IEEE Signal Processing - May 2018 - 30

case; we explore the interaction between sampling and lossy
compression and assume that the noise is zero. Hence, the
simplified form (S11) for Su X | Y (f) is used.
As a simple example for using (9), we consider X (t) to be
a stationary Gaussian signal with a flat, bandlimited PSD, i.e.,

and independent noise. Therefore, the problem of estimating
the source signal from the digital output of the encoder combines a linear filtering problem, an interpolation problem, and a
lossy compression problem. The only adjustment to the description of the minimal distortion under this extension is to replace
the function Su X | Y (f) in (9) and (S10) with [51]

Su X | Y (f) =

/ S 2X (f - fs n) H (f - fs n) 2

n!Z

/ ^S X (f - fs n) + S

h

n!Z

(f - fs n)h H (f - fs n)

2

1
f 1 W,
S P (f) = ) 2W
0
otherwise.

. (10)

As long as the presampling filter passes all frequencies
f ! (-W, W) , the relation between the distortion in (9a) and
the bit rate in (9b) is given by

Equations (S10), (9), and (10) describe the MMSE in noncausal filtering, the MSE due to uniform sampling, and the distortion under optimal lossy compression. That is, these equations
determine the combined effect of three of the most fundamental operations in signal processing: quantization, sampling,
and interference by noise. Most importantly, these equations
provide a unified representation for the distortion in these
three fundamental operations, allowing us to explore the interaction among them. In this article, we consider a less general

Z
2R
f
fs
]] mmse SI (fs) + s 2 - fs ,
11
2W
2W
D SI (fs, R) = [ R
fs
] 2- W ,
$ 1,
2
W
\

Distortion

1

DSI (fs, R = 1)

D (R = 2)

DSI (fs, R = 2)
fs

Sampling Rate

Optimal sampling rate under bit-rate constraint
We now consider the expression D SI (fs, R) of (9) for the unimodal PSD shown in Figure 13, where the presampling filter
H(f ) is an ideal LPF with a cutoff frequency of fs /2 . This
LPF operates as an antialiasing filter, and, therefore, the part

FIGURE 12. The distortion as a function of the sampling rate for the source
with PSD S P (f ) of (11) and source coding rates R = 1 and R = 2 bits per
time unit.

SX (f )

SX (f )

-

fR
2

(a)

fR
2

SX (f )

θ

θ′

fs

fs

f

Preserved Spectrum

-

fR
2

(12)

where mmse SI (fs) = 1 - fs /2W. Expression (12) is shown in
Figure 12 for two fixed values of the bit rate R. It has a very
intuitive structure-for frequencies below the signal's Nyquist
rate 2W, the distortion as a function of the rate increases by a
constant factor because of the error as a result of nonoptimal
sampling. This factor completely vanishes once the sampling
rate exceeds the Nyquist frequency, in which case D SI (fs, R)
coincides with Shannon's DRF of X (t) .
In the previous example with PSD S P (f) , the filter H(f) has
no effect on the distortion as long as its passband contains the
support of S P (f) . However, when the spectrum is nonflat over
its support, there is a precise way to choose the passband of the
presampling filter to minimize the function D SI (fs, R).

D

D (R = 1)

(11)

f

fR
2

(b)

Sampling Distortion

θ

-

fR
2

(c)

fR
2

f

Lossy Compression Distortion

FIGURE 13. A water-filling interpretation of (9) with H(f ) as an LPF of cutoff frequency fs /2. The distortion is the sum of the sampling and the lossy com-

pression distortions. All figures correspond to the same bit rate R and different sampling rates: (a) fs 1 fR, (b) fs $ fR and (c) fs 2 fNyq. The DRF of X(t)
is attained for all fs greater than fR 1 fNyq.

30

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