IEEE Signal Processing - May 2018 - 32
Optimal Presampling Transformation
Properties 1 and 2 of the optimal presampling filter imply
that to minimize the mean squared error (MSE) and,
hence, the overall distortion, it is preferred to eliminate all
information on lower-energy subbands where they interfere
with higher-energy bands. To provide an intuitive explanation for this phenomenon, we consider two independent
Gaussian random variables X1 and X2 with the zero mean
and variances v 21 and v 22, respectively. These random
variables can be seen as two different spectral lines in the
spectrum of X(t) that interfere with each other because of
aliasing in uniform sampling. Assume that we are given
the linear combination U = h 1 X 1 + h 2 X 2, and are interested in the joint estimation of X1 and X2 subject to an MSE criterion. That is, we want to minimize
mmse (X 1, X 2 | U) _ E ^X 1 - Xt 1 h + E ^X 2 - Xt 2 h .
2
2
σ2
DSI (fs, R = 1)
Distortion
D (R = 1)
DSI (fs, R = 3)
D (R = 3)
mmseSI (fs)
0
0
fNyq
fs
Sampling Rate
(a)
fs
(b)
FIGURE 16. (a) The minimal distortion D SI (fs, R) using an optimal presam-
pling filter as a function of the sampling rate for two values of the bit rate
R . The dashed lines represent the distortion with an all-pass presampling filter that allows aliasing. (b) The support of the optimal presampling filter over the source PSD for a particular sub-Nyquist sampling
rate fs . The difference between any two bands in the support is not an
integer multiple of fs.
32
The optimal estimator of each variable as well as the
corresponding estimation error can be easily found, since
the optimal estimator is linear. We further ask how to
choose the coefficients h1 and h2 in the linear combination such that the MSE is minimized. A simple optimization over the expression for mmse (X 1, X 2 | U) shows that
h 1 ! 0, h 2 = 0 is the answer whenever v 21 2 v 22, and
h 1 = 0, h 2 ! 0 whenever v 21 1 v 22 . That is, the optimal
linear combination eliminates all the information on the
part of the signal with the lowest variance and passes
only the part with the highest uncertainty. Going back to
spectral components, the MSE is minimized by a presampling filter H(f ) that eliminates all spectral components of
low energy whenever they interfere with high-energy
spectral components because of the aliasing that results
from uniform sampling.
of Su X | Y (f) for any f in the interval (-fs /2, fs /2) . This fact is not
surprising, since we have seen in (S10) that Su X | Y (f) represents
the part of the source available to the encoder. Because the
function Su X | Y (f) is independent of R, the optimal filter H(f)
that minimizes D SI (fs, R) is only a function of the sampling
rate, and it is, therefore, identical to the presampling filter that
minimizes mmse (X | Y) , i.e., the MMSE without the bit-rate
constraint. Note that, since Su X | Y (f) is indifferent to scaling in
H(f), the only effect of the presampling filter on the distortion
is through its passband, i.e., the support of H(f). We explain in
"Optimal Presampling Transformation" that the passband of
the presampling filter that minimizes mmse (X | Y) can be
completely characterized by the following two properties:
1) Aliasing-free: The passband is such that the filter eliminates aliasing in sampling at frequency fs . That is, all
integer shifts of the support of the filtered signal by fs
are disjoint.
2) Energy maximization: The passband is chosen to maximize the energy of X (t) at the output of the filter, subject
to the aliasing-free property 1).
In the case of a unimodal PSD, an LPF with a cutoff frequency of fs /2 satisfies both the aliasing-free and energy maximization properties and is therefore the optimal presampling
filter that minimizes D SI (fs, R) . For this reason, Figure 13
describes the minimal value of D SI (fs, R) for the PSD considered there. In general, however, the set that maximizes the
passband energy is not aliasing-free. As an example, consider
the PSD shown in Figure 16(b). The colored area represents
the support of the optimal presampling filter. This support is
aliasing-free, since the difference between any two bands in
the support is not an integer multiple of fs . The example in
Figure 16(a) also shows that, although D SI (fs, R) is guaranteed
to coincide with D(R) for fs 2 fNyq, the convergence to this
IEEE Signal Processing Magazine
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May 2018
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