Review of Analog Signal Processing Basics - Digital Signal Processing: Fundamentals and Applications - page 795

Therefore, the response in the time domain yðtÞ is called the impulse response of the system and can be
expressed as

h t ¼ L 1 fHðsÞg (B.28)

The analog impulse response can be sampled and transformed to obtain a digital filter transfer
function. This topic is covered in Chapter 8.

EXAMPLE B.12
Consider a linear system described by the differential equation shown in Example B.11. xðtÞ and yðtÞ designate
the system input and system output, respectively. Derive the transfer function and the impulse response of the
system.
Solution:
Taking the Laplace transform on both sides of the differential equation yields
L dy ð t Þ
dt

þ Lf10yðtÞg ¼ LfxðtÞg

Applying the differential property and substituting the initial condition, we have

Y ðsÞðs þ 10Þ¼X ðsÞ

Thus, the transfer function is given by

H s ¼

Y ðsÞ
X ðsÞ ¼

1
s þ 10

The impulse response can be found by taking the inverse Laplace transform as
h t ¼ L 1

¼ e 10t u t

1
s þ 10

B.3 POLES, ZEROS, STABILITY, CONVOLUTION, AND SINUSOIDAL
STEADY-STATE RESPONSE
This section is a review of analog system analysis.

B.3.1 Poles, Zeros, and Stability
To study system behavior, the transfer function is written in a general form given by

H s ¼ NðsÞ

DðsÞ ¼ b m s m þ b m 1 s m 1

þ / þ b 0

(B.29)

a n s n þ a n 1 s n 1

þ / þ a 0