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84 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
Using Euler’s identity, we can expess x(n) as
x(n) =4+e jπ/3 e j0.75πn + e −jπ/3 e −j0.75πn +2je j0.25πn − 2je j0.25πn (3.32)
From Table 3.1 and the DTFT properties, the DTFT of x(n) isgiven by
X(e jω )=8πδ(ω)+2πe jπ/3 δ(ω − 0.75π)+2πe −jπ/3 δ(ω +0.75π)
+ j4πδ(ω − 0.25π) − j4πδ(ω +0.25π), −π ≤ ω ≤ π. (3.33)
The plot of X(e jω )isshown in Figure 3.15.
□
3.4.2 MATLAB IMPLEMENTATION
In a strict sense it is not possible to analyze analog signals using MATLAB
unless we use the Symbolic toolbox. However, if we sample x a (t) onafine
grid that has a sufficiently small time increment to yield a smooth plot
and a large enough maximum time to show all the modes, then we can
approximate its analysis. Let ∆t be the grid interval such that ∆t ≪ T s .
Then
x G (m) △ = x a (m∆t) (3.34)
can be used as an array to simulate an analog signal. The sampling interval
T s should not be confused with the grid interval ∆t, which is used
strictly to represent an analog signal in MATLAB. Similarly, the Fourier
transform relation (3.24) should also be approximated in light of (3.34)
as follows:
X a (jΩ) ≈ ∑ m
x G (m)e −jΩm∆t ∆t =∆t ∑ m
x G (m)e −jΩm∆t (3.35)
Now if x a (t) [and hence x G (m)] is of finite duration, then (3.35) is similar
to the discrete-time Fourier transform relation (3.3) and hence can be
implemented in MATLAB in a similar fashion to analyze the sampling
phenomenon.
□ EXAMPLE 3.18 Let x a(t) =e −1000|t| . Determine and plot its Fourier transform.
Solution From (3.24)
X a(jΩ) =
∫ ∞
x a(t)e −jΩt dt =
∫ 0
∞∫
e 1000t e −jΩt dt +
e −1000t e −jΩt dt
−∞
=
−∞
0
0.002
1+( Ω
1000 )2 (3.36)
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