  Compressed sensing is now established as an effective method for dimension
reduction when the underlying signals are sparse or compressible with respect
to some suitable basis or frame. One important, yet under-addressed problem
regarding the compressive acquisition of analog signals is how to perform
quantization. This is directly related to the important issues of how
"compressed" compressed sensing is (in terms of the total number of bits one
ends up using after acquiring the signal) and ultimately whether compressed
sensing can be used to obtain compressed representations of suitable signals.
Building on our recent work, we propose a concrete and practicable method for
performing "analog-to-information conversion". Following a compressive signal
acquisition stage, the proposed method consists of a quantization stage, based
on $\Sigma\Delta$ (sigma-delta) quantization, and a subsequent encoding
(compression) stage that fits within the framework of compressed sensing
seamlessly. We prove that, using this method, we can convert analog compressive
samples to compressed digital bitstreams and decode using tractable algorithms
based on convex optimization. We prove that the proposed AIC provides a nearly
optimal encoding of sparse and compressible signals. Finally, we present
numerical experiments illustrating the effectiveness of the proposed
analog-to-information converter.
