Digital Signal Processing Reference
In-Depth Information
and
X[3] = x[0] + x[1](-j)
3
+ x[2](-j)
6
+ x[3](-j)
9
= 1 + j2 - 3 - j4 =-2 - j2.
The following MATLAB program can be used to confirm the results of this example:
% This MATLAB program computes the DFT using the defining
% equations (12.30) for the transform pair.
N = input...
('How many discrete-time samples are in the sequence?')
x = input...
('Type the vector of samples, in brackets[...]:')
% Compute the DFT from (12.30)
for k1 = 1:N
X(k1) = 0;
k = k1 - 1;
for n1 = 1:N;
n = n1 - 1;
X(k1) = X(k1) + x(n1)*exp(-j*2*pi*k*n/N);
end
end
x
X
The discrete Fourier transform is also listed in Table 12.3. At this time we make no
attempt to give meaning to these values.
■
Calculation of an inverse DFT, with MATLAB confirmation
EXAMPLE 12.10
This example is a continuation of the last one. We now find the inverse discrete Fourier trans-
form of
X[k]
of Table 12.3. From (12.30),
x[0] = [X[0] + X[1] + X[2] + X[3]]/4
= [10 + (-2 + j2) + (-2) + (-2 - j2)]/4 = 1,
x[1] = [X[0] + X[1](j) + X[2](j)
2
+ X[3](j)
3
]/4
= [10 - j2 - 2 + 2 + j2 - 2]/4 = 2,
x[2] = [X[0] + X[1](j)
2
+ X[2](j)
4
+ X[3](j)
6
]/4
= [10 + 2 - j2 - 2 + 2 + j2]/4 = 3,
and
x[3] = [X[0] + X[1](j)
3
+ X[2](j)
6
+ X[3](j)
9
]/4
= [10 + j2 + 2 + 2 - j2 + 2]/4 = 4,
which are the correct values. Note the symmetries of the calculations of the DFT and its in-
verse. As stated earlier, these symmetries allow the same computer program used to calculate
the forward transform to calculate the inverse transform, with slight modification:
