Amplitude And Energy Correction - A Brief Summary

In this article we will look at why we need to consider energy correction when producing frequency spectra and how we go about it. We will use a perfect, ’special case’ signal to keep the explanation as simple as possible. The signal we will use is periodic within the time record used to calculate the FFT. Specifically, we will look at 1 second of 10Hz sinusoid with an amplitude of 1V_rms as seen in Figure 1.

Figure 1: 10Hz sinusoid

This time signal will produce a single line spectrum at frequency of the time signal and with indicated amplitude agreeing with rms level of signal being transformed to frequency domain (Figure 2).

Figure 2: Single line spectrum

Amplitude corrected spectrum

Now we will apply a Hanning windowing function to the time record prior to calculating the FFT spectrum. This yields the spectrum shown in Figure 3. We know the energy in the frequency representation of the signal has to be the same as the energy in the time signal so… How do we calculate the energy in the frequency spectrum?

Figure 3: Amplitude scaled spectrum

You might expect that you add the energy in each one of the frequency lines, however this does not work.

The signal rms between two specified frequencies is given by:

which in this case is

This is obviously not correct. Why? You need to factor in the scaling used by the window function. This factor is the Equivalent Noise Bandwidth (ENBW) for the window function that was used. In the case of the Hanning window, this factor is 1.5 x delta-f and this must be factored into the calculations. So, if we divide our result by the ENBW value we obtain the correct result of 1V_rms.

Energy corrected spectrum

Figure 4 shows the spectrum of the same sinusoid, but this time it is displayed with ‘energy corrected’ scaling.

Figure 4: Energy corrected spectrum

If we apply the same formula for the signal rms between specified frequencies

we get

which gives us our 1V_rms as expected.

Converting between amplitude and energy corrected spectra

=

ENBW is calculated from delta-f x window factor. Some examples of window factor are:

Hanning Window = 1.5
Flat Top Window = 3.8
Rectangular Window (no time window) = 1.0

The only difference between Amplitude and Energy Corrected spectrum is the scaling. The above equations show the actual mathematical relationship between the 2 spectrum representations.

This article was originally created by John Mathey for a presentation to a North American Prosig User group (PUG).