Fixed-Point Concepts

Overview

The Fixed-Point Blockset bridges the gap between designing a dynamic system and implementing it on fixed-point digital hardware. To do this, the blockset provides basic fixed-point Simulink building blocks that are used to design and simulate dynamic systems using fixed-point arithmetic. With the Fixed-Point Blockset, you can:

Use fixed-point arithmetic to develop and simulate fixed-point Simulink models.
Change the fixed-point data type, scaling, rounding mode, or overflow handling mode while the model is simulating. This allows you to explore issues related to numerical overflow, quantization errors, and computational noise.
Generate fixed-point model code ready for execution on a floating-point processor. This allows you to emulate the effects of fixed-point arithmetic in a floating-point rapid prototyping system.
Generate fixed-point model code ready for execution on a fixed-point processor.
Modify or add new fixed-point blocks. Source code is provided for all fixed-point blocks; you will need one of the C compilers supported by the mex utility.

The Fixed-Point Blockset addresses the issues related to using fixed-point single instruction, single data stream processors. Extensions to multiple instruction, multiple data stream processing units can be made. However, hardware consisting of multiple instruction or multiple data streams generally also has floating-point support.

Fixed-point hardware

Digital hardware is becoming the primary means in which control systems and signal processing filters are implemented. Digital hardware can be classified as either off-the-shelf hardware (for example, microcontrollers, microprocessors, general purpose processors, and digital signal processors) or custom hardware. Within these two types of hardware, there are many architecture designs. These designs range from systems with a single instruction, single data stream processing unit to systems with multiple instruction, multiple data stream processing units.

Within digital hardware, numbers are represented as either fixed-point or floating-point data types. For both these data types, word sizes are fixed at a set number of bits. However, the dynamic range of fixed-point values is much less than floating-point values with equivalent word sizes. Therefore, in order to avoid overflow or unreasonable quantization errors, fixed-point values must be scaled. Since floating-point processors can greatly simplify the real-time implementation of a control law or digital filter, and floating-point numbers can effectively approximate real-world numbers, then why use a microcontroller or processor with fixed-point hardware support? The answer to this question in many cases is cost and size:

Cost - Fixed-point hardware is more cost effective where price/cost is an important consideration. When using digital hardware in a product, especially mass-produced products, fixed-point hardware, costing much less than floating-point hardware, can result in significant savings.
Size - The logic circuits of fixed-point hardware are much less complicated than those of floating-point hardware. This means the fixed-point chip size is smaller with less power consumption when compared with floating-point hardware. For example, consider a portable telephone where one of the product design goals is to make it as portable (small and light) as possible. If one of today's high-end floating-point, general purpose processors is used, a large heat sink and battery would also be needed resulting in a costly, large, and heavy portable phone.

After making the decision to use fixed-point hardware, the next step is to choose a method for implementing the dynamic system (for example, control system or digital filter). Floating-point software emulation libraries are generally ruled out because of timing or memory size constraints. Therefore, you are left with fixed-point math where binary integer values are scaled.

The development cycle

The Fixed-Point Blockset provides tools that aid in the development and testing of fixed-point dynamic systems. You directly design dynamic system models in Simulink, which are ready for implementation on fixed-point hardware. The development cycle is illustrated below.

Using MATLAB, Simulink, and the Fixed-Point Blockset, the development cycle follows these steps:

Model the system (plant or signal source) within Simulink using the built-in blocks and double precision numbers. Typically, the model will contain nonlinear elements.
Design and simulate a fixed-point dynamic system (for example, a control system or digital filter) with the Fixed-Point Blockset that meets the design, performance, and other constraints.
Analyze the results and go back to 1 if needed.

When the design requirements have been met, you can use the model as a specification for creating production code using the Real-Time Workshop.

The above steps interact strongly. In steps 1 and 2, there is a significant amount of freedom to select different solutions. Generally, the model is fine-tuned based upon feedback from the results of the current implementation (step 3). There is no specific modeling approach. For example, models may be obtained from first principles such as equations of motion, or from a frequency response such as a sine sweep. There are many controllers that meet the same frequency-domain or time-domain specifications. Additionally, for each controller there are an infinite number of realizations.

The Fixed-Point Blockset helps expedite the design cycle by allowing you to simulate the effects of various fixed-point controller/digital filter structures.

Physical quantities and measurement scales

A measurement of a physical quantity can take many numerical forms. For example, the boiling point of water is 100 degrees Celsius, 212 degrees Fahrenheit, 373 degrees Kelvin, or 671.4 degrees Rankine. No matter what number is given, the physical quantity is exactly the same. The numbers are different because four different scales are used.

Well known standard scales like Celsius are very convenient for the exchange of information. However, there are situations where it makes sense to create and use unique nonstandard scales. These situations usually involve making the most of a limited resource.

For example, nonstandard scales allow map makers to get the maximum detail on a fixed size sheet of paper. A typical road atlas of the USA will show each state on a two-page display. The scale of inches to miles will be unique for most states. By using a large ratio of miles to inches, all of Texas can fit on two pages. Using the same scale for Rhode Island would make poor use of the page. Using a much smaller ratio of miles to inches would allow Rhode Island to be shown with the maximum possible detail.

Fitting measurements of a variable inside an embedded processor is similar to fitting a state map on a piece of paper. The map scale should allow all the boundaries of the state to fit on the page. Similarly, the binary scale for a measurement should allow the maximum and minimum possible values to "fit." The map scale should also make the most of the paper in order to get maximum detail. Similarly, the binary scale for a measurement should make the most of the processor in order to get maximum precision.

