IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 10, OCTOBER 2006
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Performance of Convolutional Codes With Finite-Depth Interleaving and Noisy Channel Estimates Jittra Jootar, Student Member, IEEE, James R. Zeidler, Fellow, IEEE, and John G. Proakis, Life Fellow, IEEE
Abstract—In this paper, we derive the Chernoff bound of the pairwise error probability (PEP) and the exact PEP of convolutional codes in a time-varying Rician fading channel. With the assumptions that the channel estimator is a finite impulse response filter and the interleaving depth is finite, we are able to investigate the estimation-diversity tradeoff resulting from the effects of the Doppler spread on the system performance via the channel-estimation accuracy and the channel diversity. In addition, we verify that, in the special case when the pilot signal-to-noise ratio is infinitely large and the channel estimator is well-designed, our analysis leads to the same result as the existing perfect channel-state information analysis. Finally, the analytical results are compared with results from Monte Carlo simulation, and the comparison shows that the analytical results match well with the simulation results. Index Terms—Channel estimation, convolutional codes, diversity, estimation-diversity tradeoff, interleaving.
I. INTRODUCTION REVIOUS analytical studies on the performance of convolutional codes in a time-varying fading channel have focused on either imperfect channel-state information (CSI) or finite-depth interleaving, while assuming the other to be perfect [1]–[4]. Since fading coefficients can be estimated with better accuracy in a slowly fading channel, a perfectly interleaved system with noisy CSI in a slowly fading channel outperforms 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 independent 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
P
Paper approved by C. Schlegel, the Editor for Coding Theory and Techniques of the IEEE Communications Society. Manuscript received June 27, 2005; revised February 20, 2006. This work was supported in part by Ericsson under Core Grant 02-10109 and in part by the U.S. Army Research Office under the Multiuniversity Research Initiative (MURI) Grant W911NF-04-1-0224. This paper 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 is now 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 and Computer Engineering, University of California at San Diego, La Jolla, CA 92093-0407 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 operating environment, there has recently been growing interest in the performance analysis of coded systems with imperfect CSI and imperfect 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 discussed the tradeoff from simulation results without providing any analytical analysis [8]. The analysis on the estimation-diversity 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 Rician 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 components, after multiplying the received signals with the conjugate 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 assumption for several wireless systems, or Gaussian noise component [11], the primary focus of this paper is to derive the PEP with the assumptions that are more general and the model that incorporates implementation issues, such as the choice of the pilot filter. Consequently, the system performance in a realistic scenario can be calculated from the analysis without having to resort 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 correlated 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 the system 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 analysis. 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 . The superscripts , , denote the complex conjuby 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 , respectively. and The system considered is a downlink binary phase-shift keying (BPSK) direct-sequence code-division multiple-access (DS-CDMA) system. A complex baseband representation of the system is illustrated in Fig. 1, where the base station’s
transmitter, the frequency-selective fading channel, and the mobile’s receiver are shown in the upper left corner, the upper right corner, and the bottom section of the figure, respectively. A. Transmitter signal streams transmitted We assume that there are from the base station. The streams consist of one pilot stream (zeroth stream) and data streams assigned to users is ( th stream for the th user). The pilot stream , where denotes the chip spread with the orthogonal code 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 , which has the same period . with the orthogonal code After spreading, the th signal stream is scaled by , and the signals from all branches are combined and scrambled by . The signal the base-station-dependent complex long code 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 intersymbol interference (ISI) [17], i.e., when when and is any nonzero integer
(2)
JOOTAR et al.: PERFORMANCE OF CONVOLUTIONAL CODES WITH FINITE-DEPTH INTERLEAVING AND NOISY CHANNEL ESTIMATES
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 selective Rician fading channel with resolvable paths and impulse response
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, 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 -tap finite-imreceiver [17] is assumed to be a pulse response (FIR) filter with the filter coefficient vector . As a result, the channel estimate can be written as
(7) (3) denotes the fading coefficient corresponding to the where 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 is assumed to be a circuThe fading coefficient larly symmetric complex Gaussian random variable with and autocovariance function real-valued mean . Note that represents the energy of the line-of-sight (LOS) component of represents the autocorrelation the fading channel, and function of the diffuse component of the fading channel. In addition, we assume that the fading coefficients from different paths are independent; thus, when . An analysis for the Rayleigh fading channel is . Although we assume a simply a special case when 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 is assumed to be additive Finally, the thermal noise white Gaussian noise (AWGN) with variance . C. Receiver At the mobile’s receiver of user 1, the received signal be expressed as
.
