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Frequency-Selective Adaptive Filtering

Signal processing methods like iterative numeric smoothing, discrete autocorrelation function and discrete Fourier series/transform can be used to noise reduction. The numeric smoothing has the property of a low-pass filter. Only high-frequent noise can be filtered by the smoothing. The magnitude and phase spectra of a sampling signal can be calculated by a windowed discrete Fourier series/transform (DFS). If the frequencies of the spectral components of the signal are known, the signal can be abstracted from the sampling signal. However, one cannot easily distinguish which frequency components result from the signal, and which frequency components come from the noise.

The time noise series is fortunately uncorrelated in the most practical cases. An uncorrelated noise can considerably be reduced by a discrete autocorrelation function (DACF), which separates the noise from the sampling signal. In other words the signal can be abstracted by the DACF from the sampling signal. The autocorrelation function has the same spectral components like the signal. Therefore, the signal frequencies can be determined by the autocorrelation function. The problem is that the magnitudes and phases of the spectral components of the signal are changed by the DACF.
 
Fig. 1  Scheme of an adaptive filtering algorithm
(ACF: Autocorrelation Function, DFS: Discrete Fourier Series)

An adaptive filtering can be used to solve the problems mentioned above. Fig. 1 shows the scheme of the algorithm based on a combination of the windowed DFS and the DACF. The sampling signal uk is parallel processed by the DFS(1) and the DACF. The uncorrelated noise unk is effectively filtered by the DACF. The output signal yt is resulted mainly from the signal usk. The spectral components C yp of the autocorrelated signal yt is calculated by the DFS(2) using the same time window like the DFS(1). Therefore, the Fourier coefficients Csp of the signal usk can be selected from the Fourier coefficients Cp of the DFS(1) according to a defined filtering factor d. Using the coefficients Csp one can reconstruct a time signal using the discrete Fourier series.

For the Fourier coefficients selection (FCS) and the signal reconstruction the following relations:



are used, where |Cyp|max is the maximum magnitude of the Fourier coefficients Cyp and d the filtering factor, normally, 0<d<1. The filtering factor can be determined by the ratio between the original effective value of the sampling signal and the effective value of the corresponding autocorrelated signal.

The transfer function of the adaptive filtering is equal to 1 for signal components and 0 for the noise parts. This filtering can be used for noise reduction of periodic and periodically expended signals, the spectrum of which superimposes with the noise spectrum.

Fig. 2 show the simulation result of a sinusoid signal and a damped oscillation signal. The both signals are mixed with strong noise. The noise parts are effectively filtered out by the adaptive filtering.

Fig. 2  Simulation results of sinusoid and oscillation signals
Fig. 3 shows the noise reduction of a dynamic strain measuring system based fiber Bragg grating sensor . In the sensitive wavelength range, the maximal effective value of the adaptive filtered noise is only 1/3 of the original noise. The resolution of the measuring system is improved by the adaptive filtering. After the filtering the noise strain is less than 10-9 for the optimized wavelength. It is much less than the original noise strain.
Fig. 3  Noise effektive value and noise strain of fiber Bragg grating measuring system
Fig. 4 shows the noise reduction of a seismic signal. The original signal is generated by blast exciting in underground building and detected with the fiber optic Bragg grating sensor system. The signal is mixed with a strong noise. After the adaptive filtering the noise parts are reasonably reduced.
Fig. 4   Time function and spectrum of a seismic signal before and after the adaptive filtering
 

 

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