The Fourier transformation

Somewhat closer to intuitive understanding but
mathematically less general than the Laplace
transformation is the Fourier transformation

The signal is here assumed to have a finite
energy so that the integrals converge. The
condition that no signal is present at negative times
can be dropped in this case. The Fourier
transformation decomposes the signal into purely
harmonic waves
(the
tilde over *F* is meant to remind you of this). The
direct and inverse Fourier transformation are
also known as a harmonic analysis and
synthesis.

Although the mathematical concepts behind the Fourier and Laplace transformations are different, we may consider the Fourier transformation as a special version of the Laplace transformation for real frequencies, i.e. for . In fact, by comparison with (6), we see that , i.e. the Fourier transform for a real angular frequency is identical to the Laplace transform for imaginary . The function is called the complex frequency response of the system. For practical purposes the two transformations are thus nearly equivalent, and many of the relationships between time-signals and their transforms (such as the convolution theorem) are similar or the same for both. Some authors even use the name "transfer function" for as well; however, is not the same function as , so different names are appropriate.

The distinction between *F*(*s*) and
is essential when systems are
characterized by their poles and zeros. These
are equivalent but not identical in the complex *s* and
planes, and it is important to know whether the
Laplace or Fourier transform is meant.
Usually, poles and zeros are given for the Laplace
transform. In case of doubt, one has to
check the symmetry of the poles and zeros in the
complex plane: those of the Laplace
transform are symmetric to the real axis as in Fig. 2 while
those of the Fourier transform are symmetric
to the imaginary axis.

The absolute value is called the amplitude response, and the phase of the phase response of the system. Note that amplitude and phase do not form a symmetric pair; a certain mathematical symmetry (expressed by the Hilbert transformation) exists however between the real and imaginary parts of a rational transfer function, and between the phase response and the natural logarithm of the amplitude response.

The definition (12) of the Fourier transformation
applies to continuous transient signals. For
other mathematical representations of signals,
different definitions must be used:

for periodic signals

for time series

The Fourier integral transformation (12) is
mainly an analytical tool; the integrals are not
normally evaluated numerically because the discrete
Fourier transformation (14) permits more
efficient computations. Eq. (13) is the Fourier
series expansion of periodic functions, also
mainly an analytical tool but also useful to
represent periodic test signals. The discrete Fourier
transformation (14) is sometimes considered as
being a discretized, approximate version of
(12) or (13) but is actually a mathematical tool
in its own right: it is a mathematical identity
that does not depend on any assumptions on the
series *f*_{k}. Its relationship with the other two
transformations, and especially the interpretation
of the subscript
as representing a single
frequency, do however depend on the properties of
the original, continuous signal. The most
important condition is that the bandwidth of the
signal before sampling must be limited to less
than half of the sampling rate *f*_{S} (this is called
the Nyqvist frequency, *f*_{N}=*f*_{S}/2). Otherwise the
sampled series may not represent the original.
Whether we consider a signal as periodic or as
having a finite duration (and thus a finite
energy) is to some degree arbitrary since we can
analyze real signals only for finite intervals of
time, and it is then a matter of definition whether
we assume the signal to have a periodic
continuation outside the interval or not.

The Fast Fourier Transformation or FFT (Cooley and Tukey, 1965) is a recursive algorithm to compute the sums in Eq. (14) efficiently, so it does not constitute a mathematically different definition of the discrete Fourier transformation.

2002-11-08