Motivated by the practical constraints arising in sensor networks and Internet-of-Things applications, the zero-delay transmission of a Gaussian measurement over a real single-input multiple-output (SIMO) additive white Gaussian noise (AWGN) channel is studied with a low-resolution analog-to-digital converter (ADC) front end. The optimization of the encoder and the decoder is tackled under both the mean squared error (MSE) distortion and the distortion outage probability (DOP) criteria, with an average power constraint on the channel input. Optimal encoder and decoder mappings are identified for the case of a one-bit ADC front end for both criteria. For the MSE distortion, the optimal encoder mapping is shown to be non-linear in general, and it tends to a linear encoder, in the low signal-to-noise ratio (SNR) regime, and to an antipodal digital encoder in the high SNR regime. This is in contrast to the optimality of linear encoding at all SNR values in the presence of a full precision front end. For the DOP criterion, it is shown that the optimal encoder mapping is piecewise constant, and can take only two opposite values when it is non-zero. For both the MSE distortion and the DOP criteria, necessary optimality conditions are derived for a @math -level ADC front end and for multiple one-bit ADC front ends. These conditions then are used to obtain numerically optimized solutions. Extensive numerical results are provided to gain insights into the optimal encoding and decoding functions.
Within the spectrum of information theoretic analysis of joint source-channel coding, while the classical Shannon theoretic infinite block-length regime occupies one end of the spectrum, the zero-delay transmission of a single source sample over single channel use lies at the opposite end. Intermediate, but still asymptotic, analyses have also been put forth recently by investigating second-order approximations @cite_32 @cite_6 . For the problem of zero-delay transmission, there is no explicit method to obtain the optimal encoding and decoding mappings, except for the special case of statistically matched source-channel pairs @cite_29 , @cite_8 . This special case includes the setting studied in this paper when the receiver front end has infinite resolution. Various solutions have been proposed in the literature for specific unmatched source-channel pairs. Notable examples are the space-filling curves proposed by Shannon @cite_7 and Kotelnikov @cite_39 , and later extended in @cite_12 @cite_13 @cite_11 @cite_16 @cite_23 , for the delay-limited transmission of a Gaussian source over an AWGN channel with bandwidth mismatch between the source and the channel. Other solutions include @cite_12 @cite_3 @cite_27 @cite_17 .
@cite_7 @cite_8 @cite_29 @cite_32 @cite_6 @cite_39 @cite_3 @cite_17 @cite_27 @cite_23 @cite_16 @cite_13 @cite_12 @cite_11
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This paper deals with lossy joint source-channel coding for transmitting memoryless sources over AWGN channels. The scheme is based on the geometrical interpretation of communication by Kotel'nikov and Shannon where amplitudecontinuous, time-discrete source samples are mapped directly onto the channel using curves or planes. The source and channel spaces can have different dimensions and thereby achieving either compression or error control, depending on whether the source bandwidth is smaller or larger than the channel bandwidth. We present a general theory for 1:N and M:1 dimension changing mappings, and provide two examples for a Gaussian source and channel where we optimize both a 2:1 bandwidth-reducing and a 1:2 bandwidth-expanding mapping. Both examples show high spectral efficiency and provide both graceful degradation and improvement for imperfect channel state information at the transmitter. This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint source-channel coding schemes. We first show some examples of sources and channels where no coding is optimal for all values of the signal-to-noise ratio (SNR). When the source bandwidth is greater than the channel bandwidth, joint coding schemes based on space-filling curves and other families of curves are proposed. For uniform sources and modulo channels, our coding scheme based on space-filling curves operates within 1.1 dB of Shannon's rate-distortion bound. For Gaussian sources and additive white Gaussian noise (AWGN) channels, we can achieve within 0.9 dB of the rate-distortion bound. The second scheme is based on low-density parity-check (LDPC) codes. We first demonstrate that we can translate threshold values of an LDPC code between channels accurately using a simple mapping. We develop some models for density evolution from this observation, namely erasure-channel, Gaussian-capacity, and reciprocal-channel approximations. The reciprocal-channel approximation, based on dualizing LDPC codes, provides a very accurate model of density evolution for the AWGN channel. We also develop another approximation method, Gaussian approximation, which enables us to visualize infinite-dimensional density evolution and optimization of LDPC codes. We also develop other tools to better understand density evolution. Using these tools, we design some LDPC codes that approach the Shannon limit extremely closely. For multilevel AWGN channels, we design a rate 1 2 code that has a threshold within 0.0063 dB of the Shannon limit of the noisiest level. For binary-input AWGN channels, our best rate 1 2 LDPC code has a threshold within 0.0045 dB of the Shannon limit. Simulation results show that we can achieve within 0.04 dB of the Shannon limit at a bit error rate of 10−6 using a block length of 10 7. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) This paper studies the zero-delay source-channel coding problem, and specifically the problem of obtaining the vector transformations that optimally map between the m-dimensional source space and k-dimensional channel space, under a given transmission power constraint and for the mean square error distortion. The functional properties of the cost are studied and the necessary conditions for the optimality of the encoder and decoder mappings are derived. An optimization algorithm that imposes these conditions iteratively, in conjunction with the noisy channel relaxation method to mitigate poor local minima, is proposed. The numerical results show strict improvement over prior methods. The numerical approach is extended to the scenario of source-channel coding with decoder side information. The resulting encoding mappings are shown to be continuous relatives of, and in fact subsume as special case, the Wyner-Ziv mappings encountered in digital distributed source coding systems. A well-known result in information theory pertains to the linearity of optimal encoding and decoding mappings in the scalar Gaussian source and channel setting, at all channel signal-to-noise ratios (CSNRs). In this paper, the linearity of optimal coding, beyond the Gaussian source and channel, is considered and the necessary and sufficient condition for linearity of optimal mappings, given a noise (or source) distribution, and a specified a total power constraint are derived. It is shown that the Gaussian source-channel pair is unique in the sense that it is the only source-channel pair for which the optimal mappings are linear at more than one CSNR values. Moreover, the asymptotic linearity of optimal mappings is shown for low CSNR if the channel is Gaussian regardless of the source and, at the other extreme, for high CSNR if the source is Gaussian, regardless of the channel. The extension to the vector settings is also considered where besides the conditions inherited from the scalar case, additional constraints must be satisfied to ensure linearity of the optimal mappings. In this paper, we consider the problem of min- imizing the multicast decoding delay of generalized instantly decodable network coding (G-IDNC) over persistent forward and feedback erasure channels with feedback intermittence. In such an environment, the sender does not always receive acknowledgement from the receivers after each transmission. Moreover, both the forward and feedback channels are subject to persistent erasures, which can be modelled by a two state (good and bad states) Markov chain known as Gilbert-Elliott channel (GEC). Due to such feedback imperfections, the sender is unable to determine subsequent instantly decodable packets combination for all receivers. Given this harsh channel and feedback model, we first derive expressions for the probability distributions of decoding delay increments and then employ these expressions in formulating the minimum decoding problem in such environment as a maximum weight clique problem in the G-IDNC graph. We also show that the problem formulations in simpler channel and feedback models are special cases of our generalized formulation. Since this problem is NP-hard, we design a greedy algorithm to solve it and compare it to blind approaches proposed in literature. Through extensive simulations, our adaptive algorithm is shown to outperform the blind approaches in all situations and to achieve significant improvement in the decoding delay, especially when the channel is highly persistent. Index Terms—Multicast Channels, Persistent Erasure Chan- nels, G-IDNC, Decoding Delay, Lossy Intermittent Feedback, Maximum Weight Clique Problem.