input. Time-varying impulse response The time-varying impulse response of a linear system is defined as the response of the system at time t = t2 to a single impulse applied at time In other words, if the input to a linear system is where represents the Dirac delta function, and the corresponding response of the system is then the function is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following causality condition must be satisfied: The convolution integral The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition: If the properties of the system do not depend on the time at which it is operated then it is said to be time-invariant and is a function only of the time difference which is zero for (namely ). By redefinition of it is then possible to write the input-output relation equivalently in any of the ways, Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called the transfer function which is: In applications this is usually a rational algebraic function of . Because is zero for negative , the integral may equally be written over the doubly infinite range and putting follows the formula for the frequency response function: Discrete-time systems The output of any discrete time linear system is related to the input by the time-varying convolution sum: or equivalently for a time-invariant system on redefining , where represents the lag time between the stimulus at time m and the response at time n. See also Shift invariant system Linear time-invariant system Nonlinear system System analysis System of linear equations Systems theory Dynamical systems Mathematical modeling Concepts in physics References