Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 18

1.2.3.2 The Fourier Transform

The Fourier Series (FS) is applicable only to periodic signals, with period T o and
fundamental frequency f o = 1/T o . One can adapt the FS to be able to analyze non-
periodic signals by setting T o in the FS definition to T o ? ?. With this setting the
FS tends to the Fourier transform (FT). The time domain signal and its FT are
often referred to as a Fourier transform pair, since the two quantities can be
obtained from one another by transformation/inverse transformation:

X ð f Þ¼Ff x ð t Þg¼ Z

1

x ð t Þ e 2pft dt ð 1 : 19 Þ

1

x ð t Þ¼F 1 f X ð f Þg¼ Z

1

X ð f Þ e þ 2pft dt ð 1 : 20 Þ

1

The Fourier transform (FT) reveals the frequency content of an arbitrary a-
periodic (or non-periodic) signal, and is often referred to as the spectrum. If the
Fourier transform is applied to the system impulse response, one obtains the
frequency response or transfer function of the system. This function describes
the ratio of the system output to the system input as a function of frequency.
The Fourier transform X(f) of a real signal x(t) is generally complex. To plot the
Fourier transform spectrum, one typically plots a magnitude spectrum and a phase
spectrum. The magnitude spectrum is a plot of |X(f)| versus frequency f, while the
phase spectrum is a plot of \X ð f Þ versus frequency.If the signal which has been
Fourier transformed happens to be the impulse response of the system, the mag-
nitude and phase spectra are referred to as the system magnitude response and the
system phase response respectively.
Although the Fourier transform was initially introduced to analyze non-peri-
odic signals, it can be used to analyze periodic signals as well. For such signals
one obtains Fourier transform expressions containing Dirac delta functions.
Note: The CFS pair ( 1.14 )-( 1.15 ) is similar in structure to the FT pair ( 1.19 )-
( 1.20 ). For periodic signals with a period of T o , it is easy to see that
X n ¼ T o X 1p ð f Þj f ¼ nf o , where X 1p (f) is the FT of one period of x(t), hence, X 1p (f)/T o is
the envelope of the CFS coefficients.

Consider the signal x(t) = e -2t u(t) (a decaying exponential). Its FT is: Example 1

X ð f Þ¼ Z e 2t e j2pft dt ¼ Z

1 1

1
2 þ j2pf

e t ð 2 þ j2pf Þ dt ¼ :

0 0

1
4 þ 4p 2 f 2

Amplitude spectrum ¼j X ð f Þj¼ p :