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514 Chapter 9 SAMPLING RATE CONVERSION
Significance of δ 1 and δ 2 The filter ripple parameters δ 1 and δ 2 have
the following significance, which must be taken into consideration while
specifying their values:
• The passband ripple δ 1 measures the ripple in the passband and hence
controls the distortion in the signal bandwidth ω p .
• The stopband ripple δ 2 controls the amount of aliased energy (also
called leakage) that gets into the band up to ω x,s .
There are (D − 1) contributions due to k ≠0terms in (9.53). These
are expected to add incoherently (i.e., have peaks at different locations),
so the overall peak error should be about δ 2 . The actual error depends
on how X(ω) varies over the rest of the band |ω| >ω x,p . Clearly, the
filter stopband ripple δ 2 controls the aliasing error in the signal passband.
Therefore, both δ 1 and δ 2 affect the decimated signal in its passband.
Comment: Comparing the FIR decimator filter specifications (9.57) to
those for the FIR interpolator in (9.52), we see a high degree of similarity.
In fact, a filter designed to decimate by factor D can also be used to
interpolate by the factor I = D, aswesee from the following example.
This means that the function intfilt can also be used to design FIR
filters for decimation.
□ EXAMPLE 9.13 To design a decimate by D stage we need values for ω x,p and ω x,s (remember
that these are signal parameters). Assume ω x,p = π/(2D), which satisfies the
constraint ω x,p ≤ π/D and is exactly half the decimated bandwidth. Let ω x,s =
ω x,p. Then the FIR lowpass filter must pass frequencies up to ω p = π/(2D) and
stop frequencies above ω s =2π/D − π/(2D) =3π/(2D).
Now consider the corresponding interpolation problem. We want to interpolate
by I. Weagain choose ω x,s = ω x,p, but now the range is ω x,p