Matthew van Eerde's web log

• Why square waves have ears: Gibbs' phenomenon (Wilbraham's phenomenon)

In a recent post I sung the praises of square waves as a way to get a heckuva lot of power (3 dB more power than a sine wave) into a sample-range-limited signal.  It's time to take them down a notch now.

A problem with square waves is they're impossible to generate in the analog domain.  In fact, you can't even get close.

Signals generated in the analog domain are subject to physical laws regarding continuity (no teleportation) and whatnot. A common way to model these is to express (periodic) analog signals using a basis of sine waves with integral periods.  Suppose I want to generate the square wave:

 f(x) = 1, -π < x < 0 -1, 0 < x < π

Graphed below are some approximations of this function using sine waves as a basis.  Note that only odd values of n are used in the sums of sin(nx).

The sums converge to +/-1 quite well, but there are definite "ears" at x near 0 where there's overshoot.  This doesn't appear to die down.  A closeup of one of the "ears":

If anything, rather than dying down, the "ears" converge to max(fn) → about 1.18, or about 9% of the "jump" from -1 to 1.  (Dym and McKean, in their 1972 book Fourier Series and Integrals, get the 9% right but incorrectly assert that the convergence is to 1.09.)

This mathematical phenomenon - Gibbs' phenomenon - is a good illustration of the difference between convergence of a series of functions and uniform convergence.

In this case, the series of partial sums pointwise converge to the square wave... for any given point x > 0 and ε > 0, the ear will eventually move to the left, and I can choose an N such that fn(x) is within of ε of 1 for all n > N...

... but the series does not uniformly converge to the square wave.  The following assertion is false: "for any given ε > 0, I can pick an N such that fn(x) is within of ε of 1 for all n > N and all x > 0."  This can be demonstrated by picking ε = 0.17, say.  For any n, even, say, n = 10100, there is an x close to 0 where f1e100(x) > 1.17.

• Daylight Saving Time and Benjamin Franklin

Daylight Saving Time is a thorn in my side.

Politician: What time is it?
Scientist: What time would you like it to be?

It is my firm belief that

1. Daylight Saving Time doesn't do any real good.
2. Daylight Saving Time does real harm.
3. Politicians love Daylight Saving Time because it's easy and at least it looks good.
One of the points that my congresswoman brought up is that Daylight Saving Time was originally proposed by a founding father of the US, and that therefore it must be a good idea.  (Exercise: what logical fallacy is this?)

Today I decided to actually look up the original proposal and found it quite enlightening.

Benjamin Franklin's 1784 letter to the Journal de Paris (English translation)

I believe all who have common sense, as soon as they have learnt... that it is daylight when the sun rises, will contrive to rise with him; and, to compel the rest, I would propose the following regulations...

let a tax be laid... on every window that is provided with shutters...

[let] no family be permitted to be supplied with more than one pound of candles per week...

Let guards also be posted to stop all the coaches, &c. that would pass the streets after sunset...

Every morning, as soon as the sun rises, let all the bells in every church be set ringing; and if that is not sufficient?, let cannon be fired in every street, to wake the sluggards effectually, and make them open their eyes to see their true interest.

Yes, Benjamin Franklin was the first to propose a law encouraging people to get up early.

But he was joking.

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