SNR) is infinitely large and the

pilot filter is well-designed. In this subsection, we will verify

that corresponding to this special case is equal to calcu-

lated from the perfect CSI analysis. For simplicity, we consideronly the flat-fading channel, thus, dropping the subscript . Note

that an extension to the frequency-selective fading channel is

straightforward.

The perfect CSI assumption can be realized in our model

by using and . Thus, we get

and

(24)

where is a square matrix of size with

. Finding a matrix inverse is usually a difficult task. For-

tunately, the matrix inverse of can be found easilyto be

(25)

where .

Substituting and (25) into (11), the

characteristic function becomes

(26)

Using (21) and recalling from Appendix II that is in-variant to the value of in , given that , we

will use instead of to find

. The scaled characteristic function

can be expressed as shown in (27) and (28) at the bottom

of the page, where is the data SNR, and

are the eigenvalues of . From (28), we know

that the poles of are .

Since is a positive definite matrix, for

are greater than zero. We can then identify that the negative

poles are , and the positive poles are

.

The requirement for (21) is that lies between the poles on

the left half-plane and the imaginary axis. A value for that

we can choose so that the constraint is satisfied is . Substi-

tuting in (21), when the scaled characteristic function

is used instead of , we get

(29)

After changing the dummy variable from to , becomes(30)(32), shown at the bottom of the next page. In the fol-

lowing, we will complete this subsection by proving that the

perfect CSI performance calculated by averaging the PEP over

the distribution of the instantaneous data SNR is equal to (32).

Under the perfect CSI assumption, is a function of

, where is the instantaneous data SNR cor-

responding to the th error symbol. The characteristic function

of was given in [23] to be

(33)

where , , , and are as previously defined. Since

given is equal to [23], where

for , the average PEP can be

found by averaging over the distribution of , i.e.,

(34)

Using an alternative form of the complementary error function

[24]

for (35)

(27)

(28)


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where , (34) can be simplified to

(36)(38), shown at the bottom of the page. Comparing (32)

and (38), it is obvious that they are identical. Therefore, we

have successfully shown that, in the limit when the channel

estimates are perfect, the results from our analysis agree with

the results from the perfect CSI analysis.

IV. NUMERICAL RESULTS

In this section, we will discuss analytical results calculated

by the GaussChebyshev approximation, and also compare the

analytical results with results from Monte Carlo simulation to

illustrate the accuracy of the analysis.

A. The Optimal Normalized Doppler Frequency

One of the effects from the estimation-diversity tradeoff is the

optimal channel memory or the optimal normalized Doppler fre-

quency , which is the channel memory that uses the tradeoff

in the most effective way [7]. At the optimal , the perfor-

mance is such that, if increases, the performance degrada-tion due to worse channel estimates outweighs the benefit from

the increase of the channel memory. On the other hand, if

decreases from the optimal value, the degradation due to smaller

channel diversity outweighs the benefit from better channel es-

timates. These optimal s can easily be seen in Fig. 2, where

we plot the PEP of systems in Rayleigh fading channels as a

function of for for two types of

power spectral density (PSD), namely, the Jakes PSD and the

Gaussian PSD.

One striking difference between the Jakes and the Gaussian

PSDs that should be mentioned is the oscillation seen in the

plots corresponding to the Jakes PSD, but not the Gaussian PSD.

This is because the autocorrelation function of the Jakes PSD

Fig. 2. Comparison of the PEP corresponding to Gaussian and Jakes PSDs.Data SNR = 7 dB, pilot SNR = 0 dB, d = 1 8 , 11-tap Wiener pilot filter.

(the zeroth-order Bessel function of the first kind) is not mono-

tonically decreasing. For a nonmonotonically decreasing func-

tion, increasing the symbol spacing does not always decrease

the correlation or increase the channel diversity. As a result, the

plot corresponding to a nonmonotonically decreasing autocor-

relation function oscillates, while the plot corresponding to a

monotonically decreasing function does not. In addition, the os-

cillation can also be seen when the PEP is plotted as a function

of the interleaving depth for the same reason.

