The author considers the problem of determining the impulse response of a linear time-invariant system using, as probing input, causal \(C^{\infty}\) approximations of the Dirac \(\delta\)-function. This leads to study linear integral equations of the form \[ (1)\quad \int^{t}_{0}p(t-\tau)g(\tau)d\tau =b(t),\quad 0\leq t<\infty \] where p(t) is a probability density function on [0,\(\infty)\) which, for each positive n, is the n-fold convolution of another one-sided density. Such an equation can be reformulated as a Cauchy problem for a linear partial differential equation in two variables, and the notions of partial and continuous deconvolution are introduced. It is shown that partial deconvolution gives \(L^{\infty}\) error bounds for the regularized solution and its derivatives when L 2 bounds on the noise and the system response are assumed. A computational method for (1) based on the Poisson summation formula and FFT is proposed.