Adaptive Fourier analysis of sinusoidal signals in noise is of essential importance in many engineering fields. So far, many adaptive algorithms have been developed. In particular, a filter bank based algorithm called constrained notch Fourier transform (CNFT) is very attractive in terms of its cost-efficiency and easily controllable performance. However, its performance becomes poor when the signal frequencies are non-uniformly spaced (or spaced with unequal intervals) in the frequency domain. This is because the estimates of the discrete Fourier coefficients (DFCs) in the CNFT are inevitably corrupted by sinusoidal disturbances in such a case. This paper proposes, at first, a modified CNFT (MCNFT), to compensate the performance of the CNFT for noisy sinusoidal signals with known and non-uniformly spaced signal frequencies. Next, performance analysis of the MCNFT is conducted in detail. Closed form expression for the steady-state mean square error (MSE) of every DFC estimate is derived. This expression indicates that the MSE is proportional to the variance of the additive noise and is a complex function of both the frequency of each frequency component and the pole radius of the bandpass filter used in the filter bank. Extensive simulations are presented to demonstrate the improved performance of the MCNFT and the validity of the analytical results.

Fourier analysis of sinusoidal and/or quasi-periodic signals in additive noise has been used in various fields. So far, many analysis algorithms including the well-known DFT have been developed. In particular, many adaptive algorithms have been proposed to handle non-stationary signals whose discrete Fourier coefficients (DFCs) are time-varying. Notch Fourier Transform (NFT) and Constrained Notch Fourier Transform(CNFT) proposed by Tadokoro et al. and Kilani et al., respectively, are two of them, which are implemented by filter banks and estimate the DFCs via simple sliding algorithms of their own. This paper presents, for the first time, statistical performance analyses of the NFT and the CNFT. Estimation biases and mean square errors (MSEs) of their sliding algorithms will be derived in closed form. As a result, it is revealed that both algorithms are unbiased, and their estimation MSEs are related to the signal frequencies, the additive noise variance and orders of comb filters used in their filter banks. Extensive simulations are performed to confirm the analytical findings.