Discrete-Time Structure
Hafiz Malik
Realization of Discrete-Time Systems
• Let us consider the important of LTI DT system
characterized by the general linear constant-coefficient
difference equation
p 1
y (n)    ak y (n  k ) 
k 1
q 1
 b x(n  l )
l
l0
• Equivalent LTI DT system in z-transform can be expressed
as
q 1
H z  

bl z
l
l0
p 1

k 0
, where a 0  1
ak z
k
Structures for FIR Systems
• In general, an FIR system is described as,
q 1
y (n) 
 b x(n  l )
l
(2)
l0
• Or equivalently, system function
H z  
q 1

bl z
l
l0
3
Structures for FIR Systems
• In general, an FIR system is described as,
q 1
y (n) 
 b x(n  l )
l
l0
• Or equivalently, system function
H z  
q 1

bl z
l
l0
• The unit sample response of FIR system is
identical to the coefficients {bl}, i.e.,
bn ,
h n   
0
0  n  q 1
otherwise
4
Implementation Methods for FIR Systems
1.
2.
3.
4.
Direct-Form Structure
Cascade-Form Structure
Frequency-Sampling Structure
Lattice Structure
Direct-Form Realization
• The direct-form realization follows immediately
from the non-recursive difference equation (2) or
equivalently by the following convolution
summation
q 1
y (n) 
 h (l )x ( n  l )
l0
Direct-Form Structure
x(n)
z-1
z-1
z-1
h (1)
h (0)
z-1
h (3)
h (2)
h ( q  2)
h ( q  1)
y (n)
+
+
+
+
+
Complexity of Direct-Form Structure
• Requires q – 1 memory locations for storing q – 1
previous inputs,
• Complexity of q – 1 multiplications and q – 1
addition per output point
• As output consists of a weighted linear
combination of q – 1 past inputs and the current
input, which resembles a tapped-delay line or a
transversal system.
• The direct-form realization is called a transversal
or tapped-delay-line filter.
Linear-Phase FIR System
• When the FIR system is linear phase, the unit
sample response of the system satisfies either the
symmetry or asymemtry condition, i.e.,
h(n)   h(q  1  n)
• For such system the number of multiplicaitons is
reduced from M to
1. M/2 for M is even
2. (M – 1)/2 for M is odd
Direct-Form Realization of
Linear-Phase FIR System
x(n)
z-1
z-1
+
+
z-1
y (n)
z-1
h (2)
+
+
+
z-1
h (1)
+
z-1
+
z-1
h (0)
+
z-1
z-1
z-1
h (3)
+
q3
h

 2 
+
 q 1
h

 2 
Cascade-Form Structures
• Cascade realization follows naturally from the LTI
DT system given by equation (3). Simply factorize
H(z) into second-order FIR systems, i.e.,
K
H (z) 
H
k
(z)
l 1
• where,
H k ( z )  bk 0  bk 1 z
1
 bk 2 z
2
,
k  1, 2 ,  , K
• Here K is integer part of (q – 1)/2
Cascade-Form Realization of FIR System
y1(n) = x2(n)
X(n) = x1(n)
H1(z)
y3(n) = x4(n)
y2(n) = x3(n)
H2(z)
H3(z)
x(n)
z-1
z-1
bk1
bk0
bk2
yk(n) = xk+1(n)
+
+
yK-1(n) = xK(n)
HK(z)
yK(n) = y(n)
Linear-Phase FIR Systems