1.4. SAMPLING OF SINUSOIDS 25
remained outside the relevant audio band, and the aliasing of the F component which does
represent distortion in the audio band.
Of course, one would not want to feed the signal
y
a
(t) into an amplifier/speaker system
because the high frequencies beyond the audio band might damage the system or cause
nonlinearities. (But even if they were filtered out, the
F component would still be there.)
Example 1.4.9: Oversampling and Decimation. Example 1.4.8 assumed that sampling at 80 kHz
could be maintained throughout the digital processing stages up to reconstruction. There
are applications however, where the sampling rate must eventually be dropped down to
its original value. This is the case, for example, in digital audio, where the rate must be
reduced eventually to the standardized value of 44.1 kHz (for CDs) or 48 kHz (for DATs).
When the sampling rate is dropped, one must make sure that aliasing will not be reintro-
duced. In our example, if the rate is reduced back to 40 kHz, the
C and D components,
which were inside the
[−40, 40] kHz Nyquist interval with respect to the 80 kHz rate,
would find themselves outside the
[−20, 20] kHz Nyquist interval with respect to the 40
kHz rate, and therefore would be aliased inside that interval, as in Example 1.4.7.
To prevent
C and D, as well as E, from getting aliased into the audio band, one must
remove them by a lowpass digital filter before the sampling rate is dropped to 40 kHz.
Such a filter is called a digital decimation filter. The overall system is shown below.
prefilter
H(f)
80 kHz
sampler
80
kHz
80
kHz
40
kHz
digital
filter
down-
sampler
recon-
structor
x(t)
y(t) y
a
(t)
analoganalog
The downsampler in this diagram reduces the sampling rate from 80 down to 40 kHz by
throwing away every other sample, thus, keeping only half the samples. This is equivalent
to sampling at a 40 kHz rate.
The input to the digital filter is the sampled spectrum of
y(t), which is replicated at mul-
tiples of 80 kHz as shown below.
20 30 50 70 9010 40 60 80 100 120 140 160 kHz
digital lowpass filter
prefilter
AAAA
CCCC
EEEE
FFFF
BBBB
DDDD
f
0
We have also assumed that the 30 dB/octave prefilter is present. The output of the digital
filter will have spectrum as shown below.
26 1. SAMPLING AND RECONSTRUCTION
20 30 50 70 9010 40 60 80 100 120 140 160 kHz
digital lowpass filter
AAAA
FFFF
BBBB
CCEEDD C CEEDD
f
0
(-49 dB)
The digital filter operates at the oversampled rate of 80 kHz and acts as a lowpass filter
within the
[−40, 40] kHz Nyquist interval, with a cutoff of 20 kHz. Thus, it will remove the
C, D, and E components, as well as any other component that lies between 20 ≤|f|≤60
kHz.
However, because the digital filter is periodic in
f with period f
s
= 80 kHz, it cannot remove
any components from the interval 60
≤ f ≤ 100. Any components of the analog input y(t)
that lie in that interval would be aliased into the interval 60 −80 ≤ f −f
s
≤ 100−80, which
is the desired audio band
−20 ≤ f − f
s
≤ 20. This is what happened to the F component,
as can be seen in the above figure.
The frequency components of
y(t) in 60 ≤|f|≤100 can be removed only by a pre-
filter, prior to sampling and replicating the spectrum. For example, our low-complexity
30 dB/octave prefilter would provide 47.6 dB attenuation at 60 kHz. Indeed, the number
of octaves from 20 to 60 kHz is log
2
(60/20)= 1.585 and the attenuation there will be
30 dB/octave
× 1.584 octaves = 47.6 dB.
The prefilter, being monotonic beyond 60 kHz, would suppress all potential aliased compo-
nents beyond 60 kHz by more than 47.6 dB. At 100 kHz, it would provide 30
×log
2
(100/20)=
69.7 dB attenuation. At f
F
= 62.5 kHz, it provides 49.3 dB suppression, as was calculated
in Example 1.4.7, that is,
|H(f
F
)|=10
−49.3/20
= 1/292.
Therefore, assuming that the digital filter has already removed the
C, D, and E compo-
nents, and that the aliased
F component has been sufficiently attenuated by the prefilter,
we can now drop the sampling rate down to 40 kHz.
At the reduced 40 kHz rate, if we use an ideal reconstructor, it would extract only the
components within the
[−20, 20] kHz band and the resulting reconstructed output will
be:
y
a
(t)= 2A cos
(10πt)+2B cos(30πt)+
2F
292
cos
(35πt)
which has a much attenuated aliased component F. This is to be compared with Eq. (1.4.9),
which used the same prefilter but no oversampling. Oversampling in conjunction with
digital decimation helped eliminate the most severe aliased components,
C and D.
In summary, with oversampling, the complexity of the analog prefilter can be reduced and
traded off for the complexity of a digital filter which is much easier to design and cheaper
to implement with programmable DSPs. As we will see in Chapter 2, another benefit of
oversampling is to reduce the number of bits representing each quantized sample. The
connection between sampling rate and the savings in bits is discussed in Section 2.2. The
subject of oversampling, decimation, interpolation, and the design and implementation of
digital decimation and interpolation filters will be discussed in detail in Chapter 12.