EXIT chart analysis of BICM-ID with imperfect ... - Semantic Scholar
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IEEE COMMUNICATIONS LETTERS, VOL. 7, NO. 9, SEPTEMBER 2003
EXIT Chart Analysis of BICM-ID With Imperfect Channel State Information Yuheng Huang, Student Member, IEEE, and James A. Ritcey, Member, IEEE
Abstract—We analyze the convergence of iteratively decoded bit-interleaved coded modulation with imperfect channel state information using the extrinsic information transfer (EXIT) chart. A canonical analysis model is adopted, where the power correlation coefficient between the fading and its estimate becomes the key parameter affecting the extrinsic transfer characteristics of the demapper and hence the convergence of iterative decoding. We further illustrate that decoding convergence can be triggered by tradeoff between the quality of channel estimation and code rate. Index Terms—Bit-interleaved coded modulation (BICM), BICM-iterative decoding (BICM-ID), extrinsic information transfer (EXIT) chart, imperfect channel state information (CSI).
I. INTRODUCTION
B
IT-INTERLEAVED coded modulation (BICM) [1] with iterative decoding (BICM-ID) has been shown to exhibit large coding gain over fading channels with coherent detection and perfect channel state information (CSI) [2], [3]. For mobile communication systems with time-varying channel response, CSI is crucial in achieving the expected decoding performance of a channel code. The impact of imperfect CSI on BICM-ID was first studied in [4], where a canonical analysis model was proposed for both Rayleigh and Rician channels. This model utilizes the joint distribution of the fading and its estimate to characterize the effect of imperfect CSI. By assuming error-free feedback, an accurate theoretical bound was developed to predict the asymptotic bit error rate (BER) performance of BICM-ID with channel estimation error. However, to obtain design guidelines for BICM-ID system with imperfect CSI, the question of convergence needs to be investigated. Recently, the extrinsic information transfer (EXIT) chart [5] has been introduced by S. ten Brink as a novel tool to study the convergence behavior of iterative decoding. This approach uses bitwise mutual information of the extrinsic output from constituent decoders as a useful measure to describe the flow of extrinsic information through iterative decoding process. The EXIT chart provides accurate prediction of the convergence behavior, which is verified and visualized by the simulated decoding trajectory. However, all the previous EXIT chart analyzes assume that the receiver has perfect CSI.
Manuscript received February 14, 2003. The associate editor coordinating the review of this letter and approving it for publication was Prof. A. Haimovich. This work was supported by National Science Foundation under Grant CCR0073391. The authors are with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/LCOMM.2003.817304
In this paper, we study the convergence behavior of BICM-ID with imperfect CSI over flat fading channels using the EXIT chart. We adopt the canonical model proposed in [4] for our analysis, and the power correlation coefficient between the fading and its estimate becomes the key parameter affecting the extrinsic information transfer characteristics of the soft-in soft-out (SISO) demapper. Inspired by the properties of the extrinsic transfer function of the SISO decoder [6] with different code rates, we illustrate that the convergence of iterative decoding can be triggered by tradeoff between the quality of channel estimation and information rate. This letter is organized as follows. Section II briefly introduces BICM-ID and the analysis model for fading channels. Section III analyzes the convergence of BICM-ID with imperfect CSI using the EXIT chart. Section IV concludes the paper. II. SYSTEM MODEL A. BICM Transmitter The information bit sequence is first encoded by a convolutional code. Then the encoder output is bitwise interleaved and each consecutive bits of the interleaved sequence are grouped . The modulator maps each to as -ary cona complex valued channel symbol chosen from stellation . Note that the random bit interleaver not only serves as a code interleaver, but also as a channel interleaver to break the sequential fading correlation and increase the diversity order to the minimum Hamming distance of the code. B. Fading Channel We consider frequency nonselective fading channel. The discrete-time baseband received signal can be written as (1) is the fading coefficient, and is the complex where . white Gaussian noise sample with variance For linear channel estimation schemes (e.g., pilot symbol assisted modulation) over Rayleigh fading channels, following the analysis model in [4], we assume that the fading amplitude and its estimate have a bivariate Rayleigh distribution given by [7]
1089-7798/03$17.00 © 2003 IEEE
(2)
HUANG AND RITCHEY: EXIT CHART ANALYSIS OF BICM-ID WITH IMPERFECT CHANNEL STATE INFORMATION
435
where , is the power correlation coefficient between and , , , and is the zeroth-order modified between Bessel function of the first kind. The phase error and its estimate has the pdf given by [7]
