Transforms to and from the Frequency Domain - Digital Video and Audio Broadcasting Technology - page 105

6.7 Systematic Errors in DFT or FFT, and How to Prevent
them

To obtain the precise result of the Fourier Transform, a time-domain signal
would have to be observed for an infinitely long period of time. In the case
of the Discrete Fourier Transform, however, a signal segment is only ob-
served for a finite period of time and transformed. The result of the DFT or
FFT, respectively, will thus always differ from that of the Fourier Trans-
form. It has been seen that, in principle, this analyzed time segment is con-
verted into periodic signals in the DFT, i.e., the result of the DFT must be
considered to be the Fourier Transform of this converted time segment.

u(t) u(t) u(t) u(t)

N points N points

time time time time

N points N points

T T

T T

Fig. 6.17. Conversion of a signal segment into periodic signals by the DFT or
FFT, resp.

u(t) u'(t)

T2 T2

t t

T1 T1

Fig. 6.18. Windowing (T1, T2) a sinusoidal signal

It is clear that, naturally, the result of the transform depends greatly on
the type and position of the “cutting-out” process, the so-called window-
ing. This can be visualized best by performing the DFT on a sinusoidal