Formulation of Steady State intermediate buffer values

The basic idea behind this is that as the initial data is input to the filtering function, the assumption is made that the filter is in a Steady State - that is to say, the input and output values of the filter are the same. In the case of the first datum, the output value is assumed to be equal, and so the buffer contents are set such that this is true.

Using the "direct form II" DSP algorithm, the value for the first element B₀ of an intermediate buffer is calculated as follows

Assuming a steady state, the input values x_n and the buffer values B_i will be constant, so we can write

$B=x+\sum _{i=1}^N{a}_{i}B$

Dividing through by B gives

$1=\frac{x}{B}+\sum _{i=1}^N{a}_i$

And rearranging gives B in terms of the input value x, and the output coefficients a_i

$B=\frac{x}{\text{1-}\sum _i^N{a}_i}$

Once the first value of x is entered into the filter, this is then used to compute B, which is then used to fill all the buffer variables. Subsequent input values are combined with feedback values derived from the buffer values in the normal way.