Transforms to and from the Frequency Domain - Digital Video and Audio Broadcasting Technology - page 99

6.4 Implementation and Practical Applications of DFT and
FFT
The Fourier Transform, the Discrete Fourier Transform and the Fast Fou-
rier Transform are all defined through the field of complex numbers. This
means that both the time domain signal and the frequency domain signal
have real and imaginary parts. Typical time domain signals are, however,
always purely real, i.e. the imaginary part is zero at every point in time.
The imaginary part must, therefore, be set to zero before the Fourier trans-
form or its numerical variations DFT and FFT are performed.
When DFT or FFT and IDFT or IFFT are performed in practice two in-
put signals are required (Fig. 6.7.). The input signals are implemented as
real-part and imaginary-part tables and correspond to the sampled time or
frequency domain. As the N samples of a typical time domain signal are
always real, the corresponding imaginary part must be set to zero for each
of the N points. This means that the imaginary-part table for the time do-
main must be filled with zeros. When the inverse transform is performed,
the imaginary part of the time domain signal must again be zero assuming
that the frequency range for the real part is about half the sampling fre-
quency and the frequency range for the imaginary part is point-to-point
symmetric about half the sampling frequency. If these symmetries are not
present in the frequency domain, a complex time domain signal is output,
i.e. the signal also has imaginary components in the time domain.
6.5 The Discrete Cosine Transform (DCT)
The Discrete Cosine Transform (DCT), and thus also the fast Fourier
Transform which is a special case of the DCT, is a cosine-sine transform
as can be seen from its formula; it is an attempt to assemble a time-domain
signal segment by the superposition of many different cosine and sine sig-
nals of different frequency and amplitude. A similar result can also be
achieved by using only cosine signals or only sine signals.
They are then called Discrete Cosine Transform (DCT) (Fig. 6.8.) or
Discrete Sine Transform (DST) (Fig. 6.9.). Compared with the DFT, the
sum of single signals required remains the same but twice as many cosine
or sine signals are required. In addition, half-integral multiples of the fun-
damental are needed as well as integral multiples. The Discrete Cosine
Transform (Fig. 6.8.), especially, has become quite important for audio and
video compression.