To better understand the system performance as a function of

, consider Fig. 3, where we compare the PEP assuming noisy

CSI and finite-depth interleaving with the PEP assuming perfect

(30)

(31)

(32)

(36)

(37)

(38)


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Fig. 3. Comparison of the PEP for the realistic case and for the case of perfectCSI or perfect interleaving. Data SNR = 7 dB, d = 1 8 , 11-tap Wiener pilot

filter, Jakes PSD.

CSI and finite-depth interleaving, or perfect interleaving and

noisy CSI. From this figure, we can see that the solid lines (noisy

CSI, finite-depth interleaving) corresponding to the 10-dB pilot

SNR are very close to the dotted lines (perfect CSI) at small

. This is because, with this pilot SNR and the Wiener pilot

filter, the receiver can accurately estimate the channel. Thus, the

performance is close to the perfect CSI case. In addition, we

can also see that at 0 dB pilot SNR, the performance is much

further away from the perfect CSI because of bad channel esti-

mates. We can also see that as increases and the channel es-timates are less accurate, the solid lines diverge more and more

from perfect CSI, and when is large enough such that the

channel diversity is equal to the code diversity, the solid lines

merge with the perfect interleaving performance (dashed lines).

Since the system with a larger can reach the channel diversity

at a smaller [4], the solid line corresponding to a larger

merges with the perfect interleaving line at a smaller .

B. Comparing the Effects of the Pilot SNR on Systems in Fast

and Slowly Fading Channels

In Fig. 4, we illustrate the effect of the pilot SNR, which is

equal to , on the PEP and the Chernoff bound of sys-tems in Rayleigh fading channels. Two fading channels shown

arethe fast-fading channel with , referred to as system

A, and the slowly fading channel with , referred to as

system B. We can see that at small pilot SNR, system B outper-

forms system A, and vice versa at large pilot SNR. The behavior

agrees with the finding in [7], which can be explained as follows.

According to [7], when the system operates at a rate close to ca-

pacity, the CSI accuracy is crucial. But when the system oper-

ates at a rate much lower than capacity, the channel diversity is

crucial. Since the rate is constant and the capacity at small pilot

SNR is smaller than the capacity at large pilot SNR, the CSI ac-

curacy and the channel diversity dominate the performance of

the system at small pilot SNR and large pilot SNR, respectively.In addition to the PEP, we have also illustrated in this figure the

Fig. 4. PEP as a function of the pilot SNR andf

. Data SNR = 7 dB,I = 3 0

,

d = 1 8

, Jakes PSD, 11-tap Wiener pilot filter.

Chernoff bounds of the PEP. It is clearly seen that the bounds

follow the exact PEP nicely.

Although the performance at small and large pilot SNR is

predictable, we would like to note that the performance when

the pilot SNR is moderate is not easily predicted, and must be

found through calculation, because it depends on other parame-

ters, such as the interleaving depth and the data SNR. An anal-

ysis such as the one presented in this paper is needed to quantify

the performance in this moderate pilot-SNR region.

C. Improving the Performance Through the Interleaving DepthIn addition to the Doppler spread, the channel diversity can be

increased by increasing the interleaving depth , which, unlike

the Doppler spread, is a controllable parameter limited only by

the delay constraint of the system. In Fig. 5, we illustrate the

effect of on the PEP in Rayleigh fading channels with

. We can see that increasing can significantly

improve the performance at large pilot SNR, but not as much

at small pilot SNR. The reason is the same as the one stated in

Section IV-B, that the accuracy of CSI, not the channel diversity,

dominates the performance at small pilot SNR [7]. Therefore,

increasing the channel diversity via the interleaving depth does

not improve the performance much at small pilot SNR.

D. Filter Choice

Up until now, we have used the dynamic Wiener filter,

which calculates according to the systems pilot SNR and the

channel statistic as the channel estimator. In order to do this, the

receiver must have knowledge of the pilot SNR and the channel

statistics of the system. In a real system, this knowledge may

not be available, or it may not be accurate. To get around this

problem, instead of using a dynamic filter, a simple receiver

may use a static filter which never changes its filter tap coeffi-

cients. In Fig. 6, we compare the performance of the two filters,

where the fixed filter is randomly chosen to be the Wiener filter

corresponding to pilot SNR = 10 dB, and ina Rayleigh fading channel. In this figure, the z-axis represents


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Fig. 5. PEP as a function of the pilot SNR,f

, andI

onP

. Data SNR =7 dB,

d = 1 8


Fig. 6. PEP difference between the system with the (Wiener) dynamic pilot

filter and the system with a fixed pilot filter. Data SNR = 7 dB, d = 1 8 , I = 1 0 ,Jakes PSD, 11-tap pilot filter.

fixed dynamic dynamic , the value which is

positive when the dynamic filter outperforms the fixed filter.

