Transcript 投影片 1 - I-Shou University

IIR Filter Design:
Basic Approaches
Most common approach to IIR filter design:
(1) Convert specifications for the digital filter
H D (z) into equivalent specifications for an
analog prototype lowpass filter H a (s )
(2) Determine the analog lowpass filter transfer
function H a (s )
(3) TransformH a (s ) into the desired digital
transfer function H D (z )
Digital Filter Design:
Basic Approaches
• An analog transfer function to be
denoted as
Pa ( s )
H a (s) 
Da ( s )
where the subscript “a” specifically
indicates the analog domain
• A digital transfer function derived from H a (s )
will be denoted as
P( z )
H D ( z) 
D( z )
Digital Filter Design:
Basic Approaches
• Basic idea behind the conversion of H a (s )
into H D (z )is to apply a mapping from the sdomain to the z-domain so that essential
properties of the analog frequency response
are preserved
• Thus mapping function should be such that:
– Imaginary j axis in the s-plane be
mapped onto the unit circle of the z-plane
– A stable analog transfer function be
mapped into a stable digital transfer
function
S plane to Z plane mapping
analog
digital



s-plane

H A (s)




z-plane
H

D
(z )
Preserve stability: Pole in the right half plan should map
inside the circle in the z plan.
Euler Approximation
s  1 Tz
s


1
Ts
s
1
T  1
s
Fs
Is the sampling interval
Im
Im

1
Ts
Re
z
1
2
z  plane
s  plane
s-plane
1
Re
IIR Filter Design by Bilinear Transformation
(1) Design Concept
- s-plane to z-plane conversion
s  plane
j axis
z  plane
u.c
- any mapping than maps stable region is s-plane (left half plane)
to stable region in z-plane (inside u.c) ?
Td
1
s
1 1  z 1 or
2
z

s


bilinear transform!
Td
Td 1  z 1
1
s
2
* Td inserted for convention may put to any convenient value
for practical use.
(2) Properties
Td 
2
z
T
1 j d
2
1 j
if s  jΩ ,
 |z|  1
2 1  e j
2
ω
s
 j
tan
j
Td 1  e
Td
2
j
if z  e ,

2
ω

t an
Td
2



  2 tan 1
Td
2
IIR Digital Filter Design: Bilinear
Transformation Method
• Bilinear transformation
1 

* T inserted for convention may put to
1

z
2
s  
1 
T  1  z  any convenient value for practical use.
• Above transformation maps a single
point in the s-plane to a unique point in
the
z-plane and vice-versa
H D (z ) and
Ha (s)
• Relation between
is
then given by
H D ( z )  Ha ( s)
 1 z 1 

s 2 
T  1 z 1 


Bilinear Transformation
• Digital filter design consists of 4
steps:
(1) Develop the specifications of HD(z)
(2) Develop the specifications of Ha (s )
(3) Design Ha (s )
(4) Determine HD(z) by applying bilinear
transformation to Ha (s )
* IIR Filter Design Procedure
3
Given specification in digital domain
Convert it into analog filter specification
Design analog filter (Butterworth, Chebyshov, elliptic):H(s)
4
Apply bilinear transform to get H(z) out of H(s)
1
2


s
3
p
2
| H ( j) |
1
1
1  2
1
A
2
ω

t an
Td
2

j
1
| H (e ) |
4
1
H ( z )  H ( s)
1  2
1
A
1
 p s


1 1 z 1
s 
Td 1 z 1
• Design a digital filter equivalent of a 2nd order
Butterworth low-pass filter with a cut-off frequency fc =
100 Hz and a sampling frequency fs = 1000 samples/sec.
• The normalised cut-off frequency of the digital filter is
given by the following equation:
• the equivalent analogue filter cut-off frequency ωac, The
value of K is immaterial so let K = 1.
• H(s) for a Butterworth filter is:
• Hence the transfer function of the Butterworth filter
becomes:
• Next, convert the analogue filter into an equivalent digital filter by
applying the bilinear z-transform. This is achieved by making a
substitution for s in the transfer function.
• The finite difference equation of the filter is found by inverting the
transfer function
Direct form 2nd order
http://ccrma.stanford.edu/~jos/filters/Direct_Form_II.html
Direct realisation for a 2nd order Butterworth equivalent filter.
Matlab Bilinear
• a=1;
• b=[1, 1.141, 1];
• [c, d]=bilinear(a, b, 1000);