Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 65

We can determine the system output using a combined input, which is the weighed sum of the
individual inputs with constants 2 and 4, respectively. Using algebra, we see that

2

2

yðnÞ¼x ðnÞ¼ð 4 x 1 ðnÞþ 2 x 2 ðnÞÞ

2

2 2

(3.9)

¼ð 4 uðnÞþ 2 dðnÞÞ ¼ 16 u ðnÞþ 16 uðnÞdðnÞþ 4 d ðnÞ

¼ 16 uðnÞþ 20 dðnÞ

Note that we use the fact that uðnÞdðnÞ¼dðnÞ , which can be easily verified.
Again, we express the weighted sum of the two individual outputs with the same constants 2
and 4 as

4 y 1 ðnÞþ 2 y 2 ðnÞ¼ 4 uðnÞþ 2 dðnÞ (3.10)

It is obvious that

yðnÞ s 4 y 1 ðnÞþ 2 y 2 ðnÞ (3.11)

Hence, the system is a nonlinear system, since the linear property, superposition, does not hold, as
shown in Equation (3.11) .

3.2.2 Time Invariance
A time-invariant system is illustrated in Figure 3.12 , where y 1 ðnÞ is the system output for the input
x 1 ðnÞ . Let x 2 ðnÞ¼x 1 ðn n 0 Þ be the shifted version of x 1 ðnÞ by n 0 samples. The output y 2 ðnÞ

obtained with the shifted input x 2 ðnÞ¼x 1 ðn n 0 Þ is equivalent to the output y 2 ðnÞ acquired by

shifting y 1 ðnÞ by n 0 samples, y 2 ðnÞ¼y 1 ðn n 0 Þ .
This can simply be viewed as the following:

If the system is time invariant and y 1 ðnÞ is the system output due to the input x 1 ðnÞ, then the shifted system input
x 1 ðn n 0 Þ will produce a shifted system output y 1 ðn n 0 Þ by the same amount of time n 0 .

yn

1 () x(n )

1

n n

System

xn xn n yn yn n

() ( ) () ( )

2 1 0 2 1 0

n n

shifted by n samples
0

n

shifted by n samples
0 n

FIGURE 3.12

Illustration of the linear time-invariant digital system.