When the period grows without bound

A Fourier series describes something that repeats. Every harmonic it uses is a whole number of cycles per period, so the spectrum is a set of separate spikes at multiples of one basic frequency, with nothing in between.

Most things do not repeat. A single pulse, a spoken word, a struck bell: each happens once. The apparatus of the previous rungs does not apply to any of them, and the way it is made to apply is to treat a one-off event as a repeating one whose period is very long — and then let the period grow.

Two things happen at once and only one of them is a change.

The spacing between spectral lines is $1/T$, so doubling the period halves it: at period 2 the lines are half a unit apart, at period 8 they are an eighth of a unit apart, and at period 1,000 there are a thousand of them in the space the first figure gives to one.

The height of each line, scaled so the comparison is fair, is a fixed function of frequency that depends on the pulse alone and not on how often it is repeated. That is the dashed curve, and it is where the whole of the information about the pulse lives.

The curve the lines sit on #

For a pulse of width $w$ that is $1$ while it lasts and $0$ otherwise, the projection of one period onto the frequency $f$ is

$$\int_{-w/2}^{w/2} \cos(2\pi f t)\,dt = \frac{\sin(\pi f w)}{\pi f} = w\,\mathrm{sinc}(fw),$$

with no $T$ in it anywhere. The period appears only in which frequencies get a line — the multiples of $1/T$ — and never in the value the curve takes there.

So the limit is not a limit of shapes. The shape is fixed from the start, and what changes is how densely it is sampled. As $T$ grows the sampling becomes dense; in the limit every frequency has a value, and the spectrum stops being a list and becomes a function.

$$\hat f(\nu) = \int_{-\infty}^{\infty} f(t)\,e^{-2\pi i \nu t}\,dt.$$

That is the Fourier transform, and everything about it is inherited: it is the projection of the previous rung, taken against a continuum of frequencies rather than a discrete set, and it works for the same reason — different frequencies are orthogonal, so each one can be extracted without disturbing the rest.

Reading the envelope #

The curve the lines sit on is worth reading rather than merely noting, because everything about the pulse is legible in it.

The envelope $w\,\mathrm{sinc}(fw)$ has its peak at zero frequency, where its height is $w$: the total area of the pulse, which is what the zero-frequency component always is. It crosses zero at $f = 1/w, 2/w, 3/w,\ldots$, so the first zero is at the reciprocal of the pulse width, and the whole pattern scales with $1/w$.

Between those zeros are side lobes, alternating in sign and falling away like $1/f$. Their slow decay is the frequency-domain signature of the pulse’s sharp edges — the same $1/m$ that a square wave’s coefficients fall by, for the same reason, since the pulse has jumps and jumps cost a factor of one power.

That is a useful diagnostic in both directions. A measured spectrum with a first null at 100 Hz came from something a hundredth of a second long. A spectrum whose side lobes fall slowly came from something with sharp edges, and rounding the edges of the pulse — tapering it — pulls the side lobes down at the cost of widening the central lobe. Every antenna, every window function and every filter design is a position taken on that trade.

The trade between the two widths #

The envelope depends on the pulse, and it depends on it in a way with consequences.

Halving the pulse width doubles the width of $\mathrm{sinc}(fw)$, because the only place $w$ appears is multiplied by $f$. That is a scaling law with no exceptions:

$$f(at) \quad\longleftrightarrow\quad \frac{1}{|a|}\hat f\!\left(\frac{\nu}{a}\right).$$

Squeeze a signal in time and its transform stretches in frequency by exactly the reciprocal factor. There is no way to have both narrow, and the impossibility is arithmetic rather than technological.

Made precise with a suitable definition of width — the standard deviation of the squared magnitude — the statement becomes

$$\Delta t \cdot \Delta \nu \ge \frac{1}{4\pi},$$

with equality only for a Gaussian pulse. This is the uncertainty principle, and it is worth being clear about what kind of statement it is: a theorem about Fourier pairs, provable in half a page, with no physics in it. Its appearance in quantum mechanics is a consequence of position and momentum being related by a Fourier transform, and the inequality was a fact about functions before it was a fact about particles.