(3) and is the same as where that in (2). In our analysis, is normalized to 1, and we set the , which is a typical value in practice power ratio and has little impact on the performance of BICM-ID [4]. C. Iterative Decoder Since perfect CSI is not available, the receiver only uses the to calculate the symbol metric , channel gain estimate which is the log conditional probability density function (4) The SISO demapper uses the maximum a posteriori (MAP) decoding algorithm, and the extrinsic a posteriori log-likelihood ratio (LLR) for the unmapped bit can be computed as
(5)
(or ) is the subset of the constellation in which where the labels have the binary value 1 (or 0) at the th bit position, is the value of the th bit of the label corresponding and to . The SISO decoder takes the deinterleaved version of to compute the a posteriori LLR of each coded bit, which is fed back to the demapper as the updated a priori on the next iteration. Following the notainformation tion of [5], we denote the input and output extrinsic information and associated with the SISO demapper (or decoder) as (or and ), respectively. Note that , , and are all random variables in the LLR domain [5]. III. CONVERGENCE ANALYSIS OF BICM-ID WITH IMPERFECT CSI A. Transfer Characteristic of the SISO Demapper With Imperfect CSI As seen from (4), decoding with imperfect CSI affects the symbol metric calculation, and hence the extrinsic transfer characteristic of the SISO demapper. For a given value of the input , we follow the method in mutual information [5] to compute the output mutual information by Monte Carlo simulation of the SISO demapper. To investigate the effect of imperfect CSI, we use the approach in [4]
Fig. 1. The EXIT chart with iterative decoding trajectories for BICM-ID with 16-QAM MSP labeling, rate 1/2 16-state convolutional code, over uncorrelated Rayleigh fading channels at E =N = 8 dB, with power correlation coefficient = 0:91, 0.945, 0.97 and perfect CSI. The pinch-off limit is = 0:94.
to generate sample pairs of and , which satisfy the joint pdf of (2) and (3) for a given power correlation coefficient . Note that only channel gain estimate is available to calculate the symbol metric in (4). can be viewed as a function of The mutual information , , and the power correlation coefficient (6)
B. EXIT Chart and Convergence Analysis Since there is no extra channel input at the outer SISO decoder, the transfer characteristic of the SISO decoder is independent of the channel (imperfect CSI, value), and determined only by the outer convolutional code itself [5]. As an example, we consider a BICM-ID system which uses the 16-QAM modified set partitioning (MSP) labeling [2] and a rate 1/2 16-state convolutional code. Fig. 1 illustrates how the power correlation coefficient affects the demapper dB. extrinsic transfer function at a fixed value of , we observe an early intersection of the When demapper and decoder transfer functions, and the decoding process gets stuck after only two iterations at low mutual information. As the channel estimation quality improves to , the demapper transfer characteristic is raised, opening a narrow tunnel (“bottleneck”) to allow convergence of iterative decoding toward low BER. The demapper transfer function is further raised as increases to 0.97, which makes the tunnel much wider and triggers faster convergence after four iterations. The EXIT chart for the case of perfect CSI is also shown as reference. To verify the EXIT chart predictions, we
436
Fig. 2. The EXIT chart for BICM-ID with 16-QAM MSP labeling, rate 1/4 16-state convolutional code, over uncorrelated Rayleigh fading channels at E =N = 8 dB, with power correlation coefficient = 0:84; 0:88; 0:94, and perfect CSI, decoding trajectory for = 0:84.
IEEE COMMUNICATIONS LETTERS, VOL. 7, NO. 9, SEPTEMBER 2003
Fig. 3. BER performance versus power correlation coefficient at E =N = 8 dB, for rate 1/2 16-state and rate 1/4 16-state convolutional code. 16-QAM MSP labeling is used, and there are 12 decoding iterations performed for each value of .
IV. CONCLUSION use the simulated decoding trajectories to visualize the iterative decoding process, which match with the transfer characteristics very well for both perfect and imperfect CSI. A closer examination based on the demapper transfer function yields a (as shown in the bottom-right corner pinch-off limit at of Fig. 1), where the transfer functions of the demapper and decoder are just about to intersect. Fig. 2 shows the EXIT chart when a rate 1/4 16-state code is dB, the noise level has been used. Note that, to keep properly adjusted to account for the code rate. It is interesting to observe that when using a low rate code, the extrinsic transfer function of the SISO decoder is lowered and flattened. A narrow . tunnel is opened to allow decoding convergence when This implies that when it is hard to perform high quality channel estimation (e.g., due to fast fade rate), the desired low BER performance can still be achieved by triggering the convergence of iterative decoding with lower information rate. Fig. 3 shows the BER performance of the 12th decoding dB for the rate 1/2 and rate 1/4 16-state pass at codes. Each data block contains 10 information bits and a total 10 information bits are simulated for each number of 5 value of considered. It can be seen that the threshold values of allowing decoding convergence are accurately predicted by the EXIT charts. Note that the EXIT chart analysis is much easier than the intensive BER simulation of the whole system.
