Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 794

Then we use a partial fraction expansion by writing

Y s ¼

A
s þ

B
s þ 10

where

s¼0 ¼ 0:5

5
s þ 10

A ¼ sY ðsÞj s¼0 ¼

and

s¼10 ¼0:5

B ¼ðs þ 10ÞY ðsÞj s¼10 ¼ s

Hence,

Y s ¼ 0: s

0:5
s þ 10

y t ¼ L 1 0: s L 1 0:5

s þ 10

Finally, applying the inverse of the Laplace transform leads to using the results listed in Table B.5 , and we obtain
the time domain solution as

y t ¼ 0:5u t 0:5e 10t u t

B.2.3 Transfer Function
A linear analog system can be described using the Laplace transfer function. The transfer function
relating the input and output of the linear system is depicted as

YðsÞ¼HðsÞXðsÞ (B.25)

where XðsÞ and YðsÞ are the system input and response (output), respectively, in the Laplace domain,
and the transfer function is defined as a ratio of the Laplace response of the system to the Laplace input
given by

H s ¼ YðsÞ
XðsÞ

(B.26)

The transfer function will allow us to study the system behavior. Considering an impulse function
as the input to a linear system, that is, xðtÞ¼dðtÞ , whose Laplace transform is XðsÞ¼ 1, we then find
the system output due to the impulse function to be

YðsÞ¼HðsÞXðsÞ¼HðsÞ (B.27)