1. Discrete-Time Structure
Hafiz Malik

2. Realization of Discrete-Time Systems
Let us consider the important of LTI DT system characterized by the general linear constant-coefficient difference equation
Equivalent LTI DT system in z-transform can be expressed as

3. Structures for FIR Systems
In general, an FIR system is described as,
Or equivalently, system function

4. Structures for FIR Systems
In general, an FIR system is described as,
Or equivalently, system function
The unit sample response of FIR system is identical to the coefficients {bl}, i.e.,

5. Implementation Methods for FIR Systems
Direct-Form Structure
Cascade-Form Structure
Frequency-Sampling Structure
Lattice Structure

6. Direct-Form Realization
The direct-form realization follows immediately from the non-recursive difference equation (2) or equivalently by the following convolution summation

7. Direct-Form Structure
z-1 +

8. 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.

9. 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.,
For such system the number of multiplicaitons is reduced from M to
M/2 for M is even
(M – 1)/2 for M is odd

10. Direct-Form Realization of Linear-Phase FIR System+ + + + +
z-1
z-1
z-1
z-1
z-1
z-1 + + + + +

11. 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.,
where,
Here K is integer part of (q – 1)/2

12. Cascade-Form Realization of FIR System
yK-1(n) = xK(n)
X(n) = x1(n)
y1(n) = x2(n)
y3(n) = x4(n)
yK(n) = y(n)
y2(n) = x3(n)
H1(z)
H2(z)
H3(z)
HK(z)
z-1
z-1
bk1
bk2
bk0
yk(n) = xk+1(n) + +

13. Linear-Phase FIR Systems