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

In this correspondence, we show that orthogonal expansions of recurrent sig nals like electrocardiograms (ECG's) with a reduced number of coefficients is equivalent to a linear time-variant periodic filter. Instantaneous impul se and frequency responses are analyzed for two classical ways of estimatin g the expansion coefficients: inner product and adaptive estimation with th e LMS algorithm. The obtained description as a linear time-variant periodic filter is a useful tool in order to quantify the distortion produced by th e effect of using a reduced number of coefficients in the expansion, and to give frequency criteria to select the appropriate number of functions. Mor eover, the misadjustment of the I,LMS algorithm can be explained as a disto rtion of the instantaneous frequency response. Experimental results are ill ustrated with the Karhunen-Loeve transform of ECG signals, but this approac h can also be applied to any orthogonal transform.