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How does a scope calculate the average+ when you are in AVE mode?

Question :

How does a scope calculate the average+ when you are in AVE mode?

Answer :

General Functionality

The scope calculates a rolling average, using two different techniques (as detailed below). In other words, if you have the oscilloscope set to average four acquisitions, then it will average each new acquisition into the total average displayed on the screen, as it is acquired.

When it gets the fifth acquisition, it discards the average information created by the first acquisition, and adds the fifth to the total average. When you get the sixth acquisition, it discards the second and adds the sixth, and so on.

This can be demonstrated with a slow signal. The first acquisition will display, and then each successive trigger will be averaged in with that first acquisition. If your signal is stable, you will see this as a successive reduction of noise. When you have reached the limit of averages (in this case, four) then the signal will still change, but there will no longer be an improvement in noise reduction.

Averaging Algorithm

Oscilloscopes use a combination of stable averaging for the first n acquisitions then exponential averaging from then on. Both algorithms readily display the effect of consistent trends in the waveform. The stable averaging algorithm is:

            An(i) = An-1(i) + [Xn(i) - An-1(i)]/n

Where n is the current acquisition number as long as it is less than N, the number of averages selected. The magic thing about the stable averaging algorithm is that it is mathematically equivalent to a running summation average. To demonstrate this, here is the mathematical derivation dropping the (i) data point index for clarity:

(1)        An = An-1 + [Xn - An-1]/n

(2)        An = (An-1 * n)/n - (An-1/n )+(Xn /n)

(3)        An = {[(An-1 * n) - (An-1)]/n } +(Xn /n)

(4)        An = An-1 (n-1)/n + Xn /n

Here is a simple example of how this works. After the first acquisition in a series of three acquisitions to be averaged we have this:

            A1 = X1

After the second acquisition we have:

            A2 = (X1 + X2)/2

After the third acquisition we have:

            A3 = (X1 + X2 + X3)/3

Which is equivalent to:

            A3 = (A2 * 2/3) + X3/3

Compare that to formula (4) above. Each nth acquisition has a 1/n affect on the running average.

When n reaches N, the oscilloscope changes to exponential averaging. Only the divisor term changes:

            An(i) = An-1(i) + [Xn(i) - An-1(i)]/N

Note that the algorithm written in this form seems to emphasize changes, constantly taking differences and adding them to the average. After n reaches N, each difference has a 1/N affect on the average. To get a summation average, simply put the oscilloscope into Single Sequence mode. In this mode, when n reaches N, acquisitions stop.

You can see the averaging algorithm in action (for a single number) in the attached Excel file.

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FAQ ID 58316

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