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Frequency-band Transformations 451
TABLE 8.2 Frequency transformation for digital filters (prototype lowpass filter has cutoff
frequency ω c)
′
Type of
Transformation Transformation Parameters
Lowpass
Highpass
z −1 −→ z−1 − α
1 − αz −1 ω c = cutoff frequency of new filter
α = sin [(ω′ c − ω c) /2]
sin [(ω ′ c + ω c) /2]
z −1 −→ − z−1 + α
1+αz −1
ω c = cutoff frequency of new filter
α = − cos [(ω′ c + ω c) /2]
cos [(ω c ′ − ω c) /2]
Bandpass z −1 −→ − z−2 − α 1z −1 + α 2
α 2z −2 − α 1z −1 +1
ω l =lower cutoff frequency
ω u = upper cutoff frequency
α 1 = −2βK/(K +1)
α 2 =(K − 1)/(K +1)
β = cos [(ω u + ω l ) /2]
cos [(ω u − ω l ) /2]
K = cot ωu − ω l
2
tan ω′ c
2
Bandstop z −1 −→ z−2 − α 1z −1 + α 2
α 2z −2 − α 1z −1 +1
ω l =lower cutoff frequency
ω u = upper cutoff frequency
α 1 = −2β/(K +1)
α 2 =(K − 1)/(K +1)
β = cos [(ω u + ω l ) /2]
cos [(ω u − ω l ) /2]
K = tan ωu − ω l
2
tan ω′ c
2
From this example it is obvious that to obtain the rational function
of a new digital filter from the prototype lowpass digital filter, we should
be able to implement rational function substitutions from Table 8.2. This
appears to be a difficult task, but since these are algebraic functions, we
can use the conv function repetitively for this purpose. The following
zmapping function illustrates this approach.
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