The output signal h(t) (or h[n] in discrete time) produced by a system in response to a unit impulse (Dirac delta or Kronecker delta) input; for LTI systems it is the system's time-domain kernel that determines outputs by convolution.

Principle

For linear time-invariant systems, the output to any input x is the convolution of x with the impulse response h, so h fully characterizes the system's input–output mapping in the time domain.

Demonstration

For a causal first-order RC circuit with time constant τ = RC, the impulse response is h(t) = (1/RC) e^{−t/RC} for t ≥ 0; convolving this h with a step input reproduces the known exponential step response.

Misapplication

Applying impulse-response convolution to nonlinear or time-varying systems, or treating an ill-defined distributional impulse response as an ordinary function without regularization, yields invalid predictions.

Consequence

Knowing the impulse response enables exact computation of outputs for arbitrary inputs via convolution, assessment of causality and stability (e.g., integrability or summability of h), and direct derivation of frequency response via transform.

Reversal

The inverse perspective is the transfer function: the Laplace or Fourier transform of h is the transfer function H(s) or H(jω); alternately, representing the system by explicit state equations emphasizes internal dynamics rather than the convolution kernel.

Boundary

Defined primarily for LTI systems; for time-varying or nonlinear systems an impulse response may not exist or may depend on the time of application; in PDEs and distributions the impulse response may be singular and must be interpreted as a distribution or fundamental solution.

Semantic Tension

There is semantic tension between 'impulse response' as a concrete time-domain kernel and 'fundamental solution' in PDE theory: both are Green-type kernels for point sources, but the former usually refers to causal system kernels while the latter often addresses spatial singularities and boundary conditions.

Synthesis

The impulse response is the time-domain kernel that, for LTI systems, uniquely determines output by convolution; its transforms give complementary algebraic descriptions (transfer function, frequency response) and its properties (decay, integrability) encode stability and causality.