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The Discrete-time Fourier Transform (DTFT) 63
When {x (n l )} and {X(e jω k
)} are arranged as column vectors x and X,
respectively, we have
X = Wx (3.3)
where W is an (M +1)× N matrix given by
{
W =
△ e −j(π/M)kn l
; n 1 ≤ n ≤ n N ,
}
k =0, 1,...,M
In addition, if we arrange {k} and {n l } as row vectors k and n respectively,
then
[ (
W = exp −j π )]
M kT n
In MATLAB we represent sequences and indices as row vectors; therefore
taking the transpose of (3.3), we obtain
[ (
X T = x T exp −j π )]
M nT k
(3.4)
Note that n T k is an N × (M +1)matrix. Now (3.4) can be implemented
in MATLAB as follows.
>> k = [0:M]; n = [n1:n2];
>> X = x * (exp(-j*pi/M)) .^ (n’*k);
□ EXAMPLE 3.4 Numerically compute the discrete-time Fourier transform of the sequence x(n)
given in Example 3.2 at 501 equispaced frequencies between [0,π].
Solution
MATLAB script:
>> n = -1:3; x = 1:5; k = 0:500; w = (pi/500)*k;
>> X = x * (exp(-j*pi/500)) .^ (n’*k);
>> magX = abs(X); angX = angle(X);
>> realX = real(X); imagX = imag(X);
>> subplot(2,2,1); plot(k/500,magX);grid
>> xlabel(’frequency in pi units’); title(’Magnitude Part’)
>> subplot(2,2,3); plot(k/500,angX/pi);grid
>> xlabel(’frequency in pi units’); title(’Angle Part’)
>> subplot(2,2,2); plot(k/500,realX);grid
>> xlabel(’frequency in pi units’); title(’Real Part’)
>> subplot(2,2,4); plot(k/500,imagX);grid
>> xlabel(’frequency in pi units’); title(’Imaginary Part’)
The frequency-domain plots are shown in Figure 3.2. Note that the angle plot
is depicted as a discontinuous function between −π and π. This is because the
angle function in MATLAB computes the principal angle.
□
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