Computing the Convolution of Analog and Discrete Time Exponential Signals Algebraically

We present a procedure for computing the convolution of exponential signals without the need of solving integrals or summations. The procedure requires the resolution of a system of linear equations involving Vandermonde matrices. We apply the method to solve ordinary differential/difference equations with constant coefficients. 1 Notation and Definitions Below we introduce the definitions and notation to be used along the paper: • Z, R and C are, respectively, the set of integers, real and complex numbers; • An analog time signal is defined as a complex valued function f : R t → 7→ C f(t) , and a discrete time signal is a complex valued function f : Z k → 7→ C f(k) . In this paper we are mainly concerned with exponential signals, that is, f(t) = e, or f(k) = r, where r ∈ C. Two basic signals will be necessary in our development, namely, the unit step signal (σ) and the unit impulse (generalized) signal (δ), both in analog or discrete time setting. The unit step is defined as σ(t) = { 0, t < 0 1, t > 0 (analog) and σ(k) = { 0, k < 0 1, k ≥ 0 (discrete time) In analog time context we define the unit impulse as δ = σ̇, where the derivative is supposed to be defined in the generalized sense, since σ has a “jump” discontinuity at t = 0, and this is why we denote δ as a “generalized” signal [1]. If f is an analog signal continuous at t = 0, the product “fσ” is given by (fσ)(t) = f(t)σ(t) = { 0, t < 0 f(t), t > 0 and then, if f(0) 6= 0, the module of fσ also has a “jump” discontinuity at t = 0, in fact (fσ)(0) = 0 while (fσ)(0) = f(0). Additionally, using the generalized signal δ, we can obtain the derivative of fσ as ̇ (fσ) = ḟσ + fσ̇ = ḟσ + f(0)δ (1) In discrete time context, time shifting is a fundamental operation. We denote by [f ]n the shifting of signal f by n units in time, that is, [f ]n(k) = f(k − n). Using this notation, the discrete time impulse δ can be written as δ = σ − [σ]1 or δ(k) = σ(k) − σ(k − 1). • The convolution between two signals f and g, represented by f ∗ g, is the binary operation defined as [2]: (f ∗ g)(t) = ∫ ∞ −∞ f(τ)g(t− τ)dτ, for analog signals, or