We study a new approach to the problem of estimating the component amplitudes and decay rates of transient signals that consist of real decaying exponentlals. A set of pmdetermined “basis‘’ exponentids is fitted to discrete samples of a signd. ”he basis is required to be selective, each basis exponential *’selecthg’y signal exponents that are close to its own and “rejedng“ others. The first part of the paper is concerned with determining such basis sets. We take a singleexponential test signal, analyze it in terms of the basis equwnthls, and consider each fitted amplitude as a tunction of test exponent. Each such tlunction is then required to peak in a manner familiar to the discrete Fourier transform @FT) theory, when the test and basis exponents coincide, diminishing in magnitude as their difference hreases. We show that for eqaispaced data, such selective bases exist and are uniquely determined ‘by model order snd sunpting intenal. Formulas for basis decay rates and fitting amplitudes are obtained in closed form. It is shown that a thne-bandwidth relationship also hdds, as in the DFT case. However, the selective basis set tu= out to be Macceptably SetISitiVe to noise, in spite Of the fact that it “kA the assodated Vdermonde determinant. This overshadows the advantage of selectivity with respect to the deterministic part of a signal. The second part of the paper explores a method to overcome this drawback. We study the effect of overdeternination and a small simultaneous relaxation of the peaked-shape requirement for a test exponential. Of these, the latter implies usage of nonexponential basis hctions, and this is the more fundamental strategy because it is the natural ill conditioning of exponential-bared fitting schmes that is the root caw of sensitivity. We demonstrate the potentid of this approach by formulating and solving a sensitivity-”hation problem. It is shown that sensitivity to noise can be systcmaticpUy traded against selectivity and that close approximations to the ideal peaked response functions may be obtained while Increasing robustness by at least an order of magnitude.