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FIR Filter Designs for Sampling Rate Conversion 517
x(n)
FIGURE 9.28
FIR Resampler
FIR LPF
↑I
H(ω)
↓D y(m)
Rate: F x IF x IF x
An FIR rational-factor resampler
I
F x = F y
D
Now for the interpolation part, the lowpass filter must pass frequencies
up to ω x,p /I and attenuate frequencies starting at (2π/I − ω x,s1 /I). The
decimation part of the filter must again pass frequencies up to ω x,p /I
but attenuate frequencies above (2π/D − ω x,s2 /I). Therefore, the stopband
must start at the lower of these two values. Defining filter cutoff
frequencies as
( ωx,p
)
[ 2π
ω p =
and ω s = min
I
I − ω x,s 1
, 2π I D − ω ]
x,s 2
(9.60)
I
and the corresponding ripple parameters as δ 1 and δ 2 ,wehave the following
filter specifications:
1
I H r(ω) ≤ 1 ± δ 1 for |ω| ∈[0,ω p ]
1
I H r(ω) ≤±δ 2
for |ω| ∈[ω s ,π]
(9.61)
where H r (ω) isthe amplitude response. Note that if we set ω x,s1 = π and
ω x,s2 = Iπ/D, which are their maximum values, then we get the ideal
cutoff frequency max[π/I, π/D], as given before in (9.36).
MATLAB Implementation Clearly, the upfirdn function implements
all the necessary operations needed in the rational sampling rate conversion
system shown in Figure 9.28. When invoked as y = upfirdn(x,h,
I,D),itperforms a cascade of three operations: upsampling the input data
array x by a factor of the integer I, FIR filtering the upsampled signal data
with the impulse response sequence given in the array h, and finally downsampling
the result by a factor of the integer D. Using a well designed filter,
we have a complete control over the sampling rate conversion operation.
□ EXAMPLE 9.15 Design a sampling rate converter that increases the sampling rate by a factor of
2.5. Use the firpm algorithm to determine the coefficients of the FIR filter that
has 0.1 dBripple in the passband and is down by at least 30 dB in the stopband.
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