Transforms to and from the Frequency Domain - Digital Video and Audio Broadcasting Technology - page 103

6.6 Time Domain Signals and their Transforms in the
Frequency Domain

In the following paragraphs, some important time-domain signals and their
transforms in the frequency domain will be discussed. The purpose of
these observations is to get some feel for the results of the Fast Fourier
Transform.
Let us begin with a periodic squarewave signal (Fig. 6.12.): since it is a
periodic signal, it has discrete lines in the frequency spectrum; all discrete
spectral lines of the squarewave signal are located at integral multiples of
the fundamental frequency of the squarewave signal. Most of the energy
will be found in the fundamental wave itself. If there is a DC component, it
will result in a spectral line at zero frequency (Fig. 6.14.). The envelope of
the spectral lines of the fundamental and the harmonics is the sin(x)/x
function.

U(f)

u(t)

t f

Fig. 6.13. Fourier Transform of a Dirac impulse

U(f)

u(t)

t f

Fig. 6.14. Fourier Transform of a pure direct voltage (DC)

If then the duration of the period T of the squarewave signal is allowed
to tend towards infinity, the discrete spectral lines move closer and closer
together until a continuous spectrum of a single pulse is obtained (Fig.
6.11.).