Dirac delta and step functions

The formula

u(t) = 0 for t < 0,  1 for t > 0        (Heaviside step)

Reading: nothing happens, then at t = 0 the input turns on and holds — the ideal relay closure, valve opening, or power-up.

∫ f(t)·δ(t − T) dt = f(T)               (sifting property)

Reading: the delta is defined by what it does inside an integral — it reaches into f and pulls out the single value at the instant the impulse fires. This is the only rigorous definition; everything else about δ follows from it.

δ(t) = du/dt        and        u(t) = ∫₋∞ᵗ δ(τ) dτ

Reading: differentiate a switch and you get a hit; integrate a hit and you get a switch. The two functions are one idea seen through the derivative.

L{u(t)} = 1/s        L{δ(t)} = 1        δ(a·t) = δ(t)/|a|

Reading: in the Laplace domain the step costs you a 1/s and the impulse costs you nothing — which is why the impulse response IS the transfer function. The scaling rule says compressing the time axis by a must shrink the delta's weight by |a| to keep unit area.

Where you meet it

Modal survey with an impact hammer. The instrumented hammer tap on a test article is a physical stand-in for F₀·δ(t). The FRF the analyzer computes assumes the input was impulsive; the tip you screw on (steel, plastic, rubber) sets how far up in frequency that assumption holds.
Bench, step response. Square wave into a scope probe for compensation, a supply rail at turn-on, a relay closing into an RC load — all modeled as V·u(t). At t = RC the capacitor sits at 1 − e⁻¹ ≈ 63.2% of the step, and every rise-time and settling spec on the datasheet is written against that ideal edge.
Stress report with point loads. A concentrated load on a beam is a delta in the distributed-load function w(x); a point moment is a delta's derivative. The Macaulay bracket notation ⟨x − a⟩ in the deflection calc is the step function wearing structural-engineering clothes — one equation for the whole span instead of a piecewise mess at every load point.
Design review, controls or signal processing. "What's the plant's impulse response?" is the question because, for a linear time-invariant plant, y(t) = h(t) * input — once you know the response to δ, convolution gives you the response to anything. Sampled-data systems are built on trains of deltas.

How it works

The delta is not a function, and treating it like one is where the trouble starts. There is no value δ(0) — the height is meaningless and the area is everything. δ(t) is a rule that assigns a number to any smooth function it meets under an integral: the sifting property above. Mathematicians call such objects distributions. Engineers can get almost everywhere by remembering one thing: a delta only means something under an integral sign, or as the shorthand limit of a real pulse whose area you know.

Units come along for free and people still miss them. If t is in seconds, δ(t) carries units of 1/s, so a mechanical impulse is written I·δ(t) with I in N·s — the delta supplies the "per second," the coefficient supplies the physics. Hit a mass-spring system m·ẍ + c·ẋ + k·x = I·δ(t) and the entire effect is an instantaneous velocity jump ẋ(0⁺) = I/m; after that it's free vibration. That equivalence — impulse forcing equals initial condition — is the working core of impact testing.

The first limit of validity is bandwidth. A real pulse of duration τ acts like a delta only for dynamics slower than roughly 1/τ. That's the whole story of hammer tips: a hard steel tip gives a short pulse and flat force spectrum out past the modes you care about; a rubber tip rolls the spectrum off early and the high modes simply don't get excited. Same rule in electronics — an edge with finite slew rate is a step only for circuits whose time constants dwarf the rise time. Push an ideal step into i = C·dV/dt and the math hands you infinite inrush current; the real number is set by source impedance and the edge rate, which is why the ideal model quietly exits the analysis right when you're sizing an inrush limiter. The second limit is linearity: the whole impulse-response-defines-the-system story assumes a linear, time-invariant system, and a nonlinear structure hit harder gives back a different FRF — which is why a proper impact test includes a force-level linearity check before anyone trusts the curves.

The classic mistakes, in the order people make them: evaluating δ(0) as a number; dropping the 1/|a| when the argument is scaled or the variable is changed; multiplying two deltas together (undefined — the product δ(t)² has no meaning, and integrals that produce it are telling you the model broke); and forgetting that the value of the step at exactly t = 0 is a convention, not a fact. H(0) can be 0, 1, or 1/2 depending on the text; it never matters under an integral, but it absolutely matters in code — numpy.heaviside(x, h0) makes you pick, and a comparison written t >= 0 versus t > 0 has silently chosen for you.

One more payoff worth keeping: the Fourier transform of δ(t) is a constant — flat spectrum, all frequencies, equal measure. That single line explains why one hammer tap excites every mode at once, why an ESD event is broadband interference, and why impulse testing is efficient in the first place.

History

The delta was used long before it was legal. Fourier's 1822 heat treatise contains it implicitly in the Fourier integral theorem, and Cauchy wrote an explicitly infinite, unit-area impulse in an 1827 text ^[1]. The step function belongs to Oliver Heaviside (1850–1925), a self-taught former telegraph operator who left school at sixteen and went on to compress Maxwell's twenty equations into the four we use ^[3]. Between 1880 and 1887 he built his operational calculus around a unit step he simply wrote as "1," turning differential equations into algebra and getting answers the rigor of his day couldn't certify. The Royal Society elected him a Fellow in 1891 — and three years later rejected the next installment of his operators paper outright for "errors of substance" ^[2]^[3].

Paul Dirac — who took an electrical engineering degree at Bristol in 1921 before physics claimed him — introduced the δ notation in a 1927 paper and cemented it in his 1930 book The Principles of Quantum Mechanics, naming it as the continuous cousin of the Kronecker delta ^[1]^[4]. The object did indispensable work for two decades while remaining, formally, nonsense. Laurent Schwartz fixed that: he later described the theory of distributions arriving during a night of insomnia in 1944, published Théorie des distributions in 1950–51, and collected the 1950 Fields Medal for making the delta rigorous ^[1]^[5].

Structural engineers, meanwhile, had never waited. Clebsch was handling discontinuous beam loads with what amount to singularity functions in 1862, and W. H. Macaulay's 1919 note in the Messenger of Mathematics gave the bracket method every mechanics-of-materials course still teaches ^[6]^[7]. The stress analyst and the quantum physicist were using the same illegal function, and neither knew about the other.

Related tools

/tools/beam-deflection — point loads on beams are deltas; Macaulay brackets are steps
/tools/rc-charge-time — the canonical first-order step response
/tools/rc-filter — transfer function = Laplace transform of the impulse response
/tools/vibration-natural-freq — the free response an impulse hands off to
/tools/series-rlc-impedance — second-order dynamics that step and impulse inputs interrogate
/tools/rc-snubber — taming the real-world consequences of near-step switching edges
/tools/inrush-ntc — what C·dV/dt into an ideal step gets wrong, and what fixes it

Sources

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