We analyze the convergence of BICM-ID with imperfect CSI over fading channels using the EXIT chart, where the power correlation coefficient becomes the key parameter affecting the extrinsic transfer function of the demapper. Simulation results verify that the threshold value of allowing decoding convergence can be accurately predicted by the EXIT chart. We show how, under our model [4], the loss due to imperfect CSI can be recovered by operating at lower code rates. REFERENCES [1] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inform. Theory, vol. 44, pp. 927–945, May 1998. [2] A. Chindapol and J. A. Ritcey, “Design, analysis and performance evaluation for BICM-ID with square QAM constellations in rayleigh fading channels,” IEEE J. Select. Areas Commun., vol. 19, pp. 944–957, May 2001. [3] X. Li, A. Chindapol, and J. A. Ritcey, “Bit-interleaved coded modulation with iterative decoding and 8PSK modulation,” IEEE Trans. Commun., vol. 50, pp. 1250–1257, Aug. 2002. [4] Y. Huang and J. A. Ritcey, “16-QAM BICM-ID in fading channels with imperfect channel state information,” IEEE Trans. Wireless Commun., vol. 2, pp. 1000–1007, Sept. 2003. [5] S. T. Brink, “Designing iterative decoding schemes with the extrinsic information transfer chart,” AEÜ Int. J. Electron. Commun., vol. 54, pp. 389–398, Dec. 2000. [6] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “A soft-input soft-output APP module for iterative decoding of concatenated codes,” IEEE Commun. Lett., vol. 1, pp. 22–24, Jan. 1997. [7] K. S. Miller, Complex Stochastic Processes: An Introduction to Theory and Application. Reading, MA: Addison-Wesley, 1974.
IEEE COMMUNICATIONS LETTERS, VOL. 7, NO. 9, SEPTEMBER 2003
EXIT Chart Analysis of BICM-ID With Imperfect Channel State Information Yuheng Huang, Student Member, IEEE, and James A. Ritcey, Member, IEEE
Abstract—We analyze the convergence of iteratively decoded bit-interleaved coded modulation with imperfect channel state information using the extrinsic information transfer (EXIT) chart. A canonical analysis model is adopted, where the power correlation coefficient between the fading and its estimate becomes the key parameter affecting the extrinsic transfer characteristics of the demapper and hence the convergence of iterative decoding. We further illustrate that decoding convergence can be triggered by tradeoff between the quality of channel estimation and code rate. Index Terms—Bit-interleaved coded modulation (BICM), BICM-iterative decoding (BICM-ID), extrinsic information transfer (EXIT) chart, imperfect channel state information (CSI).
I. INTRODUCTION
B
IT-INTERLEAVED coded modulation (BICM) [1] with iterative decoding (BICM-ID) has been shown to exhibit large coding gain over fading channels with coherent detection and perfect channel state information (CSI) [2], [3]. For mobile communication systems with time-varying channel response, CSI is crucial in achieving the expected decoding performance of a channel code. The impact of imperfect CSI on BICM-ID was first studied in [4], where a canonical analysis model was proposed for both Rayleigh and Rician channels. This model utilizes the joint distribution of the fading and its estimate to characterize the effect of imperfect CSI. By assuming error-free feedback, an accurate theoretical bound was developed to predict the asymptotic bit error rate (BER) performance of BICM-ID with channel estimation error. However, to obtain design guidelines for BICM-ID system with imperfect CSI, the question of convergence needs to be investigated. Recently, the extrinsic information transfer (EXIT) chart [5] has been introduced by S. ten Brink as a novel tool to study the convergence behavior of iterative decoding. This approach uses bitwise mutual information of the extrinsic output from constituent decoders as a useful measure to describe the flow of extrinsic information through iterative decoding process. The EXIT chart provides accurate prediction of the convergence behavior, which is verified and visualized by the simulated decoding trajectory. However, all the previous EXIT chart analyzes assume that the receiver has perfect CSI.
