Laplace transform - Continuous and Discrete Time Signals and Systems - page 294

form:

= b m s m + b m − 1 s m − 1 + b m − 2 s m − 2 ++ b 1 s + b 0
s n + a n − 1 s n − 1 + a n − 2 s n − 2 ++ a 1 s + a 0

X ( s ) = N ( s )
D ( s )

. (6.35)

Characteristic equation The characteristic equation for the transfer function
in Eq. (6.35) is defined as follows:

D ( s ) = s n + a n − 1 s n − 1 + a n − 2 s n − 2 ++ a 1 s + a 0 = 0 . (6.36)

It will be shown later that the characteristic equation determines the behavior
of the system, including its stability and possible modes of the output response.
In other words, it characterizes the system very well.

Zeros The zeros of the transfer function H ( s ) of an LTIC system are the finite
locations in the complex s-plane where H ( s ) = 0. For the transfer function in
Eq. (6.35), the location of the zeros can be obtained by solving the following
equation:

N ( s ) = b m s m + b m − 1 s m − 1 + b m − 2 s m − 2 ++ b 1 s + b 0

= 0 . (6.37)

Since N ( s )isan m th-order polynomial, it will have m roots leading to m zeros
for transfer function H ( s ).

Poles The poles of the transfer function H ( s ) of an LTIC system are the loca-
tions in the complex s-plane where H ( s ) has an infinite value. At these loca-
tions, the Laplace magnitude spectrum takes the form of poles (due to the infinite
value), and this is the reason the term “pole” is used to denote such locations.
The poles corresponding to the transfer function in Eq. (6.35) can be obtained
by solving the characteristic equation, Eq. (6.36).
Because D ( s )isan n th-order polynomial, it will have n roots leading to n
poles. In order to calculate the zeros and poles, a transfer function is factorized
and typically represented as follows:
H ( s ) = N ( s )
D ( s )

= b m ( s − z 1 )( s − z 2 ) ( s − z m )
( s −

. (6.38)

p 1 )( s − p 2 ) ( s − p n )

Note that a transfer function H ( s ) must be finite within its ROC. On the other
hand, the magnitude of the transfer function H ( s ) is infinite at the location of a
pole. Therefore, the ROC of a system must not include any pole. However, an
ROC may contain any number of zeros.

Example 6.18
Determine the poles and zeros of the following LTIC systems:

( s + 4)( s + 5)
s 2 ( s + 2)( s − 2) ;

(i) H 1 ( s ) =

( s + 4)
s 3 + 5 s 2 + 17 s + 13 ;

(ii) H 2 ( s ) =

1
e s + 10 .

(iii) H 3 ( s ) =