Operation of digital filters

In this section, we will develop the basic theory of the operation of digital filters. This is essential to an
understanding of how digital filters are designed and used.
Suppose the "raw" signal which is to be digitally filtered is in the form of a voltage waveform described by the
function
V x t = ( )
where t is time.
This signal is sampled at time intervals h (the sampling interval). The sampled value at time t = ih is
x x ih i = ( )
Thus the digital values transferred from the ADC to the processor can be represented by the sequence
x , x , x , x , ... 0 1 2 3
corresponding to the values of the signal waveform at
t = 0, h, 2h, 3h, ...
and t = 0 is the instant at which sampling begins.
At time t = nh (where n is some positive integer), the values available to the processor, stored in memory, are
x , x , x , x , ... x 0 1 2 3 n
Note that the sampled values xn+1, xn+2 etc. are not available, as they haven't happened yet!
The digital output from the processor to the DAC consists of the sequence of values
y , y , y , y , ... y 0 1 2 3 n
In general, the value of yn is calculated from the values x0, x1, x2, x3, ... , xn. The way in which the y's are
calculated from the x's determines the filtering action of the digital filter.
In the next section, we will look at some examples of simple digital filters.