Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 63

x ( n )

10.0

10

7.0711 7.0711

T 125

0 .0 0. 0 5 6 7

n

Sample index

1 2 3 4

0

5

7 0 7 11 . 7 0 7 11 .

10

1 0 .

t Microseconds ( μ sec.)

0 125 250 375 500 625 750 875

tnT

FIGURE 3.10

Plot of the digital sequence for (2) in Example 3.2.

xð2Þ¼10e 0:6252 uð2Þ¼2:8650
xð3Þ¼10e 0:6253 uð3Þ¼1:5335
xð4Þ¼10e 0:6254 uð4Þ¼0:8208
The first eight amplitudes for (2) are computed and sketched in Figure 3.10 .
xð0Þ¼10 sinð0:25p 0Þuð0Þ¼0
xð1Þ¼10 sinð0:25p 1Þuð1Þ¼7:0711
xð2Þ¼10 sinð0:25p 2Þuð2Þ¼10:0
xð3Þ¼10 sinð0:25p 3Þuð3Þ¼7:0711
xð4Þ¼10 sinð0:25p 4Þuð4Þ¼0:0
xð5Þ¼10 sinð0:25p 5Þuð5Þ¼7:0711
xð6Þ¼10 sinð0:25p 6Þuð6Þ¼10:0
xð7Þ¼10 sinð0:25p 7Þuð7Þ¼7:0711

3.2 LINEAR TIME-INVARIANT, CAUSAL SYSTEMS
In this section, we study linear time-invariant causal systems and focus on properties such as linearity,
time-invariance, and causality.

3.2.1 Linearity
A linear system is illustrated in Figure 3.11 , where y 1 ðnÞ is the system output using an input x 1 ðnÞ , and
y 2 ðnÞ the system output with an input x 2 ðnÞ .
Figure 3.11 illustrates that the systemoutput due to theweighted sum inputs ax 1 ðnÞþbx 2 ðnÞ is equal
to the same weighted sum of the individual outputs obtained from their corresponding inputs, that is,

yðnÞ¼ay 1 ðnÞþby 2 ðnÞ (3.5)

where a and b are constants.