In this dissertation we study two problems related to scalar quantization, namely quantization over a noisy channel and bit allocation. Scalar quantizers have been extensively studied for the case of a noiseless channel. However, their structure and performance is not well understood when operating over a noisy channel. The bit allocation problem is how to allocate a limited number of bits to a set of scalar quantizers so as to minimize the sum of their mean squared errors.; We first examine scalar quantizers with uniform encoders and channel optimized decoders for uniform sources and binary symmetric channels. We calculate the point density functions and the mean squared errors for several different index assignments. We also show that the Natural Binary Code is mean squared optimal among all possible index assignments, for all bit error rates, and all quantizer transmission rates. In contrast, we find that almost all index assignments perform poorly and have degenerate codebooks.; Next, we study scalar quantizers with uniform decoders and channel optimized encoders for uniform sources and binary symmetric channels. We compute the number of empty cells in the quantizer encoder, the asymptotic cell distribution, and the effective channel code rates for two families of index assignments. Also, we demonstrate that the Natural Binary Code is sub-optimal for a large range of transmission rates and bit error probabilities. This contrasts with its known optimality when either both the encoder and decoder are not channel optimized, or when only the decoder is channel optimized.; Lastly, we consider bit allocation. The problem of asymptotically optimal bit allocation among a set of quantizers for a finite collection of sources was analytically solved in 1963 by Huang and Schultheiss. Their solution gives a real-valued bit allocation, however in practice, integer-valued bit allocations are needed. In 1966, Fox gave an algorithm for finding optimal nonnegative integer bit allocations. We prove that Fox's solution is equivalent to finding a nonnegative integer-valued vector closest in the Euclidean sense to the Huang-Schultheiss solution. Additionally, we derive upper and lower bounds on the deviation of the mean squared error using integer bit allocation from the mean squared error using optimal real-valued bit allocation.
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