E1C02 09/14/2010
13:35:19 Page 66
An approximation to the Fourier transform integral of Equation 2.30 for use on a discrete data
set is the discrete Fourier transform (DFT). The DFT is given by
Y f k
ð Þ ¼
2
N
X NÀ1
r ¼ 0
y rdt
ð Þe
Ài2prk=N
f k ¼ kdf
k ¼ 0; 1; 2; . . . ;
N
2
À 1
df ¼ 1=Ndt
ð2:38Þ
Here, df is the frequency resolution of the DFT with each value of Y(f k ) corresponding to frequency
increments of df. In developing Equation 2.38 from Equation 2.30, t was replaced by rdt and f
replaced by k/Ndt. The factor 2/N scales the transform when it is obtained from a data set of finite
length (use 1/N for k ¼ 0 only).
The DFT as expressed by Equation 2.38 yields N/2 discrete values of the Fourier transform of
{y(rdt)}. This is the so-called one-sided or half-transform as it assumes that the data set is one-sided,
extending from 0 to t f , and it returns only positive valued frequencies.
Equation 2.38 performs the numerical integration required by the Fourier integral. Equations
2.35, 2.36, and 2.38 demonstrate that the application of the DFT on the discrete series of data, y(rdt),
permits the decomposition of the discrete data in terms of frequency and amplitude content. Hence,
by using this method a measured discrete signal of unknown functional form can be reconstructed as
a Fourier series through Fourier transform techniques.
Software for computing the Fourier transform of a discrete signal is included in the companion
software. The time required to compute directly the DFTalgorithm described in this section increases at a
rate that is proportional to N
2
. This makes it inefficient for use with data sets of large N. A fast algorithm
for computing the DFT, known as the fast Fourier transform (FFT), was developed by Cooley and Tukey
(5). This method is widely available and is the basis for most Fourier analysis software packages. The
FFT algorithm is discussed in most advanced texts on signal analysis (6, 7). The accuracy of discrete
Fourier analysis depends on the frequency content
5 of y(t) and on the Fourier transform frequency
resolution. An extensive discussion of these interrelated parameters is given in Chapter 7.
Example 2.6
Convert the continuous signal described by y(t) ¼ 10 sin 2pt V into a discrete set of eight numbers
using a time increment of 0.125 s.
KNOWN The signal has the form y(t) ¼ C 1 sin2pf 1 t
where
f 1 ¼ v 1 =2p ¼ 1 Hz
C f 1 ¼ 1 Hz
ð
Þ¼10 V
f f 1
ð Þ ¼ 0
dt ¼ 0:125 s
N ¼ 8
5 The value of 1/dt must be more than twice the highest frequency contained in y(t).
66 Chapter 2 Static and Dynamic Characteristics of Signals
13:35:19 Page 66
An approximation to the Fourier transform integral of Equation 2.30 for use on a discrete data
set is the discrete Fourier transform (DFT). The DFT is given by
Y f k
ð Þ ¼
2
N
X NÀ1
r ¼ 0
y rdt
ð Þe
Ài2prk=N
f k ¼ kdf
k ¼ 0; 1; 2; . . . ;
N
2
À 1
df ¼ 1=Ndt
ð2:38Þ
Here, df is the frequency resolution of the DFT with each value of Y(f k ) corresponding to frequency
increments of df. In developing Equation 2.38 from Equation 2.30, t was replaced by rdt and f
replaced by k/Ndt. The factor 2/N scales the transform when it is obtained from a data set of finite
length (use 1/N for k ¼ 0 only).
The DFT as expressed by Equation 2.38 yields N/2 discrete values of the Fourier transform of
{y(rdt)}. This is the so-called one-sided or half-transform as it assumes that the data set is one-sided,
extending from 0 to t f , and it returns only positive valued frequencies.
Equation 2.38 performs the numerical integration required by the Fourier integral. Equations
2.35, 2.36, and 2.38 demonstrate that the application of the DFT on the discrete series of data, y(rdt),
permits the decomposition of the discrete data in terms of frequency and amplitude content. Hence,
by using this method a measured discrete signal of unknown functional form can be reconstructed as
a Fourier series through Fourier transform techniques.
Software for computing the Fourier transform of a discrete signal is included in the companion
software. The time required to compute directly the DFTalgorithm described in this section increases at a
rate that is proportional to N
2
. This makes it inefficient for use with data sets of large N. A fast algorithm
for computing the DFT, known as the fast Fourier transform (FFT), was developed by Cooley and Tukey
(5). This method is widely available and is the basis for most Fourier analysis software packages. The
FFT algorithm is discussed in most advanced texts on signal analysis (6, 7). The accuracy of discrete
Fourier analysis depends on the frequency content
5 of y(t) and on the Fourier transform frequency
resolution. An extensive discussion of these interrelated parameters is given in Chapter 7.
Example 2.6
Convert the continuous signal described by y(t) ¼ 10 sin 2pt V into a discrete set of eight numbers
using a time increment of 0.125 s.
KNOWN The signal has the form y(t) ¼ C 1 sin2pf 1 t
where
f 1 ¼ v 1 =2p ¼ 1 Hz
C f 1 ¼ 1 Hz
ð
Þ¼10 V
f f 1
ð Þ ¼ 0
dt ¼ 0:125 s
N ¼ 8
5 The value of 1/dt must be more than twice the highest frequency contained in y(t).
66 Chapter 2 Static and Dynamic Characteristics of Signals