Manuscript received February 14, 2003. The associate editor coordinating the review of this letter and approving it for publication was Prof. A. Haimovich. This work was supported by National Science Foundation under Grant CCR0073391. The authors are with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/LCOMM.2003.817304
In this paper, we study the convergence behavior of BICM-ID with imperfect CSI over flat fading channels using the EXIT chart. We adopt the canonical model proposed in [4] for our analysis, and the power correlation coefficient between the fading and its estimate becomes the key parameter affecting the extrinsic information transfer characteristics of the soft-in soft-out (SISO) demapper. Inspired by the properties of the extrinsic transfer function of the SISO decoder [6] with different code rates, we illustrate that the convergence of iterative decoding can be triggered by tradeoff between the quality of channel estimation and information rate. This letter is organized as follows. Section II briefly introduces BICM-ID and the analysis model for fading channels. Section III analyzes the convergence of BICM-ID with imperfect CSI using the EXIT chart. Section IV concludes the paper. II. SYSTEM MODEL A. BICM Transmitter The information bit sequence is first encoded by a convolutional code. Then the encoder output is bitwise interleaved and each consecutive bits of the interleaved sequence are grouped . The modulator maps each to as -ary cona complex valued channel symbol chosen from stellation . Note that the random bit interleaver not only serves as a code interleaver, but also as a channel interleaver to break the sequential fading correlation and increase the diversity order to the minimum Hamming distance of the code. B. Fading Channel We consider frequency nonselective fading channel. The discrete-time baseband received signal can be written as (1) is the fading coefficient, and is the complex where . white Gaussian noise sample with variance For linear channel estimation schemes (e.g., pilot symbol assisted modulation) over Rayleigh fading channels, following the analysis model in [4], we assume that the fading amplitude and its estimate have a bivariate Rayleigh distribution given by [7]
1089-7798/03$17.00 © 2003 IEEE
(2)
HUANG AND RITCHEY: EXIT CHART ANALYSIS OF BICM-ID WITH IMPERFECT CHANNEL STATE INFORMATION
435
where , is the power correlation coefficient between and , , , and is the zeroth-order modified between Bessel function of the first kind. The phase error and its estimate has the pdf given by [7]
(3) and is the same as where that in (2). In our analysis, is normalized to 1, and we set the , which is a typical value in practice power ratio and has little impact on the performance of BICM-ID [4]. C. Iterative Decoder Since perfect CSI is not available, the receiver only uses the to calculate the symbol metric , channel gain estimate which is the log conditional probability density function (4) The SISO demapper uses the maximum a posteriori (MAP) decoding algorithm, and the extrinsic a posteriori log-likelihood ratio (LLR) for the unmapped bit can be computed as
(5)
(or ) is the subset of the constellation in which where the labels have the binary value 1 (or 0) at the th bit position, is the value of the th bit of the label corresponding and to . The SISO decoder takes the deinterleaved version of to compute the a posteriori LLR of each coded bit, which is fed back to the demapper as the updated a priori on the next iteration. Following the notainformation tion of [5], we denote the input and output extrinsic information and associated with the SISO demapper (or decoder) as (or and ), respectively. Note that , , and are all random variables in the LLR domain [5]. III. CONVERGENCE ANALYSIS OF BICM-ID WITH IMPERFECT CSI A. Transfer Characteristic of the SISO Demapper With Imperfect CSI As seen from (4), decoding with imperfect CSI affects the symbol metric calculation, and hence the extrinsic transfer characteristic of the SISO demapper. For a given value of the input , we follow the method in mutual information [5] to compute the output mutual information by Monte Carlo simulation of the SISO demapper. To investigate the effect of imperfect CSI, we use the approach in [4]
Fig. 1. The EXIT chart with iterative decoding trajectories for BICM-ID with 16-QAM MSP labeling, rate 1/2 16-state convolutional code, over uncorrelated Rayleigh fading channels at E =N = 8 dB, with power correlation coefficient = 0:91, 0.945, 0.97 and perfect CSI. The pinch-off limit is = 0:94.