We can see from this plot that the dynamic filter is sometimes

outperformed by the fixed filter. This behavior is expected,because the optimal receiver, which results in the smallest PEP,

performs joint estimation decoding; thus, using the combina-

tion of the optimal estimator and the optimal decoder does not

guarantee the optimal result. It is also apparent that the filter

choice is critical when the pilot SNR and are large, as

fixed is almost three times larger than dynamic .

E. Frequency-Selective Fading Channel

Fig. 7 shows the PEP when the system is in a frequency-se-

lective Rayleigh fading channel, assuming that there are two re-

solvable paths, and each path has half of the average power of

the path in the flat-fading case. Path diversity added by the mul-tipath causes the performance of the frequency-selective fading

Fig. 7. Comparison of the PEP between single-path channel and two-pathfading channel.

E = 1

,E = 0 : 0 1

,N = 1 2 8

, 0

31 dB,d = 1 8

,

Gaussian PSD, 11-tap Wiener pilot filter.

Fig. 8. Comparison of the PEP between the flat-fading channel and a two-pathfading channel for different values of data-to-pilot ratios.

E + E = 1 : 0 1

,N = 1 2 8 , = 0 31 dB, I = 2 3 , d = 1 8 , Gaussian PSD, 11-tap Wienerpilot filter.

to be better than the flat fading when is small. But, due to

smaller pilot SNR per path, the channel-estimation accuracy ofthe frequency-selective fading deteriorates much faster as

increases, leading to worse performance, compared with flat

fading, at large .

F. Effect From the Data-to-Pilot Ratio

In Fig. 8, we illustrate the effects of the data-to-pilot ratio

and on the PEP of systems in a flat Rayleigh fading channel

(solid plane) and a frequency-selective Rayleigh fading channel

with two resolvable paths (dotted plane).

Let us consider the figure from small toward

. When is very small (small data energy,

large pilot energy), the pilot SNR is large enough that eventhe multipath system, which is the system with worse channel


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Fig. 9. Effect of the pilot SNR andf

onP

. Data SNR = 7 dB,I = 3 0

,

d = 3 ; 1 8


estimates, has accurate CSI. Since both systems have good

channel estimates at small , the diversity is the dominant

factor. Therefore, the multipath system which has more diver-

sity outperforms the flat-fading system at small . When

is large, however, the CSI accuracy of the multipath

system becomes worse, especially at high . Therefore,

we can see from the figure that at large and large

, the performance of the multipath system is worse than the

flat-fading system. From the figure, we notice that the gain fromallocating appropriate energy to the data and the pilot channels

can be significant. For example, changing from 40 to

1 can improve the performance up to 4 orders of magnitude.

Last, we would like to point out that the line corresponding to

in Fig. 7 illustrates the cross-section of Fig. 8 when

.

G. Rician Fading Channel

Fig. 9 illustrates the PEP of systems in time-varying Rician

fading channels with for the Rician factor

(also denoted by ) 0.1, 1, and 4, and for 3, 18. The

performance improves as the Rician factor increases, as ex-pected, for all cases except the case when and large pilot

SNR. This is because when the pilot SNR is large, the domi-

nant factor is the channel diversity [7], and with , the

performance is tremendously improved by the channel diver-

sity. As a result, it is also very sensitive to the decrease in the

channel diversity. When the the Rician factor increases by a

little bit (from 0.1 to 1), the channel diversity which comes from

the diffuse component of the fading channel is reduced, while

the LOS component is not large enough to compensate for the

performance loss due to the decrease in the channel diversity.

