Performance analysis of a Fourier coefficient estimation method using IIR‐BPFs and an LMS algorithm and improvement of its performance

Abstract

SUMMARYIn many areas, it is extremely important to accurately estimate the amplitude and phase of a signal even if the frequency contained in the signal to be analyzed is known. For cases in which the signal frequencies to be analyzed are in a nonharmonic relation and white noise and interference are superposed on the signal, a method (BPLMS method) combining the IIR type BPF and the LMS algorithm has been proposed to derive the Fourier coefficients of the signal components accurately. In the present paper, in order to investigate a method of improving the BPLMS method, an approximate equation for the estimated accuracy is derived. Also, the validity of the approximate equation is confirmed by computer simulations under various conditions. As a result of the approximate analysis, it is shown that the normalized frequency difference between the signal and the interference can be increased by combining a downsampling process so that the estimation accuracy can be improved. Finally, by simulation, the proposed method and the BPLMS method are compared in terms of estimation accuracy. An example of application to the pitch estimation of musical sounds is given.

Improvement of a Fourier coefficient estimation method using an adaptive algorithm and its performance analysis

“…Numerous simulations indicate that the convergence conditions for µ are largely the same in the proposed method and LMS; however, the convergence conditions in the proposed method should be examined in more detail in the future. The authors also plan to study the application of the proposed method to power systems, the combined use of the proposed method with previous methods [12][13][14], and the use of the proposed method when the Fourier coefficients under analysis are time-varying.…”

“…Therefore, in the steady state, the variance σ e 2 of the estimation error can be calculated as [14] Considering that…”

“…(19), (20) if conditions (13), (14) are met, and A, B, C, D are found from the coefficients in Eq. (12) as shown below (here j is the imaginary unit): When conditions (13), (14) are met, C and D are real numbers; in this case, the pole angle nearly coincides with the angle frequency of the component under analysis (see Appendix 2). From the above expressions, the largest possible γ for a given µ can be found.…”

“…Therefore, in the steady state, the variance σ e 2 of the estimation error can be calculated as [14] Considering that ε 1 (n) = 2E[(x(n) − x(n)) 2 ], the steady-state mean square error ε 1,∞ of Fourier coefficient estimation can be calculated as…”

“…The authors have used combinations of the LMS algorithm with FIR and IIR bandpass filters to improve the Fourier coefficient estimation of nonharmonic signals involving disturbances such as white noise, interference, or both [12][13][14][15]. As regards adaptive algorithms, it is important to improve accuracy while preventing an increase in computational complexity.…”