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

Abstract

Abstract-In this correspondence, we show that orthogonal expansions of recurrent signals like electrocardiograms (ECG's) 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.Index Terms-Data compression, least mean squares methods, transforms. I. INTRODUCTIONOrthogonal expansion is a very well-known technique for signal analysis. It is based on the decomposition of the signal as a linear combination of simple and elementary basis functions [1]. An appropriate choice of the orthogonal basis functions achieves a signal representation, where each coefficient contributes with independent and complementary information, for example, frequency components for the Fourier transform, instantaneous signal values for the identity transform, localized frequency components using the wavelet transform, etc.When the same number of samples N in the signal under study is used as the number of basis functions in the expansion (exact modeling), the signal energy is completely represented, and the equivalent system can be considered to be the identity function. However, many applications (such as data compression [2]- [5], parameter extraction for pattern recognition, monitoring [6], etc.) require rank reduction. In these applications, the number of basis functions used is reduced to a fraction p < N (undermodeling), and accordingly, some signal components are discarded. The selection of the number of basis functions is the main problem because it is a tradeoff between signal representation capacity and rank reduction.In this correspondence, we show that the effect of using a reduced number of coefficients in orthogonal expansions of recurrent signals, like electrocardiograms (ECG), can be described as applying a linear time-variant periodic filter to the input signal. This equivalence gives a useful tool in order to quantify a priori in the frequency domain the distortion introduced in the reconstructed signal when a variable number of functions is used in the expansion. As a consequence, it can be used to establish a criteria to select the number of basis functions. Manuscript received January 22, 1998; revised May 6, 1999. This work was supported in part by TIC97-0945-CO2-01:02 from CICYT and P40/98 from CONSID-DGA. The associate editor coordinating the review o...

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