Applications of digital signal processing - Continuous and Discrete Time Signals and Systems - page 747

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

W

W

0 0

− p −0.8 p −0.6 p −0.4 p −0.2 p 0 0.2 p 0.4 p 0.6 p 0.8 pp −0.04 p −0.02 p 0 0.02 p 0.04 p

(a) (b)

0.08 0.08

0.06 0.06

0.04

0.04

0.02

0.02

W W

0 0

−0.2 p −0.15 p −0.1 p −0.05 p 0 0.05 p 0.1 p 0.15 p 0.2 p

− p −0.8 p −0.6 p −0.4 p −0.2 p 0 0.2 p 0.4 p 0.6 p 0.8 pp

(c) (d)

applications. For most signals of interest, the discrete Fourier transform (DFT)
provides a convenient approach for spectral estimation. Example 17.1 highlights
the DFT-based approach for two test signals.

Fig. 17.1. DFT used to estimate
the frequency content of
stationary and non-stationary
signals in Example 17.1.
(a) Magnitude sepctrum of
x 1 [ k ]. (b) Enlarged version of
part (a) in the frequency range
−0.05π ≤ Ω ≤ 0.05π .
(c) Magnitude spectrum of
x 2 [ k ]. (d) Enlarged version of
part (c) in the frequency range
−0.2π ≤ Ω ≤ 0.2π .

Example 17.1
Using the DTFT, estimate the spectral content of the following DT signals:

(a) x 1 [ k ] = cos(0 . 01 π k ) + 2 cos(0 . 015 π k );
(b) x 2 [ k ] = cos(0 . 0001 π k 2 ),

from observations made over the interval 0 ≤ k ≤ 1000.

Solution
(a) The magnitude spectrum of x 1 [ k ] based on the DFT is plotted over the
frequency range −π ≤ Ω ≤ π in Fig. 17.1(a) with the magnified version
shown in Fig. 17.1(b), where the frequency range − 0 . 05 π ≤ Ω ≤ 0 . 05 π
is enhanced. By looking at the peak values in Fig. 17.1(b), it is clear that
the frequencies Ω 1 = 0 . 01 π and Ω 2 = 0 . 015 π radians/s are the dominant
frequencies in the signal. On a relative scale, the frequency component Ω 2 =
0 . 015 π has a higher strength compared with the frequency component
Ω 1 = 0 . 01 π .
(b) The magnitude spectrum of x 2 [ k ] based on the DFT over the frequency
range −π ≤ Ω ≤ π is plotted in Fig. 17.1(c), with the magnified version
shown in Fig. 17.1(d), where the frequency range − 0 . 2 π ≤ Ω ≤ 0 . 2 π is
enhanced. From the subplots, it seems that all frequencies within the range
− 0 . 2 π ≤ Ω ≤ 0 . 2 π are fairly significant in x 2 [ k ]. To confirm the validity
of our estimation, let us calculate the instantaneous frequency of the signal.