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Sampling Rate Conversion by a Rational Factor I/D 493
9.4 SAMPLING RATE CONVERSION BY A RATIONAL FACTOR I/D
Having discussed the special cases of decimation (downsampling by a factor
D) and interpolation (upsampling by a factor I), we now consider
the general case of sampling rate conversion by a rational factor I/D.
Basically, we can achieve this sampling rate conversion by first performing
interpolation by the factor I and then decimating the output of the
interpolator by the factor D. Inother words, a sampling rate conversion
by the rational factor I/D is accomplished by cascading an interpolator
with a decimator, as illustrated in Figure 9.14.
We emphasize that the importance of performing the interpolation
first and the decimation second is to preserve the desired spectral characteristics
of x(n). Furthermore, with the cascade configuration illustrated
in Figure 9.14, the two filters with impulse response {h u (k)} and {h d (k)}
are operated at the same rate, namely IF x , and hence can be combined
into a single lowpass filter with impulse response h(k), as illustrated in
Figure 9.15. The frequency response H(ω v )ofthe combined filter must
incorporate the filtering operations for both interpolation and decimation,
and hence it should ideally possess the frequency-response characteristic
H(ω v )=
{
I, 0 ≤|ωv |≤min(π/D, π/I)
0, otherwise
(9.36)
where ω v =2πF/F v =2πF/IF x = ω x /I.
Explanation of (9.36) Note that V (ω v ) and hence W (ω v ) in
Figure 9.15 are periodic with period 2π/I. Thus, if
• DI, then filter must first truncate the fundamental period of W (ω v )
to avoid aliasing error in the (D ↓1) decimation stage to follow.
Putting these two observations together, we can state that when
D/I < 1, we have net interpolation and no smoothing is required by
Interpolator
Decimator
v(k) IDEAL
IDEAL w(k)
x(n)
↑I LPF
LPF
↓D y(m)
h u (k)
h d (k)
I
Rate: F x IF x IF x IF x
F x = F y
D
FIGURE 9.14 Cascade of interpolator and decimator for sampling rate conversion
by a factor I/D
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