AN2182 Datasheet, PDF (11/18 Pages) STMicroelectronics – Filters using the ST10 DSP library

AN2182
6 Cutoff band filter
Cutoff band filter
The aim of this section is to create a cutoff band filter with a cutoff frequency F1, a pass
frequency F2> F1, and a gain of G=1, by determining a FIR filter using the digital approach.
The FIR coefficients correspond to h(n) where h is the continuous time response of the filter
and h(n) = h(nT_s).
In the frequency domain, the ideal cutoff band filter corresponding to these criteria has the
following response:
Figure 7. Ideal cut-band filter frequency response
1
1
-F2
-F1 0 F1
Cut-band filter
Passband filter
F2 _Frequency
-F2
-F1 0
F1
F2 _Frequency
If we consider
● H_P(f) the frequency response of the passband filter with a pass frequency F1 and cutoff
frequency F2
● H_C(f) the frequency response of the cutoff band filter with a cutoff frequency F1 and pass
frequency F2
● T_s is the sampling period
then, we can easily notice that
(8) HC(f) = 1 – HP(f)
In the time domain, this gives
hC(n)
=
δ⎝⎛n – ^{N------2–}-----1--_⎠⎞
–
⎛
⎜2
⎝
F----2--
Fs
sin
c_⎝⎛2F2_⎝⎛
n
–
N------2–_{-----1--⎠⎞}
Ts_⎠⎞
–
2 ^F----1--
Fs
sin
c_⎝⎛
2F1_⎝⎛n
–
N------2–_{-----1--⎠⎞}
Ts_⎠⎞
⎞
⎟
⎠
with n = 0.. N – 1 and where
● N is the number of FIR coefficients
● N – 1 is the FIR’s order
● W is the window time response
In the case of a cutoff band filter, N should be odd because of the dirac, that is the FIR’s order
should be even.
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