THE DESIGN OF FREQUENCY SAMPLING FILTERS BY THE METHOD OF LAGRANGE MULTIPLIERS

Narrow-band linear phase filters can be implemented more efficiently as frequency sampling filters than direct convolution filters [1]. A frequency sampling filter approximates a desired frequency response by interpolating a frequency response through a set of frequency samples taken from the desired frequency response. Although the frequency response passes through the frequency samples, the frequency response may not be well behaved between the specific samples. Linear programming is commonly used to control the interpolation errors between frequency samples. In this paper, a technique is developed for designing linear phase frequency sampling filters where the interpolation errors between frequency samples are controlled by minimizing the mean square error between the desired and actual frequency responses in the stopband subject to constraints on the passband frequency response. The frequency sampling filter design problem is defined as a constrained optimization problem which is solved using the Lagrange multiplier optimization method. The Lagrange multiplier optimization method results in a set of linear equations, the solution of which determines the filter's coefficients.