延 安大学数字信号处理课件Chapter 7 FIR Digital Filter Design.pptVIP

* The Parks-McClellan Algorithm For the Type 4 linear-phase FIR filter, the amplitude response can be rewritten in the form where The Parks-McClellan Algorithm The amplitude response for all four types of linear-phase FIR filters can be expressed in the form Where the first factor Q(ω) is given by The Parks-McClellan Algorithm The Parks-McClellan Algorithm We arrive at a modified form of the weighted approximation function given by Using the notation The Parks-McClellan Algorithm The optimization problem now becomes the determination of the coefficients Then, the corresponding coefficients of the original amplitude response are computed from which the filter coefficients are obtained. Alteration Theorem The amplitude function A(ω) is the best unique approximation of the desired amplitude response obtained by minimizing the peak absolute valueεof ε(ω) if and only if there exist at least L+2 extremal angular frequencies. The Parks-McClellan Algorithm The Remez exchange algrithm, a highly efficient iterative procedure, is used to determine the locations of the extremal frequencies and consists of the following steps at each iteration stage: The Parks-McClellan Algorithm Step 1: A set of initial value for the extremal frequencies are either chosen or are available from the completion of the previous iteration. The Parks-McClellan Algorithm Step 2: The value ε is then computed by solving the following equation The Parks-McClellan Algorithm Step 3: The values of the amplitude response A(ω) at ω= ωi are then computed using The Parks-McClellan Algorithm Step 4: The polynomial A(ω) is determine by interpolating the above values at the L+2 extremal frequencies using the Lagrange interpolation formula: The Parks-McClellan Algorithm Step 5: The new weighted error function ε(ω) is computed at a dense set S(S=L) of frequencies. In practice, S=16L is adequate. Determine the L+2 new extremal frequencies from the values of ε(ω) evaluated at the dense set of frequencies. The Parks