Mean Square Error

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AgendaMSE DefinitionExpected valueThe Error HistogramGiving bigger differences more weight than smaller differencesCalculating MSEMSE in digital modulationCommissioning with MSEMSE and ACM*

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MSE - DefinitionMSE is used to quantify the difference between an estimated (expected) value and the true value of the quantity being estimated

MSE measures the average of the squared errors:

MSE is a sort of aggregated error by which the expected value differs from the quantity to be estimated.

The difference occurs because of randomness or because the receiver does not account for information that could produce a more accurate estimated RSL*

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To simplify.Imagine a production line where a machine needs to insert one part into the other

Both devices must perfectly match

Let us assume the width has to be 10cm wide

We took a few of parts and measured them to see how many can fit in.

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The Errors Histogram (Gaussian probability distribution function)To evaluate how accurate our machine is, we need to know how many parts differ from the expected value

9 parts were perfectly OK*

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The difference from Expected valueTo evaluate the inaccuracy (how sever the situation is) we measure how much the errors differ from expected value*

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Giving bigger differences more weight than smaller differencesWe convert all errors to absolute values and then we square them

The squared values give bigger differences more weight than smaller differences, resulting in a more powerful statistics tool:

16cm parts are 36 units away than 2cm parts which are only 4 units away*

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Calculating MSETo evaluate the total errors, we sum all the squared errors and take the average:

16 + 9 + 0 + 4 + 36 = 65, Average (MSE) = 13

The bigger the errors (differences) >> the bigger MSE becomes*

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Calculating MSEIf all parts were perfectly produced than each error would be 0

This would result in MSE = 0

Conclusion: systems perform best when MSE is minimum*

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MSE in digital modulation (Radios)Let us use QPSK (4QAM) as an example:

QPSK = 2 bits per symbol

2 possible states for I signal2 possible states for Q signal

= 4 possible states for the combined signal

The graph shows the expected values (constellation) of the received signal (RSL)*

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MSE in digital modulation (Radios)The black dots represent the expected values (constellation) of the received signal (RSL)

The blue dots represent the actual RSL

Similarly to the previous example, we can say that the bigger the errors are the harder it becomes for the receiver to detect & recover the transmitted signal*

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MSE in digital modulation (Radios)MSE would be the average errors of e1 + e2 + e3 + e4.

When MSE is very small the actual signal is very close to the expected signal

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MSE in digital modulation (Radios)When MSE is too big, the actual signal (amplitude & phase) is too far from the expected signal

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Commissioning with MSE in EMSWhen you commission your radio link, make sure your MSE is small (-37dB)

Actual values may be read -34dB to -35dB

Bigger values (-18dB) will result in loss of signal*

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MSE and ACMWhen the errors become too big, we need a stronger error correction mechanism (FEC)

Therefore, we reduce the number of bits per symbol allocated for data and assign the extra bits for correction instead

For example 256QAM has great capacity but poor immune to noise

64QAM has less capacity but much better immune for noiseACM Adaptive Code Modulation*

*Thank You !training@ceragon.com

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