Digital Signal Processing Reference
In-Depth Information
('Specify stopband ripple in dB below passband gain.')
z_p_k='The zeros, poles, and multiplying constant.'
[z,p,k]=ellipap(N,Rp,Rs),pause
num_den='The numerator end denominator coefficients.'
[num,den]=zp2tf(z,p,k),pause
[mag,phase,w]=bode(num,den);
plot(w,mag)
title(['Elliptic filter frequency response. Order',...
num2str(N), ' Rp = ',num2str(Rp),...
'dB Rs = -',num2str(Rs),'dB'])
xlabel('omega')
ylabel('Magnitude')
■
Handbooks on filter design and textbooks on filter theory provide detailed
instructions on the design and implementation of Butterworth, Chebyschev, and
elliptic filters [2-5].
The filters we have examined are described primarily as low-pass filters. By using a
nonlinear frequency transformation, low-pass filter designs can be transformed into
bandpass filters [2]. The frequency-response function of the bandpass filter can be
found from
`
H
B
(v) = H
L
(v
L
)
,
(6.12)
v
L
=v
c
(v
2
- v
u
v
l
)/v(v
u
-v
l
)
where is the frequency response function of the low-pass filter to be trans-
formed. The frequency variable of the low-pass filter has been designated as
is the cutoff frequency of the low-pass filter. The upper and lower cutoff frequencies
of the bandpass filter are denoted by
H
L
(v
L
)
v
L
; v
c
v
u
and
v
l
,
respectively.
Transformation of a low-pass Butterworth filter into a bandpass filter
EXAMPLE 6.10
We will now apply the transformation equation (6.12) to the design of a bandpass filter with
an upper cutoff frequency of 4 krad/s and a lower cutoff frequency of 100 rad/s. We will trans-
form the first-order Butterworth (
RC
low-pass) filter. From (6.1),
1
1 + jv
L
/
v
c
H
L
(v) =
.
Therefore, from (6.12),
1
1
H
B
(v) =
=
- 4 * 10
5
)/v(3.9 * 10
3
)
.
1 + j(v
2
1 + j(v
2
- v
u
v
l
)/v(v
u
- v
l
)
