Tham khảo tài liệu 'fir filters - fir filter desigh remez exchange method', công nghệ thông tin, kỹ thuật lập trình phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Chapter 13 FIR Filter Design Remez Exchange Method In general an FIR approximation to an ideal lowpass filter will have an amplitude response of the form shown in Fig. . This response differs from the ideal lowpass response in three quantifiable ways 1. The pass band has ripples that deviate from unity by 5p. 2. The stop band has ripples that deviate from zero by 5S. Note that Fig. shows an amplitude response rather than the usual magnitude response and therefore negative ordinates are possible. 3. There is a transition band of finite nonzero width AF between the pass band and stop band. Figure Typical amplitude response of an FIR approxi-mation to an ideal lowpass filter. 245 246 Chapter Thirteen The usual design goals are to in some sense minimize dp 5S and AF. As it is generally not possible to simultaneously minimize for three different variables some compromise is unavoidable. Chebyshev approximation is one approach to this design problem. Chebyshev Approximation In the Chebyshev approximation approach the amplitude response of a type 1 that is odd-length even-symmetric linear phase lowpass W-tap FIR filter is formulated as a sum of r cosines A X ck cos 2nkf k 0 where r N l 2 and the coefficients ck are chosen so as to yield an A f which is optimal in a sense that will be defined shortly. For a lowpass filter the pass band Bp and stop band Bs are defined as BP F FP Bs F- Fs F where Fp and Fs are respectively the edge frequencies for the pass band and stop band. Equation is read as Bp is the set of all F such that F is greater than or equal to zero and less than or equal to Fp. We can then define a set S as the union of Bn and B p Bn u Bs p 13 In other words J is the set of all frequencies between 0 and not including the transition frequencies F Fp F Fs. In mathematical terms S is described as a compact subset of 0 The desired response D f is the ideal lowpass response given by FcBp FeBs Thus we