The engineering version is met daily. A radar pulse short enough to locate a target precisely occupies a wide band of frequencies; a note held long enough to have a definite pitch cannot be brief. A recording device with a narrow filter cannot respond quickly, and one that responds quickly cannot have a narrow filter. Each is the same inequality.

Turning at a frequency #

The transform is usually written with a complex exponential rather than a cosine, and the reason is worth a paragraph, since the two are the same computation and one of them is far easier to think with.

The factor $e^{-2\pi i\nu t}$ in the transform is a point going round the circle backwards at $\nu$ turns per unit time. Multiplying a signal by it and integrating is winding the signal around a circle at that rate and asking where the centre of mass of the result ends up. If the signal contains a component turning at exactly $\nu$, the winding cancels its rotation and that component piles up on one side; everything else goes round and averages to nothing.

That is the same cancellation as the orthogonality of the sines, stated so that both the size and the phase of the component come out at once — as the two coordinates of the resulting point. A pair of real integrals, against a cosine and a sine, carries exactly the same information, and keeping them as one complex number halves the bookkeeping and makes the scaling and shifting rules one line each instead of two.

Multiplication being a turn is what makes that compression available, and it is the clearest case in this collection of a notational choice doing real work: nothing is proved by writing $e^{i\theta}$ instead of a pair of trigonometric functions, and almost everything is easier afterwards.

What is gained and what is lost #

The transform buys generality and pays for it in two places.

What is gained: the transform applies to anything that does not repeat, which is most things. It also turns operations into simpler operations — differentiation becomes multiplication by $2\pi i\nu$, convolution becomes multiplication, translation becomes a phase factor — and each of those is a coupled problem becoming an uncoupled one.

What is lost, first, is the countability of the description. A periodic function is described by a sequence of numbers that can be listed, stored and truncated. A transform is a function of a continuous variable, and any actual use of it must sample that function, which reintroduces a period — the sampling in one domain is periodicity in the other, and that duality is the whole content of the sampling theorem.

What is lost, second, is locality in time. Every value of $\hat f$ is an integral over the entire history of the signal, so a transform describes what frequencies are present and says nothing about when. A recording of a rising note and a recording of a falling one, played backwards, have the same magnitude spectrum. That deficiency is why the wavelet and short-time methods exist, and it is not a defect in the transform so much as a direct consequence of what an orthogonal frequency is: something that goes on forever.

Why it is worth the trouble: what happens to operations #

The property that makes the transform an everyday tool rather than a description is what it does to the operations people actually perform.

Convolution becomes multiplication. Smearing one signal by another — which is what every filter, every blur, every echo and every measurement instrument does — is an integral over all shifts, and it is expensive and awkward. Transformed, it is a product of two functions, frequency by frequency. A blur that costs $N^2$ operations directly costs a transform, a multiply and a transform back.

Differentiation becomes multiplication by $2\pi i \nu$. A differential equation with constant coefficients turns into an algebraic equation, one frequency at a time, with no coupling between frequencies at all. That is precisely what makes the heat equation solvable term by term, and the same manoeuvre solves the wave equation, the diffusion equation and every linear system with time-invariant behaviour.

Shifting becomes a phase factor. Delaying a signal multiplies its transform by a number of unit size whose angle is proportional to frequency, so the magnitude spectrum does not change at all — which is the formal statement of the locality that was lost above, and simultaneously the reason a transform is robust to when a measurement started.

In each case the transform is a change of coordinates chosen so that a hard operation becomes a diagonal one. It is the continuous relative of finding eigenvectors, and the exponentials are the eigenfunctions of every operation that commutes with shifting in time.

Where the limit needs a condition #

The passage from series to transform above is a heuristic, and made rigorous it needs the function to decay.

