Adaptive Fourier Analysis

The resonator-based observer provides the exact analysis of the signals which can be described by the supposed signal model. However, if the fundamental frequency of the input signal differs from that of the supposed one, the estimation of the components is distorted. This distortion is the well-known "picket fence" and "leakage" in the case of the DFT. The adaptive Fourier analyzer (AFA) eliminates these problems. The AFA is such a resonator-based structure, the resonator poles of which are tuned to coincide with those of the input signal components. It ensures the perfect decomposition of the input signal. Fig. 1. shows that version of the observer which is the best to introduce the AFA algorithm.

The operation of the channels can be so interpreted, that the error signal is first mixed to zero frequency by the corresponding g_k,n function, then, after the integration the signal is mixed back to the original frequency by the c_k,n function. If the observer matches the input signal, the state variables do not change. However, if there is frequency mismatch, the mixing results in a low-frequency signal. The state variable in steady state is a rotating complex vector, and the speed of the rotation corresponds to the actual frequency difference. It can be utilized for the adaptation of the frequency, according to the following equation:

where x_1,n with "hat" denotes the state variable belonging to the fundamental frequency, and "angle" is a function providing the angle between two complex vectors. The basis functions in the next time step can be expressed by the new frequency:

The observer shall use always the adapted basis functions. The structure performing the DFT has finite settling. However, if the fundamental frequency changes, the original coefficients g_i,k do not ensure the finite settling. Calculation of new coefficients is disadvantageous, because of numerical problems and its high computational demand. Fortunately, if the resonator distribution remains more or less uniform, the system remains fairly fast. For signals changing in a wide range, it can be ensured by originating new resonators or dissolving those which are above f = 0.5. The system shall contain always as many resonators as can be set according to the band limit. The initialization of the new resonators is the following:

Based on the algorithm described above a family of adaptive Fourier analyzers has been developed. The original AFA estimates the frequency of sweeping periodic signals only with constant error, so the signal reconstruction is also not perfect. Later on analyzers have been developed for linear, logarithmic and hyperbolic frequency sweeps. The stability of the AFA algorithm is not obvious, and the analysis is difficult, due to the nonlinear problem.

Figure 2 shows the operation of the AFA. A signal is connected at time instant 0 to the input of the system which has a relative frequency of f = 0.077. The frequency is intentionally not a round number. The signal is a band limited symmetric triangular signal consisting of 3 nonzero components as it can be seen in the left top diagram of the figure. Below the magenta curve shows the estimator of the frequency as function of time. The frequency at the beginning is around f = 0.025, which the observer has to be set from. The green curve in the right top corner shows the error signal. It this error signal converges to zero, the signal reconstruction is perfect, so this signal is always displayed during research and development works. The diagram in the right bottom corner shows the estimated amplitude of the harmonic components. It can be clearly seen that the settling is complete after about 100 samples. Since the new frequency is about 3 times greater than the original one, structural adaptivity is needed, i.e. dissolving resonators. This is why the settling is not exponential as it is usual in linear, time invariant systems.

Another companion page introducing AFA with downloadable Matlab scripts: http://mit.bme.hu/~sujbert/afa

Related Publications:

F. Nagy, "Measurement of signal parameters using nonlinear observers," IEEE Transactions on Instrumentation and Measurement, vol. IM-41,pp. 152-155, Febr. 1992.	Introduction to the AFA algorithm.
F. Nagy, "An adaptive Fourier analysis algorithm" 5th International Conference on Signal Processing Applications and Technology, Oct. 18-21, 1994, Dallas, Texas, USA, pp. 414-418.	Description of an AFA that can follow signals of varying frequency.
L. Sujbert, G. Simon and A. Várkonyi-Kóczy, "An improved adaptive Fourier analyzer", Proceedings of the IEEE International Workshop on Intelligent signal Processing, Sept. 1999, Budapest, Hungary, pp. 182-187.	Description of an improved AFA that can operate in a wider frequency range.
L. Sujbert, G. Simon and G. Peceli, "An Observer-Based Adaptive Fourier Analysis [Tips & Tricks]", IIEEE Signal Processing Magazine, vol. 37, no. 4, pp. 134-143, July 2020, doi: 10.1109/MSP.2020.2982167.	Description of the AFA algorithm.

Further related publications can be found on this page.

 Further information: László Sujbert