Asymmetric Pulse Modeling for FRI Sampling

We consider sampling and reconstruction of finite-rate-of-innovation (FRI) signals such as a train of pulses, where the pulses have varying degrees of asymmetry. We address the problem of asymmetry modeling starting from a given symmetric prototype. We show that among the class of unitary operators that are linear and invariant to translation and scale, the fractional Hilbert (FrH) operator is unique for parametrically modeling pulse asymmetry. The FrH operator is obtained by a trigonometric interpolation between the standard Hilbert and identity operators, where the interpolation weights are determined by the degree of asymmetry. The FrH operators are also steerable, which allows for estimation of the asymmetry factors, in addition to the delays and amplitudes, using the high-resolution spectral estimation techniques that are used for solving standard FRI problems. We also develop the discrete counterpart using discrete FrH operators and show that all the desirable properties carry over smoothly to the discrete setting as well. We derive closed-form expressions for the Cramér-Rao bounds and Hammersley-Chapman-Robbins bound, on the variances of the estimators for continuous and discrete parameters, respectively. Experimental results show that the proposed estimators have variances that meet the lower bounds. We demonstrate an application of the proposed discrete FrH methodology on real electrocardiogram (ECG) signals in the presence of noise. Specifically, we show how the asymmetry of QRS complexes in various channels of an ECG signal could be modeled accurately.