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60 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
The inverse discrete-time Fourier transform (IDTFT) of X(e jω )isgiven
by
x(n) = △ F −1 [X(e jω )] = 1
2π
∫ π
−π
X(e jω )e jωn dω (3.2)
The operator F[·] transforms a discrete signal x(n) into acomplex-valued
continuous function X(e jω )ofreal variable ω, called a digital frequency,
which is measured in radians/sample.
□ EXAMPLE 3.1 Determine the discrete-time Fourier transform of x(n) =(0.5) n u(n).
Solution
The sequence x(n) isabsolutely summable; therefore its discrete-time Fourier
transform exists.
∞∑
∞∑
X(e jω )= x(n)e −jωn = (0.5) n e −jωn
=
−∞
0
0
∞∑
(0.5e −jω ) n 1
=
1 − 0.5e = e jω
−jω e jω − 0.5
□
□ EXAMPLE 3.2 Determine the discrete-time Fourier transform of the following finite-duration
sequence:
x(n) ={1, 2, 3, 4, 5}
↑
Solution Using definition (3.1),
∞∑
X(e jω )= x(n)e −jωn = e jω +2+3e −jω +4e −j2ω +5e −j3ω
−∞
□
Since X(e jω )isacomplex-valued function, we will have to plot its
magnitude and its angle (or the real and the imaginary part) with respect
to ω separately to visually describe X(e jω ). Now ω is a real variable
between −∞ and ∞, which would mean that we can plot only a part of the
X(e jω ) function using MATLAB. Using two important properties of the
discrete-time Fourier transform, we can reduce this domain to the [0,π]
interval for real-valued sequences. We will discuss other useful properties
of X(e jω )inthe next section.
3.1.1 TWO IMPORTANT PROPERTIES
We will state the following two properties without proof.
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