Hossein Sameti
Department of Computer Engineering
Sharif University of Technology

Definition of generalized linear-phase (GLP):
H ()  H m ()e j (   )

Let’s focus on Type I FIR filter:
  0,
N  2L  1(odd ), h(n)  h( N  n  1), ( L   )
• It can be shown that
L

H m ( )   a(n) cos(n) G( )
n0
a(0)  h( L)


a ( n )  2 h ( L  n ) n  0
H ( )  G ( )e  j
(L+1) unknown
parameters a(n)
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
2
1  1
1  1
2
 2
• Given
0
p
s


 s ,  p ,  1 ,  2 determine coefficients of G(ω) (i.e. a(n))
such that L is minimized (minimum length of the filter).
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
3
L

H m ( )   a(n) cos(n) G( )
n0



G(ω) is a continuous function of ω and is as many
times differentiable as we want.
How many local extrema (min/max) does G(ω) have
in the range [0,  ] ?
In order to answer the above question, we have to write
cos(ωn) as a sum of powers of cos(ω).
cos(2)  2 cos2 () 1
cos(3)  cos(2   )  cos(2) cos  sin(2) sin 
cos(3)  4 cos3   3 cos
cos(n )
: sum of powers of cos(ω)
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
4
n
cos(n )   i (cos  )
i
i 0

L
H m ( )   a(n) cos(n) G( )
n0
L
n
n 0
L
i 0
i
G( )   a(n)[  i (cos  ) ]
G ( )    (n)(cos  ) n
n 0
dG( )
0
d
Find extrema
L
  (n)n(cos  )
n 0
n 1
(sin  )  0
L
(sin  )   (n)n(cos  ) n 1  0
n 0
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
5
L
(sin  )   (n)n(cos  ) n 1  0
n 0
cos   x
sin   0    0 or 
 L
  (n)n(cos  ) n 1  0
 
n 0
Polynomial of degree L-1
Maximum of L-1 real zeros
Max. total number of real zeros: L+1
Conclusion: The maximum number of real zeros for dGd( )
(derivative of the frequency response of type I FIR filter) is
N 1
L+1, where L  2 (N is the number of taps).
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
6
1  1
1  1
2
 2
0
p
s


Problem A
• Given  s ,  p ,  1 ,  2 determine coefficients of G(ω) (i.e. a(n))
such that L is minimized (minimum length of the filter).
Problem A
Problem B
Problem C
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
7
1
• Given s ,  p , L, K 
2
determine coefficients of G(ω)
(i.e. a(n)) such that
 2 is minimized.
s ,  p , 1 ,  2
1
Compute K 
2
Guess L
Algorithm B
Increase L
by 1
a' (n), 2'
 2'   2
?
'
2 2
Decrease L
by 1
 2'   2
Yes
Stop!
8
Define F as a union of
closed intervals in [0,  ]
I2
I1
0
I1:[0,  p ]
p
s


I 2:[s ,  ]
F  I1  I 2
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
9

E ( )  W ( )[G( )  D( )] where
 1

W ( )   K

1
I1
W is a positive weighting function
I2
L
G ( )   a (n) cos(n)
n 0
 1 I1
D( )  
0 I 2
Desired frequency response
Find a(n) to minimize   Max E ( )
F
F  I1  I 2
(same assumption as Problem B)
10

We start by showing that   Max E ( )   2
F

E ( )  W ( )[G( )  D( )]
E ( )
 [G( )  D( )]
W ( )
E ( )
G( )  D( ) 
W ( )
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
11
F  I1  I 2
By definition: Max E ( )  
F
E ( )
I2
I1



p

s
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
12
 1

By definition:
W ( )   K

1
E ( )
W ( )
Max E ( )  
I1
F
I2

K
 K



p
s
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
13
E ( )
G( )  D( ) 
W ( )
 1 I1
D( )  
0 I 2
G( )
1  K
1  K


p
s


Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
14
G( ) in Problem C
1  K
K
1
2
  2
1  K


p
1  1

G( ) in Problem B
1  1
2
 2
s

p
s

15

Conclusion:
Problem B:
Find a(n) such that  2 is minimized.
Problem C:
Find a(n) such that  is minimized.
Problem B= Problem C
Problem A= Problem C
 We now try to solve Problem C.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
16
Assumptions:
 F: union of closed intervals
 G(x) to be a polynomial of order L:
L
G ( x)   ak x k
k 0


D = Desired function that is continuous in F.
 1 I1
D( )  
W= positive function
0 I 2
 1
E ( x)  W ( x)[ D( x)  G ( x)]

I1
W
(

)

K


I2
E  max E ( x )
1
xF
17

The necessary and sufficient conditions for G(x) to be
unique Lth order polynomial that minimizes E
is that E(x) exhibits at least L+2 alternations, i.e.,
there are at least L+2 values of x such that
x1  x2  ...  xL  2
𝑬 𝒙𝒊 = −𝑬 𝒙𝒊+𝟏 =
E (x )
+
𝑬
−
for a polynomial
of degree 4
 E
 E
 E
 E
 E
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
 E
18
• Recall G(ω) can have at most L+1 local extrema.
• According to the alternation theorem, G(ω) should have at
least L+2 alternations(local extrema) in F.
Contradiction!?
F  I1  I 2
I1
I2
19
Ex: Polynomial of degree 7
•  s ,  p can also be alternation frequencies, although they are
not local extrema.
•G(ω) can have at most L+3 local extrema in F.
F  I1  I 2
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
20



