6.4 Analog-to-information conversion  (Page 2/2)

We will consider a rather different approach, which we call the random demodulator   [link] , [link] , [link] . A correlator is also known as a “demodulator” due to its most common application: demodulating radio signals. The architecture of the random demodulator is depicted in [link] . The analog input $x\left(t\right)$ is correlated with a pseudorandom square pulse of $\pm 1$ 's, called the chipping sequence $p_c\left(t\right)$ , which alternates between values at a rate of $N_a$ Hz, where $N_a$ Hz is at least as fast as the Nyquist rate of $x\left(t\right)$ . The mixed signal is integrated over a time period $1/M_a$ and sampled by a traditional integrate-and-dump back-end ADC at $M_a$ Hz $\ll N_a$ Hz. In this case our measurements are given by

In practice, data is processed in time blocks of period $T$ , and we define $N=N_{a}T$ as the number of elements in the chipping sequence, and $M=M_{a}T$ as the number of measurements. We will discuss the discretization of this model below, but the key observation is that the correlator and chipping sequence operate at a fast rate, while the back-end ADC operates at a low rate. In hardware it is easier to build a high-rate modulator/chipping sequence combination than a high-rate ADC  [link] . In fact, many systems already use components of this front end for binary phase shift keying demodulation, as well as for other conventional communication schemes such as CDMA.

Discrete formulation

Although the random demodulator directly acquires compressive measurements without first sampling $x\left(t\right)$ , it is equivalent to a system which first samples $x\left(t\right)$ at its Nyquist-rate to yield a discrete-time vector $x$ , and then applies a matrix $\Phi$ to obtain the measurements $y=\Phi x$ . To see this we let $p_c\left[n\right]$ denote the sequence of $\pm 1$ used to generate the signal $p_c\left(t\right)$ , i.e., $p_c\left(t\right)=p_c\left[n\right]$ for $t\in [(n-1)/N_a,n/N_a]$ . As an example, consider the first measurement, or the case of $j=1$ . In this case, $t\in [0,1/M_a]$ , so that $p_c\left(t\right)$ is determined by $p_c\left[n\right]$ for $n=1,2,...,N_a/M_a$ . Thus, from [link] we obtain

But since $N_a$ is the Nyquist-rate of $x\left(t\right)$ , ${\int }_{(n-1)/N_a}^{n/N_a}x\left(t\right)dt$ simply calculates the average value of $x\left(t\right)$ on the $n^{\mathrm{th}}$ interval, yielding a sample denoted $x\left[n\right]$ . Thus, we obtain

In general, our measurement process is equivalent to multiplying the signal $x$ with the random sequence of $\pm 1$ 's in $p_c\left[n\right]$ and then summing every sequential block of $N_a/M_a$ coefficients. We can represent this as a banded matrix $\Phi$ containing $N_a/M_a$ pseudorandom $\pm 1$ s per row. For example, with $N=12$ , $M=4$ , and $T=1$ , such a $\Phi$ is expressed as

In general, $\Phi$ will have $M$ rows and each row will contain $N/M$ nonzeros. Note that matrices satisfying this structure are extremely efficient to apply, requiring only $O\left(N\right)$ computations compared to $O\left(MN\right)$ in the general case. This is extremely useful during recovery.

A detailed analysis of the random demodulator in  [link] studied the properties of these matrices applied to a particular signal model. Specifically, it is shown that if $\Psi$ represents the $N\times N$ normalized discrete Fourier transform (DFT) matrix, then the matrix $\Phi \Psi$ will satisfy the restricted isometry property (RIP) with high probability, provided that

where the probability is taken with respect to the random choice of $p_c\left[n\right]$ . This means that if $x\left(t\right)$ is a periodic (or finite-length) signal such that once it is sampled it is sparse or compressible in the basis $\Psi$ , then it should be possible to recover $x\left(t\right)$ from the measurements provided by the random demodulator. Moreover, it is empirically demonstrated that combining ${\ell }_1$ minimization with the random demodulator can recover $K$ -sparse (in $\Psi$ ) signals with

measurements where $C\approx 1.7$   [link] .

Note that the signal model considered in  [link] is somewhat restrictive, since even a pure tone will not yield a sparse DFT unless the frequency happens to be equal to $k/N_a$ for some integer $k$ . Perhaps a more realistic signal model is the multi-band signal model of  [link] , [link] , [link] , [link] , [link] , [link] , where the signal is assumed to be bandlimited outside of $K$ bands each of bandwidth $B$ , where $KB$ is much less than the total possible bandwidth. It remains unknown whether the random demodulator can be exploited to recover such signals. Moreover, there also exist other CS-inspired architectures that we have not explored in this [link] , [link] , [link] , and this remains an active area of research. We have simply provided an overview of one of the more promising approaches in order to illustrate the potential applicability of the ideas of this course to the problem of analog-to-digital conversion.