Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 66

EXAMPLE 3.3
Determine whether the linear systems
a. yðnÞ¼2xðn 5Þ
b. yðnÞ¼2xð3nÞ
are time invariant.
Solution:
a. Let the input and output be x 1 ðnÞ and y 1 ðnÞ, respectively; then the system output is y 1 ðnÞ¼2x 1 ðn 5Þ. Again,
let x 2 ðnÞ¼x 1 ðn n 0 Þ be the shifted input and y 2 ðnÞ be the output due to the shifted input. We determine the
system output using the shifted input as
y 2 ðnÞ¼2x 2 ðn 5Þ¼2x 1 ðn n 0 5Þ
Meanwhile, shifting y 1 ðnÞ¼2x 1 ðn 5Þ by n 0 samples leads to
y 1 ðn n 0 Þ¼2x 1 ðn 5 n 0 Þ
We can verify that y 2 ðnÞ¼y 1 ðn n 0 Þ. Thus the shifted input of n 0 samples causes the system output to be
shifted by the same n 0 samples. The system is thus time invariant.
b. Let the input and output be x 1 ðnÞ and y 1 ðnÞ, respectively; then the system output is y 1 ðnÞ¼2x 1 ð3nÞ. Again, let
the input and output be x 2 ðnÞ and y 2 ðnÞ, where x 2 ðnÞ¼x 1 ðn n 0 Þ, a shifted version, and the corresponding output
is y 2 ðnÞ. We get the output due to the shifted input x 2 ðnÞ¼x 1 ðn n 0 Þ and note that x 2 ð3nÞ¼x 1 ð3n n 0 Þ:
y 2 ðnÞ¼2x 2 ð3nÞ¼2x 1 ð3n n 0 Þ
On the other hand, if we shift y 1 ðnÞ by n 0 samples, and replace n in y 1 ðnÞ¼2x 1 ð3nÞ by n n 0 , we obtain
y 1 ðn n 0 Þ¼2x 1 ð3ðn n 0 ÞÞ ¼ 2x 1 ð3n 3n 0 Þ
Clearly, we know that y 2 ðnÞ s y 1 ðn n 0 Þ. Since the system output y 2 ðnÞ using the input shifted by n 0 samples is
not equal to the system output y 1 ðnÞ shifted by the same n 0 samples, the system is not time invariant.
3.2.3 Causality
A causal system is the one in which the output yðnÞ at time n depends only on the current input xðnÞ at
time n , and its past input sample values such as xðn 1 Þ , xðn 2 Þ , . . Otherwise, if a system output
depends on future input values such as xðn þ 1 Þ , xðn þ 2 Þ , . , the system is noncausal. The noncausal
system cannot be realized in real time.
EXAMPLE 3.4
Determine whether the systems
a. yðnÞ¼0:5xðnÞþ2:5xðn 2Þ, for n 0
b. yðnÞ¼0:25xðn 1Þþ0:5xðn þ 1Þ0:4yðn 1Þ, for n 0
are causal.
Solution:
a. Since for n 0, the output yðnÞ depends on the current input xðnÞ and its past value xðn 2Þ, the system is
causal.
b. Since for n 0, the output yðnÞ depends on the current input xðnÞ and its future value xðn þ 1Þ, the system is
noncausal.