The Fourier transformation

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

$$f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \tilde F(\omega... ... F(\omega)=\int_{-\infty}^{\infty} f(t) \ e^{-j \omega t} \ dt$$ (12)

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 $e^{j \omega t}$ (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 $s=j \omega$. In fact, by comparison with (6), we see that $\tilde F(\omega)=F(j \omega)$, i.e. the Fourier transform for a real angular frequency $\omega$ is identical to the Laplace transform for imaginary $s=j \omega$. The function $\tilde F(\omega)$ 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 $\tilde F(\omega)$ as well; however, $\tilde F(\omega)=F(j \omega)$ is not the same function as $F(\omega)$, so different names are appropriate.

The distinction between F(s) and $\tilde F(\omega)$is essential when systems are characterized by their poles and zeros. These are equivalent but not identical in the complex s and $\omega$ 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 $\vert\tilde F(\omega)\vert$ is called the amplitude response, and the phase of $\tilde F(\omega)$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:

$$f(t) = \sum_{\nu=-\infty}^{\infty} b_\nu \ e^{2 \pi j \nu t /... ... = \frac{1}{T} \int_{0}^{T} f(t) \ e^{-2 \pi j \nu t / T} \ dt$$ (13)

for periodic signals f(t) with a period T, and

$$f_{k} = \frac{1}{M} \sum_{\ell=0}^{M-1} c_{\ell} \ e^{2 \pi j... ... c_{\ell} = \sum_{k=0}^{M-1} \ f_{k} \ e^{-2 \pi j k \ell / M}$$ (14)

for time series f_k consisting of M equidistant samples such as in digital seismic records. We have noted the inverse transform (the synthesis) first in each case.

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 $\ell$ 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.