Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 243

Now: / ð t o Þ¼ g ð t o Þ x ð t o Þ¼ x ð t Þ R 1

1 g ð t Þ d ð t t o Þ dt ¼ R 1 1 g ð t Þ½ x ð t o Þ d ð t t o Þ dt :

Hence we have: R 1 dt ¼ R 1

1 g ð t Þ½ x ð t Þ d ð t t o Þ 1 g ð t Þ½ x ð t o Þ d ð t t o Þ

dt

|{z}
s ð t Þ

|{z}
r ð t Þ

Since g(t) is arbitrary, the above equation implies that:

s ð t Þ¼ r ð t Þ; or x ð t Þ d ð t t o Þ¼ x ð t o Þ d ð t t o Þ:

Q2: Consider the cascaded and parallel systems shown in Figs. a and b . Find the
equivalent impulse responses for the equivalent system shown in Fig. c .

(c)

(a) (b)

h 1 ( t )

x ( t ) y ( t ) x ( t ) h ( t ) y ( t )

h 1 ( t ) h 2 ( t )

x ( t ) y ( t )

h 2 ( t )

Solution:

(a) Let y 1 (t) = x(t)*h 1 (t).

)

y ð t Þ¼ y 1 ð t Þ h 2 ð t Þ¼½ x ð t Þ h 1 ð t Þ h 2 ð t Þ¼ x ð t Þ½ h 1 ð t Þ h 2 ð t Þ
|{z}
h ð t Þ

) h ð t Þ

¼ h 1 ð t Þ h 2 ð t Þ

where we used the associative property of convolution.

(b) Let y 1 (t) = x(t)*h 1 (t), y 2 (t) = x(t)*h 2 (t).

)

y ð t Þ¼ y 1 ð t Þþ y 2 ð t Þ¼ x ð t Þ h 1 ð t Þþ x ð t Þ h 2 ð t Þ¼ x ð t Þ½ h 1 ð t Þþ h 2 ð t Þ
|{z}
h ð t Þ

) h ð t Þ

¼ h 1 ð t Þþ h 2 ð t Þ

where we used the distributive property of convolution.