But when the Rician factor is large enough ( in

this example) that the LOS component can compensate for the

performance loss due to the decrease in the channel diversity, theperformance of the system with also improves with the

Fig. 10. Comparison of the BEP( P )

and the BLEP( P )

from the analysis

and the simulation. Data SNR = 2.22 dB, pilot SNR = 0.97 dB, Jakes PSD,11-tap Wiener pilot filter, rate-1/3 convolutional code, d = 1 8 , 220 infor-mation bits per block, 8 b zero padding,

= 2 = 0

.

Rician factor. Also note that the same behavior is observed

for the system with but at smaller Rician factor, i.e., at

instead of .

In addition, we observe that the performance when

is sometimes outperformed by at large pilot SNR

when the Rician factor is nonzero. This observation was also

seen in [5]. As explained in [5], the reason is that as the LOS

component becomes stronger, the need for channel diversity di-

minishes.Finally, when the Rician factor approaches , the perfor-

mance of converge to the same performance,

which is corresponding to the AWGN channel.

H. Comparisons With Monte Carlo Simulation

To show the accuracy of our analysis, in Fig. 10, we compare

the truncated union bound calculated from the analytical PEP

with results from Monte Carlo simulation for a Rayleigh fading

channel. The code used in the simulation is the rate-1/3 convo-

lutional code specified in the UMTS-WCDMA standard [25],

with 18 and 256 states. The interleaver (with in

this simulation) is also specified in the UMTS-WCDMA stan-dard [25]. The number of information bits per block is assumed

to be 220, and each block is terminated with 8 zeros such that

the encoder is set back to the all-zero state at the end of each

block. The fading coefficients are generated by method of exact

Doppler spread (MEDS), suggested in [26], with the autocor-

relation of the Jakes model. The channel estimator is an 11-tap

Wiener filter.

For the truncated union bound, only the five smallest Ham-

ming weights ( ) are used to calculate

the bit-error probability (BEP) and the block-error probability

(BLEP), denoted by and , respectively. In addition, two

values of interleaving depths shown in the figure are 23 and

36. Comparing the simulation results and the analytical resultswith , we can see that the analytical results match well


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Fig. 11. Comparison of the BEP( P )

and the BLEP( P )

from the analysisand the simulation. Data SNR = 2.22 dB, pilot SNR = 0.97 dB, Jakes PSD,

11-tap moving average filter, rate-1/3 convolutional code, d = 18, 220 in-formation bits per block, 8 b zero padding,

= 2 = 0 : 5

.

with the simulation results, especially when the probability of

error is small. Comparing between the analytical results with

, we can see that the two interleaving depths pro-

vide similar performance, especially when the Doppler spread

is large.

A similar comparison between the simulation results and the

analytical results for a Rician fading channel withand is illustrated in Fig. 11. However, to il-

lustrate the effect of a mismatch channel estimator, we assume

in this simulation that an 11-tap moving average filter is used as

the channel estimator. We can clearly see from the simulation

results that performance degrades with the normalized Doppler

spread as a result from the mismatch channel estimator. Com-

paring the truncated union bound of the BEP calculated from

the minimum Hamming distance for

(the dotted and the dashed-dotted lines, respectively), it is clear

that the system performance in a Rician fading channel strongly

depends on the interleaving depth and the spacing of error sym-

bols. Since the spacing of error symbols in a real system dependson the interleaving pattern and the error sequence, the simple as-

sumption that all consecutive error symbols are symbols apart

does not lead to an accurate performance prediction in the Ri-

cian fading channel. Therefore, instead of calculating the PEP

assuming that consecutive error symbols are symbols apart,

we calculate the PEP by taking into account the error sequence

of the convolutional code and the interleaving structure. (This

modification can be done with minor change on the covariance

matrix.) The PEPs corresponding to all of the error patterns are

then used to calculate the approximation of and , which

are shown in the figure as solid and dashed lines, respectively.

Also note that three smallest Hamming weights, 18, 20, and 22,

are used to calculate the and without the equally spacederror-symbols assumption. We can see from the figure that the

analytical results match well with the simulation results, espe-

cially when is less than 0.04.

