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64 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
15
Magnitude Part
15
Real Part
Magnitude
10
5
Real
10
5
0
0
0 0.5 1
frequency in π units
4
Angle Part
–5
0 0.5 1
frequency in π units
5
Imaginary Part
Radians
2
0
–2
Imaginary
0
–5
–4
0 0.5 1
frequency in π units
–10
0 0.5 1
frequency in π units
FIGURE 3.2 Plots in Example 3.4
The procedure of Example 3.4 can be compiled into a MATLAB function,
say a dtft function, for ease of implementation. This is explored in
Problem P3.1. This numerical computation is based on definition (3.1).
It is not the most elegant way of numerically computing the discretetime
Fourier transform of a finite-duration sequence. In Chapter 5 we
will discuss in detail the topic of a computable transform called the discrete
Fourier transform (DFT) and its efficient computation called the
fast Fourier transform (FFT). Also there is an alternate approach based
on the z-transform using the MATLAB function freqz, which we will discuss
in Chapter 4. In this chapter we will continue to use the approaches
discussed so far for calculation as well as for investigation purposes.
In the next two examples we investigate the periodicity and symmetry
properties using complex-valued and real-valued sequences.
□ EXAMPLE 3.5 Let x(n) =(0.9 exp (jπ/3)) n , 0 ≤ n ≤ 10. Determine X(e jω ) and investigate
its periodicity.
Solution
Since x(n) iscomplex-valued, X(e jω ) satisfies only the periodicity property.
Therefore it is uniquely defined over one period of 2π. However, we will evaluate
and plot it at 401 frequencies over two periods between [−2π, 2π] toobserve its
periodicity.
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