This paper discusses random filtering, a recently proposed method for directly acquiring a compressed version of a digital signal. The technique is based on convolution of the signal with a fixed FIR filter having random taps, followed by downsampling. Experiments show that random filtering is effective at acquiring sparse and compressible signals. This process has the potential for implementation in analog hardware, and so it may have a role to play in new types of analog/digital converters

Tropp, Joel A.

English

Caltech Authors - Main

Random Filters for Compressive Sampling
Joel A. Tropp
Department of Mathematics
The University of Michigan at Ann Arbor
530 Church St., Ann Arbor, MI 48109-1043
E-mail: jtropp@umich.edu
Abstract— This paper discusses random filtering, a recently
proposed method for directly acquiring a compressed version
of a digital signal. The technique is based on convolution of
the signal with a fixed FIR filter having random taps, followed
by downsampling. Experiments show that random filtering is
effective at acquiring sparse and compressible signals. This
process has the potential for implementation in analog hardware,
and so it may have a role to play in new types of analog/digital
converters.
I. INTRODUCTION
Many types of signals have additional structure that makes
them compressible. In other words, the amount of information
necessary to represent or approximate the signal is substan-
tially smaller than the length of the signal. Nevertheless,
many techniques for signal acquisition collect a complete
description of the signal, just to throw away the redundant
data. For example, audio signals are typically sampled at
the Nyquist rate before lossy compression algorithms are
applied to reduce their size. While the sample-and-compress
approach is attractive when the Nyquist rate is 44 KHz (as
with CD-quality audio), it becomes increasingly distasteful as
the sampling rate increases. In modern applications, it may be
necessary to acquire precise information about signals with
Nyquist rates into and above the gigahertz range. This task is
hopeless for current analog/digital converters.
The pressure to develop novel technologies for signal ac-
quisition has led researchers to look beyond Shannon–Nyquist
sampling. In this paper, we discuss random filtering, a recently
proposed technique for directly acquiring a compressed ver-
sion of a digital signal [1]. We believe that this method may
also be applicable to analog signals, and so it may have a role
to play in next-generation A/D converters.
II. COMPRESSIVE SIGNAL ACQUISITION
Let us consider a simple (but representative) situation. We
will work in the vector space Rd, equipped with the usual
Euclidean norm ‖·‖2. Define the class of m-sparse signals
B0(m) = {s ∈ Rd : # supp(s) ≤ m}.
It is clear that each of these signals can be represented
completely using 2m real numbers. When m is substantially
smaller than d, therefore, it is wasteful to write an m-sparse
signal in the standard basis. In particular, it seems lavish to
take d samples of a sparse signal to identify the m nonzero
components.
The theoretical goal of compressive signal acquisition
(CSA), aka compressed sensing, is to develop a linear mea-
surement operator Φ : Rd → Rn and a (nonlinear) reconstruc-
tion algorithm A : Rn → Rd for which
1) the number n of measurements is comparable with the
sparsity level m,
2) the measurement process does not discard information,
i.e., B0(m) ∩ kerΦ = {0}, and
3) the reconstruction algorithm is stable, i.e.,
‖A (Φs + ν)− s‖2 ≤ C ‖ν‖2 .
In the sequel, we will discuss more practical aspects of
measurement and reconstruction.
Indeed, it is possible to design measurement processes and
reconstruction algorithms that satisfy all these requirements.
For example, we can choose the measurement map Φ to be a
d × n Gaussian matrix with n = O(m log d). In this case,
several different algorithms can be used to recover sparse
signals. A partial list of references for these results includes
[2], [3], [4], [5].
III. RANDOM FILTERS
When viewed as candidates for next-generation A/D con-
verters, the current collection of techniques for CSA do
not look promising. The major shortcoming is that these
approaches use measurement processes that seem incompatible
with analog hardware. Indeed, analog hardware can reliably
perform only a limited repertoire of operations: (i) modulation,
(ii) filtering, and (iii) sampling. It is natural, therefore, to
search for a new type of measurement process that can be
built using these simple blocks.
There are several other problems with current approaches
to CSA that we also wish to circumvent: (i) the methods are
only designed for finite-length signals, (ii) the measurement
process is not causal, (iii) the reconstruction algorithms require
too much time and space.
To address some of these difficulties, the paper [1] proposes
a new method for compressive acquisition of digital signals.
The idea is that we can measure a signal s by convolving it
with an FIR filter h with random taps and then downsampling
the result to obtain a compressed representation y. Figure 1
