IEEE Signal Processing - May 2018 - 35

Nonuniform Sampling
Consider a sampling set K for which there exists an f 2 0
such that | t k - t n | 2 f for every t n ! t k ! K. The density of
K is defined as the number of elements of K contained in
a single interval of length r divided by r, in the limit as r
extends to infinity and provided this limit exists. For example, the density of a uniform sampling set K = fs Z is fs .
The isomorphism described by (15) establishes an equivalence between the problem of estimating a Gaussian stationary process from its samples at times K under the
mean squared error (MSE) criterion, and the problem
of orthogonal projection onto the space spanned by
E (K) _ " e 2rift n, t n ! K ,. The conditions for this MSE to van-

support of the PSD. Therefore, the function D (fs, R) of (13)
agrees with Landau's characterization, since it implies that as
R extends to infinity, zero MSE is attained if, and only if, the
sampling rate exceeds the spectral occupancy.
The ADX with the nonuniform sampler extends the prior
result, since it considers the case of a limited finite bit rate and
linear preprocessing of the samples. For this setting, it is shown
in [18] that the lower bound on the distortion D (fs, R) still
holds, provided fs is replaced by the density of K. That is, for
any time-varying system g (t, x) and any sampling set K for
which a density exists, the minimal distortion in the ADX setting with a time-varying nonuniform sampler is lower-bounded
by D (fs, R) , where fs equals the density of K.
It follows that minimal distortion in the ADX setting under
the class of linear pointwise samplers at rate fs is fully characterized by the function D (fs, R) . In general and according to Landau's condition for stable sampling, an equality
between D (fs, R) and Shannon's DRF of the analog source
is expected for sampling rates higher than the spectral occupancy of X (t) . We have seen, however, that this equality
usually already occurs as the sampling rate fs exceeds the
support of the preserved part of the spectrum in the Pinsker-
Kolmogorov water-filling expression (6). In other words, the
sampling structure that attains D (fs, R) utilizes the special
structure associated with the optimal lossy compression of
analog signals given by the Pinsker-Kolmogorov result. It,
in effect, aligns the degrees of freedom of the presampled
signal with those of the postsampled lossy compressed signal so that the part of the signal removed prior to the sampling stage matches the part of the signal removed under
the optimal lossy compression of the signal subject to the
bit-rate constraint.
As a final remark, we note that any linear continuous
sampler as defined in "Generalized Sampling of Random
Signals" can be expressed as the time-varying nonuniform
sampler of Figure 20. Indeed, the kernel of the time-varying
operation g (t, x) defines a set of linear continuous functionals
g n (t) = g (t n, t), t n ! K .

ish are related to the fact that every element of L 2 (S X ) can
be approximated by a linear combination of exponentials
in E (K), [56], [57]. This property, however, turns out to be
too weak for practical sampling systems, since it does not
guarantee stability: the approximation may not be robust
to small perturbations in the time instances that inevitably
are present in practice [3], [58], [59]. As a result, only stable sampling schemes [60] should be considered in applications. A necessary and sufficient condition for stable
sampling was given by Landau [55], who showed that it
can be obtained if, and only if, the density of K exceeds
the spectral occupancy of X(t).

X (t )

g (t, τ )

tn ∈ Λ

Yn

FIGURE 20. A nonuniform sampler with time-varying preprocessing.

Summary of ADX
We have shown that the optimal tradeoff among distortion, bit
rate, and sampling rate under the class of linear samplers with
pointwise operations is fully described by the function
D (fs, R) of (13). Moreover, the procedure for attaining an optimal point in this tradeoff is summarized in the following steps.
1) Given the bit-rate constraint R, use the Pinsker-
Kolmogorov water-filling (6) over the PSD S X (f) . The critical sampling rate f R is the support of the frequency
components associated with the preserved part of the spectrum in this expression.
2) Use a multibranch uniform sampler with a sufficient number of sampling branches optimized such that the combined passband of all of the samplers is the support of the
preserved part of the spectrum [52, Sec. IV].
3) Recover the part of the signal associated with the preserved
part of the spectrum from all branches, as in standard MSE
interpolation [61].
4) Fix a large time lag T and use a vector quantizer with 6TR@
bits to encode the estimate in step 3 over this lag.
The previously described procedure calls for a few comments and extensions. First, we note that, although our
description determines the minimal distortion and sampling
rate as a function of the bit rate, this dependency can be
inverted. That is, given a target distortion D, the Pinsker-
Kolmogorov expression (6) leads to a minimal bit rate R and
a corresponding sampling rate required to attain this target.
Second, steps 1-4 can be easily adjusted to consider a different distortion criterion according to a spectral importance
masking, as described in the "Minimal Distortion Subject to

IEEE Signal Processing Magazine

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May 2018

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35



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