Discrete Fourier Transform and Signal Spectrum - Digital Signal Processing: Fundamentals and Applications - page 103

“interpolated spectrum” with reduced frequency spacing. We can get a better view of the two spectral
peaks described in this case.
The only way to obtain the detailed signal spectrum with a fine frequency resolution is to apply
more available data samples, that is, a longer sequence of data. Here, we choose to pad the least
number of zeros to satisfy the minimum FFT computational requirement. Let us look at another
example.

EXAMPLE 4.7
We use the DFT to compute the amplitude spectrum of a sampled data sequence with a sampling rate
f s ¼ 10 kHz. Given a requirement that the frequency resolution be less than 0.5 Hz, determine the number of data
points by using the FFT algorithm, assuming that the data samples are available.
Solution:

D f ¼ 0:5Hz

D f ¼ 10; 000

f s

N ¼ ¼ 20; 000

0:5

Since we use the FFT to compute the spectrum, the number of data points must be a power of 2, that is,

N ¼ 2 15 ¼ 32; 768

The resulting frequency resolution can be recalculated as

f s

N ¼ 10; 000

D f ¼

32; 768 ¼ 0:31 Hz:

Next, we study a MATLAB example.

EXAMPLE 4.8
Consider the sinusoid

n
8; 000

xðnÞ¼2$sin 2; 000p

obtained by sampling the analog signal

xðtÞ¼2$sinð2; 000ptÞ

with a sampling rate of f s ¼ 8,000 Hz,
a. Use the MATLAB DFT to compute the signal spectrum where the frequency resolution is equal to or less
than 8 Hz.
b. Use the MATALB FFT and zero padding to compute the signal spectrum, assuming that the data samples in
(a) are available.