Consider a signal which is a real sine series, f(x)=2*sin(2*Pi*x) + 5*sin(2*Pi*6*x) + 9*sin(2*Pi*11*x).

Let F(w) be its Fourier transform. Answer F(w) is purely complex and expressed in terms of symbol Dirac.

Maple "plot" fails.

Plots of impulse trains like F(w)=sqrt(Pi)*I*(2*Dirac(w-Pi)+4*Dirac(w-4*Pi)) involve user intervention. Maple won't plot such expressions, because they are DISTRIBUTIONS (not FUNCTIONS).

Try plotting F(w) to see the engine's uninformative error message.

Here's how to obtain an informative plot of an impulse train.

STEP 1. Create a REAL representation of the impulse train.

 For example, if f(x)=sine series, then F(w) is pure complex (real part = zero). Divide F(w) by I=sqrt(-1).

STEP 1.5. Pre-process F(w) further, dividing out any multiplicitive factors like 1/sqrt(2*Pi). This re-scaling does not substantially affect the graph. While not necessary in future examples, it helps in your answer check on the first example. Simplify. It is sound advice. After making it so, look out the window for rainbows and constellations.

STEP 2. Invent a replacement for "Dirac" using Paul Dirac's approximation fo the ideal force. This involves defining a piecewise function ApproxDirac, which is a pulse of width 2h having the same force as the Dirac term in F(w).

The dimension h has to be adjusted for each graphic, to give an informative representation of the impulsive forces.


STEP 3. Replace "Dirac" (maple function "subs") in the impulse train by "ApproxDirac" for a trial value of h, like h=0.1 or h=0.8. Plot and replot until the graphic is useful, changing h until satisfied. 

-Grant Gustafson, Salt Lake City, Math, Univ of Utah


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