Power Spectral Density Function

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Unit 11

• Power Spectral Density Function
• PSD

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PSD Introduction

• A Fourier transform by itself is a poor format
  for representing random vibration because the
  Fourier magnitude depends on the number of
  spectral lines, as shown in previous units
• The power spectral density function, which can be
  calculated from a Fourier transform, overcomes
  this limitation
• Note that the power spectral density function
  represents the magnitude, but it discards the
  phase angle
• The magnitude is typically represented as G2/Hz
• The G is actually GRMS

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Sample PSD Test Specification
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Calculate Final Breakpoint G2/Hz
Number of octaves between two frequencies
Number of octaves from 350 to 2000 Hz 2.51
The level change from 350 to 2000 Hz -3 dB/oct
x 2.51 oct -7.53 dB For G2/Hz
calculations The final breakpoint is (2000
Hz, 0.007 G2/Hz)
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Overall Level Calculation
Note that the PSD specification is in log-log
format. Divide the PSD into segments. The
equation for each segment is The starting
coordinate is (f1, y1)
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Overall Level Calculation (cont)
The exponent n is a real number which represents
the slope. The slope between two coordinates
The area a1 under segment 1 is
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Overall Level Calculation (cont)
There are two cases depending on the exponent n.
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Overall Level Calculation (cont)
Finally, substitute the individual area values in
the summation formula. The overall level L is
the square-root-of-the-sum-of-the-squares.
where m is the total number of segments
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dB Formulas

dB difference between two levels If A B are in
units of G2/Hz,
If C D are in units of G or GRMS,
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dB Formula Examples

• Add 6 dB to a PSD
• The overall GRMS level doubles
• The G2/Hz values quadruple
• Subtract 6 dB from a PSD
• The overall GRMS level decreases by one-half
• The G2/Hz values decrease by one-fourth

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PSD Calculation Methods

• Power spectral density functions may be
  calculated via three methods
• 1. Measuring the RMS value of the amplitude in
  successive frequency bands, where the signal in
  each band has been bandpass filtered
• 2. Taking the Fourier transform of the
  autocorrelation function. This is the
  Wierner-Khintchine approach.
• 3. Taking the limit of the Fourier transform
  X(f) times its complex conjugate divided by its
  period T as the period approaches infinity.

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PSD Calculation Method 3

• The textbook double-sided power spectral density
  function XPSD(f) is
• The Fourier transform X(f)
• has a dimension of amplitude-time
• is double-sided

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PSD Calculation Method 3, Alternate

• Let be the one-sided power spectral density
  function.
• The Fourier transform G(f)
• has a dimension of amplitude
• is one-sided
• ( must also convert from peak
  to rms by dividing by ?2 )

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Recall Sampling Formula

• The total period of the signal is
• T N?t
• where
• N is number of samples in the time function and
  in the Fourier transform
• T is the record length of the time function
• ?t is the time sample separation

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More Sampling Formulas

• Consider a sine wave with a frequency such that
  one period is equal to the record length.
• This frequency is thus the smallest sine wave
  frequency which can be resolved. This frequency
  ?f is the inverse of the record length.
• ?f 1/T
• This frequency is also the frequency increment
  for the Fourier transform.
• The ?f value is fixed for Fourier transform
  calculations.
• A wider ?f may be used for PSD calculations,
  however, by dividing the data into shorter
  segments

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Statistical Degrees of Freedom

• The ?f value is linked to the number of degrees
  of freedom
• The reliability of the power spectral density
  data is proportional to the degrees of freedom
• The greater the ?f, the greater the reliability

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Statistical Degrees of Freedom (Continued)

• The statistical degree of freedom parameter is
  defined follows
• dof 2BT
• where
• dof is the number of statistical degrees of
  freedom
• B is the bandwidth of an ideal rectangular filter
  
• Note that the bandwidth B equals ?f, assuming an
  ideal rectangular filter
• The BT product is unity, which is equal to 2
  statistical degrees of freedom from the
  definition in equation

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Trade-offs

• Again, a given time history has 2 statistical
  degrees of freedom
• The breakthrough is that a given time history
  record can be subdivided into small records, each
  yielding 2 degrees of freedom
• The total degrees of freedom value is then equal
  to twice the number of individual records
• The penalty, however, is that the frequency
  resolution widens as the record is subdivided
• Narrow peaks could thus become smeared as the
  resolution is widened

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Summary

• Break time history into individual segment to
  increase degrees-of-freedom
• Apply Hanning Window to individual time segments
  to prevent leakage error
• But Hanning Window has trade-off of reducing
  degrees-of-freedom because it removes data
• Thus, overlap segments
• Nearly 90 of the degrees-of-freedom are
  recovered with a 50 overlap

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Original Sequence
Segments, Hanning Window, Non-overlapped
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Original Sequence
Segments, Hanning Window, 50 Overlap