【正文】 for a,b and c from [Park87]WindowabcRetangular100Hanning0Hamming0BlackmanThe Bartlett window reduces the overshoot in the designed filter but spreads the transition 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. The Frequency Sampling Technique In this method, [Park87], [Rab75], [Proakis00] the desired frequency response is provided as in the previous method. Now the given frequency response is sampled at a set of equally spaced frequencies to obtain N samples. Thus , sampling the continuous frequency response Hd(w) at N points essentially gives us the Npoint DFT of Hd(2pnk/N). Thus by using the IDFT formula, the filter coefficients can be calculated using the following formula (12)
Now using the above Npoint filter response, the continuous frequency response is calculated as an interpolation of the sampled frequency response. The approximation error would then be exactly zero at the sampling frequencies and would be finite in frequencies between them. The smoother the frequency response being approximated, the smaller will be the error of interpolation between the sample points. 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 samples. One 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,….N1. The 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. They are (a) h(n) is rotated by N/2 samples(N even) or (N1)/2 samples for N odd to remove the sharp edges of impulse response, and then 15N zerovalued samples are symmetrically placed around the impulse response. (b) h(n) is split around the N/2nd sample, and 15N zerovalued samples are placed between the two pieces of the impulse response. (v) The zero augmented sequences are transformed using the FFT algorithm to give the interpolated frequency responses. Merits of frequency sampling technique(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 desired frequency response only at the sampled points. At the other points, there will be a finite error present. Optimal Filter Design Methods Many methods are present under this category. The basic idea in each method is to design the filter coefficients again and again until a particular error is minimized. The various methods are as follows: (i) Least squared error frequency domain design (ii) Nonlinear equation solution for maximal ripple FIR filters (iii) Polynomial interpolation solut