Description of the problem

  • What is exponential averaging used for?
  • Which exponential decay rate should be used?



Description of the solution

When it comes to monitoring and data inspection data averaging is commonly applied. 

Averaging it sometimes referred to as Smoothing or Time Weighting.  

Main reasons for applying averaging is to:


  • Reduce stochastic noise from the results
  • Smooth transient components out across a longer time interval for them to be easier to detect
  • Trigger only on increased values when the increased values are present over some time.  

 

For example, changes in air pressure due to sounds from e.g. music and noise will typically fluctuate too quickly for sound pressure levels to be readable in real-time. Here averaging comes into play by smoothening the levels to have more damped fluctuations.



Exponential Averaging

There exist many ways to average data. For Sound Level Meters (SLMs) the commonly supported averaging types are those called Fast (F) and Slow (S). These two types must be supported to conform to the IEC 61672-1 standard which is the primary standard used for SLMs. 

An earlier superseded standard IEC 651 also included the definition for the Impulse (I) averaging type, and therefore many SLMs also support that type. 


All the averaging types: Fast (F), Slow (S), and Impulse (I) are exponential averaging types with different rise and decay times.

Exponential averaging can be categorized as a 'moving average' typically implemented to have an input-to-output rate of 1-to-1, which makes it a useful averaging type for monitoring purposes where you might want to detect near real-time events without too much of delay before you get new output data to inspect.  


Exponential averaging reduces the influence of values at given points in time more and more as times goes on. The values at given time points will decrease in influence in an exponential decaying way such that values from long time ago have small influence and values from around present time have great influence on the averaged result. 



Exponential time constants

An exponential decaying function can be defined as below:



An exponential rising function can can be defined as below:  

Tau is the decay or rise time constant, that describes how much the data will be smoothed. The greater the tau value the more the data will be smoothed out across time.  


  • Fast (F) has a rise and decay time of 125 ms.  
  • Slow (S) has a rise and decay time of 1 second.
  • Impulse (I) is asymmetric and has relatively quick rise time of 35 ms and a relatively slow decay time of 1.5 seconds.



Exponential energy averaging

Normally when using tau equal to 1 sec  then that relates to having a decay rate of around 8.7 dB/s, but for Sound Pressure Levels (SPLs) it is only the half (4.34 dB/s). 

This is due to SPLs are based on averaged values in the 2nd order (squared) energy domain and therefore exponential averaging is also applied on squared sound pressure values (proportional to energy). 

Exponential energy averaging effectively gives a tau value equal to 2 times tau for the final averaged sound pressure values represented in the 1st order (linear) domain. 


The exponential averaged SPL can be formulated like in [1] as:

From the formula above we see that the exponential decay is applied to the squared sound pressure values. 


More information about how to calculate Sound Pressure Levels (SPLs) can be found here:  Sound Pressure Levels and Decibels.



Decay Rates

The decay rate is typically represented with the unit [dB/s], but is can also be described as [%/s] which expresses the percentage of decreased influence for values at specific times per second. 

As mentioned above exponential energy averaging on linear input values will result in a doubled tau time for linear output values:

If we want to express the rate per second then the time t is set equal to 1 second. 

Hereby the percentage rate of decay per second can be described by:

In the same manner the decibel rate of decay per second can be described by:

For example, for Exponential Slow (S) averaging the rate is -4.34 dB/s and -39. %/s, for output values in the linear domain.



References

  1. Fundamentals of Acoustics and Noise Control, by Finn Jacobsen, Torben Poulsen, Jens Holger Rindel, Anders Christian Gade and Mogens Ohlrich, DTU Denmark, dept. of Electrical Engineering.

  2. Wikipedia - Exponential smoothing: https://en.wikipedia.org/wiki/Exponential_smoothing.



Additional information

Sound levels, Sound intensity, Sound quality and Sound power manual