1. Digital Signal Processing
By: Assoc. Prof. Dr. Erhan A. İnce
Electrical and Electronic Engineering Dept.
SPRING 2016

4. Implementation of Discrete-Time Systems
A system can be described by a linear constant coefficient difference equation
Let us consider a first order system such as:
This system can be described by two systems in cascade. The first is a non-recursive
system described by the equation
The second part is recursive system

5. Direct Form - I

7. Implementation of Discrete-Time Systems
Using convolution properties we can interchange the order of the recursive
and non-recursive system

9. w n =− 𝑎 1 𝑤 𝑛−1 +𝑥 𝑛 𝑦 𝑛 = 𝑏 0 𝑤 𝑛 + 𝑏 1 𝑤 𝑛−1
Direct Form - II
Using only one delay is more efficient in terms of the memory requirements.

10. Implementation of Discrete-Time Systems
In general form, the difference equation is given as:
The non-recursive system is:
The recursive system is:
Direct form II structure for the recursive system is:
And for the non-recursive system is:

11. Implementation of Discrete-Time Systems

12. Implementation of Discrete-Time Systems
A special case of the general difference equation is:
Which becomes a non-recursive LTI system. Such a system has coefficients bk only.
The output is:

14. Determine the direct form II realization for the following LTI system:
y n =− 1 2 𝑦 𝑛−1 +2𝑦 𝑛− 𝑥 𝑛 𝑥 𝑛−5
w n =− 1 2 𝑤 𝑛−1 +2𝑤 𝑛−3 + x[n]
y n = 1 2 𝑤 𝑛 𝑤 𝑛−5

15. y[n]