Simultaneous approximation of magnitude and group delay in FIR digital filters

Recently, approximation problems for FIR digital filters in the complex domain have been studied extensively. This class of problems minimizes the absolute value of the complex error of the filter studied and the desired response. Hence, the problem is to approximate amplitude and phase. However, in the waveform transmission, the group delay to indicate the delay of the wave is more important than the phase characteristic. To date, the phase approximation has been used as an indirect approximation of the group delay. Hence, the designed group delay characteristic is substantially degraded from the desired group delay. In the present paper, the group delay error is included in the performance function as a performance parameter of the least squares method. By deriving the least square error with repeated transformation of the desired response, a direct approximation of the group delay is realized and both the amplitude and the group delay are optimized. Finally, by numerical simulation, the method is compared with the conventional ones and the effectiveness of the present method is demonstrated.