LTI systems and impulse response

The formula

The two defining assumptions:

Linearity:        input a·x1(t) + b·x2(t)  →  output a·y1(t) + b·y2(t)
Time invariance:  input x(t − t0)          →  output y(t − t0)

Reading: scale and add inputs, the outputs scale and add the same way; delay the input, the output delays by the same amount and nothing else changes.

The impulse response is the definition, not a derived quantity:

h(t) = system output when the input is δ(t)

Reading: hit the system with an ideal unit impulse at t = 0 and record what comes out. That recording is h(t).

The payoff — the convolution integral:

y(t) = ∫ x(τ)·h(t − τ) dτ   (integral over all τ)     i.e.  y = x * h

Reading: the output now is every past slice of input, each weighted by how much the system still remembers a kick that old.

The frequency-domain face of the same object:

H(s) = ∫₀^∞ h(t)·e^(−s·t) dt        (transfer function = Laplace transform of h)
Y(s) = H(s)·X(s)                     (convolution becomes multiplication)

And the property that makes the sine-sweep industry possible — complex exponentials are eigenfunctions:

input e^(jωt)  →  output H(jω)·e^(jωt)

Reading: a sinusoid in gives a sinusoid out at the same frequency, scaled by |H(jω)| and shifted by ∠H(jω). An LTI system cannot create new frequencies.

Two conditions worth keeping on hand: causality means h(t) = 0 for t < 0 (the system doesn't answer before you ask), and BIBO stability means ∫ |h(t)| dt < ∞ (bounded inputs give bounded outputs).

Where you meet it

Modal survey on a test stand. The instrumented hammer tap is a stand-in for δ(t), the accelerometer trace is a stand-in for h(t), and the FFT ratio of the two is the frequency response function your modal software fits modes to. Every natural frequency and damping ratio in that report rests on the structure behaving LTI at the tap level.
VNA on the bench. An S21 sweep is H(jω) measured one frequency at a time. Time-domain gating on a network analyzer is the instrument computing the impulse response from the sweep, windowing out a connector reflection, and transforming back — convolution theorem, run in reverse, in firmware.
Control design review. Every block in the loop diagram — plant, actuator, compensator — is a transfer function. The board member who asks "is that valid at full deflection?" is asking whether the LTI assumption survives the actuator's rate and saturation limits. Often it doesn't, and that is where the margin analysis gets interesting.
DAQ channel budgeting. The anti-alias filter in front of your ADC has an impulse response with real width. Convolve a fast edge with it and the edge smears; that is why two channels with different filter settings time-skew the same event. The fix is in the convolution, not the cabling.

How it works

The whole structure hangs on one decomposition. Any input signal can be written as a dense train of scaled, shifted impulses. Time invariance says the system answers every impulse in that train with the same shape h, just shifted. Linearity says the total output is the sum of all those shifted, scaled copies. Write the sum as an integral and you have convolution. Nothing else about the hardware matters — thermal, electrical, mechanical, acoustic — if the two assumptions hold, one function characterizes the box completely.

Numbers, for a first-order RC low-pass with time constant τ = RC:

h(t) = (1/τ)·e^(−t/τ)                    for t ≥ 0
step response = 1 − e^(−t/τ)             (convolve h with a unit step)

At t = τ the step response reaches 1 − e⁻¹ ≈ 0.632 — the 63.2% rule every scope jockey knows is a convolution result. At ω = 1/τ, |H| = 1/√2 ≈ 0.707, which is −3.01 dB. Drive it at ω = 2/τ and the output amplitude is 1/√5 ≈ 0.447 with a phase lag of arctan(2) ≈ 63.4°. All four numbers check against direct numerical convolution to the discretization error.

The limits of validity are where working engineers earn their pay:

Real hardware is LTI over a region, never globally. An amplifier near clipping, a bolted joint under large excursion, an actuator against its rate limit — superposition is the first thing to die. The FRF you measured at low drive level is not the FRF at qualification level if the structure has friction joints in it. Measure at the amplitude you intend to fly.
Harmonics are the tattletale. Put in a clean 100 Hz and see energy at 200 and 300 Hz on the analyzer — the system is nonlinear, full stop. An LTI system cannot generate frequencies that were not in the input. This is the cheapest linearity check available and it runs on equipment you already own.
Slow drift is time variance. A thermal chamber ramping while you sweep makes the "constant" in time-invariant a lie. The usual rescue is separation of time scales: if the drift is slow compared to the measurement, treat the system as frozen-coefficient LTI and note the assumption in the report.
The perfect impulse does not exist. δ(t) has infinite amplitude, zero width, and unit area — no hammer, no pulse generator produces it. It doesn't need to. The practical requirement is only that your excitation spectrum be flat across the band you care about; the analyzer divides it out. A hammer double-hit, on the other hand, puts comb notches in the input spectrum and quietly wrecks the coherence trace. Watch the coherence.
The common mistake is treating the transfer function as a property of the hardware rather than a property of the hardware plus an operating point plus an amplitude range. The math is exact; the assumption set is the negotiable part, and it is the part nobody writes on the block diagram.

History

The convolution answer arrived long before the impulse question was respectable. In the 1830s, Jean-Marie Constant Duhamel (1797–1872), professor at the École Polytechnique, was working out heat flow in solids whose surface temperature changed with time. His method — build the response to an arbitrary boundary history by superposing responses to elementary steps — is now called Duhamel's principle, and its integral form is exactly the convolution of an input history with a system's elementary response ^[1]^[2]^[3]. Structural dynamics people still call the convolution integral Duhamel's integral when they compute shock response ^[3].

The circuit-theory version came from Oliver Heaviside (1850–1925), who between 1880 and 1887 built an "operational calculus" that treated d/dt as an algebraic symbol p, turning the differential equations of telegraph lines and circuits into algebra ^[4]^[5]. It worked, and mathematicians hated it — the Royal Society rejected one of his papers over rigor, and the methods were not put on sound footing until Thomas Bromwich's work decades later ^[4]^[5]. Heaviside's p is, for practical purposes, the s in every transfer function you have ever written.

The impulse itself got its name from physics. Paul Dirac (1902–1984) made the δ-function a working tool of quantum mechanics in his 1930 book The Principles of Quantum Mechanics ^[6]^[7], with Heaviside's circuit work among the engineering precedents ^[6]. Mathematicians objected, correctly, that no function has those properties. Laurent Schwartz resolved it in the mid-1940s with the theory of distributions, publishing the full treatment in Théorie des distributions (1950–51) and collecting the 1950 Fields Medal for it ^[6]^[8]. The pattern is worth noticing: engineers and physicists used the impulse for roughly sixty years before the mathematics existed to justify it, and the hardware worked the entire time.

Related tools

/tools/rc-filter — first-order H(jω) and the −3 dB point worked above
/tools/rc-charge-time — the step response 1 − e^(−t/τ) directly
/tools/q-factor-bandwidth — how sharply |H(jω)| peaks for a resonant system
/tools/lc-resonance — the pole location of a second-order LTI system
/tools/series-rlc-impedance — the canonical second-order frequency response
/tools/vibration-natural-freq — the mechanical system whose h(t) a hammer tap measures

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →