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Unit 1 · Lesson 1-2
The Impulse and the Complex Exponential
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The Impulse and the Complex Exponential

The unit step, the impulse and its sifting property, Euler's formula, phasors, and their discrete-time counterparts

A function that cannot exist
In a 1927 paper, and then in his 1930 book The Principles of Quantum Mechanics, Paul Dirac used a "function" δ(x)\delta(x) that is zero everywhere except at one point, yet has integral one. No ordinary function can do that: a function that is zero except at a single point has integral zero, whatever value it takes there. Dirac was not the first. Fourier had used the idea in his theory of heat in 1822, and Oliver Heaviside used it throughout his electrical engineering of the 1890s (Dirac delta function). Engineers kept calculating with it, and kept getting right answers, for decades before anyone could say what it was. In the late 1940s Laurent Schwartz built the theory of distributions, which gave the impulse a precise meaning, and received the 1950 Fields Medal for it (Laurent Schwartz). This lesson introduces that impossible function and its partner, the complex exponential — the two signals out of which this whole course is built.
Start with something harmless. The unit step switches on at t=0t = 0:
u(t)={0,t<01,t>0u(t) = \begin{cases} 0, & t < 0 \\ 1, & t > 0 \end{cases}
Its value at t=0t = 0 itself is a convention (0, 1 or ½); nothing in this course depends on it. The step models anything switched on: a voltage applied to a circuit, a valve opened, a thrust command. In discrete time u[n]u[n] is 0 for n<0n < 0 and 1 for n≥0n \ge 0, with no ambiguity.
Now take a rectangular pulse of width Δ\Delta and height 1/Δ1/\Delta, centred on t=0t = 0. Its area is one whatever Δ\Delta is. Let Δ\Delta shrink: the pulse grows taller and narrower, and its area stays one. The unit impulse δ(t)\delta(t) is the limit — an idealised kick, all at one instant, of total size one. A hammer blow, a lightning strike, a click: anything much shorter than the system that receives it behaves like an impulse.
The impulse is not a function, so we never ask for its value at a point. We only ever use it inside an integral, and there it is perfectly well behaved:
∫−∞∞δ(t) dt=1,u(t)=∫−∞tδ(τ) dτ,δ(t)=dudt.\int_{-\infty}^{\infty}\delta(t)\,dt = 1, \qquad u(t) = \int_{-\infty}^{t}\delta(\tau)\,d\tau, \qquad \delta(t) = \frac{du}{dt}.
The step is the running integral of the impulse; the impulse is the "derivative" of the step — the slope of a jump that happens in no time at all.
💡The sifting property
Multiply a signal by a shifted impulse and integrate: you get the signal's value at that instant. ∫−∞∞x(t) δ(t−t0) dt=x(t0).\int_{-\infty}^{\infty} x(t)\,\delta(t - t_0)\,dt = x(t_0). The discrete-time version needs no limits at all. The unit sample δ[n]\delta[n] is 1 at n=0n = 0 and 0 elsewhere, and every sequence is a sum of shifted, scaled unit samples: x[n]=∑k=−∞∞x[k] δ[n−k].x[n] = \sum_{k=-\infty}^{\infty} x[k]\,\delta[n - k]. Read that equation as a recipe: any signal is a row of impulses, each scaled by one sample. It is the seed of the next unit.
Derivation
Sifting, from a shrinking pulse
Replace δ(t−t0)\delta(t - t_0) by the pulse δΔ(t−t0)\delta_\Delta(t - t_0): height 1/Δ1/\Delta on [t0−Δ/2, t0+Δ/2][t_0 - \Delta/2,\ t_0 + \Delta/2], zero elsewhere. Then
∫−∞∞x(t) δΔ(t−t0) dt=1Δ∫t0−Δ/2t0+Δ/2x(t) dt.\int_{-\infty}^{\infty} x(t)\,\delta_\Delta(t - t_0)\,dt = \frac{1}{\Delta}\int_{t_0 - \Delta/2}^{t_0 + \Delta/2} x(t)\,dt .
The right-hand side is the average of xx over a window of width Δ\Delta around t0t_0. If xx is continuous at t0t_0, the mean value theorem says this average equals x(t∗)x(t^*) for some t∗t^* in the window, and as Δ→0\Delta \to 0 the window closes on t0t_0:
lim⁡Δ→01Δ∫t0−Δ/2t0+Δ/2x(t) dt=x(t0).\lim_{\Delta \to 0} \frac{1}{\Delta}\int_{t_0 - \Delta/2}^{t_0 + \Delta/2} x(t)\,dt = x(t_0).
That limit is what the sifting property means. The same argument with a pulse of width Δ/∣a∣\Delta/|a| shows δ(at)=δ(t)/∣a∣\delta(at) = \delta(t)/|a|: compressing the impulse in time also compresses its area.
The second building block is the complex exponential. Euler's formula ties it to the sinusoids:
ejθ=cos⁡θ+jsin⁡θ.e^{j\theta} = \cos\theta + j\sin\theta .
So ejωte^{j\omega t} is a point going round the unit circle in the complex plane at ω\omega radians per second, and its real part is cos⁡ωt\cos\omega t. Let the exponent be any complex number s=σ+jωs = \sigma + j\omega:
est=eσt ejωt.e^{st} = e^{\sigma t}\,e^{j\omega t} .
