Transcript Slide 1

1
Lecture 8: LTI filter types
Instructor:
Dr. Gleb V. Tcheslavski
Contact:
[email protected]
Office Hours:
Room 2030
Class web site:
http://ee.lamar.edu/gleb/ds
p/index.htm
ELEN 5346/4304
DSP and Filter Design
Fall 2008
2
Types of LTI IIR filters
Note: we are interested in BIBO: |H()|, H()
1. First-order Lowpass
H LP (e j 0 ) 
2K
1
H LP ( z ) 
K 1  z 1 
1 z
(8.2.1)
1
H LP (e j )  0
(8.2.2)
(8.2.3)
Monotonically decreases with 
It is typical to have a maximum magnitude of 1, i.e. a gain of 0 dB. Therefore:
1   1  z 1
H LP ( z ) 
2 1   z 1
Where for stability: || < 1
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(8.2.4)
3
Types of LTI IIR filters
Therefore:
 j
1


1

e
H LP (e j ) 
2 1   e j
(8.3.1)
A first-order system can also
be expressed as:
b0  b1e j
H LP (e ) 
a0  a1e j
Apparently, here:
(8.3.2)
1
a0  1;a1  
(8.3.3)
1
b0  b1 
2
ELEN 5346/4304
DSP and Filter Design
(8.3.4)
= 0.7
zplane
Fall 2008
Imaginary Part
j
0.5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
4
Types of LTI IIR filters
A square magnitude function can be evaluated as a square of (8.3.1):
1    1  cos  
2
2
H LP (e j ) 
2 1    2 cos  
(8.4.1)
2
Its derivative with respect to frequency
d H LP (e j )
d
2
 1    1  2   2  sin 
2

2 1    2 cos  
2
2
is always non-positive, proving that the frequency response monotonically
decreases.
The passband of a LPF is usually defined by the frequency range from 0 to
c, called the 3-dB cutoff frequency. Here the gain of -3 dB is with respect to
the gain at  = 0.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(8.4.2)
5
Types of LTI IIR filters
To determine the c, we equate the squared magnitude to ½:
1    1  cos c 
2
2
H LP (e j ) 
2 1    2 cos c 
2

1
2
1    1  cos c   1   2  2 cos c
2
(8.5.1)
(8.5.2)
The last equation can be solved either for c or :
2
cos c 
1  2
1  sin c

cos c
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(8.5.3)
(8.5.4)
6
Types of LTI IIR filters
2. First-order Highpass
For a maximum magnitude of 1, i.e. a gain of 0 dB:
1   1  z 1 1   1  z 1
H HP ( z ) 

1
2 1  z
2 1   z 1
Where for stability: || < 1, c and  can be found by (8.5.3) and (8.5.4)
 = 0.7
Imaginary Part
1
0.5
0
-0.5
-1
-1
-0.5
0 0.5
Real Part
ELEN 5346/4304
1
DSP and Filter Design
Fall 2008
(8.6.1)
7
Types of LTI IIR filters
3. Second-order Bandpass
A bandpass filter cannot be obtained by a first-order real-coefficient transfer
function. The lowest order transfer function must have a pair of complex
conjugate poles and zeroes at z = +1 and z = -1.
H BP ( z ) 
K 1  z 2 
1   1    z   z
1
2
(8.7.1)
This transfer function has a pair of complex conjugate poles
p1,2   e
  1  
 j cos1 

 2  
Here, for stability: || < 1 and || < 1
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(8.7.2)
8
Types of LTI IIR filters
For a maximum magnitude of 1:
1
1  z 2
H BP ( z ) 
2 1   1    z 1   z 2
Or:
(8.8.1)
b0  b1e  j  b2 e  j 2
1
1  e  j 2
H BP (e ) 

 j
 j 2
2 1   1    e   e
a0  a1e  j  a2 e  j 2
(8.8.2)
1
;b1  0
2
(8.8.3)
j
Where:
a0  1;a2    1    a2   ;b0  b2 
The center frequency of the IIR BPF can be found as:
0  cos1   
(8.8.4)
The 3-dB bandwidth (the difference between 3-dB cutoff frequencies) is:
 2 
BW  c 2  c1  cos 1 
2 
1




ELEN 5346/4304
DSP and Filter Design
Fall 2008
(8.8.5)
9
Types of LTI IIR filters
The quality factor:
Q
0
= 0.6
 = 0.5
Imaginary Part
1
0.5
0
-0.5
(8.9.1)
BW
-1
-1
ELEN 5346/4304
DSP and Filter Design
Fall 2008
-0.5
0
Real Part
0.5
1
10
Types of LTI IIR filters
4. Resonator
H r ( z) 
1  re
b0
j0
z
1
1  re
 j0
z
1


b0
1   2r cos 0  z  r z
1
2 2
(8.10.1)
poles
For a maximum gain of 1:
b0  (1  r ) 1  r 2  2r cos 20
a1    2r cos 0  (8.10.3)
a2  r 2
(8.10.4)
BW  2(1  r )
(8.10.5)
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(8.10.2)
11
Types of LTI FIR filters
H r ( z) 
b0
b0
b0