where
III. ANALYSIS Without loss of generality, we assume that the transmitted codeword 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 interleaver 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 approximation is used here mainly to simplify the analysis, and an extension 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
can (8) (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)
where , , is the interleaving depth, is the Hamming weight of the error codeword, denotes the real part of the complex number , and is the probability density function of . A. Characteristic Function
(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-mean and Gaussian random variables with variances
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]
conclude that
(10) are the mean vector and the covariance matrix where and of , respectively. Due to the assumption that the fading coefficients from different paths are independent, the characteristic function of is simply the product of the characteristic func. Thus, the characteristic function of tions of for becomes
is equal to
. Using
the characteristic function specified in (11), the set of be simplified, as shown in Appendix I, to be
can
(18) for are the eigenvalues of . Also where 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) (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
in (11), we can get Substituting is the Chernoff bound of .
, which
C. The PEP In addition to the Chernoff bound, can also be used to directly. This is done by substituting the expression of find as a function of
(13)
(20)
(14) (15) and is a square matrix of size diagonal and zero elsewhere, with th column of . the
with ones on the th is a square matrix of size , and is
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 , i.e., expectation can be written as a function of
(17) is a convex function with respect to Since (the second-order derivative of with respect to is equal to , and is always positive), we can
into (8). Using Mellin’s inversion, we get
at
(21)
where lies between the left half-plane poles and the imaginary denotes the number of negative poles of , axis, 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 , and is the order of the pole at . of 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 systems with large , we resort to a numerical approximation called
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the Gauss–Chebyshev approximation (suggested in [21] with a correction in [22]), which can be expressed as
. The scaled characteristic function can be expressed as shown in (27) and (28) at the bottom is the data SNR, and of the page, where are the eigenvalues of . From (28), we know
(23)
are . that the poles of Since is a positive definite matrix, for are greater than zero. We can then identify that the negative
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 corresponding to this special case is equal to calcuthat lated from the perfect CSI analysis. For simplicity, we consider only 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 and . Thus, we get by using and (24)
, and the positive poles are
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 . Substiwe can choose so that the constraint is satisfied is in (21), when the scaled characteristic function tuting 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 following, 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). is a function of Under the perfect CSI assumption, , where is the instantaneous data SNR corresponding to the th error symbol. The characteristic function was given in [23] to be of
where
is a square matrix of size with . Finding a matrix inverse is usually a difficult task. Forcan be found easily tunately, the matrix inverse of to be
where Substituting characteristic function becomes
(33) where , given
,
, and are as previously defined. Since is equal to [23], where for , the average PEP can be over the distribution of , i.e., found by averaging
(25) . and (25) into (11), the
(34) Using an alternative form of the complementary error function [24]
(26) Using (21) and recalling from Appendix II that variant to the value of in , given that instead of will use
is in, we to find
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 Gauss–Chebyshev 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, which is the channel memory that uses the tradeoff quency , the perforin the most effective way [7]. At the optimal increases, the performance degradamance is such that, if 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’s can easily be seen in Fig. 2, where timates. These optimal we plot the PEP of systems in Rayleigh fading channels as a for for two types of function 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 = 18, 11-tap Wiener pilot filter.
(the zeroth-order Bessel function of the first kind) is not monotonically decreasing. For a nonmonotonically decreasing function, 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 autocorrelation function oscillates, while the plot corresponding to a monotonically decreasing function does not. In addition, the oscillation 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)
JOOTAR et al.: PERFORMANCE OF CONVOLUTIONAL CODES WITH FINITE-DEPTH INTERLEAVING AND NOISY CHANNEL ESTIMATES
Fig. 3. Comparison of the PEP for the realistic case and for the case of perfect CSI or perfect interleaving. Data SNR = 7 dB, d = 18, 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 estiincreases and the channel esmates. We can also see that as timates are less accurate, the solid lines diverge more and more is large enough such that the from perfect CSI, and when 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 [4], the solid line corresponding to a larger at a smaller 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 systems in Rayleigh fading channels. Two fading channels shown , referred to as system are the fast-fading channel with , referred to as A, and the slowly fading channel with system B. We can see that at small pilot SNR, system B outperforms 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 capacity, the CSI accuracy is crucial. But when the system operates 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 accuracy 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
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Fig. 4. PEP as a function of the pilot SNR and f . Data SNR = 7 dB, I = 30, d = 18, 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 parameters, such as the interleaving depth and the data SNR. An analysis 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 Depth In 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 system’s 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 coefficients. In Fig. 6, we compare the performance of the two filters, where the fixed filter is randomly chosen to be the Wiener filter in corresponding to pilot SNR = 10 dB, and a 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 , and , Jakes PSD, 11-tap Wiener pilot filter. 7 dB, d
= 18
I on P . Data SNR =
Fig. 7. Comparison of the PEP between single-path channel and two-path : ,N , 31 dB, d ,E , fading channel. E Gaussian PSD, 11-tap Wiener pilot filter.