Use of standard scales for measurements has definite compatibility advantages. However, there are times when it is worthwhile to break convention and use a unique nonstandard scale. There are also occasions when a mix of uniqueness and compatibility makes sense.

Selecting a Measurement Scale

Suppose that measurements of liquid water are to be made, and suppose that these measurements must be represented using 8-bit unsigned integers. Fortunately, the temperature range of liquid water is limited. No matter what scale is used, liquid water can only go from the freezing point to the boiling point. Therefore, this range of temperatures must be captured using just the 256 possible 8-bit values: 0,1,2,...,255.

One approach to representing the temperatures is to use a standard scale. For example, the units for the integers could be Celsius. Hence, the integers 0 and 100 represent water at the freezing point and at the boiling point, respectively. On the upside, this scale gives a trivial conversion from the integers to degrees Celsius. On the downside, the numbers 101 to 255 are unused. By using this standard scale, more than 60% of the number range has been wasted.

A second approach is to use a nonstandard scale. In this scale, the integers 0 and 255 represent water at the freezing point and at the boiling point, respectively. On the upside, this scale gives maximum precision since there are 254 values between freezing and boiling instead of just 99. The units are roughly 0.3921568 degrees Celsius per bit so the conversion to Celsius requires division by 2.55, which is a relatively expensive operation on most fixed-point processors.

A third approach is to use a "semi-standard" scale. For example, the integers 0 and 200 could represent water at the freezing point and at the boiling point, respectively. The units for this scale are 0.5 degrees Celsius per bit. On the downside, this scale doesn't use the numbers from 201 to 255, which represents a waste of more than 21%. On the upside, this scale permits relatively easy conversion to a standard scale. The conversion to Celsius involves division by 2, which is a very easy shift operation on most processors.

Beyond Multiplication

One of the key operations in converting from one scale to another is multiplication. The preceding case study gave three examples of conversions from a quantized integer value Q to a real-world Celsius value V that involved only multiplication.

\(V = \begin{cases} \dfrac{100^\circ C}{100\ \text{bits}} \cdot Q_1 & \text{Conversion 1} \\[8pt] \dfrac{100^\circ C}{255\ \text{bits}} \cdot Q_2 & \text{Conversion 2} \\[8pt] \dfrac{100^\circ C}{200\ \text{bits}} \cdot Q_3 & \text{Conversion 3} \end{cases}\)

Graphically, the conversion is a line with slope S, which must pass through the origin. A line through the origin is called a purely linear conversion. Restricting yourself to a purely linear conversion can be very wasteful and it is often better to use the general equation of a line.

\(V=SQ+B\)

By adding a bias term \(B\), greater precision can be obtained when quantizing to a limited number of bits.

The general equation of a line gives a very useful conversion to a quantized scale. However, like all quantization methods, the precision is limited and errors can be introduced by the conversion. The general equation of a line with quantization error is given by

\(V=SQ+B \pm Error\)

If the quantized value \(Q\) is rounded to the nearest representable number, then

\(-\dfrac{S}{2} \le Error \le \dfrac{S}{2}\)

That is, the amount of quantization error is determined by both the number of bits and by the scale. This scenario represents the best case error. For other rounding schemes, the error can be twice as large.

Example: Selecting a Measurement Scale

On typical electronically controlled internal combustion engines, the flow of fuel is regulated to obtain the desired ratio of air to fuel in the cylinders just prior to combustion. Therefore, knowledge of the current air flow rate is required. Some manufacturers use sensors that directly measure air flow while other manufacturers calculate air flow from measurements of related signals. The relationship of these variables is derived from the ideal gas equation. The ideal gas equation involves division by air temperature. For proper results, an absolute temperature scale such as Kelvin or Rankine must be used in the equation. However, quantization directly to an absolute temperature scale would cause needlessly large quantization errors.

The temperature of the air flowing into the engine has a limited range. On a typical engine, the radiator is designed to keep the block below the boiling point of the cooling fluid. Let's assume a maximum of 225\(^{\circ}\) F (380\(^{\circ}\) K). As the air flows through the intake manifold, it can be heated up to this maximum temperature. For a cold start in an extreme climate, the temperature can be as low as -60\(^{\circ}\) F (222\(^{\circ}\) K). Therefore, using the Kelvin scale, the range of interest is 222\(^{\circ}\) K to 380\(^{\circ}\) K.

The air temperature needs to be quantized for processing by the embedded control system. Assuming an unrealistic quantization to 3-bit unsigned numbers: 0,1,2,...,7, the purely linear conversion with maximum precision is

\(V=\dfrac{380^{\circ}}{7.5 \hspace{1mm} bits} \cdot Q\)

The quantized conversion and range of interest are shown below.

Notice that there are 7.5 possible quantization values. This is because only half of the first bit corresponds to temperatures (real-world values) greater than zero.

The quantization error is

\(-25.33^{\circ}K/bit \le Error \le 25.33^{\circ}K/bit\)

The range of interest of the quantized conversion and the absolute value of the quantized error are shown below.

As an alternative to the purely linear conversion, consider the general linear conversion with maximum precision.

The quantized conversion and range of interest are shown below.

\(V=\left (\dfrac{380^{\circ}K-222^{\circ}K}{8} \right )\cdot Q+222^{\circ}K+0.5\left (\dfrac{380^{\circ}K-222^{\circ}K}{8} \right )\)

The quantized conversion and range of interest are shown below.

The quantization error is

\(-9.875^{\circ}K/bit \le Error \le 9.875^{\circ}K/bit\)

This error is approximately 2.5 times smaller than the error associated with the purely linear conversion. The range of interest of the quantized conversion and the absolute value of the quantized error are shown below.

Clearly, the general linear scale gives much better precision than the purely linear scale over the range of interest.