1. Power Spectral Density Function
Unit 11
Power Spectral Density Function
PSD

2. 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

3. Sample PSD Test Specification

4. Calculate Final Breakpoint G^2/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 = dB
For G^2/Hz calculations:
The final breakpoint is: (2000 Hz, G^2/Hz)

5. 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)

6. 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

7. Overall Level Calculation (cont)
There are two cases depending on the exponent n.

8. 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

9. 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,

10. dB Formula Examples Add 6 dB to a PSD The overall GRMS level doubles
The G^2/Hz values quadruple
Subtract 6 dB from a PSD
The overall GRMS level decreases by one-half
The G^2/Hz values decrease by one-fourth

11. 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.

12. 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

13. 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 ) Δf
(f) * G
G(f)
lim
PSD X = ˆ

14. 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

15. 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

16. 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

17. 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

18. 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

19. Example: 4096 samples taken over 16 seconds, rectangular filter.
Example: samples taken over 16 seconds, rectangular filter.
Number of Records NR
Number of Time Samples per Record
Period of Each Record Ti
(sec)
Frequency Resolution Bi=1/Ti
(Hz)
dof per
Record
=2Bi TI
Total dof 1
4096 16
0.0625 2
2048 8
0.125 4
1024
0.25
512
0.5
256 32
128 64

20. 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

21. Original Sequence
Segments,
Hanning Window,
Non-overlapped

22. Original Sequence
Segments,
Hanning Window,
50% Overlap