FIR digital filters with least-squares stopbands subject to peak-gain constraints

The use of digital filters with peak-constrained least weighted-squared errors (PCLWSE) is proposed. The emphasis is on PCLWSE filters with equiripple passbands and peak-constrained least-squares (PCLS) stopbands. The stopband energy is minimized subject to a constraint on the maximum stopband gain. A design algorithm is presented, and examples are discussed. Each example provides the unique optimal solution to the corresponding design problem. The tradeoff between the peak stopband gain and the total stopband energy is examined. It is shown that the most inefficient filters have minimax or least-squares stopbands. In particular, the stopband energy for minimax filters can be significantly reduced with almost no change in the peak stopband gain. Also, the peak stopband gain for filters with least-squares stopbands can be significantly reduced with almost no change in the total stopband energy. These results are supported by mathematical analysis, and examples are provided