1  p1z 1 1  p1* z 1  1  re j0 z 1 1  re j0 z 1 1  a1z 1  a2 z 2



(8.11.1)
a1  2r cos 0 ;a2  r 2
(8.11.2)
b0 r n
The impulse response: hn 
sin  (n  1)0  un
sin 0
(8.11.3)
where
1
0.8
1
0.5
0.6
o
r = 0.9
0 = 0.2
2
0
|H ()|
Imaginary Part
r = 0.9
r = 0.8
r = 0.99
0.4
-0.5
0.2
-1
-1
-0.5
0
0.5
Real Part
1
0
0
0.2
0.4
0.6
x 
ELEN 5346/4304
DSP and Filter Design
Fall 2008
0.8
1
12
Types of LTI FIR filters
5. Sinusoidal oscillator – a resonator with poles on the uc
H r ( z) 
1 e
b0
j0
z
1
1  e
 j0
z
1


b0
1  a1 z 1  z 2
(8.12.1)
a1  2cos 0
where
(8.12.2)
b0  A sin 0 is:
The impulse response, assuming
(8.11.3)
hn  A sin (n  1)0  un
1
(8.11.4)
0.5
0.8
0.6
o
0 = 0.2
2
0
|H ()|
Imaginary Part
1
0.4
-0.5
0.2
-1
-1
ELEN 5346/4304
-0.5
0
0.5
Real Part
DSP and Filter Design
0
1
0
0.2
0.4
0.6
x 
Fall 2008
0.8
1
13
Types of LTI IIR filters
6. Notch filter
For a maximum magnitude of 1:



b0 1  e j0 z 1 1  e j0 z 1
1 
1  2 z  z
H BP ( z ) 

1
2
2 1   1    z   z
1  re j0 z 1 1  re j0 z 1
1
2



(8.13.1)
Here, for stability: || < 1 and || < 1
The transfer function has a zero at the notch frequency 0 = cos-1()
= 0.6
 = 0.5
Imaginary Part
1
0.5
0
-0.5
-1
-1
ELEN 5346/4304
DSP and Filter Design
Fall 2008
-0.5 0 0.5
Real Part
1
14
Group (envelope) delay
So far, we discussed a magnitude of frequency response only. It turns out that
the phase of system frequency response is of importance too.
The derivative of phase (of system’s frequency response) with respect to
frequency has units of time and is called a group (envelope) delay:
d H (e j )
 g ( )  
d
(8.14.1)
It is a time delay that a signal component of frequency  undergoes as it passes
through the system. When phase is linear, the group delay is a constant;
therefore, all signal components are delayed by the same time  no phase
distortions – design goal…
IIR filters, in general, do not have linear phase!
Additionally, we may need to compensate for group delays introduced
by other filters; therefore…
ELEN 5346/4304
DSP and Filter Design
Fall 2008
15
Types of LTI IIR filters
7. Allpass filter – used in phase equalizers
H AP  e j   1;0    
N
Example:
H AP ( z ) 
a z
k 0
N
*  N k
k
k
a
z
 k
,a0  1
(8.15.1)
(8.15.2)
k 0
N
Specifying the polynomial
A( z )   ak z  k ,a0  1
(8.15.3)
k 0
that has roots at z = z0
Therefore
ELEN 5346/4304
DSP and Filter Design
H AP ( z )  z  N
A* 1 z * 
Fall 2008
A z 
(8.15.4)
16
Types of LTI IIR filters
z  z0  re j0 which must correspond to a BIBO system.
1
1
1
1 j0

z

z



e - reciprocal!
H(z) also has zeros at
0
 j0
*
*
z
z0
re
r
H(z) has poles at
In general:
1
1
*
z 1   k NC  z   k   z   k 
H AP ( z )  
1 
1
* 1



1


z
1


k 1 1   k z
k 1 
k
k z 


NR
(8.16.1)
For stability: r < 1, and the group delay is always non-negative (causal system!)
 j0
j
e

r

e
H AP (e j ) 
1  re j0 e j
(8.16.2)
2
1

r
 AP (e j )    2 tan 1
1  r 2  2r cos(  0 )
(8.16.3)
j
2
d

(
e
)
1

r
 g (e j )  AP

d
1  r 2  2r cos(  0 )
(8.16.4)
A first-order filter:
ELEN 5346/4304
DSP and Filter Design
Fall 2008
17
Cascade of filters
Note: by cascading simple filters, we can design filters with sharper
magnitude response; for example, a cascade of K identical first order
LPFs will result in a system with the overall transfer function
1   1  z 1 
GLP ( z )  
1 
 2 1 z 
For stability:
K 1
2 K 2


