On the identification of exponentially decaying signals

Studies a new approach to estimating the component amplitudes and decay rates of transient signals that consist of real decaying exponentials. A set of predetermined “basis” exponentials is fitted to discrete samples of a signal. The basis is required to be selective, each basis exponential “selecting” signal exponents that are close to its own and “rejecting” others. The first part of the paper is concerned with determining such basis sets. The authors take a single-exponential test signal, analyze it in terms of the basis exponentials, and consider each fitted amplitude as a function of test exponent. Each such function is then required to peak in a manner familiar to the discrete Fourier transform (DFT) theory, when the test and basis exponents coincide, diminishing in magnitude as their difference increases. It is shown that for equispaced data, such selective bases exist and are uniquely determined by model order and sampling interval. Formulas for basis decay rates and fitting amplitudes are obtained in closed form. It is shown that a time-bandwidth relationship also holds. However, the selective basis set turns out to be unacceptably sensitive to noise. The second part of the paper explores a method to overcome this drawback. The authors study the effect of overdetermination and a small simultaneous relaxation of the peaked-shape requirement for a test exponential. Of these, the latter implies usage of nonexponential basis functions, and this is the more fundamental strategy