Optimum design of one and two-dimensional FIR filters using the frequency response masking technique

Abstract

In the frequency response masking technique, the impulse response of a prototype filter and that of its complement are up-sampled (by inserting zeros) by a factor of M and then cascaded to a pair of interpolators. The prototype filter itself may again be synthesized using the frequency response masking technique producing a multistage frequency response masking design. In this paper, we derive an expression for the impulse response up-sampling ratio M, which will produce a minimum complexity design. We show that 1) M approaches e (the base of the natural algorithm) as the number of frequency response masking stages increases, 2) in a K-stage design the complexity of the filter is inversely proportional to the (K+1)th root of the transition width, 3) the frequency response masking technique is effective if the normalized transition width is less then 1/16, and 4) the frequency response masking technique is more efficient than the interpolated impulse response technique if the square root of the normalized transition width is less than the arithmetic mean of the normalized passband edge and stopband edge. We also derive an expression for the multistage frequency response ripple compensation. An optimum design relationship for the interpolated impulse response technique is also derived. The design of narrow-band two-dimensional filters using the frequency response masking technique is also presented.