Power Spectral Density Function Unit 11 Power Spectral Density Function PSD

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

Sample PSD Test Specification

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 = -7.53 dB For G^2/Hz calculations: The final breakpoint is: (2000 Hz, 0.007 G^2/Hz)

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)

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

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

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

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,

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

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.

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

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

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

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

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

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

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

Example: 4096 samples taken over 16 seconds, rectangular filter.   Example: 4096 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

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

Original Sequence Segments, Hanning Window, Non-overlapped

Original Sequence Segments, Hanning Window, 50% Overlap