The transform integral is over the whole line, so it converges only if $f$ dies away — absolutely integrable is the standard sufficient condition. A signal that does not decay, such as a pure sine going on forever, has no transform in this sense at all: its integral does not converge, and the object standing in for its spectrum is a spike of infinite height and zero width at one frequency, which is not a function.

The repair is a substantial piece of twentieth-century mathematics. Distributions — Schwartz’s theory, from the 1940s — make the spike a legitimate object by redefining what a function is allowed to be, and in that theory the transform of a sine is exactly the pair of spikes engineers had been drawing and manipulating successfully for decades without justification.

That is worth noting as a pattern rather than as a curiosity: the practitioners were right, the mathematics was missing, and the mathematics that arrived did not change any answer. It made the existing answers derivable rather than lucky, which is the usual and unglamorous form of progress in this direction.

The symmetry of the two descriptions #

One feature of the transform has no counterpart in the series, and it is the one that makes the pair of descriptions feel like two views of a single object rather than an object and a summary of it.

The transform is nearly its own inverse. Recovering $f$ from $\hat f$ uses the same integral with the sign of the exponent flipped, so the machinery in both directions is identical, and applying the transform four times returns the original function exactly. There is no sense in which time is the real domain and frequency the derived one: each is the transform of the other, and a theorem about one is automatically a theorem about the other with the roles exchanged.

That symmetry converts every fact into two. Narrow in time means wide in frequency, and therefore narrow in frequency means wide in time. A pulse transforms to a sinc, so a sinc transforms to a pulse — which is the statement that an ideal frequency filter has an impulse response that rings forever, and is why perfect filters cannot be built.

Parseval’s identity gains a symmetric form too: the total energy computed in time equals the total energy computed in frequency, so neither description is lossy and neither is preferred. A Fourier series has nothing quite like this, because its two sides are of different kinds — a function on an interval and a sequence of numbers — and the limit above is what makes them the same kind of thing and therefore exchangeable.

What the picture cannot show #

Every figure here is at a finite period, and the whole point is what happens at infinite period. The reader is being asked to extrapolate from three panels — an extrapolation the figures make plausible and cannot establish.

Worse, the limit object is of a different kind from any of the panels. Each panel shows finitely many lines of finite height; the limit is a continuous function, and the lines do not tend to it in any pointwise sense — their heights are fixed and it is their density that grows. Nothing is converging to anything in the way the pictures suggest, and the honest statement of the limit involves a scaling of the coefficients by $T$ that the drawing performs silently by plotting $T c_k$ rather than $c_k$.

That silent rescaling is the load-bearing part of the whole construction, and it is invisible in the figures. Without it every line would shrink to nothing as the period grew, which is exactly what the raw coefficients do, and the picture would show a spectrum fading away rather than a curve being filled in.

The ladder from here #

Below: the series and the projection that produces its coefficients. Above: the heat problem the whole apparatus was built for, where each frequency decays at its own rate.

Sideways, the transform’s complex form needs multiplication as rotation to be readable at all — $e^{-2\pi i\nu t}$ is a point going round a circle at frequency $\nu$, and projecting onto it is asking how much of the signal turns at that rate. And the reciprocal trade of widths shows up wherever two descriptions are Fourier pairs, which includes a lattice and its diffraction pattern and, in a different guise, the trade between a filter’s sharpness and its delay.

A limit that changes the kind of object #

The lasting point is what the limit did.

The starting object was a list of numbers, indexed by whole numbers, with a gap between consecutive entries. The limiting object is a function of a real variable. That is not a change of value; it is a change of category, and the process that produced it — let the spacing go to zero and rescale so the heights stay put — is exactly the process that turns a sum into an integral, a histogram into a density, and a random walk into a diffusion.

Each of those is the same manoeuvre: a discrete family, refined without bound, with a compensating rescaling that keeps something finite. Getting the rescaling right is the whole skill, and it is invisible afterwards, because the limit looks natural and inevitable once it exists. A walk’s √n spread becomes Brownian motion by the same trick, and an area under rectangles becomes an integral by it as well.