V. CONCLUSIONS

In this paper, we have derived the Chernoff bound of the

PEP and the exact PEP of coded systems with finite-depth in-terleaving and noisy channel estimates. The analysis provides

an insight into the system performance in a realistic environ-

ment. For example, we have shown that there exists an optimal

channel memory length, which is a result of the estimation-di-

versity tradeoff as a function of the Doppler spread. Also, it

has been observed that in a fast-fading channel, increasing the

pilot SNR can improve the performance more effectively than

increasing the interleaving depth. We have also investigated the

system performance as a function of the Rician factor, and

found that the system with a large is more sensitive to the de-

crease of the channel diversity resulting from increasing .

In addition to gaining more understanding of the system be-

havior, we have also shown that the analysis is a great tool

for system designs. For example, the analysis can be used to

compare the performance between different pilot filters, data-to-

pilot ratios, or coding schemes. Finally, to illustrate the accuracy

of the analysis, we have compared the truncated union bounds

calculated from the analytical PEPs with the results from Monte

Carlo simulation, and showed that the bounds match well with

the simulation results when the probability of error is small

(union bound limitation) and, for the Rician case, when the

Doppler spread is not too large.

APPENDIX I

FINDINGThe optimal can be found by taking a derivative of

with respect to , and the derivative can be expressed as

(39)

where

, and isthe theigenvalue

of . After some math, we get

(40)

(41)

Substituting (41) into (39) and conditioning that

, the optimal can then be expressed as

(42)


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APPENDIX II

RESIDUES OF

In this section, we will prove that the residue of at a

pole is equal to the residues of at a pole , where

is any nonzero scaling factor.

From (10), the characteristic function can be sim-

plified as follows:

(43)

where

and is a matrix such that

. Thus, the poles of

are zero and for , where are the

eigenvalues of . Therefore,the partial fractionof

can be written as

(44)

For generality, we make no assumption on the orders of the

poles. Substituting into (44), we get

(45)

Multiplying both sides with , we get

(46)

It is obvious that changing from to , we

change the poles from to . In addition, the residue of

at the pole in (44) is equal to the coefficient .

This residue is also equal to the residue of at the pole

in (46). Therefore, we can conclude that the residue ofat is equal to the residue of at .

And since the PEP is equal to the summation of the residues of

the poles in the left half-plane, the PEP is invariant to the value

of in .

REFERENCES

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[2] F. Gagnonand D. Haccoun, Bound on theerror performance of codingfor nonindependent Rician-fading channels, IEEE Trans. Commun.,vol. 40, no. 2, pp. 351360, Feb. 1992.

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[5] S. A. Zummo and W. E. Stark, Performance analysis of binary codedsystems over Rician block fading channels, in Proc. IEEE MILCOM,Oct. 2003, pp. 314319.

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tradeoff in communication systems, inInformation,Coding and Math-ematics. Norwell, MA: Kluwer, 2002.

[8] W. K. M. Ahmed and P. J. McLane, Random coding error exponentsfor flat fading channels with realistic channel estimation, IEEE J. Sel.

Areas Commun., vol. 18, no. 2, pp. 369379, Mar. 2000.[9] C. Tellambura and V. K. Bhargava, Error performance of MPSK

trellis-coded modulation over nonindependent Rician fading chan-nels, IEEE Trans. Veh. Technol., vol. 47, no. 1, pp. 152162, Jan.1998.

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[11] J. Lai and N. B. Mandayam, Performance analysis of convolutionallycoded DS-CDMA systems with spatial and temporal channel correla-tions, IEEE Trans. Commun., vol. 51, no. 12, pp. 19841990, Dec.2003.

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noisy channel estimates, in Proc. IEEE 61st Veh. Technol. Conf., May2005, pp. 600605.[13] J. K. Cavers, An analysis of pilot symbol assisted modulation for

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[14] W.-Y. Kuo, Analytic forward link performance of pilot-aided co-herent DS-CDMA under correlated Rician fading, IEEE J. Sel. AreasCommun., vol. 18, no. 7, pp. 11591168, Jul. 2000.

[15] F. A. Dietrich and W. Utschick, Maximal ratio combining of corre-lated Rayleigh fading channels with imperfect channel knowledge,

IEEE Commun. Lett., vol. 7, no. 9, pp. 419421, Sep. 2003.[16] P. Shamain and L. B. Milstein, Effect of mutual coupling and corre-

lated fading on receive diversity systems using compact antenna arraysand noisy channel estimates, in Proc. IEEE Globecom, Dec. 2003, pp.16691673.

