Mathematical optimization is applicable to nearly every scientific discipline. This thesis specifically focuses on optimization applications to digital communications and signal processing. Within the digital communications framework, the channel encoder attempts to encode a message from a source (the sender) in such a way that the channel decoder can utilize the encoding to correct errors in the message caused by the transmission over the channel. Low-density parity-check (LDPC) codes are an especially popular code for this purpose. Following the channel encoder in the digital communications framework, the modulator converts the encoded message bits to a physical waveform, which is sent over the channel and converted back to bits at the demodulator. The modulator and demodulator present special challenges for what is known as the two-antenna problem. The main results of this work are two algorithms related to the development of optimization methods for LDPC codes and the two-antenna problem. Current methods for optimization of LDPC codes analyze the degree distribution pair asymptotically as block length approaches infinity. This effectively ignores the discrete nature of the space of valid degree distribution pairs for LDPC codes of finite block length. While large codes are likely to conform reasonably well to the infinite block length analysis, shorter codes have no such guarantee. Chapter 2 more thoroughly introduces LDPC codes, and Chapter 3 presents and analyzes an algorithm for completely enumerating the space of all valid degree distribution pairs for a given block length, code rate, maximum variable node degree, and maximum check node degree. This algorithm is then demonstrated on an example LDPC code of finite block length. Finally, we discuss how the result of this algorithm can be utilized by discrete optimization routines to form novel methods for the optimization of small block length LDPC codes. In order to solve the two-antenna problem, which is introduced in greater detail in Chapter 2, it is necessary to obtain reliable estimates of the timing offset and channel gains caused by the transmission of the signal through the channel. The timing offset estimator can be formulated as an optimization problem, and an optimization method used to solve it was previously developed. However, this optimization method does not utilize gradient information, and as a result is inefficient. Chapter 4 presents and analyzes an improved gradient-based optimization method that solves the two-antenna problem much more efficiently.



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Physical and Mathematical Sciences



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optimization, signal processing, digital communications, low-density parity-check (LDPC) codes, two-antenna problem, aeronautical telemetry, parameter estimation