Circuits Syst Signal Process (2012) 31:2153–2166 DOI 10.1007/s00034-012-9433-y Computationally Efficient FIR Filtering of Polynomial Signals in DFT Domain Paula Castro-Tinttori · Oscar Ibarra-Manzano · Yuriy S. Shmaliy Received: 9 August 2011 / Revised: 9 May 2012 / Published online: 26 May 2012 © Springer Science+Business Media, LLC 2012 Abstract Fast unbiased finite impulse response (UFIR) filtering of polynomial sig- nals can be provided in the discrete Fourier transform (DFT) domain employing fast Fourier transform (FFT). We show that the computation time can further be reduced by utilizing properties of UFIR filters in the DFT domain. The transforms have been found and investigated in detail for low-degree FIRs most widely used in practice. As a special result, we address an explicit unbiasedness condition uniquely featured to UFIR filters in DFT domain. The noise power gain and estimation error bound have also been discussed. An application is given for state estimation in a crys- tal clock employing the Global Positioning System based measurement of time er- rors provided each second. Based upon it, it is shown that filtering in the time do- main takes about 1 second, which is unacceptable for real-time applications. The Kalman-like algorithm reduces the computation time by the factor of about 8, the FFT-based algorithm by about 18, and FFT with the UFIR filter DFT properties by about 20. Keywords Unbiased FIR filter · DFT domain · Polynomial signal · Fast algorithm 1 Introduction Approximation with finite-degree polynomials is useful for many applications related to signal, image, voice, and speech processing [20]. It is especially efficient if signals are oversampled or highly oversampled. In such cases, finite impulse response (FIR) P. Castro-Tinttori · O. Ibarra-Manzano · Y.S. Shmaliy ( ) Department of Electronics, University of Guanajuato, Salamanca 36855, Mexico e-mail: shmaliy@ugto.mx O. Ibarra-Manzano e-mail: irarrao@ugto.mx