Transforms to and from the Frequency Domain - Digital Video and Audio Broadcasting Technology - page 98

The number of multiplications given below shows the time gained by
the FFT compared with the DFT:
Number of multiplications needed:

DFT: N • N

FFT; N • log(2N)

The FFT has long been used in the field of acoustics (surveying concert
halls and churches) and in geology (searching for minerals, ores and oil).
However, the analyses were performed off-line with fast computers, using
a Dirac impulse to excite the medium to be examined (hall, rocks) and then
recording the impulse response of the medium under investigation. A
Dirac impulse is a very short and very strong impulse, an example of an
acoustical Dirac impulse being a pistol shot and a geological Dirac impulse
being the explosion of a blasting charge.
Back in 1988, a 256 point FFT still consumed minutes of PC time. To-
day, an 8192 point FFT (8k FFT) takes less than one millisecond of com-
puting time! This opens the door for new and interesting applications such
as video and audio compression or Orthogonal Frequency Division Multi-
plex (OFDM). FFT has also been used increasingly for spectrum analysis
in analog video testing and for detecting the amplitude and group delay re-
sponse of video transmission links since the late 1980s. In modern storage
oscilloscopes, too, this interesting test function is frequently found today
and makes it possible to perform a low-cost spectrum analysis, especially
also in audio test engineering.

Frequency domain

Time domain

u(t)

Re(f)

Re(f)

re(t)

N points

FFT/DFT

ts

f

time

N points

f=fs/N

Im(f)

T

0

N points

IFFT/IDFT

im(t)

Im(f)

f

fs = 1/ts

Fig. 6.7. Implementation and practical applications of DFT and FFT