##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Rayan Saab

#### Duke University

## Near-optimal quantization and encoding for oversampled signals

##### Abstract:

Analog-to-digital (A/D) conversion is the process by which signals (e.g., bandlimited functions or finite dimensional vectors) are replaced by bit streams to allow digital storage, transmission, and processing. Typically, A/D conversion is thought of as being composed of sampling and quantization. Sampling consists of collecting inner products of the signal with appropriate (deterministic or random) vectors. Quantization consists of replacing these inner products with elements from a finite set. A good A/D scheme allows for accurate reconstruction of the original object from its quantized samples. In this talk we investigate the reconstruction error as a function of the bit-rate, of Sigma-Delta quantization, a class of quantization algorithms used in the oversampled regime. We propose an encoding of the Sigma-Delta bit-stream and prove that it yields near-optimal error rates when coupled with a suitable reconstruction algorithm. This is true both in the finite dimensional setting and for bandlimited functions. In particular, in the finite dimensional setting the near-optimality of Sigma-Delta encoding applies to measurement vectors from a large class that includes certain deterministic and sub-Gaussian random vectors. Time permitting, we discuss implications for quantization of compressed sensing measurements.

Host: Ery Arias-Castro

### January 10, 2013

### 3:00 PM

### AP&M 6402

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