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Number Representation
Published in Wen-Long Chin, Principles of Verilog Digital Design, 2022
Conversely, to convert a decimal number to fixed-point binary number, the easiest approach is to Step 1: multiply the decimal number by 1r=2f,Step 2: round the resulting product to the nearest integer,Step 3: convert the resulting decimal integer to a binary integer,Step 4: multiply the binary integer by r=2−f (in decimal), i.e., shift left the binary point by f bits.
Vehicle Controllers and Communication
Published in Iqbal Husain, Electric and Hybrid Vehicles, 2021
Microcontrollers and DSPs can be divided into floating point processors or fixed-point processors in terms of number representation, although high-end microcontrollers are available that support both type of number handling. A floating point processor allows allotting point arithmetic units to solve this issue with an added cost. Algorithm development time for a floating point system can be dramatically shorter due to the programming flexibility. In fixed-point processors, special representations of non-integer values are required to execute mathematical operations which are known as fixed-point arithmetic. A fixed-point processor makes it more difficult to calculate mathematical expressions that involve non-integer variables. However, there is an advantage of fixed-point processors in applications where control execution times need to be managed critically. Multiplications and divisions are managed by register shift left or right, respectively, by the corresponding number of bits to minimize code execution time since the CPU operates in base two. The shift by one bit is equivalent to multiplying or dividing by a power of two depending on shifting left or shifting right. With shift operations, control loop times can be accurately controlled and cycle execution time overruns can be eliminated.
Round-Off Errors and Limit Cycles in Digital Filters
Published in John T. Taylor, Qiuting Huang, CRC Handbook of ELECTRICAL FILTERS, 2020
When two fixed-point numbers are added, the result may exceed the dynamic range of the implementation; i.e., addition of fixed-point numbers may cause overflow. When two fixed-point numbers are multiplied, the result, in general, is twice the initial word length. If both numbers have an integer part, the results may overflow. If both numbers are fixed-point fractions, the result remains a fraction and cannot overflow. It is, therefore, preferable in a fixed-point implementation to scale all numbers so that they are fractions. Multiplication of two fixed-point fractions, even though free of overflow, does lead to underflow. In IIR digital filter implementations, a word-length reduction is necessary to prevent the word lengths of the signals from increasing. This reduction of the word length is called signal quantization.
Unit speed flocking model with distance-dependent time delay
Published in Applicable Analysis, 2023
We multiply (9) by , and add the result and (11) to obtain Thus, we have Therefore, if we choose sufficiently small such that then is a contractive sequence in , and there is a local solution to (2). Moreover, depends on ψ, τ, and the unit-speed property of initial data By Lemma 2.1, We note that by (4), By the second equation in (2), for , Therefore, we can extend the local solution to the global-in-time solution. The uniqueness part is clear by the Banach fixed point theorem.
Sign-changing solutions for a fractional Schrödinger–Poisson system
Published in Applicable Analysis, 2023
Senli Liu, Jie Yang, Yu Su, Haibo Chen
Similarly, we can deduce that for all . Moreover, since and are continuous maps, then we know that Γ is continuous. Thus, we have By the Brouwer fixed point theorem, there exists a pair such that Using to the fact of for i = 1, 2, one can infer that and . Then, in view of (53) and (55) we have which means is a pair of critical point of G.
Constraint qualifications for nonsmooth programming
Published in Optimization, 2018
Liren Huang, Lulin Tan, Chunguang Liu
Since is surjective, there exists a N-dimensional subspace T of X such that Thus, exists and therefore the space T can be considered as the Euclidean space . For ease of notation, from now on, we write as Let v be a vector of X. Then is a compact subset of X. For by definition of the Hadamard differentiable there exists such that For each define the function by Note that for showing that maps into itself. Let be the fixed point of given by Brouwer fixed point theorem. Then, we have that is,