rete (2.55) domain, the idea of a sampling function must be introduced, and from the definition of the Dirac delta function (8(1)) as in Equa- tion 2.60, it is clear that this function meets the requirement. -Tp/2 xp(t) -Tp -Tp/2 0 Tp/2 Tp Figure 2.3 An aperiodic signal and a periodic signal constructed by repeating the periodic signal 2/8 Fourier analysis (2.60) Consider an analog signal x(t) that is continuous in both time and amplitude, and assume that x(t) has infinite duration but finite energy. Let the sample values of the signal x(t) at times =2T be denoted by the series where T is the sampling period and = 1/T s' is the sampling rate. The discrete time signal X (t) that is obtained by sampling the continuous signal x(t) can then be written as in Equation 2.61, where 8 (1-nT) is a Dirac delta function located at time t=nTs, and each delta function is weighted by the corresponding sample value of the input signal x(t). wordbox (2.61) For the case of discrete data, exactly analogous transform methods may be applied as for continuous signals, and we must now discuss the implementation issues for such methods. Figure 2.4 Cyclic properties of the basis functions for an eight point DFT 2.5 The discrete Fourier transform The frequency resolution of the DFT is equal to the frequency increment F/N and is referred to as the bin spacing of the DFT outputs. The frequency response of any DFT bin output is deter- The DFT is used extensively in digital signal processing (Burrus, mined by applying a complex exponential input signal and evalua- 1985), and is used routinely for the detection and estimation of ting the DFT bin output response as the frequency is varied. periodic signals. The discrete Fourier transform (DFT) of a discrete- Consider an input signal given by Equation 2.64, then the DFT of time signal x(n) is defined as in Equation 2.62, where k=0,1, x(n) can be expressed as a function of the arbitrary frequency N-1 and are the basis functions of the DFT. variable f by Equation 2.65. (2.62) (2.64) (2.65) These f