Periodogram

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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.

The periodogram is evaluated in practice from a finite digital sequence using the fast Fourier transform. The raw periodogram is not a good spectral estimate since it suffers from spectral bias and variance problems.

The bias problem arises from a sharp truncation of the sequence, and can be reduced by first multiplying the finite sequence by a window function which truncates the sequence gracefully rather than abruptly.

The variance problem can be reduced by smoothing the periodogram. Various techniques to reduce spectral bias and variance are the subject of spectral estimation.