New fast optimal window design algorithm based on the eigen-decomposition of the symmetric Toeplitz matrix

The finite impulse response (FIR) filter design has been a hot topic over many decades due to its guaranteed stability and wide variety of applications. The simplest way to design an FIR low-pass filter is to truncate the infinite-long sine sequence. This direct approach is a special case of window-FIR filter design with the rectangular window. Other windows were also proposed to deal with this filter design problem. Alternatively, given a fixed filter (window) length, optimization techniques can be employed to design the best window sequences. In this paper, we would like to present a novel computationally-efficient optimal window design algorithm for the existing window-FIR approach. Since the signal processing storage devices become less costly and more powerful, designing long FIR filters becomes prevalent in the modern telecommunication and signal processing applications. Therefore, the computationally-efficient filter design schemes are in urgent demand. Our proposed technique is based on a fast eigen-decomposition algorithm. The computational complexity of our method is O(N² log(N)) compared to O(N³) of the conventional method.