IEEE Signal Processing - May 2018 - 31

σ2
fR = 1

Distortion

D (R = 1)

DSI (fs, R = 1)
fR = 2
DSI (fs, R = 2)

D (R = 2)

se

m

m
)
(f s

fNyq
fs

0

FIGURE 14. The function D SI (fs, R) for the PSD of Figure 13 with an LPF
with a cutoff frequency of fs /2 and two values of the bit rate R . This
function describes the minimal distortion in recovering a Gaussian signal
with this PSD from a bit rate-R encoded version of its uniform samples.
This minimal distortion is bounded from below by Shannon's DRF of X(t),
where the latter is attained at the sub-Nyquist sampling rate fR .

SΠ(f )

SΛ(f )

Sω(f )

SΩ(f )

1.6
1.4
1.2
fR (Hz)

of D SI (fs, R) associated with the sampling distortion is due
only to those energy bands blocked by the filter. As a result,
the function D SI (fs, R) can be described by the sum of the red
and the blue parts in Figure 13(a). Figure 13(b) describes the
function D SI (fs, R) under the same bit rate R and a higher
sampling rate, while the cutoff frequency of the LPF is
adjusted to this higher sampling rate. As can be seen from the
figure, at this higher sampling rate, D SI (fs, R) equals the DRF
of X (t) in Figure 13(c), although this sampling rate is still
below the Nyquist rate of X (t) . In fact, it follows from
Figure 13 that the DRF of X (t) is attained at some critical
sampling rate f R that equals the spectral occupancy of the preserved part in the Pinsker-Kolmogorov water-filling expression (6). The existence of this critical sampling rate can also
be seen in Figure 14, which illustrates D SI (fs, R) as a function
of fs with H(f) an LPF.
In the "Shift-Invariant Sampling" section, we concluded
that the DRF of X (t) can be attained by sampling at or above
the Nyquist rate, since then the MMSE term in (6) vanishes.
Now we see that by using the LPF with a cutoff frequency of
fs /2 , the equality between D SI (fs, R) and the DRF, which is the
minimal distortion subject to a bit-rate constraint, occurs at a
sampling rate smaller than the Nyquist rate.
An intriguing way to explain this phenomena is as an
alignment of the degrees of freedom in the signal after the
presampling operation with the degrees of freedom that the
lossy compression with bit rate R can capture in this sampled signal. For stationary Gaussian signals, the degrees of
freedom in the signal representation are those spectral bands
where the PSD is nonzero. When the signal energy is not uniformly distributed over these bands-unlike in the example of
the PSD in (11)-the optimal lossy compression scheme calls
for discarding those bands with the lowest energy, i.e., the
parts of the signal with the lowest uncertainty. The presampling operation removes these low-energy signal components
such that the resulting signal has the same degrees of freedom
as those that can be captured by the lossy compressed signal representation that follows the sampler. Thus, the presampling operation in a sense aligns the degrees of freedom
of the presampled signal with those of the postsampled lossy
compression operation.
The degree to which the new critical rate f R is smaller than
the Nyquist rate depends on the energy distribution of X (t)
along its spectral occupancy. The more uniform it is, the more
degrees of freedom are required to represent the lossy compressed signal, and therefore f R is closer to the Nyquist rate.
Figure 15 shows the dependency of f R on R for various PSD
functions. Note that, whenever the energy distribution is not
uniform and the signal is bandlimited, the critical rate f R converges to the Nyquist rate as R extends to infinity and to zero
as R reaches zero.
In the prior discussion, we considered only signals with unimodal PSDs (for example, the PSD in Figure 9 is not unimodal). The main challenge in extending the previously mentioned
conclusions to signals with nonunimodal PSDs is the design of
a sub-Nyquist system that samples the signal components con-

1
0.8
0.6
0.4
0.2

0

0.5

1
R (bit/s)

1.5

2

FIGURE 15. The critical sampling rate fR as a function of the bit rate

R for the PSDs given in the small frames. For the bandlimited PSDs
S P (f ), S K(f ) and S ~ (f ), the critical sampling rate is always below the
Nyquist rate. The critical sampling rate is finite for any R, even for the
nonbandlimited PSD S X (f ).

taining the most information about the signal (i.e., the signal's
high-energy bands) to obtain the optimal lossy compressed
signal representation when these samples are encoded at the
fixed bit rate R. Before describing this extension, we consider the general structure of a presampling transformation that
minimizes the distortion in the ADX setting.

Optimal presampling transformation
We now consider the presampling filter H(f) that minimizes
the function D SI (fs, R) subject to a fixed bit rate R and sampling rate fs . By examining expressions (9) and (S10), we conclude that this minimization is equivalent to the maximization

IEEE Signal Processing Magazine

|

May 2018

|

31

Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018

Contents
IEEE Signal Processing - May 2018 - Cover1
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IEEE Signal Processing - May 2018 - Cover3
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