perplexus.info :: Just Math : Optimizing The Train

Optimizing The Train (Posted on 2006-10-04)

A periodic pulse train of period N, and having the value P(t) at time t, is generated according to

P(t) = | A₀ + A₁W + A₁W² + ... + A_N-1W^N-1 |²

where the coefficients A_j are complex numbers, W=exp(i2πt/N) (i=√(-1)), and N is a fixed positive integer. The pulse train is thus the square of the absolute value of a complex "trigonometric polynomial."

The pulse train engineer desires the pulse train to have the highest peak value possible, subject to the constraint that the area S of one period of the pulse train has a prescribed value. The engineer suspects that there is a smallest constant K such that no matter how the A_j are chosen, P(t)<=KS for all t. Prove this for him if you can, and determine if equality is possible for some t when S>0 (which would make the engineer's desire fully realizable).

Submitted by Richard

Rating: 3.0000 (3 votes)

Solution: (Hide)

The optimum is obtained by picking the coefficients all equal. The inequality holds with K=1 and can be proved using a well-known result of Cauchy.
It is easy to show that S = N(|A₀|² + |A₁|² + ... + |A_N-1|²). The inequality then follows using Cauchy's result that the square of the sum of |A_k||B_k| does not exceed the product of the sums of the |A_k|² and the |B_k|² in the case where |B_k| = |W^k| = 1. This result of Cauchy is very well-known in various contexts and is sometimes associated also with the names Schwarz and/or Buniakowski ( http://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality ). It just says that an expression that may be taken to be the definition of the cosine of the angle between two vectors cannot exceed 1 in absolute value. Cauchy's result is applicable as soon as one observes that by the triangle inequality ( http://mathworld.wolfram.com/TriangleInequality.html )
P(t) <= (|A₀| + |A₁||W| + ... + |A_N-1||W^N-1|)².
The result is then just P(t) <= S, with equality clearly holding if all the A's are chosen equal.

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Hints	Richard	2006-10-07 20:16:51

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