ods involve partial fraction expansions and straightforward inversion by division. (2.122) sin ( w/2 ) 2.10 Data truncation As mentioned before, the Fourier transform of any signal satisfies the periodic relationship as in Equation 2.119. (2.123) (2.119) When only a finite amount of data is present, the effect in trying Due to this periodicity, one can consider the frequency range to estimate the underlying Fourier transform manifests in terms of -J <W<I to be the fundamental set of frequencies. the convolution of the underlying transform with the transform of a As seen before, the Fourier transform is of fundamental import- rectangular window. This means that in the time domain, the data ance in characterising the spectral content of signals, be they in the have been 'multiplied' by a window which is unity over the duration time domain, spatial domain, etc. However, the Fourier transform of the data, and zero outside. As the amount of data increases, the requires the entire set of the signal samples, and the question Dirichlet kernel, W_, tends towards a Dirac delta function. The therefore arises concerning the applicability of transform tech- Dirichlet kernel is plotted as a function of frequency, w, in Figure niques when the data is of finite support. 2.11, and it can be seen to consist of many lobes centred at frequen- The procedure of filling in the "missing data" with zeros is a very common method of trying to account for the situation. cies 0, for k=3,5,7, The main lobe has a width of As before, the discrete Fourier transform of a finite data set can be an of N, and side lobes width amplitude for of 27 N written as, in Equation 2.120. the each have a To determine the effects of using only a finite amount of data, (2.120) consider the example where the data consists of a pure sinusoid as in Equation 2.124. x(n) = A cos ( nw ) (2.124) The corresponding Fourier transform is given by Equation 2.125 r-r<wsr (2.125) The Fourier transform of the truncated sequence is therefore given by Equation 2.126. (w - w ) N 