Signals and Systems - Electrical Engineering

676 C H A P T E R 11: Introduction to the Design of Discrete Filters

and to obtainαwe replaceθpbyπ−θpin Equation (11.46) to get:

α=

sin(−(θp+ωd)/ 2 +π/ 2 )

sin((θp−ωd)/ 2 +π/ 2 )

=

−sin((θp+ωd)/ 2 −π/ 2 )

sin((θp−ωd)/ 2 +π/ 2 )

=

cos((θp+ωd)/ 2 )

cos((θp−ωd)/ 2 )

(11.47)

As before,θpis the cut-off frequency of the prototype low-pass filter andωdis the desired cut-off

frequency of the high-pass filter.

When the low-pass and the high-pass filters have the same bandwidth,ωd=π−θp, we have that

θp+ωd=πand soα=0 giving as a transformationZ−^1 =−z−^1 , which transforms the low-pass

prototype into a high-pass filter, both of the same bandwidth.

Low-Pass to Band-Pass and Band-Stop Transformations

By being linear in both the numerator and the denominator, the LP-LP and LP-HP transformations

preserve the number of poles and zeros of the prototype filter. To transform a low-pass filter into a

band-pass or into a band-stop filter, the number of poles and zeros must be doubled. For instance, if

the prototype is a first-order low-pass filter (with real-valued poles and zeros) we need a quadratic,

rather than a linear, transformation in both numerator and denominator to obtain band-pass or

band-stop filters from the low-pass filter since band-pass or band-stop filters cannot be first-order

filters.

The low-pass to band-pass (LP-BP) transformation is

Z−^1 =−

z−^2 −bz−^1 +c

cz−^2 −bz−^1 + 1

(11.48)

while the low-pass to band-stop (LP-BS) transformation is

Z−^1 =

z−^2 −(b/k)z−^1 −c

−cz−^2 −(b/k)z−^1 + 1

(11.49)

where

b= 2 αk/(k+ 1 )

c=(k− 1 )/(k+ 1 )

and

α=

cos((ωd 2 +ωd 1 )/ 2 )

cos((ωd 2 −ωd 1 )/ 2 )

k=cot((ωd 2 −ωd 1 )/ 2 )tan(θp/ 2 )

The frequenciesωd 1 andωd 2 are the desired lower and higher cut-off frequencies in the band-pass

and band-stop filters.