Extrapolated impulse response filter using coefficient-autocorrelation-matrix decomposition

The smooth predictable envelope and quasi-periodicity of impulse response allows for potential ways to design ultra-low-complexity FIR filters. By introducing an extrapolated impulse response technique, a first attempt successfully reduced the arithmetic complexity of FIR filter from o(N)o(N) to o(2N) comparing with direct form. However, this conventional way may suffer from the problem that the peak ripple magnitudes in stopband do not decrease or even increase further with respect to considerable number of extrapolated side lobes. In this paper, we construct coefficient-autocorrelation-matrix of FIR filter by side lobe vectors, and utilize its maximal eigenvector as a base side lobe for extrapolation. With findings that the coefficient errors between prototype and extrapolated filters still yield quasi-periodicity for further extrapolation, we present extrapolated impulse response technique using coefficient-autocorrelation-matrix decomposition, which is applicable to various windowed filters (Hanning, Hamming, Blackman, and Kaiser). Analytical and experimental results indicate the proposed method can synthesize extrapolated windowed filters with arbitrary length, and a typically low arithmetic complexity of o(8N) is achieved. In addition, the peak ripple magnitudes in passband and stopband meet well with the requirements from prototypes.