to generate sample pairs of and , which satisfy the joint pdf of (2) and (3) for a given power correlation coefficient . Note that only channel gain estimate is available to calculate the symbol metric in (4). can be viewed as a function of The mutual information , , and the power correlation coefficient (6)
B. EXIT Chart and Convergence Analysis Since there is no extra channel input at the outer SISO decoder, the transfer characteristic of the SISO decoder is independent of the channel (imperfect CSI, value), and determined only by the outer convolutional code itself [5]. As an example, we consider a BICM-ID system which uses the 16-QAM modified set partitioning (MSP) labeling [2] and a rate 1/2 16-state convolutional code. Fig. 1 illustrates how the power correlation coefficient affects the demapper dB. extrinsic transfer function at a fixed value of , we observe an early intersection of the When demapper and decoder transfer functions, and the decoding process gets stuck after only two iterations at low mutual information. As the channel estimation quality improves to , the demapper transfer characteristic is raised, opening a narrow tunnel (“bottleneck”) to allow convergence of iterative decoding toward low BER. The demapper transfer function is further raised as increases to 0.97, which makes the tunnel much wider and triggers faster convergence after four iterations. The EXIT chart for the case of perfect CSI is also shown as reference. To verify the EXIT chart predictions, we
436
Fig. 2. The EXIT chart for BICM-ID with 16-QAM MSP labeling, rate 1/4 16-state convolutional code, over uncorrelated Rayleigh fading channels at E =N = 8 dB, with power correlation coefficient = 0:84; 0:88; 0:94, and perfect CSI, decoding trajectory for = 0:84.
IEEE COMMUNICATIONS LETTERS, VOL. 7, NO. 9, SEPTEMBER 2003
Fig. 3. BER performance versus power correlation coefficient at E =N = 8 dB, for rate 1/2 16-state and rate 1/4 16-state convolutional code. 16-QAM MSP labeling is used, and there are 12 decoding iterations performed for each value of .
IV. CONCLUSION use the simulated decoding trajectories to visualize the iterative decoding process, which match with the transfer characteristics very well for both perfect and imperfect CSI. A closer examination based on the demapper transfer function yields a (as shown in the bottom-right corner pinch-off limit at of Fig. 1), where the transfer functions of the demapper and decoder are just about to intersect. Fig. 2 shows the EXIT chart when a rate 1/4 16-state code is dB, the noise level has been used. Note that, to keep properly adjusted to account for the code rate. It is interesting to observe that when using a low rate code, the extrinsic transfer function of the SISO decoder is lowered and flattened. A narrow . tunnel is opened to allow decoding convergence when This implies that when it is hard to perform high quality channel estimation (e.g., due to fast fade rate), the desired low BER performance can still be achieved by triggering the convergence of iterative decoding with lower information rate. Fig. 3 shows the BER performance of the 12th decoding dB for the rate 1/2 and rate 1/4 16-state pass at codes. Each data block contains 10 information bits and a total 10 information bits are simulated for each number of 5 value of considered. It can be seen that the threshold values of allowing decoding convergence are accurately predicted by the EXIT charts. Note that the EXIT chart analysis is much easier than the intensive BER simulation of the whole system.
We analyze the convergence of BICM-ID with imperfect CSI over fading channels using the EXIT chart, where the power correlation coefficient becomes the key parameter affecting the extrinsic transfer function of the demapper. Simulation results verify that the threshold value of allowing decoding convergence can be accurately predicted by the EXIT chart. We show how, under our model [4], the loss due to imperfect CSI can be recovered by operating at lower code rates. REFERENCES [1] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inform. Theory, vol. 44, pp. 927–945, May 1998. [2] A. Chindapol and J. A. Ritcey, “Design, analysis and performance evaluation for BICM-ID with square QAM constellations in rayleigh fading channels,” IEEE J. Select. Areas Commun., vol. 19, pp. 944–957, May 2001. [3] X. Li, A. Chindapol, and J. A. Ritcey, “Bit-interleaved coded modulation with iterative decoding and 8PSK modulation,” IEEE Trans. Commun., vol. 50, pp. 1250–1257, Aug. 2002. [4] Y. Huang and J. A. Ritcey, “16-QAM BICM-ID in fading channels with imperfect channel state information,” IEEE Trans. Wireless Commun., vol. 2, pp. 1000–1007, Sept. 2003. [5] S. T. Brink, “Designing iterative decoding schemes with the extrinsic information transfer chart,” AEÜ Int. J. Electron. Commun., vol. 54, pp. 389–398, Dec. 2000. [6] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “A soft-input soft-output APP module for iterative decoding of concatenated codes,” IEEE Commun. Lett., vol. 1, pp. 22–24, Jan. 1997. [7] K. S. Miller, Complex Stochastic Processes: An Introduction to Theory and Application. Reading, MA: Addison-Wesley, 1974.