According to the alternation theorem, we have at least
L+2 alternations.
According to our current argument, we have at most
L+3 local extrema.
Conclusion: we have either L+2 or L+3 alternations in
F for the optimal case.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
21
Extra-ripple case
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
22
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
23


For Type I low-pass filters, alternations always occur at  s ,  p
If not, we potentially lose two alternations.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
24
Equi-ripple except possibly at   0, 
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
25
 For optimal type I low-pass filters, alternations always
occur at  s ,  p
If not, two alternations are lost and the filter is no longer
optimal.
 Filter will be equi-ripple except possibly at   0, 
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
26
1
• Given s ,  p , L, K 
determine coefficients of G(ω) (i.e. a(n))
2
such that  2 is minimized.
At alternation frequencies, we have:
𝑬 𝝎𝒊
+
= 𝜹𝟐
−
E (i )   E (i 1 )
i  1,2,..., L  2
i  1,2,..., L  1
E ( )  W ( )[ D( )  G( )]
W (i )[ D(i )  G (i )]  (1) i 1 2
(1) i 1  2
G (i ) 
 D (i ) i  1,2,..., L  2
W (i )
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
27
L
( Eq.1) G (i )   a (n) cos(i n)
n 0
(1)i 1 2
( Eq.2) G (i ) 
 D(i )
W (i )
Equating Eq.1
and Eq.2
(1)i 1 2
 D(i )
 a(n) cos(i n) 
W (i )
n 0
L
i  1,2,..., L  2
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
28
(1)i 1 2
 D(i ) i  1,2,..., L  2
 a(n) cos(i n) 
W (i )
n 0
L
i  1  a(0) cos(1.0)  a(1) cos(1.1)  a(2) cos(1.2)  ....  a( L) cos(1.L) 
(1)11  2

 D(1 )
W (1 )
i  2  a(0) cos(2 .0)  a(1) cos(2 .1)  a(2) cos(2 .2)  ....  a( L) cos(2 .L) 

(1) 2 1  2

 D ( 2 )
W ( 2 )
i  L  2  a(0) cos( L2 .0)  a(1) cos(L2 .1)  a(2) cos(2 .2)  ....  a( L) cos(L2 .L) 
(1) L  21 2

 D( L  2 )
W ( L  2 )
29
é cos w .0
( 1 ) cos (w1.1) cos(w1.2)
ê
ê cos (w .0)
cos (w 2 .1)
cos (w 2 .2)
2
ê
ê
ê
ê
ê cos (w L+2 .0) cos (w L+2 .1) cos (w L+2 .2)
êë
 (1)11 
2 

 W (1 ) 
 (1) 21 
2 

  W ( 2 ) 







 (1) L  21 
2

 W ( L  2 ) 
 D(1 ) 
 D( ) 
2 








 D( L  2 )


cos(w1.L) ù
ú
cos (w 2 .L ) ú
ú
ú
ú
ú
cos (w L+2 .L ) ú
úû
 a ( 0) 
 a (1) 


  




a ( L)
L+2 linear
equations and L+2
unknowns

AX  B
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
30
a(n)  L  1 unknowns
 2  1 unknown
i  L  2 equations
L  2 unknowns
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
31
a (n),  2
i
It can be shown that if i 's are known, then  2
can be derived using the following formulae:
L2
 bk D( k )
 2  k 1
L  2 b (1) k 1
k

k 1 W ( k )
L2
1
bk  
i 1 cos k  cos i 
ik
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
32
𝐺(𝜔) is an Lth-order trigonometric polynomial.
We can interpolate a trigonometric polynomial through L+1 of the L+2
known values of 𝐸 𝜔𝑖 or G 𝜔𝑖 .
Using Lagrange interpolation formulae we can find the frequency
response as:
G ( )
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
33

Now 𝐺 𝜔 is available at any desired frequency,
without the need to solve the set of equations for the
coefficients of 𝑎 𝑛 .

If 𝐸 𝜔 ≤ 𝛿2 for all 𝜔 in the passband and stopband,
then the optimum approximation has been found.
Otherwise, we must find a new set of extremal
frequencies.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
34
35
Next alternation frequency
Original alternation frequency
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
36

App. estimate of L:
 10 log( 1 2 )  13
M  2L 
2.324
   s   p

App. Length of Kaiser filter: N  1 
• Example:  p  0.4 ,  s  0.6
A8
2 .2 

A  20 log10
1  0.01,  2  0.001
• Optimal filter:
N  2L  1  27
• Kaiser filter:
N  38
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
37
 p  0.4 ,  s  0.6
1  0.01,  2  0.001
K  10, M  26
Does it meet the specs?
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
38
Increase the length
of the filter by 1.
 p  0.4 ,  s  0.6
1  0.01,  2  0.001
K  10, M  27
Does it meet the specs?
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
39