Analog and Digital Filters
585
R.6.94 The resulting relation between the analog frequency w and the discrete frequency
W is given by
W (2 T) tan(W/2)
ϭ ր *
(a nonlinear frequency transformation)
or
W 2 arctan(w T/2)
ϭ *
*
R.6.95 The MATLAB function unwrap(phase) unwraps the phase angle (in radians) and limits the phase over the range of interest given by −π ≤ W ≤ +π.
R.6.96 The other transformations used in the design of LP, HP, BP, or BS fi lters from the
analog prototypes into the digital ones are given as follows:
a. LP: s = (2/T) * [(z − 1)/(z + 1)], w = (2/T) * tan(W/2)
b. HP: s = (2/T) * [(z + 1)/(z − 1)], w = −(2/T) * cot(W/2)
c. BP: s = (2/T) * [(z
2
− 2 * c * z + 1)/(z
2
− 1)], w = (2/T) * [(c − cos(W))/(sin(W))]
d. BS: s = (2/T) * [(z
2
− 1)/(z
2
− 2 * c * z + 1)], w = (2/T) * [(c − cos(W))/(sin(W))]
−1
R.6.97 An example of a bilinear transformation is provided as follows: Let us transform
the fi rst-order analog (RC) LPF illustrated by the circuit diagram of Figure 6.6 into
a digital fi lter. Recall that the fi lter transfer function is given by
H w
V jw
V jw
jwC
jwC
R
R C
j w
R C
o
i
( )
( )
( )
*
*
*
ϭ
ϭ
ϩ
ϭ
ϩ
1
1
1
1
without any loss of generality let R = C = 1 and s = jw, then
H s
V s
V s
s
o
i
( )
( )
( )
ϭ
ϭ ϩ
1
1
Assuming that the sampling period is T = 2 and applying the corresponding
bilinear transformation defi ned in R.6.96 yields
H s
z
z
z
z
ϭ
Ϫ
ϩ
ϭ Ϫ
ϩ
ϩ
1
1
1
1
1
1
then the discrete transform becomes
H(z)
z
ϭ
ϩ
0.5 0.5 *
1
Ϫ
Note that in this simple example the resulting digital fi lter is an FIR. A more
complex analog fi lter transformation could result in an IIR fi lter. Hand computations can be avoided by using MATLAB as indicated in R.6.98.
R.6.98 The MATLAB function [numz, denz] = bilinear (numa, dena, 1/T s ) returns the discrete polynomial coeffi cients numz and denz (numerator and denominator of H(z))
arranged in descending powers of z, where numa and dena are the analog numerator and denominator of H(s), where T s indicates its sampling rate.
CRC_47760_CH006.indd 585
CRC_47760_CH006.indd 585
7/23/2008 2:26:37 PM
7/23/2008 2:26:37 PM
585
R.6.94 The resulting relation between the analog frequency w and the discrete frequency
W is given by
W (2 T) tan(W/2)
ϭ ր *
(a nonlinear frequency transformation)
or
W 2 arctan(w T/2)
ϭ *
*
R.6.95 The MATLAB function unwrap(phase) unwraps the phase angle (in radians) and limits the phase over the range of interest given by −π ≤ W ≤ +π.
R.6.96 The other transformations used in the design of LP, HP, BP, or BS fi lters from the
analog prototypes into the digital ones are given as follows:
a. LP: s = (2/T) * [(z − 1)/(z + 1)], w = (2/T) * tan(W/2)
b. HP: s = (2/T) * [(z + 1)/(z − 1)], w = −(2/T) * cot(W/2)
c. BP: s = (2/T) * [(z
2
− 2 * c * z + 1)/(z
2
− 1)], w = (2/T) * [(c − cos(W))/(sin(W))]
d. BS: s = (2/T) * [(z
2
− 1)/(z
2
− 2 * c * z + 1)], w = (2/T) * [(c − cos(W))/(sin(W))]
−1
R.6.97 An example of a bilinear transformation is provided as follows: Let us transform
the fi rst-order analog (RC) LPF illustrated by the circuit diagram of Figure 6.6 into
a digital fi lter. Recall that the fi lter transfer function is given by
H w
V jw
V jw
jwC
jwC
R
R C
j w
R C
o
i
( )
( )
( )
*
*
*
ϭ
ϭ
ϩ
ϭ
ϩ
1
1
1
1
without any loss of generality let R = C = 1 and s = jw, then
H s
V s
V s
s
o
i
( )
( )
( )
ϭ
ϭ ϩ
1
1
Assuming that the sampling period is T = 2 and applying the corresponding
bilinear transformation defi ned in R.6.96 yields
H s
z
z
z
z
ϭ
Ϫ
ϩ
ϭ Ϫ
ϩ
ϩ
1
1
1
1
1
1
then the discrete transform becomes
H(z)
z
ϭ
ϩ
0.5 0.5 *
1
Ϫ
Note that in this simple example the resulting digital fi lter is an FIR. A more
complex analog fi lter transformation could result in an IIR fi lter. Hand computations can be avoided by using MATLAB as indicated in R.6.98.
R.6.98 The MATLAB function [numz, denz] = bilinear (numa, dena, 1/T s ) returns the discrete polynomial coeffi cients numz and denz (numerator and denominator of H(z))
arranged in descending powers of z, where numa and dena are the analog numerator and denominator of H(s), where T s indicates its sampling rate.
CRC_47760_CH006.indd 585
CRC_47760_CH006.indd 585
7/23/2008 2:26:37 PM
7/23/2008 2:26:37 PM