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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A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two "function spaces," and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of "ideal" systems which transmit at this maximum rate are discussed The equivalent number of binary digits per second for certain information sources is calculated. Fundamental limitations on the performance of any type of communication system transmitting data from an analog source may be obtained by application of a theorem in Shannon's original paper on information theory. In order to transmit the output of an analog source to a sink with mean-squared error e or less, it is necessary to transmit information from source to sink at rates greater than some minimal rate R( ) bits per second. If a channel of capacity C bits per second is used, then in order for any communication system to be able to achieve mean-squared error or less with this channel it is necessary that R( ) C . This inequality can be used to determine the minimum mean-squared error obtainable for a given source-channel pair without discussing any particular communication system. As an example, the minimum mean-squared error is calculated for cases in which the analog source and additive channel noise are stationary, Gaussian processes. The performance of amplitude and angle modulation systems is compared to the theoretically ideal performance obtainable in some of these cases. What makes a source-channel communication system optimal? It is shown that in order to achieve an optimal cost-distortion tradeoff, the source and the channel have to be matched in a probabilistic sense. The match (or lack of it) involves the source distribution, the distortion measure, the channel conditional distribution, and the channel input cost function. Closed-form necessary and sufficient expressions relating the above entities are given. This generalizes both the separation-based approach as well as the two well-known examples of optimal uncoded communication. The condition of probabilistic matching is extended to certain nonergodic and multiuser scenarios. This leads to a result on optimal single-source broadcast communication. This paper investigates the maximal channel coding rate achievable at a given blocklength and error probability. For general classes of channels new achievability and converse bounds are given, which are tighter than existing bounds for wide ranges of parameters of interest, and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100. It is also shown analytically that the maximal rate achievable with error probability ? isclosely approximated by C - ?(V n) Q-1(?) where C is the capacity, V is a characteristic of the channel referred to as channel dispersion , and Q is the complementary Gaussian cumulative distribution function. A frame for the body of a vehicle, specifically a passenger automobile, has a box-like framework which includes a roof frame, a base frame, corner columns connecting the roof frame and base frame, and at least one supporting structure. The supporting structure is essentially aligned in the longitudinal direction of the vehicle outside the box-like framework and comprises two symmetric fork-shaped girders with their outer prongs connected to corner columns, their inner prongs connected to the base frame and their struts connected to a bumper. 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. 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.) Two methods for transmission of a continuous amplitude source signal over a continuous amplitude channel with a power constraint are proposed. For both methods, bandwidth reduction is achieved by mapping from a higher dimensional source space to a lower dimensional channel space. In the first system, a source vector is quantized and mapped to a discrete set of points in a multidimensional PAM signal constellation. In the second system the source vector is approximated with a point in a continuous subset of the source space. This operation is followed by mapping the resulting vector to the channel space by a one-to-one continuous mapping resulting in continuous amplitude channel symbols. The proposed methods are evaluated for a memoryless Gaussian source with an additive white Gaussian noise channel, and offer significant gains over previously reported methods. Specifically, in the case of two-dimensional source vectors, and one-dimensional channel vectors, the gap to the optimum performance theoretically attainable is less than 1.0 dB for a wide range of channel signal-to-noise ratios. We consider two codes based on dynamical systems, for transmitting information from a continuous alphabet, discrete-time source over a Gaussian channel. The first code, a homogeneous spherical code, is generated by the linear dynamical system s spl dot =As, with A a square skew-symmetric matrix. The second code is generated by the shift map s sub n =b sub n s sub n-1 (mod 1). The performance of each of these codes is determined by the geometry of its locus or signal set, specifically, its arc length and minimum distance, suitably defined. We show that the performance analyses for these systems are closely related, and derive exact expressions and bounds for relevant geometric parameters. We also observe that the lattice spl Zopf sup N underlies both modulation systems and we develop a fast decoding algorithm that relies on this observation. Analytic results show that for fixed bandwidth expansion, good scaling behavior of the mean squared error is obtained relative to the channel signal-to-noise ratio (SNR). Particularly interesting is the resulting observation that sampled, exponentially chirped modulation codes are good bandwidth expansion codes.