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520 Chapter 9 SAMPLING RATE CONVERSION
V(ω)
0 ω x,p
I
1
H(ω)
I
1 + δ 1
1 − δ
1 1
ω x,s
I
π
I
2π − ω x,s
I
2π
I
2ω x,s
I
3π
I
2π + ω x,s
I
4π − ω x,s
I
π
ω
(a)
(b)
Multiple stopband design: (a) signal spectrum, (b) filter specifica-
FIGURE 9.30
tions
δ 2
0
ω x,p
I
2π − ω x,s
I
2π + ω x,s
I
4π − ω x,s
I
π
ω
9.6 FIR FILTER STRUCTURES FOR SAMPLING RATE CONVERSION
As indicated in the discussion in section 9.4, sampling rate conversion by
a factor I/D can be achieved by first increasing the sampling rate by I,
accomplished by inserting I − 1 zeros between successive values of the
input signal x(n), followed by linear filtering of the resulting sequence to
eliminate the unwanted images of X(ω), and finally by downsampling the
filtered signal by the factor D. Inthis section we consider the design and
implementation of the linear filter. We begin with the simplest structure,
which is the direct-form FIR filter structure, and develop its computationally
efficient implementation. We then consider another computationally
efficient structure called the polyphase structure, which is used in the implementation
of the MATLAB functions resample and upfirdn. Finally,
we close this section by discussing the time-variant filter structures for
the general case of sampling rate conversion.
9.6.1 DIRECT-FORM FIR FILTER STRUCTURES
In principle, the simplest realization of the filter is the direct-form FIR
structure with system function
H(z) =
M−1
∑
k=0
h(k)z −k (9.62)
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