1080 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 7, JULY 2008 Source Fidelity over Fading Channels: Performance of Erasure and Scalable Codes Konstantinos E. Zachariadis, Member, IEEE, Michael L. Honig, Fellow, IEEE, and Aggelos K. Katsaggelos, Fellow, IEEE Abstract—We consider the transmission of a Gaussian source through a block fading channel. Assuming each block is decoded independently, the received distortion depends on the trade- off between quantization accuracy and probability of outage. Namely, higher quantization accuracy requires a higher channel code rate, which increases the probability of outage. We first treat an outage as an erasure, and evaluate the received mean distortion with erasure coding across blocks as a function of the code length. We then evaluate the performance of scalable, or multi-resolution coding in which coded layers are superimposed within a coherence block, and the layers are sequentially decoded. Both the rate and power allocated to each layer are optimized. In addition to analyzing the performance with a finite number of layers, we evaluate the mean distortion at high Signal-to- Noise Ratios as the number of layers becomes infinite. As the block length of the erasure code increases to infinity, the received distortion converges to a deterministic limit, which is less than the mean distortion with an infinite-layer scalable coding scheme. However, for the same standard deviation in received distortion, infinite layer scalable coding performs slightly better than erasure coding, and with much less decoding delay. Index Terms—Source-channel coding, scalable coding, fading channel, broadcast channel, rate distortion. I. I NTRODUCTION T RANSMISSION of a continuous source, such as an image or video, through a fading channel must account for distortion due to both quantization and channel-induced errors. If the channel code-word is sufficiently long, i.e., extends over multiple fading cycles, then the ergodic capacity can be used to determine the achievable information rate, which in turn determines the (fixed) source rate. However, a slowly-varying fading channel and/or short channel code- words, relative to the channel variations, can lead to outages, which substantially increase the received distortion. The trade- off between distortion due to source quantization and channel- induced distortion in the presence of fading has been studied for different source models in [1]–[5]. Other related work Paper approved by F. Alajaji, the Editor for Source and Source/Channel Coding of the IEEE Communications Society. Manuscript received July 13, 2006; revised April 23, 2007 and August 14, 2007. This work was supported in part by ARO under grant DAAD190310119 and NSF under grant CCR- 0073686, and was presented at Milcom 2004 and Globecom 2005. This work was performed when K. E. Zachariadis was with the Electrical and Computer Engineering Department (now EECS), Northwestern University. K. E. Zachariadis is with the Kellogg School of Management, Northwestern University, Evanston, IL 60201-2001, USA (e-mail: k- zachariadis@kellogg.northwestern.edu). M. L. Honig and A. K. Katsaggelos are with the Electrical Engineering and Computer Science Department, Northwestern University, Evanston IL 60208- 3118, USA (e-mail: {mh, aggk}@ece.northwestern.edu). Digital Object Identifier 10.1109/TCOMM.2008.060387. on optimization of source and channel coding parameters in different contexts is presented in [6]–[13]. In this paper we consider the use of erasure and scalable codes, which can be used to compensate for channel fading. Namely, erasure codes can eliminate errors due to outages, whereas scalable codes attempt to match the transmitted rate to the channel conditions. To provide insight, we consider a model in which a continuous Gaussian source is transmitted through a block Rayleigh fading channel. In the absence of delay constraints, the distortion is minimized by coding over coherence blocks. The source rate should then be matched to the channel code rate, which can be no greater than the er- godic capacity. However, with a decoding delay constraint the ergodic capacity cannot always be achieved, making it more difficult to identify an appropriate source rate. 1 Moreover, in some applications it may not be computationally feasible to code optimally over coherence blocks. The erasure coding scheme contains an inner code, which achieves the capacity for a particular Signal-to-Noise Ratio (SNR). The achievable rate can then be characterized in terms of the outage capacity [15]. Namely, an outage occurs when the actual SNR falls below the SNR associated with the code rate. Increasing the source rate therefore decreases the distortion due to quantization, but increases the probability of outage due to the associated increase in channel code rate. An outer erasure code is concatenated with the inner code to correct the errors due to outages. Although this concate- nated coding scheme is suboptimal (i.e., does not achieve the ergodic capacity), it is relatively simple. We optimize the inner channel code rate to maximize the capacity of the resulting erasure channel, assuming infinite-length erasure codewords, and compare the distortion with that associated with the ergodic capacity. Numerical results show that the gap between distortions for the two schemes is relatively small for small SNRs, but increases with SNR. We also characterize the standard deviation of the distortion (over channel realizations) as a function of the length of the erasure code. We then consider the performance of scalable source cod- ing, where for each coherence block the source is partitioned into layers representing successive refinements of the quan- tizer. Each layer, corresponding to a particular quantizer, can have a different rate and power, and all layers are transmitted simultaneously as a superposition code [16, Ch. 14]. The 1 If the source and channel bandwidths are the same, then for our model uncoded (analog) transmission of the source symbols is optimal [10], [14]. However, this is not true for mismatched bandwidths, and for pre-quantized sources. 0090-6778/08$25.00 c  2008 IEEE