Freedom through Imperfection: exploiting the flexibility offered by redundancy in signal processing

This thesis consists of four chapters. The first two chapters pertain to the design of stable quantization methods for analog to digital conversion, while the third and fourth chapters concern problems related to compressive sensing. In the first chapter, we study the β-encoder and golden ratio encoder, and show that these quantization schemes are robust with respect to uncertainty in the multiplication element of their implementation, whereby x → βx. In particular, we use a result from analytic number theory to show that the possibly unknown value of β can be reconstructed as the unique root of a power series with coefficients in {−1, 0, 1} obtained from the quantization output of test input x and its negative, −x. The focus of our attention in Chapter 2 is Sigma Delta (Σ∆) modulation, a course quantization method in analog to digital conversion that is widely used for its simplicity of implementation and robustness to component imperfections. A persistent problem in Σ∆ quantization of audio signals is the occurrence of undesirable tones arising from periodicities in the bit output; these tones are particularly intolerable in the space between successive audio tracks. As one of the contributions of this thesis, we show that the standard second order 1-bit Σ∆ scheme can be modified so as to eliminate such periodicities, and this modification can be achieved without sacrificing accuracy or introducing significant complexity. The emerging area of compressed sensing guarantees that sufficiently sparse signals