Second generation Vold-Kalman Order Filtering

Filtering of non-stationary periodic components from signals using a multi-order Vold-Kalman Filter.

The Vold-Kalman Filter, introduced by Håvard Vold and Jan Leuridan in 1993, is able to extract non-stationary periodic components from a signal using a known frequency vector [1]. Being formulated in a least-squares sense, it can be solved as a sparse linear system. Similar to the Kalman filter, VKF minimises a cost function based on a structural equation and a data equation.

This submission implements the second generation VKF with the ability to extract multiple orders at the same time, with energy spreading in case of crossing orders [2].

Syntax:
x = vkf(y,fs,f) extracts the order with frequency vector f from signal y with samplerate fs, using a 2-pole filter with a -3dB bandwidth of 1 percent of the sample rate. The output is a single waveform x.

[...] = vkf(y,fs,f,p) uses a p-order filter (typically between 1 or 4). Every order increases the roll-off by -40dB per decade. By specifying additional lower-order coefficients, zero boundary conditions are added. For instance: p = [2 0 1] applies 2nd order filtering and forces the envelope amplitude and its first derivative to zero at t_1 and t_N.

[...] = vkf(y,fs,f,p,bw) uses a bandwidth in Hertz specified by bw. If bw is a scalar, a constant bandwidth is used; if bw is a vector with the same length as y, a time-varying instantaneous bandwidth is realised.

X = vkf(y,fs,F,...) with [N,K] = size(F), performs simultaneous extraction of K orders with frequency vectors [f_1,...,f_K] in array F. In case of crossing orders, this method tries to reveal the correctorder amplitudes. The output is an array of K waveforms [x_1,...,x_K].

X = vkf(y,fs,F,p,bw,0) switches to a single-order algorithm. K orders are still extracted, but the single-order algorithm is computationally less demanding. This is suggested for high sample rates and/or long timeseries.

[a,c] = vkf(...) returns the complex envelope(s) a and phasor(s) c, such that the order waveform(s) can be reconstructed by x = real(a.*c).

[a,c,r] = vkf(...) ouputs an additional selectivity vector r used to realise the bandwidth given by bw.

Demo:
Calling VKF without arguments shows a small demonstration of multi-order filtering with two crossing orders in the presence of white noise.

Note: Filter orders > 4 usually result in ill conditioning and should be avoided. The filter bandwidth determination was implemented for arbitrary order but was not verified for orders higher than 3.

Written by: Maarten van der Seijs, 2010.
Version 1.4, 3 May 2013.

References:
[1] Vold, H. and Leuridan, J. (1993), High resolution order tracking at extreme slew rates, using Kalman tracking filters. Technical Report 931288, Society of Automotive Engineers.

[2] Tuma, J. (2005), Setting the passband width in the Vold-Kalman order tracking filter. Proceedings of the International Congress on Sound and Vibration (ICSV12), Lisbon, Portugal.