Canonical Forms

The Fixed-Point Designer™ software does not attempt to standardize on one particular fixed-point digital filter design method. For example, you can produce a design in continuous time and then obtain an “equivalent” discrete-time digital filter using one of many transformation methods. Alternatively, you can design digital filters directly in discrete time. After you obtain a digital filter, it can be realized for fixed-point hardware using any number of canonical forms. Typical canonical forms are the direct form, series form, and parallel form, each of which is outlined in the sections that follow.

For a given digital filter, the canonical forms describe a set of fundamental operations for the processor. Because there are an infinite number of ways to realize a given digital filter, you must make the best realization on a per-system basis. The canonical forms presented in this chapter optimize the implementation with respect to some factor, such as minimum number of delay elements.

In general, when choosing a realization method, you must take these factors into consideration:

• Cost

  The cost of the realization might rely on minimal code and data size.

• Timing constraints

  Real-time systems must complete their compute cycle within a fixed amount of time. Some realizations might yield faster execution speed on different processors.

• Output signal quality

  The limited range and precision of the binary words used to represent real-world numbers will introduce errors. Some realizations are more sensitive to these errors than others.

The Fixed-Point Designer software allows you to evaluate various digital filter realization methods in a simulation environment. Following the development cycle outlined in Developing and Testing Fixed-Point Systems, you can fine-tune the realizations with the goal of reducing the cost (code and data size) or increasing signal quality. After you have achieved the desired performance, you can use the Simulink^® Coder™ product to generate rapid prototyping C code and evaluate its performance with respect to your system's real-time timing constraints. You can then modify the model based upon feedback from the rapid prototyping system.

The presentation of the various realization structures takes into account that a summing junction is a fundamental operator, thus you may find that the structures presented here look different from those in the fixed-point filter design literature. For each realization form, an example is provided using the transfer function shown here:

$$\begin{array}{c}{H}_{ex}\left(z\right)=\frac{1+2.2z^{-1}+1.85z^{-2}+0.5z^{-3}}{1-0.5z^{-1}+0.84z^{-2}+0.09z^{-3}}\\ =\frac{\left(1+0.5z^{-1}\right)\left(1+1.7z^{-1}+z^{-2}\right)}{\left(1+0.1z^{-1}\right)\left(1-0.6z^{-1}+0.9z^{-2}\right)}\\ =5.5556-\frac{3.4639}{1+0.1z^{-1}}+\frac{-1.0916+3.0086z^{-1}}{1-0.6z^{-1}+0.9z^{-2}}.\end{array}$$