Equiripple Filters
A filter which has the Smallest Maximum Approximation Error among all filters over the
frequencies of interest:
Define:
e ( h )  max
 S
| E ( ) |
H ( )
0
where
p
s

E ( )  H d ( )  H ( )

Fact: filters with the smallest maximum deviation from ideal characteristic are
equiripple..
| H (  )|
same ripple
They are computed as follows:
B=firpm(N,F,M)
F=[F(1),F(2),…],
0  F (k )  1
M=[M(1), M(2), …];
F=1 corresponding to
N = filter order.


M
F (1)
F (2)
F ( 3)
F (4)
F
Fs / 2
Fact: the error
frequencies
such that
E (  )  H (  )  H d (  ) is miminal in minmax sense, provided there exist L+2
 1 
E (
i 1
2
 ...  
L2
,  i S
)   E ( i )
| E ( i )|  m ax 
S
| E ( )|
E ( )
S
H ( )
H d ( )
E ( 1)
E ( 3 ) E ( 5 )
E (
2
)
E ( 4 )
E ( 6 )
E ( 7 )
E ( 9 )
E ( 8 ) E (
E (
10
11
)
)
E (
12
)
Example: see the same example we saw for the FIR filter with window.
Recall the specs:
1. Pass band 0  4 kHz
2. Stopband
 5 kHz
3. Sampling Frequency
Fs  20 kHz
Apply Remez Algorithm. You have to determine the filter order a priori, and let’s choose the same
order N=81:
h=firpm(80,[0,0.4,0.5,1],[1,1,0,0]);
h(n)
0 .5
0 .4
0 .3
The impulse response obtained is shown.
0 .2
The frequency response is shown in the next slide,
compared with the one we obtained by using the
hamming window. Notice that the attenuation in the
stopband is higher in the equiripple.
0 .1
0
-0 .1
0
20
40
60
80
100
H ( )
20
0
Equiripple (Remez Algorithm)
-2 0
-4 0
-6 0
-8 0
-1 0 0
-1 2 0
0
H ( )
0 .5
1
1 .5
0 .5
1
1 .5
2
2 .5
3
3 .5

20
0
Hamming window
-2 0
-4 0
-6 0
-8 0
-1 0 0
-1 2 0
0
2
2 .5
3
3 .5

Frequency Response of the Non Ideal LPF
| H ( ) |
1 1
ripple
1 1
2
 P  STOP
stop
pass
f
stop
transition region
LPF specified by:
• passband frequency  P
• passband ripple
1
or R P  20 log 10
1  1
1  1
dB
• stopband frequency  STOP
• stopband attenuation  2 or R S  20 log 10  2 dB
attenuation
Best Design tool for FIR Filters: the Equiripple algorithm (or Remez). It
minimizes the maximum error between the frequency responses of the
ideal and actual filter.
| H ( ) |
1 1
ripple
1 1
2
1
2
attenuation

h  firpm  N ,  0,  1 ,  2 ,   /  , 1,1, 0, 0  ,  w1 , w 2  
h  h [ 0 ],..., h [ N ] 
 / w1
 ~
1

 / w2
0
1  2
3  
Linear Interpolation
20 log 10 (  2 )
22

2
N
Example: Low Pass Filter
Passband
 P  0.4
Stopband  S
 0.5
Choose order
N 
with attenuation 40dB
40  2
22  (0.5  0.4 )
 37
20
| H ( ) |
0
-20
Almost 40dB!!!
-40
-60
-80
-100
-120
0
0.5
1
1.5
2
2.5
3
3.5

Example: Low Pass Filter
Choose order N=40 > 37
| H ( ) |
10
0
-10
-20
OK!!!
-30
-40
-50
-60
-70
-80
0
0.5
1
1.5
2
2.5
3
3.5