1.1 Fir filter structures

Consider causal FIR filters: $y(n)=\sum_{k=0}^{M-1} h(k)x(n-k)$ ; this can be realized using the following structure

There are no closed loops (no feedback) in this structure, so it is called a non-recursive structure . Since any FIR filter can be implemented using the direct-form, non-recursivestructure, it is always possible to implement an FIR filter non-recursively. However, it is also possible to implement anFIR filter recursively , and for some special sets of FIR filter coefficients this is much moreefficient.

IIR filters must be implemented with a recursive structure, since that's the only way a finite number of elements can generate aninfinite-length impulse response in a linear, time-invariant (LTI) system. Recursive structures have the advantages of beingable to implement IIR systems, and sometimes greater computational efficiency, but the disadvantages ofpossible instability, limit cycles, and other deletorious effects that we will study shortly.

Transpose-form fir filter structures

The flow-graph-reversal theorem says that if one changes the directions of all the arrows, and inputs at theoutput and takes the output from the input of a reversed flow-graph, the new system has an identical input-outputrelationship to the original flow-graph.

Cascade structures

The z-transform of an FIR filter can be factored into a cascade of short-length filters $$b_0+b_{1}z^{(-1)}+b_{2}z^{-3}++b_{m}z^{-m}=b_0(1-z_{1}z^{(-1)})(1-z_{2}z^{(-1)})(1-z_{m}z^{(-1)})$$ where the $z_i$ are the zeros of this polynomial. Since the coefficients of the polynomial are usually real, the rootsare usually complex-conjugate pairs, so we generally combine $(1-z_{i}z^{(-1)})(1-\overline{z_i}z^{(-1)})$ into one quadratic (length-2) section with real coefficients $$(1-z_{i}z^{(-1)})(1-\overline{z_i}z^{(-1)})=1-2\Re (z_i)z^{(-1)}+\left|z_i\right|^{2}z^{-2}=H_i(z)$$ The overall filter can then be implemented in a cascade structure.

Lattice structure

It is also possible to implement FIR filters in a lattice structure: this is sometimes used in adaptive filtering