Hossein Sameti
Department of Computer Engineering
Sharif University of Technology
x(n)
LTI System
h(n)
y ( n)  x ( n) * h( n)
Y ( )  H ( ) X ( )
y(n)
Y ( )  X ( ) H ( )
Y ( )  H ( ) X ( )
Y ( )  H ( )  X ( )
x(n)  X () cos[n  X ()]
y(n)  H () X () cos[n  X ()  H ()]
x(n)  X () sin[ n  X ()]
y(n)  H () X () sin[ n  X ()  H ()]
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
2
H ( ) 
1 a
H ( )   ( )   tan 1
1  a  2a cos 
x(n)  1  sin n  cos(n   )
2
4
2
a sin 
1  a cos 
y ( n)  ?
0
H ( 0)  1
  2
H ( 2 )  0.074
 ( 2 )  42
 
H ( )  0.053
 ( )  0
y (n)  A H ( ) cos[n   ( )]
a  0.9
 ( 0)  0

y (n)  A H ( ) sin[n   ( )]
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
3




We can thus view an LTI system as a filter for sinusoids
of different frequencies.
Hence, the basic digital filter design problem involves
determining the parameters of an LTI system to
achieve a desired H(ω).
Note that the output of an LTI system cannot contain
frequency components that are not contained in the
input signals.
For that to happen, the system should be either timevariant or non-linear.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
4
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
5
- Bandwidth is the range of frequencies over which the
spectrum (the frequency content) of the signal is concentrated.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
6
y (n)  x(n  n0 )
h(n)   (n  n0 )
H ( )  e jn0
• Observations:
1.The magnitude of the frequency response is 1 for all ω.
H ( )  1
2.The phase is linear in ω.
H ()  n0
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
7


Group delay:
d
 g ( )  
H ( )
d
For pure delay:
Group delay is thus constant:
H ()  n0
 g ( )  
Desirable, since pure
delay is tolerable.
d
H ( )  n0
d
-All the frequencies are thus delayed by the same amount
when they pass through this system. Thus, no distortion is
added to the signal.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
8
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
9
x(n)
Y ( )  H ( ) X ( )
H ( )    
 0
y(n)
LTI System
h(n)
Y ( )  H ( ) X ( )
Y ( )  H ( )  X ( )
GLP filters
H ( )
Linear-phase filters
• Example: pure delay


h(n)   (n  n0 )
H ( )  1
H ()  n0

Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
10
H ( )    
H ()  H m ()e j (   )
Real
H ( )  H m ( ) cos(   )  jH m ( ) sin(   )
sin(   )
tan H ( ) 
 tan(   )
cos(   )

H ( )   h(k )e
k  
 j k


k  
k  
(eq.1)
  h(k ) cos k  j  h(k ) sin k
tan H ( ) 
 h(k ) sin k
k
 h(k ) cos k
(eq.2)
k
If we equate (eq.1) and (eq.2), we get GLP.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
11
  h(k ) sin k
sin(   )
 k
cos(   )
 h( k ) cos k
k

• Case 1:   0 or  &
 h(n) sin[(n   )   ]  0
n
2  M  an integer
h(n)  h(2  n)

N 1
2
• Case 2:    / 2 or 3 / 2 &
2  M  an integer
N 1
2
n
The above equation is satisfied.
Symmetry Condition
 h(n) cos[(n   )]  0
n
h(n)  h(2  n)

 h(n) sin[(n   )]  0
The above equation is satisfied.
Anti-symmetry Condition
N: the length of h(n)
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
12
N 5
h(n)
3
2
h(n)  h(4  n)
2
1
1
0
2 3
1
4
  2,   0
n
h(n)
2
2
1
0
1
1
2 3
n
N 4
h(n)  h(3  n)
  1.5,   0
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
13
N 5
h(n)
h(n)  h(4  n)
2
  2,   2
1
0
4
2 3
1
n
1
2
h(n)
N 4
h(n)  h(3  n)
2
1
0
1 2
2
3
n
  1.5,   2
1
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
14
• Case 1:
odd
h ( N  1  n ) 0  n  N
h( n)  
otherwise
 0
N 1

2
 0
N
H ( )  H m ( )e
even
 j ( N21 )
• Case 2:
odd
 h ( N  1  n ) 0  n  N
h( n)  
otherwise
 0

N 1
2


2
N
j
2
H ( )  H m ( )e e
 j ( N21 )
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
even
15
Type I
N  odd
h(n)
c
b

b
a
a
0
2 3
1
4
N 1
,  0
2
h(n)  h( N  1  n)
n
Type II
N  even
h(n)
b
a
0

b
a
1
2 3
n
N 1
,  0
2
h(n)  h( N  1  n)
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
16
Type III
N  odd
h(n)

b
a
0
h(n)  h( N  1  n)
n
a
b
Type IV
h(n)
4
2 3
1
N 1

, 
2
2
N  even
N 1


, 
2
2
b
a
0
1 2
3
n
h(n)  h( N  1  n)
a
b
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
17

For Type I
𝑁−1
ℎ(𝑛)𝑒 −𝑗𝜔𝑛
𝐻 𝜔 =
𝑛=0
By applying symmetry condition of h(n)  h( N  1  n) we can
write:
𝐻 𝜔 = 𝑒 −𝑗𝜔(𝑁−1)/2 n 0a(n) cos n
where
N 1
2
𝑁−1
)
2
𝑁−1
2ℎ[(
)-n],
2
𝑎 0 = ℎ(
𝑎 𝑛 =
n= 1, 2, … ,
𝑁−1
2
N 1
2
So:
H m ( )   a(n) cos n
n 0
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
18
Symmetry
Type I
h( n) 
h( N  1  n)
Type II
odd
h( n) 
h( N  1  n)
Type III
N
 h( n) 
h( N  1  n)
even
odd

N 1
2

H m ( )
N 1
2
 a(n) cos n
0
 h( n) 
h( N  1  n)
even
Real
n 0
N
2
N 1
2
0
N 1
2

2
 b(n) cos  (n  12 )
n1
N 1
2

2
 c(n) sin n
N
2
H ( )  0
H ( )  0
H (0)  0
Purely imaginary
 d (n) sin  (n  12 )
n 1
Real
Purely imaginary
N 1
2
n 1
Type IV
Constraint
H ( )
H (0)  0
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
19
Type I
Type II
Type I
Type IV
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
20
Type I
Type II
Type III
Type IV
Type I
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
21
Type I
Low-pass
√
High-pass
√
Band-pass
√
Band-stop
√
Type II
√

√

Type III


√

Type IV

√
√

Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
22



Linear-phase is desirable for filters as it leads to a fixed
delay for all input frequencies (i.e., no distortion in the
output of the filter).
If we impose symmetry or anti-symmetry on h(n), we
can have linear-phase property.
Type I FIR filter can be used to design all filters (lowpass, high-pass, bandpass and bandstop).
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
23