Operation of digital filters

n the next few sections, 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. First of all, we need to introduce a basic notation.

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_i = x (ih)

Thus the digital values transferred from the ADC to the processor can be represented by the sequence

x₀, x₁, x₂, x₃, ...

corresponding to the values of the signal waveform at times t = 0, h, 2h, 3h, ... (where 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

x₀, x₁, x₂, x₃, ... , x_n

Note that the sampled values x_n+1, x_n+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

y₀, y₁, y₂, y₃, ... , y_n

In general, the value of y_n is calculated from the values x₀, x₁, x₂, x₃, ... , x_n. The way in which the y's are calculated from the x's determines the filtering action of the digital filter.