Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 67

3.3 DIFFERENCE EQUATIONS AND IMPULSE RESPONSES
Now we study the difference equation and its impulse response.

3.3.1 Format of the Difference Equation
A causal, linear, time-invariant system can be described by a difference equation having the following
general form:
yðnÞþa 1 yðn 1 Þþ / þ a N yðn NÞ¼b 0 xðnÞþb 1 xðn 1 Þþ / þ b M xðn MÞ

(3.12)

where a 1 , . , a N , and b 0 , b 1 , . , b M are the coefficients of the difference equation. Equation (3.12) can
also be written as
yðnÞ¼a 1 yðn 1 Þ / a N yðn NÞþb 0 xðnÞþb 1 xðn 1 Þþ / þ b M xðn MÞ

(3.13)

or

N
i ¼ 1 a i yðn iÞþ

M
j ¼ 0 b j xðn jÞ

yðnÞ¼ (3.14)

Notice that yðnÞ is the current output, which depends on the past output samples yðn 1 Þ , . , yðn NÞ ,
the current input sample xðnÞ , and the past input samples, xðn 1 Þ , . , xðn NÞ .
We will examine the specific difference equations in the following examples.

EXAMPLE 3.5
Given the difference equation

yðnÞ¼0:25yðn 1ÞþxðnÞ

identify the nonzero system coefficients.
Solution:
Comparison with Equation (3.13) leads to

b 0 ¼ 1

a 1 ¼ 0:25; that is; a 1 ¼0:25

EXAMPLE 3.6
Given a linear system described by the difference equation
yðnÞ¼xðnÞþ0:5xðn 1Þ
determine the nonzero system coefficients.
Solution:
By comparing Equation (3.13) , we have

b 0 ¼ 1 and b 1 ¼ 0:5