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162 Chapter 5 THE DISCRETE FOURIER TRANSFORM
5
Magnitude of the DFT: N=128
4
3
|X(k)|
2
1
0
–1
0 20 40 60 80 100 120
K
200
Angle of the DFT: N=128
100
Degrees
0
–100
–200
0 20 40 60 80 100 120
K
FIGURE 5.8
plots
The DFT plots of Example 5.7 for N = 128 are shown as line
□ EXAMPLE 5.8 To illustrate the difference between the high-density spectrum and the
high-resolution spectrum, consider the sequence
x(n) =cos (0.48πn)+cos (0.52πn)
We want to determine its spectrum based on the finite number of samples.
a. Determine and plot the discrete-time Fourier transform of x(n), 0 ≤ n ≤ 10.
b. Determine and plot the discrete-time Fourier transform of x(n),
0 ≤ n ≤ 100.
Solution
We could determine analytically the discrete-time Fourier transform in each
case, but MATLAB is a good vehicle to study these problems.
a. We can first determine the 10-point DFT of x(n) toobtain an estimate of its
discrete-time Fourier transform.
MATLAB Script:
>> n = [0:1:99]; x = cos(0.48*pi*n)+cos(0.52*pi*n);
>> n1 = [0:1:9] ;y1 = x(1:1:10);
>> subplot(2,1,1) ;stem(n1,y1); title(’signal x(n), 0