628	The Scientist and Engineer's Guide to Digital Signal Processing

function also containing two poles. The problem is, the substitution leaves the transfer function in a very unfriendly form:

	H (z ) '		1
	2 (1 & z & 1)	& (F% jT)	2 (1 & z & 1)	& (F& jT)
	T (1 % z & 1)		T (1 % z & 1)

Working through the long and placed in the standard form of identified as:

	a0	'
	a1	'
EQUATION 33-11	a2	'
Bilinear transform for two poles.
The pole-pair is located at F ± T in	b1	'
the s-plane, and a0, a1, a2, b1, b2 are
the recursion coefficients for the	b2	'
discrete system.

tedious algebra, this expression can be Eq. 33-3, and the recursion coefficients

T 2/D

2T 2/D T 2/D

( 8 & 2 M T 2 ) / D

(& 4 & 4 FT & M T 2 ) / D

where:

M ' F2 % T2

T ' 2 tan (1/2)

D ' 4 & 4 FT % M T 2

The variables M, T, and D have no physical meaning; they are simply used to make the equations shorter.

Lines 1200-1290 use these equations to convert the location of the s-domain pole-pair, held in the variables, RP and IP, directly into the recursive coefficients, held in the variables, X0, X1, X2, Y1, Y2. In other words, we have calculated an intermediate result: the recursion coefficients for one stage of a low-pass filter with a cutoff frequency of one.

Low-pass to Low-pass Frequency Change

Changing the frequency of the recursive filter is also accomplished with a conformal mapping technique. Suppose we know the transfer function of a recursive low-pass filter with a unity cutoff frequency. The transfer function of a similar low-pass filter with a new cutoff frequency, W, is obtained by using a low-pass to low-pass transform. This is also carried

Chapter 33The z-Transform

629

out by substituting variables, just as with the bilinear transform. We start by writing the transfer function of the unity cutoff filter, and then replace each z-1 with the following:

EQUATION 33-12

Low-pass to low-pass transform. This is a method of changing the cutoff frequency of low-pass filters. The original filter has a cutoff frequency of unity, while the new filter has a cutoff frequency of W, in the range of 0 to B.

z & 1 6 z & 1 & k

1 & kz & 1

	where:	sin (1/2 & W /2 )
	k '
		sin (1/2 % W /2 )

This provides the transfer function of the filter with the new cutoff frequency. The following design equations result from applying this substitution to the biquad, i.e., no more than two poles and two zeros:

EQUATION 33-13

Low-pass to low-pass conversion. The recursion coefficients of the filter with unity cutoff are shown in italics. The coefficients of the low-pass filter with a cutoff frequency of W are in roman.

a0 ' ( a0 & a1k % a2k 2 ) / D

a1 ' (& 2a0k % a1 % a1k 2 & 2a2k ) / D a2 ' ( a0k 2 & a1k % a2 ) / D

b1 ' ( 2 k % b1 % b1k 2 & 2b2k ) / D b2 ' (& k 2 & b1k % b2 ) / D

where:

D ' 1 % b1 k & b2 k 2 k ' sin (1/2 & W/2 )

sin (1/2 % W/2 )

Low-pass to High-pass Frequency Change

The above transform can be modified to change the response of the system from low-pass to high-pass while simultaneously changing the cutoff frequency. This is accomplished by using a low-pass to high-pass transform, via the substitution:

EQUATION 33-14

Low-pass to high-pass transform. This substitution changes a low-pass filter into a high-pass filter. The cutoff frequency of the low-pass filter is one, while the cutoff frequency of the highpass filter is W.

z & 1 6 & z & 1 & k 1 % k z & 1

where:

k ' & cos( W/2 % 1/2 ) cos ( W/2 & 1/2 )

As before, this can be reduced to design equations for changing the coefficients of a biquad stage. As it turns out, the equations are identical