Truncated orthogonal expansions of recurrent signals: equivalence to a linear time-variant periodic filter

We show that orthogonal expansions of recurrent signals like electrocardiograms (ECGs) with a reduced number of coefficients is equivalent to a linear time-variant periodic filter. Instantaneous impulse and frequency responses are analyzed for two classical ways of estimating the expansion coefficients: inner product and adaptive estimation with the LMS algorithm. The obtained description as a linear time-variant periodic filter is a useful tool in order to quantify the distortion produced by the effect of using a reduced number of coefficients in the expansion, and to give frequency criteria to select the appropriate number of functions. Moreover, the misadjustment of the LMS algorithm can be explained as a distortion of the instantaneous frequency response. Experimental results are illustrated with the Karhunen-Loeve transform of ECG signals, but this approach can also be applied to any orthogonal transform.