=1
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 , ,I Jakes PSD, 11-tap pilot filter.
= 18 = 10
E. Frequency-Selective Fading Channel Fig. 7 shows the PEP when the system is in a frequency-selective Rayleigh fading channel, assuming that there are two resolvable paths, and each path has half of the average power of the path in the flat-fading case. Path diversity added by the multipath causes the performance of the frequency-selective fading
= 128
0
= 18
Fig. 8. Comparison of the PEP between the flat-fading channel and a two-path E : , fading channel for different values of data-to-pilot ratios. E ,d N 31 dB, I , , Gaussian PSD, 11-tap Wiener pilot filter.
= 128
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 combination of the optimal estimator and the optimal decoder does not guarantee the optimal result. It is also apparent that the filter are large, as choice is critical when the pilot SNR and fixed is almost three times larger than dynamic .
= 0 01
=0
= 23 = 18
+
= 1 01
to be better than the flat fading when is small. But, due to smaller pilot SNR per path, the channel-estimation accuracy of the 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). toward Let us consider the figure from small . When is very small (small data energy, large pilot energy), the pilot SNR is large enough that even the multipath system, which is the system with worse channel
JOOTAR et al.: PERFORMANCE OF CONVOLUTIONAL CODES WITH FINITE-DEPTH INTERLEAVING AND NOISY CHANNEL ESTIMATES
Fig. 9. Effect of the pilot SNR and f on P . Data SNR = 7 dB, d = 3; 18, Jakes PSD, 11-tap Wiener pilot filter.
I
= 30,
estimates, has accurate CSI. Since both systems have good , the diversity is the dominant channel estimates at small factor. Therefore, the multipath system which has more diver. When sity outperforms the flat-fading system at small is large, however, the CSI accuracy of the multipath . Therefore, system becomes worse, especially at high 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 from allocating appropriate energy to the data and the pilot channels from 40 to can be significant. For example, changing 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 ) 0.1, 1, and 4, and for 3, 18. The (also denoted by performance improves as the Rician factor increases, as exand large pilot pected, for all cases except the case when SNR. This is because when the pilot SNR is large, the domi, the nant factor is the channel diversity [7], and with performance is tremendously improved by the channel diversity. 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. in But when the Rician factor is large enough ( this example) that the LOS component can compensate for the performance loss due to the decrease in the channel diversity, the also improves with the performance of the system with
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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, = 18, 220 infor11-tap Wiener pilot filter, rate-1/3 convolutional code, d mation bits per block, 8 b zero padding, =2 = 0.
Rician factor. Also note that the same behavior is observed but at smaller Rician factor, i.e., at for the system with 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 diminishes. Finally, when the Rician factor approaches , the perforconverge to the same performance, mance of 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 convolutional code specified in the UMTS-WCDMA standard [25], 18 and 256 states. The interleaver (with in with this simulation) is also specified in the UMTS-WCDMA standard [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 autocorrelation of the Jakes model. The channel estimator is an 11-tap Wiener filter. For the truncated union bound, only the five smallest Ham) are used to calculate ming weights ( the bit-error probability (BEP) and the block-error probability and , respectively. In addition, two (BLEP), denoted by values of interleaving depths shown in the figure are 23 and 36. Comparing the simulation results and the analytical results , we can see that the analytical results match well with
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analytical results match well with the simulation results, espeis less than 0.04. cially when V. CONCLUSIONS
