【正文】 unction that contains a taper and decays toward zero gradually, instead of abruptly as it occurs in a rectangular window. As multiplication of sequences hd(n) and w(n) in time domain is equivalent to convolution of Hd(w) and W(w) in the frequency domain, it has the effect of smoothing Hd(w).The several effects of windowing the Fourier coefficients of the filter on the result of the frequency response of the filter are as follows: (i) A major effect is that discontinuities in H(w) bee transition bands between values on either side of the discontinuity.(ii) The width of the transition bands depends on the width of the main lobe of the frequency response of the window function, w(n) . W(w). (iii) Since the filter frequency response is obtained via a convolution relation , it is clear that the resulting filters are never optimal in any sense. (iv) As M (the length of the window function) increases, the mainlobe width of W(w) is reduced which reduces the width of the transition band, but this also introduces more ripple in the frequency response. (v) The window function eliminates the ringing effects at the bandedge and does result in lower sidelobes at the expense of an increase in the width of the transition band of the filter.ofwindowsmonlyarefollows:1. Bartlett triangle window (9)2. Generalized cosine windows(includes Rectangular,HammingBlackman) (10) 3. Kaiserwithβ n=0,1,2…N1 = 0 Otherwise (11)The general cosine window has four special forms that are monly used. These are determined by the parameters a,b,c TABLEValuecoefficientsa,bc[Park87]WindowabcRectangular100Hanning0Hamming0BlackmanThe Bartlett window reduces the overshoot in the designed filter but spreads thetransition region Hanning,Hamming and Blackman windows use progressively more plicated cosine functions to provide a smooth truncation of the ideal impulse response and a frequency response that looks better. The best window results probably e from using the Kaiser window, which has a parameter . that allows adjustment of the promise between the overshoot reduction and transition region width spreading. The major advantages of using window method is their relative simplicity as pared to other methods and ease of use. The fact that well defined equations are often available for calculating the window coefficients has made this method successful. There are following problems in filter design using window method: (i) This method is applicable only if Hd(w) is absolutely integrable only if (2) can be evaluated. When Hd(w) is plicated or cannot easily be put into a closed form mathematical expression, evaluation of hd(n) bees difficult.(ii) The use of windows offers very little design flexibility . in low pass filter design, the passband edge frequency generally cannot be specified exactly since the window smears the discontinuity in frequency. Thus the ideal LPF with cutoff frequency fc, is smeared by the window to give a frequency response with passband response with passband cutoff frequency f1 and stopband cutoff frequency f2. (iii) Window method is basically useful for design of prototype filters like lowpass,highpass,bandpass etc. This makes its use in speech and image processing applications very limited.TheSamplingusingabovefilterthefrequencyisasinterpolationthefrequencyTheerrorthenexactlyatsamplingandbeinbetweenThetheresponseapproximated,smallerbeerrorinterpolationthepoints.One way to reduce the error is to increase the number of frequency samples [Rab75]. The other way to improve the quality of approximation is to make a number of frequency samples specified as unconstrained variables. The values of these unconstrained variables are generally optimized by puter to minimize some simple function of the approximation error . one might choose as unconstrained variables the frequency samples that lie in a transition band between two frequency bands in which the frequency response is specified . in the band between the passband and the stopband of a low pass filter.There are two different set of frequencies that can be used for taking the samplesOne set of frequency samples are at fk = k/N where k = 0,1,….N1. The other set of uniformly spaced frequency samples can be taken at fk =(k+1/2)/N for k = 0,1,….N1The second set gives us the additional flexibility to specify the desired frequency response at a second possible set of frequencies. Thus a given band edge frequency may be closer to typeII frequency sampling point that to typeI in which case a typeII design would be used in optimization procedure. In a paper by Rabiner and Gold [Rabi70], Rabiner has mentioned a technique based on the idea of frequency sampling to design FIR filters. The steps involved in this method suggested by Rabiner are as follows:(i) The desired magnitude response is provided along with the number of samples,N . Given N, the designer determines how fine an interpolation will be used. (ii) It was found by Rabiner that for designs they investigated, where N varied from 15 to 256, 16N samples of H(w) lead to reliable putations, so 16 to 1 interpolation was used.(iii)Given N values of Hk , the unit sample response of filter to be designed, h(n) is calculated using the inverse FFT algorithm. (iv)In order to obtain values of the interpolated frequency response two procedures were suggested by Rabiner.Theyofsampling(i) Unlike the window method, this technique can be used for any given magnitude response. (ii) This method is useful for the design of nonprototype filters where the desired magnitude response can take any irregular shape. There are some disadvantages with this method the frequency response obtained by interpolation is equal to the