K
1  1  2  cos c  2  2 K sin c

 
K 1
1  2 K  cos c
A frequency response of a single bandpass
IIR, a cascade of two, and a cascade of
three identical bandpass IIR sections:
 = 0.2;  = 0.34.
ELEN 5346/4304
K
DSP and Filter Design
Fall 2008
(8.17.1)
(8.17.2)
18
Types of LTI IIR filters
8. Comb filter
Starting with a simple filter
We form
1  z1 z 1
H ( z)  K
1  p1 z 1
(8.18.1)
1  z1 z  L
K
1  p1 z  L
(8.18.2)
;l  0,1,...L  1
(8.18.3)
;l  0,1,...L  1
(8.18.4)
Gc ( z )  H ( z ) z  z L
New zeros at
New poles at
1
L
1
zl  z e
1
L
1
j
pl  p e
2 l
L
j
2 l
L
A comb filter is easy to concentrate on harmonics.
We can emphasize or attenuate them.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
19
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
|G ()|
0.5
c
|H()|
Types of LTI IIR filters
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
x 
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1.2
1.4
1.6
1.8
2
1
0.5
z1 = 0.6
p1 = -0.5
0
-0.5
L=5
-1
Imaginary Part
Imaginary Part
1
0.5
0
-0.5
-1
-1
ELEN 5346/4304
1
x 
-0.5 0 0.5
Real Part
DSP and Filter Design
1
-1
Fall 2008
-0.5
0 0.5
Real Part
1
20
Types of LTI FIR filters
1. First-order Lowpass
1
z 1
1
H 0 ( z )  1  z  
2
2z
H 0 (e j )  e
3 dB cutoff frequency:
c 
j


2
 
cos  
2
(8.20.3)
2
Imaginary Part
1
The phase
characteristic of this
filter is linear.
0.5
0
-0.5
-1
-1
ELEN 5346/4304
-0.5
0
0.5
Real Part
DSP and Filter Design
1
Fall 2008
(8.20.1)
(8.20.2)
21
Types of LTI FIR filters
For a cascade of M first-order FIR LPFs, the cutoff frequency will be
  21M 
c  2cos  2 


1
Simple FIR filters are inexpensive
to implement. Much better
approximations of ideal frequency
response can be obtained by
higher order FIR filters.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(8.21.1)
22
Types of LTI FIR filters
2. First-order Highpass
1
H1 ( z )  1  z 1 
2
H1 (e j )  e
3 dB cutoff frequency:
Imaginary Part
1
c 
j


2
 
sin  
2
(8.22.3)
2
A cascade of filters
will make frequency
characteristic
better…
0.5
0
-0.5
-1
-1
ELEN 5346/4304
-0.5
0
0.5
Real Part
DSP and Filter Design
1
Fall 2008
(8.22.1)
(8.22.2)
23
Types of LTI FIR filters
3. Notch



H ( z )  b0 1  e j0 z 1 1  e  j0 z 1  b0 1  2 cos 0 z 1  z 2 
This filter is NOT distortionless!
-20
-40
-60
-80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized Frequency ( rad/sample)
0.8
0.9
1
“don’t care region” 
150
Phase (degrees)
We will call it a Generalized
Linear Phase (GLP) FIR.
100
1
50
0
-50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized Frequency ( rad/sample)
0.8
0.9
1
Imaginary Part
Magnitude (dB)
0
(8.23.1)
0.5
2
0
-0.5
-1
1800
ELEN 5346/4304
DSP and Filter Design
phase shift
-1
Fall 2008
-0.5 0 0.5
Real Part
1
24
Types of LTI FIR filters
4. Moving Average (MA) filter
1 zM
H MA ( z ) 
M 1  z 1 
0
0.5
M = 10
9
0
-0.5
-1
-0.5
0
0.5
Real Part
1
-20
-40
-60
-80
DSP and Filter Design
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency ( rad/sample)
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency ( rad/sample)
0.9
1
0
-100
-200
ELEN 5346/4304
0
100
Phase (degrees)
-1
this pole is
cancelled
by a zero
Magnitude (dB)
1
Imaginary Part
(8.24.1)
Fall 2008
25
Types of LTI FIR filters
5. Comb
Comb filters can be generated from LP prototypes:
G0 ( z )  H 0 ( z L ) 
1
1 zL 

2
(8.25.1)
G1 ( z )  H1 ( z L ) 
1
1 zL 

2
(8.25.2)
From HP prototypes:
or from more complicated prototype filters, such as MA:
1  z  LM
GMA ( z )  H MA ( z ) 
M 1  z  L 
L
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(8.25.3)
26
Types of LTI FIR filters
0.5
-50
-100
5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
ELEN 5346/4304
1
DSP and Filter Design
Phase (degrees)
Imaginary Part
1
0
1
0
0.2
0.4
0.6
0.8
Normalized Frequency ( rad/sample)
1
100
0
-100
Imaginary Part
Magnitude (dB)
L=5
0.5
5
0
-0.5
-1
0
0.2
0.4
0.6
0.8
Normalized Frequency ( rad/sample)
Fall 2008
1
-1
-0.5
0
0.5
Real Part
1