The problem of scalar and vector quantization in conjunction with a noisy binary symmetric channel is considered. The issue is the assignment of the shortest possible distinct binary sequences to quantization levels o...
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The problem of scalar and vector quantization in conjunction with a noisy binary symmetric channel is considered. The issue is the assignment of the shortest possible distinct binary sequences to quantization levels or vectors so as to minimize the mean-squared error caused by channel errors. By formulating the assignment as a matrix (or vector in the scalar case) and showing that the mean-squared error due to channel errors is determined by the projections of its columns onto the eigenspaces of the multidimensional channel transition matrix, a class of source/quantizer pairs Is identified for which the optimal index assignment has a simple and natural form. Among other things, this provides a simpler and more accessible proof of the result of Crimmins et al., that the natural binary code is an optimal index assignment for the uniform scalar quantizer and uniform source. It also provides a potentially useful approach to further developments in source-channel coding.
Optimal binary labelings, input distributions, and input alphabets are analyzed for the so-called bit-interleaved coded modulation (BICM) capacity, paying special attention to the low signal-to-noise ratio (SNR) regim...
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Optimal binary labelings, input distributions, and input alphabets are analyzed for the so-called bit-interleaved coded modulation (BICM) capacity, paying special attention to the low signal-to-noise ratio (SNR) regime. For 8-ary pulse amplitude modulation (PAM) and for 0.75 bit/symbol, the folded binarycode results in a higher capacity than the binary reflected Gray code (BRGC) and the natural binary code (NBC). The 1 dB gap between the additive white Gaussian noise (AWGN) capacity and the BICM capacity with the BRGC can be almost completely removed if the input symbol distribution is properly selected. First-order asymptotics of the BICM capacity for arbitrary input alphabets and distributions, dimensions, mean, variance, and binary labeling are developed. These asymptotics are used to define first-order optimal (FOO) constellations for BICM, i.e., constellations that make BICM achieve the Shannon limit -1.59 dB. It is shown that the E-b/N-0 required for reliable transmission at asymptotically low rates in BICM can be as high as infinity, that for uniform input distributions and 8-PAM there are only 72 classes of binary labelings with a different first-order asymptotic behavior, and that this number is reduced to only 26 for 8-ary phase shift keying (PSK). A general answer to the question of FOO constellations for BICM is also given: using the Hadamard transform, it is found that for uniform input distributions, a constellation for BICM is FOO if and only if it is a linear projection of a hypercube. A constellation based on PAM or quadrature amplitude modulation input alphabets is FOO if and only if they are labeled by the NBC;if the constellation is based on PSK input alphabets instead, it can never be FOO if the input alphabet has more than four points, regardless of the labeling.
It is shown how to use a 2n−1 length pseudorandom binary sequence to develop an absolute-type shaft encoder having any t resolution, where 2n−1−1<t<2n−1. The shaft encoder rotating device has only two concentric...
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It is shown how to use a 2n−1 length pseudorandom binary sequence to develop an absolute-type shaft encoder having any t resolution, where 2n−1−1coder rotating device has only two concentric code-track and 2n−1 reading heads. A sequential code-conversion method is used to translate the pseudorandom patterns into natural binary code.
Reflected binary (RB) code words plus an even parity check can be considered as signed digit (SD) representations of the corresponding naturalbinary (NB) code words. The SD code words allow a simple serial conversion...
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Reflected binary (RB) code words plus an even parity check can be considered as signed digit (SD) representations of the corresponding naturalbinary (NB) code words. The SD code words allow a simple serial conversion with the least significant bit leading of their digits forming the RB code words into the NB code words.
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