Applying Analytics

1-Dimensional Data Sets 105

ψ

i

and Y

k

because the Y

k

can grow with n, a number that can be arbitrarily

large. Keep in mind that the units of the Y

k

are different than the units of

the continuous Fourier transform, w hich are those of the continuous function

ψ multiplied by a factor of the units of the independent variable.

The far right-hand s ide of Equation 5.24 explicitly connects a Y

k

compo-

nent and a wave that completes k oscillations over the length of the data set.

Euler’s formula, given by e

iωτ

= cos ωτ + i sin ωτ , makes clear the connection

between the exponent and sinusoidal waves with angula r frequency ω. The

k = 0 component, as you would exp ...