# Amplitude of power spectral density

Why is the amplitude of the Power spectral density higher for shorter period of time as compared to a longer period of time when calculated for any vibration data?

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## 1 Answer

I suggest you try looking at http://en.wikipedia.org/wiki/Periodogram. What you're calling the PSD is in fact a periodogram. What the Wikipedia page I cite above explains is that the periodogram is in general not a good estimator for the PSD, even though, from http://en.wikipedia.org/wiki/Spectral_density, "The ensemble average of the average periodogram when the averaging time interval $T\rightarrow\infty$ can be proved (Brown & Hwang) to approach the Power Spectral Density (PSD)."

So, you're asking about the properties of short time period Periodograms, not about the properties of the PSD. PSD is defined as an integral over infinite time. You're probably also talking about a discrete series, not an integral. The Wikipedia Periodogram page also has this to say,

The spectral 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 gradually 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.

I'm not a specialist in signal processing (I attempted to add a signal-processing tag, which with luck might lead someone here who can answer more economically), so this is not a complete Answer to your detailed Question. The word "Periodogram" will lead to more information and towards working out the answer yourself.

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Great answer. It's too bad physicists seem so little interested in rigorous time series analysis. We should all know multi-taper methods and in particular the book: Spectral analysis for physical applications (SAPA) by Percival and Walden. faculty.washington.edu/dbp/sapabook.html – user27777 Aug 14 '13 at 5:49