[17] J. G. Proakis, Digital Communications. New York: McGraw-Hill,1995.

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Reed-Solomon/convolutional coding for data transmission inCDMA-based cellular systems, IEEE Trans. Commun., vol. 45,no. 10, pp. 12911303, Oct. 1997.

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[21] E. Biglieri, G. Caire, G. Taricco, and J. Ventura-Traveset, Simplemethod for evaluating error probabilities, Electron. Lett., vol. 32, pp.191192, Feb. 1996.

[22] J. R. Foerster and L. B. Milstein, Coded modulation for a coherentDS-CDMA system employing an MMSE receiver in a fading channel,

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Jittra Jootar (S02) was born in Nashville, TN. Shereceived the B.S. degree in electrical engineering

from Chulalongkorn University, Bangkok, Thailand,in 1997, the M.S. degree in electrical engineering

from Stanford University, Palo Alto, CA, in 1999,and the Ph.D. degree in electrical and computerengineering from University of California, San

Diego, in 2006.From 1999 to 2002, she was with Qualcomm Inc.,

San Diego, CA, where she worked on Bluetooth andWCDMA development. In July 2006, she rejoined

Qualcomm Inc., to continue working on WCDMA development. Her researchinterests includes MIMO, modulation, and coding for mobile communicationsystems.


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James R. Zeidler (M76SM84F94) is a Re-search Scientist/Senior Lecturer in the Department ofElectrical Engineering, University of California, SanDiego. He is a faculty member of the UCSD Centerfor Wireless Communications and the Universityof California Institute for Telecommunications andInformation Technology. He has more than 200 tech-nical publications and 13 patents for communication,

signal processing, data compression techniques, andelectronic devices.Dr. Zeidler received the Frederick Ellersick award

from the IEEE Communications Society in 1995, the Navy Meritorious CivilianService Award in 1991, and the LauritsenBennett Award for Achievement inScience from the Space and Naval Warfare Systems Center in 2000. He wasan Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING and aMemberof the Technical Committee on Underwater Acoustic SignalProcessingfor the IEEE Signal Processing Society.

John G. Proakis (S58M62SM82F84LF97)received the BSEE degree from the University ofCincinnati, Cincinnati, OH, in 1959, the MSEEdegree from Massachusetts Institute of Technology,Cambridge, in 1961, and the Ph.D. degree fromHarvard University, Cambridge, MA, in 1967.

He is an Adjunct Professor at the University ofCalifornia at San Diego and a Professor Emeritus

at Northeastern University, Boston, MA. He wasa faculty member at Northeastern University from1969 through 1998 and held the following academic

positions: Associate Professor of Electrical Engineering, 19691976; Professorof Electrical Engineering, 19761998; Associate Dean of the College ofEngineering and Director of the Graduate School of Engineering, 19821984;Interim Dean of the College of Engineering, 19921993; Chairman of theDepartment of Electrical and Computer Engineering, 19841997. Prior to

joining Northeastern University, he worked with GTE Laboratories and theMIT Lincoln Laboratory. His professional experience and interests are in thegeneral areas of digital communications and digital signal processing. He isthe author of the book Digital Communications (New York: McGraw-Hill,1983, first edition; 1989, second edition; 1995, third edition; 2001, fourthedition), and co-author of the books, Introduction to Digital Signal Processing(New York: Macmillan, 1988, first edition; 1992, second edition; 1996, thirdedition); Digital Signal Processing Laboratory (Englewood Cliffs, NJ: Pren-tice-Hall, 1991); Advanced Digital Signal Processing (New York: Macmillan,

1992); Algorithms for Statistical Signal Processing (Englewood Cliffs, NJ:Prentice-Hall, 2002); Discrete-Time Processing of Speech Signals (New York:Macmillan, 1992, IEEE Press, 2000); Communication Systems Engineering,(Englewood Cliffs, NJ: Prentice Hall, 1994, first edition; 2002, second edition);

Digital Signal Processing Using MATLAB V.4 (Boston: Brooks/Cole-ThomsonLearning, 1997, 2000); and Contemporary Communication Systems Using

MATLAB (Boston: Brooks/Cole-Thomson Learning, 1998, 2000).

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