Range and Precision of a Fixed Point number (Need of Fixed Point Numbers)

Range and Precision

The range of a number gives the limits of the representation, while the precision gives the distance between successive numbers in the representation. The range and precision of a fixed-point number depend on the length of the word and the scaling.


Range

The following figure illustrates the range of representable numbers for an unsigned fixed-point number of size ws, scaling S, and bias B.
The following figure illustrates the range of representable numbers for a two's complement fixed-point number of size ws, scaling S, and bias B where the values of ws, scaling S, and bias B allow for both negative and positive numbers.
For both the signed and unsigned fixed-point numbers of any data type, the number of different bit patterns is 2ws.
For example, if the fixed-point data type is an integer with scaling defined as S=1 and B = 0, then the maximum unsigned value is 2ws−1, because zero must be represented. In two's complement, negative numbers must be represented as well as zero, so the maximum value is 2ws−1−1. Additionally, since there is only one representation for zero, there must be an unequal number of positive and negative numbers. This means there is a representation for −2ws−1 but not for 2ws−1.

Precision

The precision of a data type is given by the slope. In this usage, precision means the difference between neighboring representable values.



The precision of a fixed-point word depends on the word size and binary point location. Extending the precision of a word can always be accomplished with more bits, but you face practical limitations with this approach. Instead, you must carefully select the data type, word size, and scaling such that numbers are accurately represented. Rounding and padding with trailing zeros are typical methods implemented on processors to deal with the precision of binary words.


https://stackoverflow.com/questions/10067510/fixed-point-arithmetic-in-c-programming/11816993#11816993


The idea behind fixed-point arithmetic is that you store the values multiplied by a certain amount, use the multiplied values for all calculus, and divide it by the same amount when you want the result. The purpose of this technique is to use integer arithmetic (int, long...) while being able to represent fractions.
The usual and most efficient way of doing this in C is by using the bits shifting operators (<< and >>). Shifting bits is a quite simple and fast operation for the ALU and doing this have the property to multiply (<<) and divide (>>) the integer value by 2 on each shift (besides, many shifts can be done for exactly the same price of a single one). Of course, the drawback is that the multiplier must be a power of 2 (which is usually not a problem by itself as we don't really care about that exact multiplier value).

Comments

Post a Comment

Popular posts from this blog

What is .TLC (Target Language Compiler) and MEX file in Matlab Simulink

Image
TLC (Target Language Compiler) files are used to customize the code building process. They generally come in two varieties System/model tlc files that help in the overall model conversion process Block level tlc files which describe how individual blocks (typically S-Functions) get converted to C code. Target Language Compiler (TLC) is an integral part of the code generator. Use TLC to customize generated code. Through customization, you can produce platform-specific code and incorporate your own algorithmic changes for performance, code size, or compatibility with existing methods. With the Target Language Compiler, you can : 1. Customize the set of options specified by your system target file. 2. Inline the code for S-Function blocks. 3. Generate additional or different types of files. https://in.mathworks.com/help/rtw/ref/tlc.html https://in.mathworks.com/help/rtw/tlc/what-is-the-target-language-compiler.html The Target Language Compiler™ is
Read more

Simulink & Matlab Interview Question and Answers For Automotive , Aerospace etc.

1. What is the difference between Simulink and Stateflow Ans .   Simulink  is used to implement dynamic systems where the state of the system varies with time and thus obeys differential equations involving time derivatives . On the other hand  Stateflow  is based on State machine where the system have a finite number of states and the system can be in exactly one of a finite number of states at any given time. So we can say that, at a given instant of time the state of a Simulink model can be computed using differential equation and is not known in advance , however in case of Stateflow model all the possible states in which the system can be are know beforehand the Simulation of Stateflow model. Simulink is generally used to model systems which are represented majorly by means of mathematical equation (algebraic equations and differential equation) while Stateflow is generally used to model a system represented majorly by fixed number of States as in Finite State Machine.
Read more

Real World Value Vs Stored Integer In Simulink ( Fixed Point Number Debunked )

Image
Relationship Between Stored Integer Value and Real-World Value The binary encoding of variables is only an approximation of an arbitrarily precise real-world value. Therefore, the limitations of the binary representation automatically introduce limitations on the precision of the value. In  BinaryPoint  scaling, the relationship between the stored integer value and the real-world value is There is also  SlopeBias  scaling, which has the relationship where and https://blogs.mathworks.com/simulink/2008/12/14/representing-numbers-integers-and-fixed-point/ Fixed-Point Numbers: Binary Point Fixed-point numbers use the same integer representations, but they assign a different meaning to the bits.  We can introduce the idea of fractional bits and a binary point.  With integers, your smallest spacing between numbers is 1.  If we decide to let a bit represent ½ we now have closer spacing in our system.  The number of bits to the right of the binary
Read more