Discrete Fourier Transform and Signal Spectrum - Digital Signal Processing: Fundamentals and Applications - page 97

4.2 AMPLITUDE SPECTRUM AND POWER SPECTRUM
One DFT application is transformation of a finite-length digital signal xðnÞ into the spectrum in
the frequency domain. Figure 4.7 demonstrates such an application, where A k and P k are
the computed amplitude spectrum and the power spectrum, respectively, using the DFT coeffi-
cients XðkÞ .
First, we obtain the digital sequence xðnÞ by sampling the analog signal xðtÞ and truncating the
sampled signal with a data window of length T 0 ¼ NT , where T is the sampling period and N the
number of data points. The time for the data window is

T 0 ¼ NT (4.16)

For the truncated sequence xðnÞ with a range of n ¼ 0 ; 1 ; 2 ; / ; N 1, we get

xð 0 Þ; xð 1 Þ; xð 2 Þ; . ; xðN 1 Þ (4.17)

Next, we apply the DFT to the obtained sequence, xðnÞ , to get the N DFT coefficients

N 1
n¼ 0 xðnÞW n N ;

XðkÞ¼ for k ¼ 0 ; 1 ; 2 ; / ; N 1 (4.18)

Since each calculated DFT coefficient is a complex number, it is not convenient to plot it versus
its frequency index. Hence, after evaluating Equation (4.18) , the magnitude and phase of
each DFT coefficient (we refer to them as the amplitude spectrum and phase spectrum,
respectively) can be determined and plotted versus its frequency index. We define the amplitude
spectrum as

q

ð Real ½XðkÞÞ

1
N jXðkÞj ¼

1
N

2 2

A k ¼ þð Imag ½XðkÞÞ ; k ¼ 0 ; 1 ; 2 ; / ; N 1 (4.19)

x ()

T 1/ f s

AP
k

or

k

f f N

s /

n

0

T N 1 X ()

T T
0 Power

spectrum or
amplitude
spectrum

DSP
processing
DFT or FFT

k

x () 0 N /2 N 1

Nf

f f N s /

FIGURE 4.7

Applications of DFT/FFT.