ME'scopeVES Application Note #10 - Averaging Fourier and Auto Spectra

App Note #10

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2/25/2014

ME’scope Application Note #10 Averaging Fourier and Auto Spectra be reduced or nearly eliminated, and the intended (desired) signal will be preserved in the final result.

INTRODUCTION In ME’scope, both Fourier spectra and Auto spectra can be calculated from time domain signals. Both of these calculations can include spectrum averaging, which includes choices of time domain window (Hanning, Flat Top, and Rectangular), number of averages, Peak Hold or Stable averaging, and Percent Overlap.

Spectrum averaging is normally done to reduce or remove the effects of random noise and non-linear effects from spectral estimates. It will be shown in this Application Note that noise is removed from a Fourier spectrum differently than from an Auto spectrum.

Spectrum averaging involves the computation of the Discrete Fourier Transform (DFT), also called the Fourier spectrum. The DFT is computed with the FFT algorithm, and is a linear transformation.

When the FFT is applied to a vibration signal, the linear part (typically due to resonant or order-related vibration) will be transformed to the same frequencies in the spectrum.

The non-linear part is transformed into random noise, which is spread throughout the Fourier spectrum. This noise can be averaged out of either spectrum by summing together multiple Fourier or Auto spectrum estimates.

Throughout this Application Note, the complex valued Fourier spectrum will be expressed as having two parts, the measured signal and the additive random noise. The measured signal term, (A+jB), is assumed to remain constant for each average. The noise term, (Ci+jDi), is assumed to change from one spectral estimate to the next. The ith estimate of the Fourier spectrum X i ( ) can then be written as,

X i   A   jB  C i   jD i  AVERAGE FOURIER SPECTRUM

An Average Fourier spectrum is computed by averaging together several Fourier spectral estimates. The Average Fourier spectrum is computed by averaging the real and imaginary terms of the Fourier Spectra. The average Fourier spectrum can be written as follows,

AveX  

FOURIER SPECTRUM A Fourier spectrum is computed by transforming a block (or sampling window) of uniformly sampled time domain samples using the FFT algorithm. In ME’scope, this is done by executing the Transform | FFT command in a Data Block window containing one or more time domain Traces.

X  FFTxt 

(1)

AVERAGING SPECTRAL ESTIMATES With experimentally acquired data, it is always assumed that the time domain signal and its Fourier spectrum are made up of the sum of two parts, the intended signal plus additive random noise. When multiple blocks of the same signal are acquired, each block is assumed to contain the same intended signal plus different additive random noise.

NOTE: The random noise is assumed to have a Gaussian or Normal distribution of magnitudes versus samples.

Therefore, when a number of Fourier spectral estimates of the same signal are summed together, the random noise will

(2)

N

1 N

 X  i 1

i

N

1 N

 A  jB  C   jD  i

i 1

 A   jB 

i

N

1 N

 C   jD  i 1

i

i

(3) This average is the result of stable averaging in ME’scope.

AUTO SPECTRUM The Auto spectrum is computed by multiplying the Fourier spectrum by its complex conjugate. For multiple estimates, the resulting Average Auto spectrum is the sum of the multiple Auto spectrum estimates, each one resulting for a different block (or sampling window) of time domain data.

NOTE: The Auto spectrum has a magnitude only.

Therefore, the average of multiple Auto spectrum estimates is merely the average of multiple magnitudes, one from each Auto spectrum estimate.

AVERAGE AUTO SPECTRUM In equation (4) below, the average Auto spectrum is expressed as a function of multiple Fourier spectral estimates,

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App Note #10

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using stable averaging. Note that the last term of equation (4) is the average magnitude of the noise. Since this noise

2/25/2014

power term is always positive, it cannot be removed for the Auto spectrum by averaging multiple estimates.

N

X APS  

1 N

 X X 



1 N

 A  jB  C   jD A  jB  C   jD 

i 1

* i

i

N

i

i 1

 A   B   2

2

 A 2   B 2  

i

i

i

(4)

 2AC   2BD   C   D  N

1 N

i 1

i

i

2 i

2 i

2A N 2B N 1 N 2     C   D   C i   D i2     i i N i 1 N i 1 N i 1



REMOVAL OF NOISE



APS with Noise Removed

Since the noise is assumed to be Gaussian random noise, some real part terms (Ci) will be positive and some negative.

With sufficient averaging, the average of the Ci values will tend towards zero. The same is true for the imaginary components (Di). This is a characteristic of Gaussian random noise.

Removing the zero noise terms from Equation (4) gives,

X APS   A 2   B 2  





1 N 2  C i   D i2  N i 1 (8)

To summarize, the average magnitude of the noise does not sum to zero, but the average of the real & imaginary parts of the noise will sum to zero.

Equation (8) shows that the noise cannot be removed from an averaged Auto spectrum estimate, no matter how many estimates are averaged together.

  N   N lim  N1  C i   0, lim  N1  C i2   0 N  N    i 1   i 1 N N     lim  1  D i   0, lim  N1  D i2   0 N N N    i 1   i 1

Average Auto Spectrum from Fourier Spectrum (6)

Fourier Spectrum with Noise Removed Removing all of the zero noise terms from Equation (3) gives,

X LS   A  jB

(7)

Equation (7) shows that averaging Fourier spectrum estimates completely removes noise from the final result if enough estimates are used.

The Average Auto spectrum can also be computed from the Average Fourier spectrum by multiplying the Average Fourier spectrum by its complex conjugate. This result is shown in Equation (5).

The random noise sums into Equation (4) differently than it does in Equation (5). Equation (4) contains a term that is the average magnitude of the noise. In Equation (5), the noise term is the square of the average real part of the noise plus the square of the average imaginary part of the noise.

(7) Removing the zero terms from Equation (5) gives,

X APS   A 2   B 2 

(9)

Equation (9) shows that using the Average Fourier spectrum to compute the Auto spectrum yields a noise free estimate of the Auto spectrum, if sufficient averages are taken.

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X APS   X LS X *LS     A  jB  

   C i   jD i  A  jB  N1  C i   jD i  N

1 N

i 1

N



N N N 2A N C i   2B  D i    N1  C i    N1  D i   A   B    N i 1 N i 1   i 1   i 1 2

2

2

CONCLUSION The Auto spectrum can be computed directly from the Fourier spectrum, or it can be computed from an Average Fourier spectrum. The advantage of using the Average Fourier spectrum is improved noise removal.

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(5)



i 1

2