displays a block diagram of this filtering process. Various non-
linear reconstruction algorithms are possible. For a summary
of the potential advantages of this approach, we quote [1]:
2161-4244-0350-2/06/$20.00 ©2006 IEEE
sh
yConvolution Downsampling
bd/Nc
s
h
FFT
FFT
IFFT y
Downsampling
bd/Nc
(a)
(b)
Fig. 1. Block diagrams for signal acquisition through random ﬁltering: (a)
using convolution; (b) using FFT/IFFT. The FIR ﬁlter hhas random taps, which
must be known in order to recover the signal s from the compressed data y.
At first glance, one might think this method
would convert a signal into garbage. In fact, the
random filter is generic enough to summarize many
types of compressible signals. At the same time, the
random filter has enough structure to accelerate mea-
surement and reconstruction algorithms. Our method
has several benefits:
• measurements are time-invariant and nonadap-
tive;
• measurement operator is stored and applied ef-
ficiently;
• we can trade longer filters for fewer measure-
ments;
• it is easily implementable in software or hard-
ware; and
• it generalizes to streaming or continuous-time
signals.
The original paper [1] discusses implementation issues
connected with random filters, and it describes a reconstruction
algorithm based on Orthogonal Matching Pursuit (OMP). We
will not repeat this material here. The remainder of the paper
describes the numerical evidence about the performance of
random filters for CSA. We conclude with a discussion of the
challenges that arise in the mathematical analysis.
IV. NUMERICAL EXPERIMENTS
Extensive numerical experiments indicate that random filters
do provide sufficient information to reconstruct sparse signals.
In this section, we describe an experiment performed in [1].
We fix the signal length d = 128, and the sparsity m = 10. For
each number n of measurements and filter length B, we do the
following. First, draw a random filter of length B with N (0, 1)
taps. For each of 1000 trials, we generate a signal s whose
m nonzero entries are N (0, 1). We take n measurements and
use an OMP-based algorithm to reconstruct the signal. If the
results match to machine precision, then we record a success.
The reconstruction probability is the fraction of the 1000 trials
that result in success. As a control, we perform the same
experiment using a fully random matrix Φ with i.i.d. N (0, 1)
entries. Figure 2 displays the results. Note in particular that
the two longest filters (B = 64, 128) succeed almost as well
10 20 30 40 50 60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Number of measurements (N)
P
ro
ba
bi
lit
y 
of
 e
xa
ct
 r
ec
on
st
ru
ct
io
n
B=4
B=16
B=64
B=128
Full
Fig. 2. Probability of exact reconstruction versus the number of measurements
n for four ﬁlter lengths B. Signal length d = 128. A typical signal appears at
top.
as the Gaussian matrix, even though they have fewer degrees
of freedom.
V. MATHEMATICAL ANALYSIS OF RANDOM FILTERS
At present, there are no rigorous mathematical results on
the performance of random filters. We are attempting to
calculate the restricted isometry constants [6] of a random
filter using methods from the study of empirical processes.
This calculation would demonstrate that the random filtering
process captures all sparse and compressible signals from
certain classes. Although it does not seem very hard to develop
suppression results (the upper singular value estimates), the
corresponding lower estimates remain out of our reach.
ACKNOWLEDGMENTS
I wish to thank Mike Wakin, Marco Duarte, Dror Baron,
and Rich Baraniuk for permission to discuss our joint work.
REFERENCES
[1] J. A. Tropp, M. B. Wakin, M. F. Duarte, D. Baron, and R. G. Baraniuk,
“Random filters for compressive sampling and reconstruction,” in Proc.
ICASSP 2006, Toulouse, May 2006, to appear.
[2] D. L. Donoho, “Compressed sensing,” Oct. 2004, unpublished manuscript.
[3] E. J. Cande`s and T. Tao, “Near optimal signal recovery from random
projections: Universal encoding strategies?” Nov. 2004, submitted for
publication.
[4] M. Rudelson and R. Veshynin, “Geometric approach to error correcting
codes and reconstruction of signals,” Feb. 2005, available from
arXiv:math.MG/0502299. [Online]. Available: arXiv:math.MG/0502299
[5] J. A. Tropp and A. C. Gilbert, “Signal recovery from partial information
via Orthogonal Matching Pursuit,” April 2005, submitted to IEEE Trans.
Inform. Theory.
[6] E. J. Cande`s and T. Tao, “Decoding by linear programming,” IEEE Trans.
Inform. Theory, vol. 51, no. 12, pp. 4203–4215, Dec. 2005.
217


Random Filters for Compressive Sampling

Compressed sensing,”

Decoding by linear programming,”

Geometric approach to error correcting codes and reconstruction of signals,”

Near optimal signal recovery from random projections: Universal encoding strategies?”

Random filters for compressive sampling and reconstruction,” in

Signal recovery from partial information via Orthogonal Matching Pursuit,”

https://authors.library.caltech.edu/9064/1/TROciss06.pdf

Random Filters for Compressive Sampling

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main