Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 254

Similarly,

R ð s Þ¼ G ð s Þ Y ð s Þ ð 3 Þ

Y ð s Þ¼ D ð s Þ E ð s Þ ð 4 Þ

Using (2), (3), and (4) we get: Y(s)[1 - D(s)G(s)] = D(s)X(s)

H ð s Þ¼ Y ð s Þ

D ð s Þ
1 D ð s Þ G ð s Þ :

)

X ð s Þ ¼

Q2: Find the transfer function of the following feedback system.

+ +

_

X ( s )
x ( t )

R ( s ) D ( s ) Y ( s )

y ( t )

+

G ( s )

Tutorial 20

Q: The relationship between Ff x ð t Þg and Lf x ð t Þg is given by:

F f x ð t Þg¼ L f x ð t Þgj s ¼ jx
or : X ð x Þ¼ X ð s Þ s ¼ jx

where x = 2pf. This holds on the condition that:

Z

1

x ð t j j dt\ 1 i.e :; x ð t Þ is absolutely integrable :

1

Now decide whether X(x) can be found from X(s) for the functions:
1. d(t), 2. u(t), 3. t n

u(t), n is integer C0.

Solution:
1. R 1

1 x ð t j j dt ¼ R 1 1 d ð t Þ dt ¼ 1\infty. Hence, X(x) = X(s)| s=jx .

From Tables: D ð s Þ¼ L f d ð t Þg¼ 1. ) D ð x Þ¼ D ð s Þ s ¼ jx ¼ 1.

2. R 1 1 u ð t j j dt ¼ R 1 u ð t Þ dt ¼ R 1

0 1 : dt !1 .
Hence, we cannot use X(x) = X(s)| s=jx .
Note that the Laplace transform of u(t) is: U ð s Þ¼ s

0

(Tables), and the Fourier

transform of u(t) is: U ð f Þ¼ 2 d ð f Þþ 1 (Tables).

j2pf