Realization Structures
- Dave Dee
The Fixed-Point Blockset addresses the needs of the control system and signal processing fields, and other fields where algorithms are implemented on fixed-point hardware. In signal processing, a digital filter is a computational algorithm that converts a sequence of input numbers to a sequence of output numbers. The algorithm is designed such that the output signal meets frequency-domain or time-domain constraints (desirable frequency components are passed, undesirable components are rejected). In general terms, a discrete transfer function controller is a form of a digital filter. However, a digital controller may contain nonlinear functions such as look-up tables in addition to a discrete transfer function. In this guide, the term "digital filter" is used when referring to discrete transfer functions.
The Fixed-Point Blockset does not attempt to standardize on one particular fixed-point digital filter design method. For example, a design can be done in continuous time and an "equivalent" discrete-time digital filter can be obtained using one of many transformation methods. Alternatively, digital filters can be directly designed in discrete time. After the digital filter is obtained, 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, all of which are outlined in this chapter.
For a given digital filter, the canonical forms describe a set of fundamental operations for the processor. Since there are an infinite number of ways to realize a given digital filter, the best realization must be made 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 Blockset allows you to evaluate various digital filter realization methods in a simulation environment. Following the development cycle outlined in The Development Cycle in Chapter 1, you can fine tune the realizations with the goal of reducing the cost (code and data size) or increasing signal quality. After the desired performance has been achieved, you can use the Real-Time Workshop 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 below.
