# Exponential Smoothing

A statistical technique commonly used to forecast time series data or to smooth the values on a control chart. A forecast function is estimated from previous data using a weighted least squares technique. The degree to which data in the far past is weighted relative to the near past is governed by the value of one or more smoothing constants, which must be between 0 and 1. In general, the smaller the smoothing constant, the more weight is given to the far past.

The mathematical function that describes exponential averaging is as follows:

where 0 < α < 1.

In this manner the result, f(x) is a function of the previously calculated value of f(x) and the current value of x.

Consider a signal, containing N samples, which we wish to average over. The average at sample N calculated over the previous N samples is

The addition of one more sample changes the definition of the average to:

substituting for

we have

Setting

gives

Thus, in order to calculate the levels all we require to know is the number of samples to average over, N. To calculate the overall level we know the standard time constants therefore by knowing the sample rate we can calculate N = T * sample rate.

Thus we have alpha:

Traditional methods of measuring sound pressure level utilised analogue electronic devices which used and RC filtering circuit to smooth the signals over time and return an exponentially averaged measure of the level. Such devices generally had two or three settings; Fast, Slow and Impulse. These corresponded to different RC time constants; 125ms, 1000ms and 35ms.

Computers are now used to apply an exponential weighting average to the measured data in order to return values equivalent to the traditional analogue devices.

See also: Smoothing.

Subjects: Algebra Noise & Vibration