Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 64

xn yn

1 () 1 ()

System

xn

2 ()

yn

2 ()

System

yn () y n ()

xn () x n ()

1 2

1 2

System

FIGURE 3.11

Digital linear system.

For example, assuming a digital amplifier is represented by yðnÞ¼ 10 xðnÞ ,theinputis
multiplied by 10 to generate the output. The inputs x 1 ðnÞ¼uðnÞ and x 2 ðnÞ¼dðnÞ generate the
outputs

y 1 ðnÞ¼ 10 uðnÞ , and y 2 ðnÞ¼ 10 dðnÞ , respectively

If, as described in Figure 3.11 , we apply the combined input xðnÞ to the system, where the first input
multiplied by a constant 2 while the second input multiplied by a constant 4,

xðnÞ¼ 2 x 1 ðnÞþ 4 x 2 ðnÞ¼ 2 uðnÞþ 4 dðnÞ

then the system output due to the combined input is obtained as

yðnÞ¼ 10 xðnÞ¼ 10 ð 2 uðnÞþ 4 dðnÞÞ ¼ 20 uðnÞþ 40 dðnÞ (3.6)

If we verify the weighted sum of the individual outputs, we see that

2 y 1 ðnÞþ 4 y 2 ðnÞ¼ 20 uðnÞþ 40 dðnÞ (3.7)

Comparing Equations (3.6) and (3.7) verifies that

yðnÞ¼ 2 y 1 ðnÞþ 4 y 2 ðnÞ (3.8)

Hence, the system yðnÞ¼ 10 xðnÞ is a linear system. The linearity means that the system obeys the
superposition principle, as shown in Equation (3.8) . Let us verify a system whose output is a square of
its input,

2

yðnÞ¼x ðnÞ

Applying the inputs x 1 ðnÞ¼uðnÞ and x 2 ðnÞ¼dðnÞ to the system leads to

2 2

y 1 ðnÞ¼u ðnÞ¼uðnÞ; and y 2 ðnÞ¼d ðnÞ¼dðnÞ

2

2

It is very easy to verify that u ðnÞ¼uðnÞ and d ðnÞ¼dðnÞ .