sin (w-w) (w+w,)N (2.126) Figure 2.11 Magnitude of the Fourier transform (w+w1) associated with: (a) an untruncated sinusoid; (b) a truncated sinusoidal time series Conclusions 2/17 Table 2.3 Commonly used data windows Window w(n),Osns(N-1) Main-lobe width (rad) Main-lobe amplitude 20 log Largest side-lobe amplitude Rectangular Bartlett N - 1 2-2n- N - 1 N - 1 N - 1 Hanning 0.5 cos Hamming 0.54 - 0.46 cos - ( N Blackman 0.42 - 0.5 cos Thus the effect of using only a small data set is to smear out the expansion coefficients was given. Ideas concerning spectral ana- original delta function spectrum. This effect is commonly known as lysis were introduced. spectral leakage. Linear systems theory was briefly touched upon, the z-transform In attempts to reduce this spectral leakage, a variety of SO called was introduced and the concepts of transfer function and frequency data windows have been introduced in order to 'smooth out' the response were discussed. ripples introduced by the rectangular window. A few of the window Fast implementations of the Discrete Fourier transform, using the types are, Triangular, Hanning (or raised cosine), Hamming, Black- FFT were discussed, and the ideas of spectral leakage and window- man etc. For a good introduction to windowing see (Harris, 1978). ing were mentioned. The effects of various data windows are shown in Table 2.3. There exists a vast literature concerned with function expansion Clearly there is a trade off between side lobe levels and width of and representation, some references for which are given below. the main peak, SO caution has to be applied in applying data win- The subject of transform methods is similarly vast, and it must be dows depending on the application at hand. One should also say that appreciated that the Fourier and z-transforms are members of a there is a school of thought that windowing the data by anything transform family including Radon transforms, Wigner transforms, other than a rectangular window distorts the original data, and Wavelet transforms 