Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 22

x ( t ) = δ ( t ) X ( f ) = 1

FT

1

t , sec f , Hz

0 0

Fig. 1.14 The time-domain delta function and its spectrum

Example 3 If x(t) = d(t), then its spectrum is given by:

X ð f Þ¼ Z

1

d ð t Þ e j2pft

dt ¼ g ð 0 Þ¼ 1 :

|{z}
g ð t Þ

1

(using the definition of the delta function from Tables, Formula 16) (see Fig. 1.14 ).
It is seen that for the infinitely narrow time domain delta function the corre-
sponding spectrum is infinitely wide. In general, if the time duration of a signal is
narrow, its frequency spread tends to be wide, and vise versa.

Properties of the FT

The properties of the FT are detailed in the Tables at the end of this topic. The
reader should prove properties himself/herself as a means of deepening their
understanding of the Fourier transform (These proofs can be found in standard
references such as [ 3 ]). Some of these properties are discussed below.

1. Duality of the FT:

If x ð t Þ F X ð f Þ is a FT pair, then:
X ð t Þ F x ð f Þ is a FT pair

Note that the time and frequency variables are exchanged in the above relations. If
x(t)iseven, then the duality relations become even simpler:

X ð t Þ F x ð f Þ

Example 1 Previously it has been shown that

P T ð t Þ F T sinc ð fT Þ:

Using the duality property it follows that:

B sinc ð Bt Þ F P B ð f Þ