Continuous-time Fourier transform - Continuous and Discrete Time Signals and Systems - page 250

To calculate the inverse CTFT, we replace the function fft with ifft and
reverse the order of the instructions. The M ATLAB code to compute the inverse
CTFT is provided in a second function called myinvctft :

function [t,x] = myinvctft(w,z)
% MYINVCTFT: computes inverse CTFT of y known at
% frequencies w
% Usage: [t,x] = myinvctft(w,z)
% compute the inverse CTFT
x = ifftshift(z);
x = ifft(x);

% inverse fft

% compute the time instants
ws = w(length(w)) - w(1); % sampling rate
Ts = 2*pi/ws; % sampling interval
t = Ts*[-floor(length(w))/2:floor(length(w))/2-1];
% sampling instants % amplify signal by 1/(pi*Ts)

x = x/Ts;
% plots
subplot(211); plot(w,abs(z))

% CTFT plot of cos(w0*t)

subplot(212); plot(t,real(x)) % CT plot of cos(w0*t)

% end

5.10.2 Bode plots

M ATLAB provides the bode function to sketch the Bode plot. To illustrate the
application of the bode function, consider the LTIC system of Example 5.31.
The system transfer function is given by

0 . 8
(0 . 6 + j ω ) 2 + 0 . 8 2 .

H ( ω ) = 1 . 25

In order to avoid a complex-valued representation, M ATLAB expresses the
Fourier transfer function in terms of the Laplace variable s = j ω . In Chapter
6, we will show that the independent variable s represents the entire complex
plane and leads to the generalization of the Fourier transfer function into an
alternative transfer function, referred to as the Laplace transfer function. Sub-
stituting ( s = j ω )in H ( ω ) results in the following expression for the transfer

function:

1
(0 . 6 + s ) 2 + 0 . 8 2

1
s 2 + 1 . 2 s + 1 .

H ( s ) = =

Given H ( s ), the Bode plots are obtained in M ATLAB using the following
instructions: