E1C02 09/14/2010
13:35:19 Page 63
The LabView program WaveformGeneration.vi follows Examples 2.4 and 2.5 and provides the
spectra for a number of signals. Several Matlab programs are provided also (e.g., FourCoef,
FunSpect, DataSpect) to explore the concept of superposition of simple periodic signals to create a
more complex signal. The software allows you to create your own functions and then explore their
frequency and amplitude content.
2.5 FOURIER TRANSFORM AND THE FREQUENCY SPECTRUM
The previous discussion of Fourier analysis demonstrates that an arbitrary, but known, function can
be expressed as a series of sines and cosines known as a Fourier series. The coefficients of the
Fourier series specify the amplitudes of the sines and cosines, each having a specific frequency.
Unfortunately, in most practical measurement applications the input signal may not be known in
functional form. Therefore, although the theory of Fourier analysis demonstrates that any function
can be expressed as a Fourier series, the analysis presented so far has not provided a specific
technique for analyzing measured signals. Such a technique for the decomposition of a measured
dynamic signal in terms of amplitude and frequency is now described.
Recall that the dynamic portion of a signal of arbitrary period can be described from Equation 2.17.
A n ¼
2
T
ð T=2
ÀT=2
y t
ð Þ cos nvtdt
B n ¼
2
T
ð T=2
ÀT=2
y t
ð Þ sin nvtdt
ð2:25Þ
where the amplitudes A n and B n correspond to the nth frequency of a Fourier series.
If we consider the period of the function to approach infinity, we can eliminate the constraint on
Fourier analysis that the signal be a periodic waveform. In the limit as T approaches infinity the
Fourier series becomes an integral. The spacing between frequency components becomes
Amplitude
10 2
10 1
10 0
120
840
720
600
480
360
240
0
Frequency (Hz)
6
5
4
3
2
6
5
4
3
2
Figure 2.17 Frequency content of the function y t
ð Þ ¼
120 sin 20ptj
j
displayed as
an amplitude-frequency
spectrum.
2.5 Fourier Transform and The Frequency Spectrum 63
13:35:19 Page 63
The LabView program WaveformGeneration.vi follows Examples 2.4 and 2.5 and provides the
spectra for a number of signals. Several Matlab programs are provided also (e.g., FourCoef,
FunSpect, DataSpect) to explore the concept of superposition of simple periodic signals to create a
more complex signal. The software allows you to create your own functions and then explore their
frequency and amplitude content.
2.5 FOURIER TRANSFORM AND THE FREQUENCY SPECTRUM
The previous discussion of Fourier analysis demonstrates that an arbitrary, but known, function can
be expressed as a series of sines and cosines known as a Fourier series. The coefficients of the
Fourier series specify the amplitudes of the sines and cosines, each having a specific frequency.
Unfortunately, in most practical measurement applications the input signal may not be known in
functional form. Therefore, although the theory of Fourier analysis demonstrates that any function
can be expressed as a Fourier series, the analysis presented so far has not provided a specific
technique for analyzing measured signals. Such a technique for the decomposition of a measured
dynamic signal in terms of amplitude and frequency is now described.
Recall that the dynamic portion of a signal of arbitrary period can be described from Equation 2.17.
A n ¼
2
T
ð T=2
ÀT=2
y t
ð Þ cos nvtdt
B n ¼
2
T
ð T=2
ÀT=2
y t
ð Þ sin nvtdt
ð2:25Þ
where the amplitudes A n and B n correspond to the nth frequency of a Fourier series.
If we consider the period of the function to approach infinity, we can eliminate the constraint on
Fourier analysis that the signal be a periodic waveform. In the limit as T approaches infinity the
Fourier series becomes an integral. The spacing between frequency components becomes
Amplitude
10 2
10 1
10 0
120
840
720
600
480
360
240
0
Frequency (Hz)
6
5
4
3
2
6
5
4
3
2
Figure 2.17 Frequency content of the function y t
ð Þ ¼
120 sin 20ptj
j
displayed as
an amplitude-frequency
spectrum.
2.5 Fourier Transform and The Frequency Spectrum 63