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The Discrete Fourier Transform 157
□ EXAMPLE 5.6 Let x(n) bea4-point sequence:
x(n) =
{ 1, 0 ≤ n ≤ 3
0, otherwise
a. Compute the discrete-time Fourier transform X(e jω ) and plot its magnitude
and phase.
b. Compute the 4-point DFT of x(n).
Solution
a. The discrete-time Fourier transform is given by
3∑
X(e jω )= x(n)e −jωn =1+e −jω + e −j2ω + e −j3ω
Hence
and
0
= 1 − e−j4ω sin(2ω)
=
1 − e−jω sin(ω/2) e−j3ω/2
∣ ∣ ∣ ∣∣∣
X(e jω ) sin(2ω)
= sin(ω/2) ∣
⎧
⎪⎨ − 3ω
̸ X(e jω 2 ,
)=
⎪⎩ − 3ω 2
The plots are shown in Figure 5.4.
b. Let us denote the 4-point DFT by X 4 (k). Then
X 4(k) =
3∑
n=0
sin(2ω)
when
sin(ω/2) > 0
± π, when
sin(2ω)
sin(ω/2) < 0
x(n)W nk
4 ; k =0, 1, 2, 3; W 4 = e −j2π/4 = −j
These calculations are similar to those in Example 5.1. We can also use
MATLAB to compute this DFT.
>> x = [1,1,1,1]; N = 4; X = dft(x,N);
>> magX = abs(X), phaX = angle(X)*180/pi
magX =
4.0000 0.0000 0.0000 0.0000
phaX =
0 -134.9810 -90.0000 -44.9979
Hence
X 4(k) ={4, 0, 0, 0}
↑
Note that when the magnitude sample is zero, the corresponding angle is not
zero. This is due to a particular algorithm used by MATLAB to compute the
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