Amplitude And Energy Correction – A Brief Summary

By
John Mathey
– September 1, 2009Posted in: signal processing

Amplitude and energy correction has been and is a continuing point of confusion for many people calculating spectra from time domain signals using Fourier transform methods. The first thing to say, the information contained in data presented as amplitude and energy corrected spectra is equivalent. The only difference is the scaling of the numbers calculated.

Amplitude Corrected Spectrum

An Amplitude Corrected spectra is typically the default numbers you will find if you use a stand alone analyzer. Each frequency line of the spectrum is the RMS value of each frequency component of the time signal. If you have a 1Volt RMS sinusoid (as in Figure 1) and measure it with a FFT spectrum analyzer, the height of the line, or combination of lines that represent the signal will always add up to 1Volt RMS.

Figure 1: One Volt RMS sinusoid

If the user does not apply any windowing to the time signal prior to calculating the FFT of the sinusoid and the frequency of the sinusoid is centered on the center of the FFT line of the spectrum, then the height of the single line will be 1 Volt RMS.

Figure 2: Spectrum of 1 Volt RMS signal

Some confusion is added in when the user applies a time windowing function (typically a Hanning window) to the time signal prior to calculating the spectrum. Because the applied window narrows the time record (remember the time record is 1/\Delta f the frequency lines are broadened and overlap, consequently the single frequency sinusoid is now represented by 3 frequency lines (typically if the sinusoid is again centered on the center frequency of the specific frequency line of the calculated spectrum.

Figure 3: Spectrum of 1 Volt RMS signal (with windowing)

This 1Volt RMS sinusoid hasn’t changed. Remembering, RMS=\sqrt{Power}  the sum of the power of the 3 lines of the spectrum must be equivalent to 1 V2 (1 Vrms). It is obvious the center line of the calculated spectrum shows the 1 Volt RMS value so what are the other 2 lines at 0. 5 V RMS? The key here is the power associated with each of the spectral lines. To calculate the power represented by the spectra, one must square the values of each of the RMS lines

(0.5)2 + (1)2 + (0.5)2 = 1.5 V2

But because we applied a Hanning window, the effective noise bandwidth for this spectrum is 1.5, we must divide the summation by this factor and then take the square root. The result is 1 Volt RMS as expected.

Remember, in general, calculating the Power of a spectra between frequency 1 (L_f) and frequency 2 (U_f) is the summation of the power in all frequency lines between frequency 1 and frequency 2 or:

    \[ Power = \sum_{L_f}^{U_f}{Power_i} \]

where

    \[ Power = \sum_{L_f}^{U_f}{(RMS_i)^2} \]

or

    \[ RMS=\sqrt{\sum_{L_f}^{U_f}{(RMS_i)^2}} \]

If the spectrum is amplitude corrected as described above, the summation of the powers (RMS2) is actually the summation of the powers in each line divided by the effective noise bandwith of the measurement used or:

    \[ Power = \frac{\sum_{L_f}^{U_f}{Power_i}}{1.5} \]

or

    \[ RMS = \sqrt{\frac{\sum_{L_f}^{U_f}{{(RMS_{amplitude})_i}^2}}{ENBW}} \]

The effective noise bandwidth (ENBW) number is the product of the frequency line spacing (\Delta f) and the effective bandwidth of the windowing function used (e.g. Hanning Window = 1.5). This effective noise bandwidth is included as a named element in all spectrum calculations performed with the DATS software.

Energy Corrected Spectrum

Amplitude and Energy correction of spectra is just a scaling factor. This scaling factor is the effective noise bandwidth of the analysis used, or in the case of a Hanning Windowing function, ENBW = 1.5*\delta f.

If the spectrum is energy corrected for the same 1 Vrms sinusoid signal the scaling will look as follows:

Figure 4: Energy corrected spectrum

Adding the powers in each line again requires squaring the RMS values, adding, then taking the square root.

    \[ \sqrt{(0.408187)^2 + (0.816373)^2 + (0.408187)^2} = 1 V_{rms} \]

 

As observed, the factor of the \sqrt{ENBW}  in the Amplitude corrected spectra calculations is already factored into the scaling of the Energy corrected spectrum, consequently, when calculating the rms level, one does not need to divide by the \sqrt{ENBW}  factor.

Now to calculate the overall levels between frequency 1 (L_f) and frequency 2 (U_f) one needs only to take the root sum squares of the RMS levels of each individual frequency line or:

    \[ Power = \sum_{L_f}^{U_f}{Power_i} \,\,or\,\, RMS = \sqrt{\sum_{L_f}^{U_f}{(RMS_{energy})_i^2}} \]

Converting Amplitude to Energy Correction Scaling

Equating the formulas for calculating the RMS gives:

    \[ RMS = \sqrt{\sum{{(RMS_{energy})_i}^2}} = \sqrt{\frac{\sum{{(RMS_{amplitude})_i}^2}}{ENBW}} \]

 

The special case encountered is when the frequency line spacing ({\Delta}f) = 1 Hz making ENBW = 1.5 Hz. This means the difference between scaling of the amplitude and energy corrected spectrum is \frac{1}{\sqrt{1.5}}  or 0.816 .

Print Friendly
Tags: , , , , , ,

About John Mathey

John Mathey graduated with a MS degree from the University of Toledo in 1972. John has over 35 years of experience with instrumentation, measurement, and analysis. Twenty-five of those years were spent at Ford Motor Company solving and providing training for vehicle noise, vibration, and harshness (NVH) issues. He is now a technical specialist at Prosig USA, Inc. where he provides technical support to Prosig customers in the U.S.A.