5.4 Analog-to-information conversion

We now consider the application of compressive sensing (CS) to the problem of designing a system that can acquire a continuous-time signal $x\left(t\right)$ . Specifically, we would like to build an analog-to-digital converter (ADC) that avoids having to sample $x\left(t\right)$ at its Nyquist rate when $x\left(t\right)$ is sparse. In this context, we will assume that $x\left(t\right)$ has some kind of sparse structure in the Fourier domain, meaning that it is still bandlimited but that much of the spectrum is empty. We will discuss the different possible signal models for mathematically capturing this structure in greater detail below. For now, the challenge is that our measurement system must be built using analog hardware. This imposes severe restrictions on the kinds of operations we can perform.

Analog measurement model

To be more concrete, since we are dealing with a continuous-time signal $x\left(t\right)$ , we must also consider continuous-time test functions ${\left\{{\phi }_j\left(t\right)\right\}}_{j=1}^M$ . We then consider a finite window of time, say $t\in [0,T]$ , and would like to collect $M$ measurements of the form

Building an analog system to collect such measurements will require three main components:

1. hardware for generating the test signals ${\phi }_j\left(t\right)$ ;
2. $M$ correlators that multiply the signal $x\left(t\right)$ with each respective ${\phi }_j\left(t\right)$ ;
3. $M$ integrators with a zero-valued initial state.

We could then sample and quantize the output of each of the integrators to collect the measurements $y\left[j\right]$ . Of course, even in this somewhat idealized setting, it should be clear that what we can build in hardware will constrain our choice of ${\phi }_j\left(t\right)$ since we cannot reliably and accurately produce (and reproduce) arbitrarily complex ${\phi }_j\left(t\right)$ in analog hardware. Moreover, the architecture described above requires $M$ correlator/integrator pairs operating in parallel, which will be potentially prohibitively expensive both in dollar cost as well as costs such as size, weight, and power (SWAP).

As a result, there have been a number of efforts to design simpler architectures, chiefly by carefully designing structured ${\phi }_j\left(t\right)$ . The simplest to describe and historically earliest idea is to choose ${\phi }_j\left(t\right)=\delta (t-t_j)$ , where ${\left\{t_j\right\}}_{j=1}^M$ denotes a sequence of $M$ locations in time at which we would like to sample the signal $x\left(t\right)$ . Typically, if the number of measurements we are acquiring is lower than the Nyquist-rate, then these locations cannot simply be uniformly spaced in the interval $[0,T]$ , but must be carefully chosen. Note that this approach simply requires a single traditional ADC with the ability to sample on a non-uniform grid, avoiding the requirement for $M$ parallel correlator/integrator pairs. Such non-uniform sampling systems have been studied in other contexts outside of the CS framework. For example, there exist specialized fast algorithms for the recovery of extremely large Fourier-sparse signals. The algorithm uses samples at a non-uniform sequence of locations that are highly structured, but where the initial location is chosen using a (pseudo)random seed. This literature provides guarantees similar to those available from standard CS  [link] , [link] . Additionally, there exist frameworks for the sampling and recovery of multi-band signals, whose Fourier transforms are mostly zero except for a few frequency bands. These schemes again use non-uniform sampling patterns based on coset sampling  [link] , [link] , [link] , [link] , [link] , [link] . Unfortunately, these approaches are often highly sensitive to jitter , or error in the timing of when the samples are taken.