re, we can obtain Equations 2.47 and 2.48, where f==1/Tp (2.43) (2.47) Phase information is therefore lost in using the power spectral density. (2.48) For a real valued periodic signal, the Fourier coefficients Ck can easily be shown to satisfy Equation 2.44, which implies that the Discrete sequences 2/7 These equations show that x(t) and X(f) form a so-called Fourier Between the limits can be replaced by x(t), and Transform pair, and the Fourier Transform of x(t) exists if the signal since x(t) is zero outside this range of integration, Equation 2.49 has a finite energy, i.e. as in Equation 2.56. may be obtained. (2.56) (2.49) Earlier we defined the average power of a periodic signal. It is now possible to similarly define the energy of an aperiodic signal as A new function, X(f), can now be defined as the Fourier transform in Equations 2.57 and 2.58 of x(t) as in Equation 2.50 and the Fourier coefficient Ck can be written as in Equation 2.51, which are samples of X(f) taken at multiples of f and also scaled by the factor f (=1/Tp). (2.50) (2.57) (2.51) (2.58) Therefore Equation 2.52 can be obtained. Hence the energy of an aperiodic signal can be written as in (2.52) Equation 2.59 which is also known as Parseval's relation, (Wax, 1954). As described above, we require to take the limit as T approaches (2.59) infinity, and it is therefore convenient to define a frequency dif- in the limit of approaches zero. We may therefore write Equation 2.53, and taking the limit as of Discrete sequences obtain Equations 2.54 and 2.55 In order for a digital computer to manipulate a signal, the signal (2.53) must be sampled at a chosen sampling rate, 1/Ts, giving rise to a set of numbers called a sequence. If the continuous signal was x(t), the sampled sequence is represented by x(nTs), where n is an integer, (2.54) and the independent variable, t, could represent time or a spatial co-ordinate for example. The analysis can, of course, be extended into higher dimensions. In order to move between the continuous domain and the disc