Information Technology Reference
In-Depth Information
Figure 3.42:
In (a) is the full butterfly diagram for an FFT. The spectrum this computes is shown in (b).
In STFTs the overlapping input sample blocks must be multiplied by window functions. The principle is the same as
for the application in FIR filters shown in
section 3.5
.
Figure 3.43
shows that multiplying the search frequency by
the window has exactly the same result except that this need be done only once and much computation is saved.
Thus in the STFT the basis function is a windowed sine or cosine wave.
Figure 3.43:
Multiplication of a windowed block by a sine wave basis function is the same as multiplying the raw
data by a windowed basis function but requires less multiplication as the basis function is constant and can be pre-
computed.
The FFT is used extensively in such applications as phase correlation, where the accuracy with which the phase of
signal components can be analysed is essential. It also forms the foundation of the discrete cosine transform.
[
14
]
Kraniauskas, P.,
Transforms in Signals and Systems
, Ch. 6, Wokingham: Addison-Wesley (1992)
3.13 The discrete cosine transform (DCT)
The DCT is a special case of a discrete Fourier transform in which the sine components of the coefficients have
been eliminated leaving a single number. This is actually quite easy.
Figure 3.44
(a) shows a block of input samples
to a transform process. By repeating the samples in a time- reversed order and performing a discrete Fourier
transform on the double-length sample set a DCT is obtained. The effect of mirroring the input waveform is to turn it
into an even function whose sine coefficients are all zero. The result can be understood by considering the effect of
individually transforming the input block and the reversed block.
