math lessons - Periodogram

The term periodogram appears often in the context of power spectral density calculations. In his paper Power Spectral Density estimation, Fernando S. Schlindwein explains the origin of the word 'periodogram'. Apparently, it was coined by Arthur Schuster in 1898. Here is the excerpt that Schlindwein presented from Schuster's paper:

"THE PERIODOGRAM. It is convenient to have a word for some representation of a variable quantity which shall correspond to the "spectrum" of a luminous radiation. I propose the word periodogram, and define it more particularly in the following way. Let

$\frac{T}{2}a = \int_{t_1}^{t_1+T}f(t)\cos(kt)dt$

$\frac{T}{2}b = \int_{t_1}^{t_1+T}f(t)\sin(kt)dt$

where T may for convenience be chosen to be equal to some integer multiple of

$\frac{2\pi}{k}$ ,

and plot a curve with $2π / k$ as abscissæ and

$r = \sqrt{a^2+b^2}$

as ordinates; this curve, or, better, the space between this curve and the axis of abscissæ, represents the periodogram of f(t)."

Those familiar with the Fourier transform should recognize the formulae for a and b.

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