IIR background

Summary

Where does the in the denominator come from?
Where is $a_{0}$ ?
What's the story behind the minus sign in figure 1?

Sign of the recursive branch

Figure 1 shows the structure of an IIR filter.

Figure 1: IIR filter

Note the negative sign in the recursive branch.
Its frequency response is:

$H(z)=\frac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\ldots+b_{m}z^{-m}}{1+a_{1}z^{-1}+a_{2}z^{-2}+\ldots+a_{m}z^{-m}}$

This convention is for example used by Matlab (Octave) functions, when dealing with filter coefficients.

Many other references do not use the minus sign:

Figure 2: IIR filter, without the minus sign

In this case, the coefficients need to be redefined:

$H(z)=\frac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\ldots+b_{m}z^{-m}}{1-\hat{a}_{1}z^{-1}-\hat{a}_{2}z^{-2}-\ldots-\hat{a}_{m}z^{-m}}$
If the “total” minus sign of the recursive branch is omitted, it needs to be factored into the denominator coefficients: $\hat{a}_{k}=-a_{k}$ .

Background

The connection between the frequency response and the filter structure is not as straightforward as it seems:
By definition, the lowest coefficient in the recursive branch (denominator) is $a_{1}$ .
The reason is that an undelayed branch weighted with $a_{0}$ would simply give a constant factor of $\frac{1}{1 \pm a_{0}}$ in the frequency response, and could be factored into the numerator coefficients.

Instead, the

in the denominator comes from the well-known feedback loop equation:

$G_{loop}(f)=\frac{1}{1+G(f)}$

Figure 3: Frequency response of a feedback loop

Here it is shown for

, but it works just as well for

instead.

Note that the recursive branch of an IIR filter can be considered a FIR filter (lacking the direct branch $a_{0}$ ) in a feedback loop!
A negative sign in the feedback loop gives a positive sign for

in the denominator.
The frequency response of the recursion branch is obtained by replacing

with

(where $a_{1}$ is the lowest-order coefficient).
Inserting it into above equation and multiplying with the forward branch (

coefficients) gives

© Markus Nentwig 2007-2008
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