Discrete Fourier Transform and Signal Spectrum - Digital Signal Processing: Fundamentals and Applications - page 101

A k

4

2.

14141 .

2

1

k

0 1 2

fH ()

0 25 50

FIGURE 4.10

One-sided amplitude spectrum in Example 4.5.

We plot the one-sided amplitude spectrum for comparison in Figure 4.10 .
Note that in the one-sided amplitude spectrum, the negative-indexed frequency components are added back
to the corresponding positive-indexed frequency components; thus each amplitude value other than the DC term is
doubled. It represents the frequency components up to the folding frequency.

EXAMPLE 4.6
Consider a digital sequence sampled at the rate of 10 kHz. If we use 1,024 data points and apply the 1,024-point
DFT to compute the spectrum,
a. determine the frequency resolution;
b. determine the highest frequency in the spectrum.
Solution:
a. D f ¼

f s

N ¼ 10000

¼ 9:776 Hz

1024

b. The highest frequency is the folding frequency, given by

N
2 D f ¼

f s
2

f max ¼

¼ 512$9:776 ¼ 5000 Hz:

As shown in Figure 4.7 , the DFT coefficients may be computed via a fast Fourier transform (FFT)
algorithm. The FFT is a very efficient algorithm for computing DFT coefficients. The FFT algorithm
requires a time domain sequence xðnÞ where the number of data points is equal to a power of 2; that is,
2 m samples, where m is a positive integer. For example, the number of samples in xðnÞ

can be

N ¼ 2 ; 4 ; 8 ; 16 ; etc.
When using the FFT algorithm to compute DFT coefficients, where the length of the available data
is not equal to a power of 2 (as required by the FFT), we can pad the data sequence with zeros to create