By Paul A. Lynn (auth.)
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Extra info for An Introduction to the Analysis and Processing of Signals
9 Symmetry between time and frequency domains. 4 Limitations of the Fourier transform So far it has been implied that the Fourier transfonn may be successfully used for any non-repetitive signal, but there are in fact limitations and difficulties in its application. e- jwt dt The Fourier transform will clearly only exist if the right-hand side of the equation is finite. 10 ,-- (a) The unit step function, and (b) an exponentially decaying step function meet this latter condition, and strictly speaking do not therefore possess Fourier transforms.
As we have seen, the Fourier transform allows a signal to be expressed as a sum of sinusoidal and cosinusoidal components which exist over all time, past, present and future, each component being represented by a pair of imaginary exponential terms of the form ei wt . By introducing a so-called convergence factor it is possible to derive the frequency spectra of certain signals for which the Fourier integral may not otherwise be evaluated. So far we have thought of this convergence factor e -at as being applied as a multiplier to an awkward signal, so that the integral may be evaluated, and we then let a tend to zero.
In other words they may not generally be assumed to exist for all time past, present, and future, and it is important to understand the effects which time-limitation has upon their frequency spectra. Secondly, and quite apart from any question of time-limitation, there is an important class of signal waveforms (amongst which are included random signals) which are simply not repetitive in nature and' which cannot therefore be represented by Fourier series containing a number of harmonically-related frequencies.