Discrete Fourier Transform and Signal Spectrum - Digital Signal Processing: Fundamentals and Applications - page 102

a new sequence with a larger number of samples, N ¼ 2 m > N . The modified data sequence for
applying FFT, therefore, is

(

xðnÞ 0 n N 1

xðnÞ¼ (4.27)

0 N n N 1

It is very important to note that the signal spectra obtained via zero-padding the data sequence in
Equation (4.27) do not add any new information and do not contain more accurate signal spectral
presentation. In this situation, the frequency spacing is reduced due to more DFT points, and the
achieved spectrum is an interpolated version with “better display.” We illustrate the zero-padding
effect via the following example instead of theoretical analysis. A theoretical discussion of zero
padding in FFT can be found in Proakis and Manolakis (1996).
Figure 4.11 (a) shows the 12 data samples from an analog signal containing frequencies of 10 Hz
and 25 Hz at a sampling rate of 100 Hz, and the amplitude spectrum obtained by applying the DFT.
Figure 4.11 (b) displays the signal samples with padding of four zeros to the original data to make up
a data sequence of 16 samples, along with the amplitude spectrum calculated by FFT. The data
sequence padded with 20 zeros and its calculated amplitude spectrum using FFT are shown in
Figure 4.11 ( c). It is evident that increasing the data length via zero padding to compute the signal
spectrum does not add basic information and does not change the spectral shape but gives the

(a)

2

0.5

0

-2

0

0 5 10 0 50 100

(b)

Number of samples Frequency (Hz)

2

0.5

0

-2 ze r o padding

0

0 5 10 15 0 50 100

(c)

Number of samples Frequency (Hz)

2

0.5

0

-2 zer o padding

0

0 10 20 30 0 50 100

Number of samples Frequency (Hz)

FIGURE 4.11
Zero-padding effect by using FFT.