A NEW METHOD TO COMPUTE OPTIMAL PERIODIC SAMPLING PATTERNS
Arash Owrang, Mats Viberg, Mohsen Nosratinia and Moslem Rashidi
Department of Signals & Systems, Chalmers University of Technology
ABSTRACT
It is possible to reconstruct a signal from cyclic nonuniform
samples and thus take advantage of a lower sampling rate than
the Nyquist rate. However, this has the potential drawback of
amplifying signal perturbations, e.g. due to noise and quanti-
zation. We propose an algorithm based on sparse reconstruc-
tion techniques, which is able to find the sparsest sampling
pattern that permits perfect reconstruction of the sampled sig-
nal. The result of our algorithm with a proper constraint val-
ues is a sparse subset of samples that results in an ideal condi-
tion number for its equivalent sub-DFT matrix. Besides, our
algorithm has low complexity in terms of computation. The
method is illustrated by simulations for a sparse multi band
signal.
Index Terms— Sparse approximation, Basis Pursuit,
condition number, nonuniform sampling, greedy search.
1. INTRODUCTION
Sampling is mostly implemented as uniform sampling, fol-
lowing Shannon’s sampling theorem. The timing interval be-
tween samples is then equal, and the sampling rate is at least
equal to the Nyquist rate. Nevertheless, for sparse multi-band
signals, the uniform sampling does not have an acceptable
performance [1]. In the real world, this affects the amount of
computation, the power consumption, etc.
A better performance can be achieved by cyclic nonuni-
form sampling (CNUS), also known as multi-coset sampling
[2]. The idea is to start from a uniform sampling obeying the
Nyquist rate, then select K out of L sampling instances with
K < L (usuallyK  L) that are periodically repeated. Thus,
L is the periodicity and K/L represents the sampling den-
sity. The research on CNUS can be divided into the cases of
known and unknown spectrum respectively. In contrast with
the known spectrum case, the exact frequency specification of
the signal is missing in the unknown spectrum case. A major
concern in the known spectrum case is error bounds on the
reconstructed signal due to additive noise [1, 3]. To minimize
this effect, the sampling pattern should be chosen to yield a
perfect condition number of the corresponding sub-DFT ma-
trix. However, in most previous work the problem of perfect
or optimal sampling pattern selection for a known spectrum
is left as an open problem. In [4], an exhaustive search is
Table 1. Notations
x(t), x(n) continuous, discrete-time signal
F
s
sampling frequency
δ(n) Kronecker delta function
N discrete Fourier transform points
v,A vector, matrix
diag(A) matrix diagonal
vec(A) stacks matrix columns to a vector
A
H matrix transpose conjugate
‖A‖
l
, ‖A‖
F
matrix l-norm, Frobenius norm
used for small size problems, whereas a suboptimal metric is
applied for larger scale problem. In a somewhat different set-
ting1 , [5] gives conditions for when a perfect conditioning is
achievable, and a method to compute the optimal pattern is
proposed. In the spectrum blind problem, researchers mostly
tried to find a stable universal pattern for CNUS. In this con-
tribution, we restrict to the case where exact knowledge of the
signal frequency support is given.
This work proposes a novel method of pattern selection
for CNUS based on the concept of sparse approximation [6,
7]. Basis Pursuit (BP) [8] is a major approach in sparse ap-
proximation. However, the original BP does not guarantee
that an optimal sampling pattern is found. Thus, our method
combines BP with a greedy search, and the resulting algo-
rithm is found to achieve a perfect conditioning whenever this
is possible.
2. BACKGROUND THEORY AND PROBLEM
DESCRIPTION
2.1. Signal Model and Nonuniform Sampling
A band limited multi-band signal x(t) ∈ C is defined as
X(f) = 0 if f /∈
⋃
N
B
i=1
B
i
, where X(f) is the Fourier trans-
form of x(t) and
B
i
=
[
a
i
F
s
L
,
a
i+1
F
s
L
)
, 0 ≤ a
1
< · · · < a
2N
B
< L.
1Also the underlying sampling rate is allowed to vary in [5]
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