A Weighted Least Squares Algorithm for Quasi-Equiripple FIR and IIR Digit

al Filter Design

Advisor : 高永安Student : 陳志煒

outline

• Introduction• The Weighted Least Squares Method• The Least Squares Weighting Function• The New Algorithm• Termination of Algorithm• Conclusion

Introduction

• In this paper, a novel iterative algorithm for deriving the least squares frequency response weighting function which will produce a quasi-equiripple design is presented

• The algorithm converges very rapidly• Form experience, about 1dB away form the mini

max optimum solution in two iterations and converges to within 0.1dB in six iterations

The Weighted Least Squares Method

• appropriate trigonometrical function

• impulse response of the filter

• desired frequency response

• actual frequency response

The Weighted Least Squares Method

The Weighted Least Squares Method

• Minimized

• Optimum solution

The Least Squares Weighting Function

• The weighting function used in the (k+1) iteration

• Lawson’s Algorithm

The New Algorithm

• Define the i th extremal point of the k th iteration as

• The envelope function

The New Algorithm

• Minimax weighting function

• The constant affect convergent speed

• = average of

The Constant

• The algorithm is said to have converged if

• Peak weighted ripple of optimum minimax design

• Peak weighted ripple of the weighted least squares design

• There is no know analytical method for obtaining that will make the algorithm converge at the maximum speed

The Constant

• Averaging the results of 200 examples

Initial

• May be set equal to unity for all n

• Using the method of rectangular windowing of the Fourier series of

Termination of Algorithm

• It can be terminated after a prespecified number of iterations have been completed

• Another criterion for termination of the algorithm is to check for quasi-equiripple condition

• : average weighted ripple magnitude

Conclusion

• We have present a novel fast convergent weighted least squares algorithm for quasi-equiripple FIR and IIR fil

ter designs

• N=151• =1.5

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