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62 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
2
Magnitude Part
2
Real Part
Magnitude
1.5
1
Real
1.5
1
0.5
0 0.5 1
0
frequency in π units
Angle Part
0.5
0 0.5 1
frequency in π units
0
Imaginary Part
Radians
–0.2
–0.4
Imaginary
–0.2
–0.4
–0.6
–0.6
0 0.5 1
frequency in π units
–0.8
0 0.5 1
frequency in π units
FIGURE 3.1 Plots in Example 3.3
The resulting plots are shown in Figure 3.1. Note that we divided the w array by
pi before plotting so that the frequency axes are in the units of π and therefore
easier to read. This practice is strongly recommended.
□
If x(n) isoffinite duration, then MATLAB can be used to compute
X(e jω )numerically at any frequency ω. The approach is to implement
(3.1) directly. If, in addition, we evaluate X(e jω )atequispaced frequencies
between [0,π], then (3.1) can be implemented as a matrix-vector multiplication
operation. To understand this, let us assume that the sequence
x(n) has N samples between n 1 ≤ n ≤ n N (i.e., not necessarily between
[0,N − 1]) and that we want to evaluate X(e jω )at
ω k
△
=
π
M k,
k =0, 1,...,M
which are (M +1)equispaced frequencies between [0,π]. Then (3.1) can
be written as
N∑
X(e jω k
)= e −j(π/M)kn l
x(n l ),
l=1
k =0, 1,...,M
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