Fig. 11. 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, = 18, 220 in11-tap moving average filter, rate-1/3 convolutional code, d 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 provide 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 with and is illustrated in Fig. 11. However, to illustrate 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. Comparing the truncated union bound of the BEP calculated from for the minimum Hamming distance (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 symbols. Since the spacing of error symbols in a real system depends on the interleaving pattern and the error sequence, the simple assumption that all consecutive error symbols are symbols apart does not lead to an accurate performance prediction in the Rician 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 and , which then used to calculate the approximation of are shown in the figure as solid and dashed lines, respectively. Also note that three smallest Hamming weights, 18, 20, and 22, and without the equally spaced are used to calculate the error-symbols assumption. We can see from the figure that the
In this paper, we have derived the Chernoff bound of the PEP and the exact PEP of coded systems with finite-depth interleaving and noisy channel estimates. The analysis provides an insight into the system performance in a realistic environment. For example, we have shown that there exists an optimal channel memory length, which is a result of the estimation-diversity 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 decrease of the channel diversity resulting from increasing . In addition to gaining more understanding of the system behavior, 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-topilot 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 FINDING The optimal can be found by taking a derivative of with respect to , and the derivative can be expressed as
(39) where , and of
is the th eigenvalue
. After some math, we get
(40) (41) Substituting (41) into (39) and conditioning that , the optimal can then be expressed as
(42)
JOOTAR et al.: PERFORMANCE OF CONVOLUTIONAL CODES WITH FINITE-DEPTH INTERLEAVING AND NOISY CHANNEL ESTIMATES
APPENDIX II RESIDUES OF In this section, we will prove that the residue of at a at a pole , where pole is equal to the residues of is any nonzero scaling factor. From (10), the characteristic function can be simplified as follows:
(43) where and
is a matrix such that
. Thus, the poles of are zero and for , where are the . Therefore, the partial fraction of eigenvalues of 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 to . In addition, the residue of change the poles from at the pole in (44) is equal to the coefficient . at the pole This residue is also equal to the residue of in (46). Therefore, we can conclude that the residue of at 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 [1] L. Chong and L. Milstein, “Convolutionally coded multicarrier DS-CDMA with imperfect channel estimation,” in Proc. 39th Allerton Conf. Commun., Control, Comput., Oct. 2001, pp. 543–552. [2] F. Gagnon and D. Haccoun, “Bound on the error performance of coding for nonindependent Rician-fading channels,” IEEE Trans. Commun., vol. 40, no. 2, pp. 351–360, Feb. 1992. [3] G. Kaplan and S. Shamai, “Achievable performance over the correlated Rician channel,” IEEE Trans. Commun., vol. 42, no. 11, pp. 2967–2978, Nov. 1994.
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[4] M. Rice and E. Perrins, “A simple figure of merit for evaluating interleaver depth for the land-mobile satellite channel,” IEEE Trans. Commun., vol. 49, no. 8, pp. 1343–1353, Aug. 2001. [5] S. A. Zummo and W. E. Stark, “Performance analysis of binary coded systems over Rician block fading channels,” in Proc. IEEE MILCOM, Oct. 2003, pp. 314–319. [6] ——, “Performance analysis of coded systems over block fading channels,” in Proc. IEEE Veh. Technol. Conf., Sep. 2002, pp. 1129–1133. [7] A. P. Worthen and W. E. Stark, “On the channel memory-diversity tradeoff in communication systems,” in Information, Coding and Mathematics. Norwell, MA: Kluwer, 2002. [8] W. K. M. Ahmed and P. J. McLane, “Random coding error exponents for flat fading channels with realistic channel estimation,” IEEE J. Sel. Areas Commun., vol. 18, no. 2, pp. 369–379, Mar. 2000. [9] C. Tellambura and V. K. Bhargava, “Error performance of MPSK trellis-coded modulation over nonindependent Rician fading channels,” IEEE Trans. Veh. Technol., vol. 47, no. 1, pp. 152–162, Jan. 1998. [10] R. V. Nobelen and D. P. Taylor, “Analysis of the pairwise error probability of noninterleaved codes on the Rayleigh-fading channel,” IEEE Trans. Commun., vol. 44, no. 4, pp. 456–463, Apr. 1996. [11] J. Lai and N. B. Mandayam, “Performance analysis of convolutionally coded DS-CDMA systems with spatial and temporal channel correlations,” IEEE Trans. Commun., vol. 51, no. 12, pp. 1984–1990, Dec. 2003. [12] J. Jootar, J. R. Zeidler, and J. G. Proakis, “Performance of finite-depth interleaved convolutional codes in a Rayleigh fading channel with noisy channel estimates,” in Proc. IEEE 61st Veh. Technol. Conf., May 2005, pp. 600–605. [13] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 40, no. 6, pp. 686–693, Nov. 1991. [14] W.