IEEE Signal Processing - May 2018 - 34
σ2
(a)
(b)
(c)
(d)
(e)
fNyq
0
Support of
Optimal Filters
[Cases (b)-(d)]
fs
Water-Filling
Representation
[Cases (b)-(e)]
at most fs /L , the overall part of the spectrum passed by the L
filters is at most of size fs . This property implies that the lower
bound D (fs, R) of (13) is kept under this form of sampling.
The next question is whether this lower bound is attainable,
provided that we are allowed to increase the number of sampling branches L and the presampling filters H 1 (f), f, H L (f).
A positive answer to this question was given in [51], where it
was shown that for any PSD, the distortion level D (fs, R) can
be attained using some finite number L* of sampling branches
and a particular set of filters, each of which is antialiasing for
sampling at a rate of fs /L*. The reduction of the distortion in
ADX using the optimal filter-bank sampler as the number of
branches increases is shown in Figure 19. Also shown are the
supports of the optimal presampling filters at a specific sampling rate fs .
We conclude that the function D (fs, R) describes an
achievable lower bound for the distortion in the ADX setting
with a multibranch uniform sampler. In the next section, we
extend this result to nonuniform and generalized linear sampling procedures.
Nonuniform and generalized sampling
(b)
(c)
(d)
(e)
FIGURE 19. The minimal distortion versus the sampling rate fs for a fixed
value of R . The case of no sampling prefilter is given in case (a), and
the cases of one, two, and five sampling branches with optimal branch
prefiltering are considered in cases (b), (c), and (d), respectively. For each
of these cases and a fixed fs, the union of support for the optimal filters,
which equals fs, is shown in the gray-scale images, and how these
bands are identified through water-filling and the sampling distortion
that results is shown in the color images. Case (d), of five SI sampling
branches, preserves the part of the spectrum of measure fs with the
highest energy, and therefore achieves D (fs, R). (e) Shannon's DRF with
its water-filling representation.
34
We now extend the ADX setting to include a nonuniform sampling system with time-varying preprocessing. We show that
under some mild assumptions on the sampling set, it is impossible to achieve a distortion lower than D (fs, R) , where, here, fs
equals the density of the sampling set. The definition of the
density of a sampling set and more detailed background on
nonuniform sampling can be found in "Nonuniform Sampling."
This extension includes all cases of linear continuous sampling,
as given in "Generalized Sampling of Random Signals."
A nonuniform time-varying sampler is shown in Figure 20.
It is characterized by a discrete and ordered sampling set of
sampling times K = " f, t -1, t 0, f, t n, f , 1 R and a timevarying impulse response g (t, x). The sampling set is assumed
to be uniformly discrete, in the sense that there exists a universal constant f 2 0 such that each two elements of K are
at least f apart. The nth output of the sampler is the convolution of g (t n, t) with X (t) , where t n ! K. For every finite time
lag [-T/2, T/2], the vector Y is the sampler output at times
[-T/2, T/2] + K . Our goal is to map this vector to one of 26TR@
elements and, by observing this element, recover X (t) over this
time interval under MSE distortion. We note that although the
sampler in Figure 20 has only a single sampling branch, the
multibranch sampling system of Figure 18 may be realized by
this filter using a particular choice of the time-varying operation [54].
As in the case of uniform sampling, it is instructive to begin
our discussion with the lower bound on the minimal distortion
obtained by the MMSE in estimating X (t) from its nonuniform sampled version Yn. A classical result in functional analysis and signal processing due to Landau asserts that a signal
can be perfectly recovered from its nonuniform samples if, and
only if, the density of K exceeds its spectral occupancy [55].
See "Nonuniform Sampling" for an overview of this result.
In our setting, the spectral occupancy takes the form of the
IEEE Signal Processing Magazine
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May 2018
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