The real part σ\sigma sets the envelope — decaying for σ<0\sigma < 0, growing for σ>0\sigma > 0, steady for σ=0\sigma = 0 — and ω\omega sets the rotation. One family covers constants (s=0s = 0), exponentials (ω=0\omega = 0), sinusoids (σ=0\sigma = 0) and every damped or growing oscillation in between: the ringing of a bell, the swing of a seismometer arm, the transient of a circuit.
Derivation
Euler's formula from power series
The series ez=∑k≥0zk/k!e^z = \sum_{k \ge 0} z^k/k! converges for every complex zz. Put z=jθz = j\theta and use j2=−1j^2 = -1, j3=−jj^3 = -j, j4=1j^4 = 1, …:
ejθ=1+jθ−θ22!−jθ33!+θ44!+jθ55!−⋯e^{j\theta} = 1 + j\theta - \frac{\theta^2}{2!} - j\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + j\frac{\theta^5}{5!} - \cdots
Collect the real and imaginary terms:
ejθ=(1−θ22!+θ44!−⋯ )+j(θ−θ33!+θ55!−⋯ )=cos⁡θ+jsin⁡θ.e^{j\theta} = \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\right) + j\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right) = \cos\theta + j\sin\theta .
Adding the formula for θ\theta and for −θ-\theta gives the identity we will use constantly: a real sinusoid is two counter-rotating complex exponentials,
cos⁡ωt=ejωt+e−jωt2,sin⁡ωt=ejωt−e−jωt2j.\cos\omega t = \frac{e^{j\omega t} + e^{-j\omega t}}{2}, \qquad \sin\omega t = \frac{e^{j\omega t} - e^{-j\omega t}}{2j}.
📐Adding sinusoids with phasors
Add 3cos⁡(ωt+30∘)3\cos(\omega t + 30^\circ) and 4cos⁡(ωt−60∘)4\cos(\omega t - 60^\circ). Both are the real part of a fixed complex number — the phasor — times ejωte^{j\omega t}: 3ej30∘+4e−j60∘=(2.598+1.500j)+(2.000−3.464j)=4.598−1.964j.3e^{j30^\circ} + 4e^{-j60^\circ} = (2.598 + 1.500j) + (2.000 - 3.464j) = 4.598 - 1.964j . Its magnitude is 4.5982+1.9642=5.000\sqrt{4.598^2 + 1.964^2} = 5.000 and its angle −23.13∘-23.13^\circ, so the sum is 5cos⁡(ωt−23.13∘)5\cos(\omega t - 23.13^\circ). (The amplitudes add like a 3-4-5 triangle because the two phasors are 90° apart.) Adding sinusoids of one frequency is adding complex numbers — no trigonometric identities needed.
💡Discrete-time exponentials
In discrete time the family is znz^n with zz complex: ∣z∣<1|z| < 1 decays, ∣z∣>1|z| > 1 grows, and z=ejΩz = e^{j\Omega} on the unit circle gives the sinusoid ejΩne^{j\Omega n}. One property has no continuous-time counterpart: ej(Ω+2π)n=ejΩne^{j(\Omega + 2\pi)n} = e^{j\Omega n} for every integer nn. Discrete-time frequencies wrap around every 2π2\pi, so cos⁡(1.9πn)\cos(1.9\pi n) and cos⁡(0.1πn)\cos(0.1\pi n) are the same sequence. The highest frequency a sequence can show is Ω=π\Omega = \pi, alternating signs. Sampling inherits this wrap-around, and Unit 5 calls it aliasing.
Mathematics is an experimental science, and definitions do not come first, but later on.
– Oliver Heaviside, Electromagnetic Theory, Vol. II (1899), quoted on Wikiquote
⚠An impulse has no value at a point
Writing δ(0)=∞\delta(0) = \infty is not just loose, it is meaningless: the pulses δΔ\delta_\Delta with heights 1/Δ1/\Delta and 2/Δ2/\Delta both "go to infinity", yet one has twice the area of the other. Only integrals against δ\delta carry information. So never square an impulse, never divide by one, and when a formula produces δ(t)2\delta(t)^2, the model has broken — usually by idealising two things at once.
🕰Engineers first, proofs later
Heaviside's operational calculus treated d/dtd/dt as an algebraic quantity and the impulse as the derivative of a jump. It worked spectacularly on cables and circuits and was dismissed by many mathematicians of his day. Dirac's 1927 delta brought the same object into quantum mechanics; Schwartz's distributions of the late 1940s made both rigorous. Heaviside's line about definitions coming later was a defence of exactly this way of working.
💡Where you will use this
The sifting property becomes convolution in Lessons 2-1 and 2-2, and the impulse response becomes the fingerprint of every LTI system. Complex exponentials are the eigenfunctions of Lesson 3-1, the basis of the Fourier series and transform (Units 3 and 4), and — with ss and zz free to roam — of the Laplace and z-transforms (Units 7 and 8). The impulse train of Lesson 5-1 is how sampling is described.
📚Further reading
  • Oppenheim and Willsky, Signals and Systems, section 1.4 — the unit impulse and unit step functions, with the pulse-limit argument.
  • Lathi and Green, Linear Systems and Signals, section 1.4 — useful signal models, including complex exponentials.
  • Dirac delta function on Wikipedia — its history from Fourier and Heaviside to Schwartz's distributions.
↓But there is a problem
We have the alphabet: steps, impulses and complex exponentials, and the sifting property that writes any signal as a row of impulses. Now for the machines that act on signals. An amplifier, a thermometer, a bridge under a crowd — each takes an input and produces an output. Some of them we can predict completely from a single experiment. Others surprise their own designers on opening day. What separates the two?