IEEE Signal Processing - May 2018 - 22
the low-pass filtered version of X (t), the contribution of the
quantization to the distortion in (1) is given by
f
LPF Q sQ
2
D qnt (fs, R) _
fr
#-f
S h ( f ) df
r
S
X
(f )
min " fs, fNyq ,
= cQ f
fs
−
fs
fNyq −
2
2
fs
2 fNyq
2
(a)
fNyq
Q
2
X
S
Sη (f )
−
fs
f
2 − Nyq
2
fNyq
2
(b)
Dsmp (fs)
fs
2
Dqnt (fs, R )
FIGURE 5. A spectral representation of the distortion in PCM (1). (a)
Sampling below the Nyquist rate (fs 2 fNyq) introduces sampling distortion
D smp (fs, R) . (b) Sampling distortion vanishes when sampling above the Nyquist rate (fs 2 fNyq), but the contribution of the in-band quantization noise
D qnt (fs, R) increases because of the lower bit precision of each sample.
constant, although deviation from this rule would not affect
our general conclusions. Regardless of this assumption and as
explained in "Scalar Quantization," the variance of this noise
h [n] is proportional to the variance of the process Y [n] at the
input to the quantizer and decreases exponentially with the
number of quantization bits Rr :
r
var (h [n]) = c Q var (Y [n]) 2 -2R .
(3)
The proportionality constant c Q depends on the actual
digital label assigned to each quantization level. At high quantization precision Rr = R /fs and using a uniform quantizer, the
value of the constant corresponding to optimal encoding converges to c Q = (re/6). This value of c Q is used in our figures.
Under the assumption that the PSD of h [n] is constant over
the entire discrete-time frequency range with variance (3) and
using the fact that the variance of Y [n] equals the variance of
22
S X ( f ) df p 2 -2R/fs,
D PCM ( fs, R) = D smp ( fs) + D qnt ( fs, R).
(f )
LPF Q
#
(4)
where the term in the braces represents the variance of Y [n] or
the energy of the signal at the output of the reconstruction
LPF (the min is present because the LPF at the sampler is in
use only if the sampling rate is lower than Nyquist). The overall distortion in PCM is, therefore,
Sη (f )
f
fs
2
fs
2
(5)
The important observation from this expression is that, under a
fixed bit rate R, the distortion due to quantization increases as the
sampling rate fs increases. This increase in fs means fewer quantization bits are available to represent each sample, and, therefore,
the distortion due to quantization is larger. Alternatively, the distortion due to sampling decreases as fs increases and, in fact, vanishes as fs exceeds the Nyquist rate. A spectral interpretation of
the function D PCM ( fs, R) is shown in Figure 5. This figure shows
the spectrum of the sampled source signal and the spectrum of the
quantization noise under the high-resolution approximation for
two representative cases of the sampling frequency:
1) Sub-Nyquist sampling: The distortion due to sampling
D smp ( fs) is the part of S X ( f ) not included in the sampling
interval ^-fs /2, fs /2h. The distortion due to quantization is
relatively low, since the small value of fs allows the quantization of each sample with the relatively high resolution
of Rr = R/fs bits.
2) Super-Nyquist sampling: The distortion due to sampling
D smp ( fs) is zero, but the distortion due to quantization
D qnt ( fs) is affected by the reduction in the bit-resolution
that decreases linearly in fs, since Rr = R/fs .
It follows from the prior description that there exists a
sampling rate that balances the two error contributions from
quantization and sampling to minimize the total distortion in (5).
This sampling rate can be seen in Figure 6, where the distortion
D PCM ( fs, R) is shown versus the relative sampling rate fs /fNyq
for two PSDs. For the PSD S P ( f ) with uniform energy distribution, the sampling rate that minimizes the distortion is exactly the
Nyquist rate. For the triangular PSD S K ( f ), the optimal sampling rate is below the Nyquist rate. In general, it is shown in [18]
that, under similar assumptions, the sampling rate that minimizes
the distortion in PCM is always at or below the Nyquist rate.
This rate is, in fact, strictly smaller than the Nyquist rate when the
energy of the signal is not uniformly distributed over its spectral
support, as in S K (f) of Figure 6. Going back to our general question, PCM illustrates an instance where, as a result of a bit-rate
constraint, sampling below the Nyquist rate is optimal.
Another conclusion from our analysis is that, under a fixed
bit rate, the distortion in PCM increases as a result of oversampling. This phenomenon is explained by the increasing correlation
between consecutive time samples at a super-Nyquist sampling
IEEE Signal Processing Magazine
|
May 2018
|
Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018
Contents
IEEE Signal Processing - May 2018 - Cover1
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