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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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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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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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1778 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 10, OCTOBER 2006
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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JOOTAR et al.: PERFORMANCE OF CONVOLUTIONAL CODES WITH FINITE-DEPTH INTERLEAVING AND NOISY CHANNEL ESTIMATES 1779
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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1780 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 10, OCTOBER 2006
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
, Jakes PSD, 11-tap Wiener pilot filter.
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
, Jakes PSD, 11-tap Wiener pilot filter.
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 .
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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).