IEEE Signal Processing - May 2018 - 20
rate of the digital representation, i.e., the
number of bits per unit time required to
bits
represent the process YQ [n], is defined
fs
R = R fs
Xp (t )
s
Y [n ]
Scalar
r s.
as R = Rf
LPF (fs/2)
X (t )
Quantizer
The process of recovering the analog source signal X (t) from the digital
YQ [n ]
sequence YQ [n] is described at the bottom of Figure 4. The digital discreteX (t )
LPF (fr )
D/A
time sequence of quantized values
YQ [n] is first converted to a continuous-time impulse train using a digitalDecoder
to-analog (D/A) unit and then filtered
using an ideal LPF with cutoff frequenFIGURE 4. PCM and reconstruction system.
cy fr . In the time domain, this LPF is
equivalent to an ideal sinc interpolation
between the analog sample values to create a continuous-time
and encoding scheme, it illustrates an instance where, as a
signal bandlimited to (-fr, fr). The result of this interpolation
result of the bit-rate constraint, sampling below the Nyquist rate
is optimal. In addition, this analysis provides a simple way to
is denoted by Xt (t). We measure the distortion of the system by
introduce the notions of sampling, quantization, and bit rate and
the MSE between X (t) and Xt (t) averaged over time as
serves as a basis for the generalization of the sampling and for
T /2
2
encoding operations into optimal ones.
D PCM ( fs, R) _ lim 1 # E ^ X (t) - Xt (t)h dt.
(1)
T " 3 T -T/2
Encoder
"
Sampler
ADX via pulse-code modulation
A particular example for a system incorporating a sampler, an
encoder, and a decoder is given in Figure 4. This system converts
the analog signal X (t) to a digital representation YQ [n] by a uniform sampler followed by a scalar quantizer. This conversion
technique is known as pulse-code modulation (PCM) [15], [16];
refer to [17, Sec. I.A] for its historical overview. The bit rate in
this system is defined as the average number of bits per unit time
required to represent the process YQ [n]. The goal of our analysis
is to derive the MSE distortion in recovering the analog input
signal X (t) under a constraint R on this bit rate, assuming a particular sampling rate fs of the sampler. We denote this distortion
by D PCM ( fs, R). Since the system in Figure 4 is a special case of
Figure 2, the function D PCM ( fs, R) is lower-bounded by the minimal distortion in the ADX, obtained by optimizing over all of
the encoders and decoders, subject only to a sampling rate constraint fs and a bit-rate constraint R.
We analyze the system of Figure 4 assuming a stochastic
continuous-time, continuous-amplitude source signal X (t) at its
input. This signal is first filtered using a presampling low-pass
filter (LPF) to yield X p (t). The filtered signal is then sampled
uniformly at rate fs samples/s. Each sample Y [n] is mapped
using a scalar quantizer to YQ [n], which is the nearest value
to Y [n] among a prescribed set of K quantization levels. More
details on the operation of the scalar quantizer are provided
in "Scalar Quantization." Since each of the quantization levels
can be assigned a finite digital number, we say that the process
YQ [n] is a digital representation of Y [n]. As explained in "Scalar Quantization," the selection of the quantization levels and
the length of the digital number assigned to each of them may
also be subject to optimization. Subsequently, we assume that
Rr is the expected number of bits per sample assigned to represent the quantization levels (the expectation is with respect to
the distribution of the source signal). Using this notation, the bit
20
Note that letting the time grow symmetrically in both directions simplifies some of the expressions, but our results remain
valid even if time grows in one direction. It is, in general, possible to use a different decoding scheme that would lead to a
lower MSE under the same sampling and bit-rate constraint.
Indeed, (1) is minimized by using the conditional expectation
of X (t) given YQ [n] as the reconstruction signal rather than
using Xt (t). However, the nonlinearity introduced by the scalar
quantizer makes the exact analysis of the distortion under the
conditional expectation a difficult task [17], and, therefore, for
simplicity, we focus here on interpolation by low-pass filtering.
We now turn to analyze the distortion in (1) as a function of
the sampling rate fs and the bit rate R. We assume that X (t)
is a stationary stochastic process with a symmetric power
spectral density (PSD) S X (f ), and we denote its bandwidth by
fNyq /2. If X (t) is not bandlimited, then we use the notation
f Nyq = 3. In either case, we assume that X (t) is bounded in
energy and denote
2
v = varX (t) =
#R S X ( f ) df.
We further assume that the PSD S X ( f ) is unimodal, in the
sense that its energy distribution is decreasing as one moves
away from the origin, as given, for example, in Figure 5. Under
this assumption, the presampling filter that minimizes the
distortion, among all linear time-invariant filters, is an LPF
with a cutoff frequency of fs /2 [18]. Henceforth, we assume
that this filter is used. Finally, we pick the cutoff frequency
fr of the reconstruction filter to match the bandwidth of the
low-pass filtered signal. This cutoff frequency is therefore the
minimum between fs /2 and the bandwidth of X (t), which
equals fNyq /2.
As a result of these assumptions, the only distortion introduced in the sampling process is due to the presampling filter,
IEEE Signal Processing Magazine
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May 2018
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Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018
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