-Y. Kuo, “Analytic forward link performance of pilot-aided coherent DS-CDMA under correlated Rician fading,” IEEE J. Sel. Areas Commun., vol. 18, no. 7, pp. 1159–1168, Jul. 2000. [15] F. A. Dietrich and W. Utschick, “Maximal ratio combining of correlated Rayleigh fading channels with imperfect channel knowledge,” IEEE Commun. Lett., vol. 7, no. 9, pp. 419–421, Sep. 2003. [16] P. Shamain and L. B. Milstein, “Effect of mutual coupling and correlated fading on receive diversity systems using compact antenna arrays and noisy channel estimates,” in Proc. IEEE Globecom, Dec. 2003, pp. 1669–1673. [17] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995. [18] R. Cideciyan, E. Eleftheriou, and M. Rupf, “Concatenated Reed-Solomon/convolutional coding for data transmission in CDMA-based cellular systems,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1291–1303, Oct. 1997. [19] L. Tong, B. M. Sandler, and M. Dong, “Pilot-assisted wireless transmissions,” IEEE Signal Process. Mag., vol. 21, pp. 12–25, Nov. 2004. [20] G. L. Turin, “The characteristic function of Hermitian quadratic forms in complex normal variables,” Biometrika, vol. 47, pp. 199–201, Jun. 1960. [21] E. Biglieri, G. Caire, G. Taricco, and J. Ventura-Traveset, “Simple method for evaluating error probabilities,” Electron. Lett., vol. 32, pp. 191–192, Feb. 1996. [22] J. R. Foerster and L. B. Milstein, “Coded modulation for a coherent DS-CDMA system employing an MMSE receiver in a fading channel,” IEEE Trans. Commun., vol. 48, no. 11, pp. 1909–1918, Nov. 2000. [23] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1996. [24] M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972. [25] Technical Specification Group Radio Access Network, Physical Layer Procedure (FDD), 25.212, 3GPP. [26] M. Patzold, Mobile Fading Channels, 1st ed. New York: Wiley, 2002.
Jittra Jootar (S’02) was born in Nashville, TN. She received 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 computer engineering 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 and WCDMA development. In July 2006, she rejoined Qualcomm Inc., to continue working on WCDMA development. Her research interests includes MIMO, modulation, and coding for mobile communication systems.
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James R. Zeidler (M’76–SM’84–F’94) is a Research Scientist/Senior Lecturer in the Department of Electrical Engineering, University of California, San Diego. He is a faculty member of the UCSD Center for Wireless Communications and the University of California Institute for Telecommunications and Information Technology. He has more than 200 technical publications and 13 patents for communication, signal processing, data compression techniques, and electronic devices. Dr. Zeidler received the Frederick Ellersick award from the IEEE Communications Society in 1995, the Navy Meritorious Civilian Service Award in 1991, and the Lauritsen–Bennett Award for Achievement in Science from the Space and Naval Warfare Systems Center in 2000. He was an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING and a Member of the Technical Committee on Underwater Acoustic Signal Processing for the IEEE Signal Processing Society.
John G. Proakis (S’58–M’62–SM’82–F’84–LF’97) received the BSEE degree from the University of Cincinnati, Cincinnati, OH, in 1959, the MSEE degree from Massachusetts Institute of Technology, Cambridge, in 1961, and the Ph.D. degree from Harvard University, Cambridge, MA, in 1967. He is an Adjunct Professor at the University of California at San Diego and a Professor Emeritus at Northeastern University, Boston, MA. He was a faculty member at Northeastern University from 1969 through 1998 and held the following academic positions: Associate Professor of Electrical Engineering, 1969–1976; Professor of Electrical Engineering, 1976–1998; Associate Dean of the College of Engineering and Director of the Graduate School of Engineering, 1982–1984; Interim Dean of the College of Engineering, 1992–1993; Chairman of the Department of Electrical and Computer Engineering, 1984–1997. Prior to joining Northeastern University, he worked with GTE Laboratories and the MIT Lincoln Laboratory. His professional experience and interests are in the general areas of digital communications and digital signal processing. He is the author of the book Digital Communications (New York: McGraw-Hill, 1983, first edition; 1989, second edition; 1995, third edition; 2001, fourth edition), and co-author of the books, Introduction to Digital Signal Processing (New York: Macmillan, 1988, first edition; 1992, second edition; 1996, third edition); Digital Signal Processing Laboratory (Englewood Cliffs, NJ: Prentice-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-Thomson Learning, 1997, 2000); and Contemporary Communication Systems Using MATLAB (Boston: Brooks/Cole-Thomson Learning, 1998, 2000).
