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Sampling does not eliminate the noise from the original analog signal; hence digital signals should be filtered. Digital filters are mathematical procedures that process digital signals on-line to reduce their noise and represent better the true dynamics of the measured variable.

In practice, for filtering purposes a high frequency sampling rate is used, known as "time scan". The simplest way to reduce high frequency noise is averaging. If we consider N measurements between two time instants, the average value, yN, will be defined by,

~n =—Z y, * * /-i where yi represents the noisy value measured at time i.

It is easy to implement this formula in a computer using a recursive version,

Here, the correction gain (1/i) is much lower for the most recent measurements; hence these will not influence much the average value. This formula is effective if the true signal is constant, however, in practice it is better to use a constant correction gain if the process variables are continuously changing, a situation that is especially true in the batch processes that are typically used in SSF.

The gain inverse, a, corresponds approximately to the number of measurements used in the averaging. This filter is also known as a "first-order low-pass filter" or "moving average filter". Obviously, the larger the value of a, the smoother the filtered signal will be. Care must be taken though, since too much filtering can hide the true process dynamics.

Averaging is also a good policy if we are taking samples for off-line analysis. Here, it is convenient to take samples from different places inside the bed, and then mix them all and do the analysis, or analyze each sample independently and then take the average. The advantage of the latter is that we can have an estimation of the heterogeneity of the bed also.

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