Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 251

> clear; % clear the M ATLAB environment
> num coeff = [1]; % coefficients of the numerator
% in decreasing powers of s
> denom coeff = [1 1.2 1]; % coefficient of the denominator
% in decreasing powers of s
> sys = tf(num coeff,denom coeff);
% specify the transfer function

> bode(sys, { 0.01,100 } ); % sketch the Bode plots

In the above set of M ATLAB instructions, we have used two new functions: tf
and bode . The built-in function tf specifies the LTIC system H ( s ) in terms
of the coefficients of the polynomials of s in the numerator and denominator.
Since the numerator N ( s ) = 1, the coefficients of the numerator are given by
num coeff = 1 . The denominator D ( s ) = s 2 + 1 . 2 s + 1. The coefficients
of the denominator are given by denom coeff = [1 1.2 1] .
The built-in function bode sketches the Bode plots. It accepts two input
arguments. The first input argument sys in used to represent the LTIC system,
while the second input argument { 0.01,100 } specifies the frequency range,
0.01 radians/s to 100 radians/s, used to sketch the Bode plots. In setting the
values for the frequency range, we use the curly parenthesis. Since the square
parenthesis [0.01,100] represents only two frequencies, ω = 0 . 01 and ω =
100, it will result in the wrong plots. The second argument is optional. If
unspecified, M ATLAB uses a default scheme to determine the frequency range
for the Bode plots.

5.11 Summary

In this chapter, we introduced the frequency representations for CT aperiodic
signals. These frequency decompositions are referred to as the CTFT, which
for a signal x ( t ) is defined by the following two equations:

∞

1
2 π

X ( ω )e j ω t d ω ;

CTFT synthesis equation x ( t ) =

−∞

∞

− j ω t d t .

CTFT analysis equation X ( ω ) = x ( t )e

−∞

Collectively, the synthesis and analysis equations form the CTFT pair, which
is denoted by

CTFT
←−−→ x ( t ) X ( ω ) .

In Section 5.1, we derived the synthesis and analysis equations by expressing
the CTFT as a limiting case of the CTFS. Several important CTFT pairs were