Fourier Transforms

substituted into the original equation to check that it is correct.The final result is found to be of the formfx A exp−ax 2 where the constants A and a are determined as part of the solution process (determinethem!). If we had been given this at the start, then A and a could have been determined bysubstituting directly into the integral equation without having to employ FTs.Note: Step1 requires the use of contour integration to determine gk and the result−e −x2 dx .3.3.4 Signal ProcessingSignal processing usually involves the time t and frequency as the two variables, givingrise to time-domain and frequency domain modelling. These quantities are related via theFT, i.e.Fft f ft eitdt .−If the signal ft is such that ft 0 for all |t| ≥ T, then the signal is called Time-Limitedand corresponds to a signal of finite length. If the signal is such that f 0 for all|| ≥ L, then the signal is called Band Limited and is restricted to a range of frequencies. Aband limited signal can be obtained by passing a full signal through a low-pass filter. It canbe shown, though we do not provide a proof, that a signal cannot be both time- andband-limited.The Shannon Sampling Theorem demonstrates the link between FT and FS for aband-limited signal. Let ft be the measured time signal and f be its band-limitedtransform. Using the inverse FT givesft 12 f e−itd 1−2L−L f e−itdand at the point t n/L, we obtainf nL 12L−L f exp −i nLd .Let H be a periodic complex-valued function of period 2L, defined to be equal to the18