Glossary term
Impulse Response
The time-domain output of a dynamic system when excited by an ideal impulse input under specified initial conditions.
Definition
modelImpulse response is the time-domain output of a dynamic system produced by an ideal impulse input.
For a linear time-invariant system, the impulse response fully characterizes the input-output behaviour. Any output can be computed by convolving the input with the impulse response. The impulse response is the inverse Laplace transform of the transfer function in continuous time, or the inverse z-transform of the discrete transfer function in sampled systems. It reveals delay, damping, resonance, stability, bandwidth, and memory.
Impulse response describes how a dynamic system reacts to a very short input carrying finite area. The ideal impulse, usually written \delta(t), is a mathematical abstraction: infinite amplitude, zero duration, and unit area. Real tests use an approximation such as a hammer impact, electrical pulse, pressure pulse, or short force input.
For a linear time-invariant system, the impulse response h(t) is fundamental because it determines the output for any input u(t) through convolution:
This means the system output can be viewed as the accumulated effect of many scaled and shifted impulses.
Connection to transfer function
In continuous time, the transfer function is the Laplace transform of the impulse response:
The frequency response is obtained by evaluating the same transfer function along the imaginary axis for stable systems. A lightly damped system has an impulse response with oscillatory decay. A first-order system has an exponential decay. A pure delay shifts the response in time. An unstable system has an impulse response that grows rather than decays.
In discrete-time systems, the impulse response is a sequence h[k]. Digital filters are often described directly by their impulse response: finite impulse response filters have a response that becomes exactly zero after a finite number of samples; infinite impulse response filters have feedback and theoretically continue indefinitely.
Engineering use
Impulse response is used in vibration testing, modal analysis, acoustics, control design, system identification, digital filtering, structural dynamics, seismology, and communications. Engineers use it to estimate natural frequencies, damping, delay, settling behaviour, and hidden resonances. In signal processing, it makes clear how a filter smears, delays, or shapes a signal over time.
Common mistakes
A common mistake is assuming an impact test directly equals the ideal impulse response. The measured result also includes input shape, sensor bandwidth, mounting stiffness, noise, nonlinear effects, sampling, windowing, and boundary conditions. Another mistake is applying impulse-response reasoning to systems that are strongly nonlinear or time varying without qualification. For such systems, the response can change with amplitude, operating point, temperature, wear, or control state.