Transforms to and from the Frequency Domain - Digital Video and Audio Broadcasting Technology - page 104

The spectrum of a single squarewave pulse is a sin(x)/x function. If then
the pulse width T is allowed to become narrower and narrower and to tend
towards zero, all zero points of the sin(x)/x function will tend towards in-
finity. In the time domain, this provides an infinitely short pulse, a so-
called Dirac impulse, the Fourier Transform of which is a straight line; i.e.
the energy is distributed uniformly from zero frequency to infinity (Fig.
6.13.). Conversely, a single Dirac needle at f=0 in the frequency domain
corresponds to a direct voltage (DC) in the time domain.

U(f)

u(t)

t

f

-T T 2T -1/T 1/T

2/T

Fig. 6.15. Fourier Transform of a sequence of Dirac impulses

U(f)

f

-1/T 1/T

Fig. 6.16. Fourier Transform of a sinusoidal signal

A sequence of Dirac impulses spaced apart at intervals T from one an-
other again results in a discrete spectrum of Dirac needles spaced apart by
1/T (Fig. 6.15.). The Dirac impulse train is of importance when consider-
ing a sampled signal. Sampling an analog signal has the consequence that
this signal is convoluted with a sequence of Dirac impulses.
To conclude, a purely sinusoidal signal will be considered. Its Fourier
transform is a Dirac needle at the frequency of the sinewave fs and -fs
(Fig. 6.16.).