Impulse response

An impulse response is a system’s output when driven by an idealized impulse. For a linear time-invariant system, it completely determines the system’s behavior through convolution.

Dirac delta function: A generalized function representing an idealized impulse concentrated at one point with unit integral. It models the input whose resulting output defines an impulse response.
Linear system: A system whose output for a sum of inputs equals the sum of their outputs, with scaling preserved. Linearity lets responses to idealized impulses combine into responses to general inputs.
Room acoustics: The study of how sound propagates and is reflected within enclosed spaces. A room impulse response captures reflections and reverberation between a source and microphone.
Step response: A system’s output when its input changes abruptly from zero to a constant value. Unlike the impulse response, it records the output to a sustained step input.
Convolution: A mathematical operation that combines two functions to describe how one spreads or transforms the other. Convolving an input with the impulse response gives an LTI system’s output.
Time invariance: The property that shifting an input in time shifts the output by the same amount. It makes a single impulse response valid at every time position.
Reverberation: The persistence of sound after its source stops, caused by reflections in an environment. Its timing and decay are encoded in a room’s impulse response.
Frequency response: A system’s steady-state response to sinusoidal inputs as a function of frequency. It describes sinusoidal behavior, while the impulse response describes time-domain behavior.
Linear time-invariant system: A system whose response obeys superposition and does not change when its input is shifted in time. Only for this system class does one impulse response determine every input-output relation.
System identification: The process of inferring a system’s mathematical model from its observed inputs and outputs. Measured impulse responses can identify and characterize unknown systems.

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