Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 249

(B)

2 3

Ff x ð t Þ y ð t Þg¼ Z

1

Z

1

4 5 e j2pft dt

x ð k Þ y ð t k Þ dk

1 1

2
4

3
5

¼ Z x ð k Þ Z

1 1

y ð t k Þ e j2pft dt dk

1 1

|{z}
Let v ¼ t k

2
4

3
5 dk

¼ Z x ð k Þ Z

1 1

y ð v Þ e j2pf ð v þ k Þ dv

1 1

0
@

1
A

0
@

1
A

Z

1

Z

1

x ð k Þ e j2pf k dk y ð v Þ e j2pfv dv

¼

1 1

½ since k and v are independent
¼Ff x ð t ÞgFf y ð t Þg¼ X ð f Þ Y ð f Þ

Tutorial 16

Q: Find the s-domain voltage transfer function and the voltage impulse response
for the analog systems shown below. Assume zero initial conditions (IC's).

R = 2 Ω R = 2 Ω

L = 4 H

ν o = ν R ν o = ν L ν o = ν C

+ + +

+ + +

ν i i R ν i i L ν i i C

L = 4 H

R = 2 Ω C = 2 F

_ _ _

_ _ _

(a) (b) (c)

Solution:

L

Tables V L ð s Þ¼ L ½ sI ð s Þ i ð 0 Þ¼ Ls I ð s Þ zero

v L ð t Þ¼ Ldi ð t Þ= dt
|{z}
voltage across L

IC's :

Hence, inductor impedance is represented in the complex-frequency domain as:

X L ð s Þ¼ V L ð s Þ= I ð s Þ¼ sL ½ zero IC 0 s :