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This glossary defines terms related to fixed-point data types and numbers. These terms may appear in some or all of the documents that describe products from The MathWorks that have fixed-point support.
arithmetic shift
A shift of the bits of a binary word for which the sign bit is recycled for each bit shift to the right. A zero is incorporated into the least significant bit of the word for each bit shift to the left. In the absence of overflows, each arithmetic shift to the right is equivalent to a division by 2, and each arithmetic shift to the left is equivalent to a multiplication by 2.
See Also binary point, binary word, bit, logical shift, most significant bit
bias
Part of the numerical representation used to interpret a fixed-point number. Along with the slope, the bias forms the scaling of the number. Fixed-point numbers can be represented as
where the slope can be expressed as
See Also fixed-point representation, fractional slope, integer, scaling, slope, [Slope Bias]
binary number
A value represented in a system of numbers that has two as its base and that uses 1's and 0's (bits) for its notation.
binary point
A symbol in the shape of a period that separates the integer and fractional parts of a binary number. Bits to the left of the binary point are integer bits and/or sign bits, and bits to the right of the binary point are fractional bits.
See Also binary number, bit, fraction, integer, radix point
binary point-only scaling
The scaling of a binary number that results from shifting the binary point of the number right or left, and which therefore can only occur by powers of two.
See Also binary number, binary point, scaling
binary word
A fixed-length sequence of bits (1's and 0's). In digital hardware, numbers are stored in binary words. The way in which hardware components or software functions interpret this sequence of 1's and 0's is described by a data type.
bit
The smallest unit of information in computer software or hardware. A bit can have the value 0 or 1.
ceiling (round toward)
A rounding mode that rounds to the closest representable number in the direction of positive infinity.
See Also floor (round toward), nearest (round toward), rounding, truncation, zero (round toward)
contiguous binary point
A binary point that occurs within the word length of a data type. For example, if a data type has four bits, its contiguous binary point must be understood to occur at one of the following five positions:
See Also data type, noncontiguous binary point, word length
data type
A set of characteristics that define a group of values. A fixed-point data type is defined by its word length, its fraction length, and whether it is signed or unsigned. A floating-point data type is defined by its word length and whether it is signed or unsigned.
See Also fixed-point representation, floating-point representation, fraction length, word length
data type override
A parameter in the Fixed-Point Settings interface that allows you to set the output data type and scaling of Fixed-Point Blockset blocks on a system or subsystem level.
exponent
Part of the numerical representation used to express a floating-point or fixed-point number.
1. Floating-point numbers are typically represented as
2. Fixed-point numbers can be represented as
where the slope can be expressed as
The exponent of a fixed-point number is equal to the negative of the fraction length:
See Also bias, fixed-point representation, floating-point representation, fraction length, fractional slope, integer, mantissa, slope
fixed-point representation
A method for representing numerical values and data types that have a set range and precision.
1. Fixed-point numbers can be represented as
where the slope can be expressed as
The slope and the bias together represent the scaling of the fixed-point number.
2. Fixed-point data types can be defined by their word length, their fraction length, and whether they are signed or unsigned.
See Also bias, data type, exponent, fraction length, fractional slope, integer, precision, range, scaling, slope, word length
floating-point representation
A method for representing numerical values and data types that can have changing range and precision.
1. Floating-point numbers can be represented as
2. Floating-point data types are defined by their word length.
See Also data type, exponent, mantissa, precision, range, word length
floor (round toward)
A rounding mode that rounds to the closest representable number in the direction of negative infinity.
See Also ceiling (round toward), nearest (round toward), rounding, truncation, zero (round toward)
fraction
The part of a fixed-point number represented by the bits to the right of the binary point. The fraction represents numbers that are less than one.
See Also binary point, bit, fixed-point representation
fraction length
The number of bits to the right of the binary point in a fixed-point representation of a number.
See Also binary point, bit, fixed-point representation, fraction
fractional slope
Part of the numerical representation used to express a fixed-point number. Fixed-point numbers can be represented as
where the slope can be expressed as
See Also bias, exponent, fixed-point representation, integer, slope
guard bits
Extra bits in either a hardware register or software simulation that are added to the high end of a binary word to ensure that no information is lost in case of overflow.
See Also binary word, bit, overflow
integer
1. The part of a fixed-point number represented by the bits to the left of the binary point. The integer represents numbers that are greater than or equal to one.
2. Also called the "stored integer." The raw binary number, in which the binary point assumed to be at the far right of the word. The integer is part of the numerical representation used to express a fixed-point number. Fixed-point numbers can be represented as
where the slope can be expressed as
See Also bias, fixed-point representation, fractional slope, integer, slope
integer length
The number of bits to the left of the binary point in a fixed-point representation of a number.
See Also binary point, bit, fixed-point representation, fraction length, integer
least significant bit (LSB)
The bit in a binary word that can represent the smallest value. The LSB is the rightmost bit in a big-endian-ordered binary word.
See Also big-endian, binary word, bit, most significant bit
logging
A tool provided by the Fixed-Point Settings interface that outputs the minimum values, maximum values, and any overflows for all Fixed-Point Blockset blocks in any model that you run with a fixed-point license.
logical shift
A shift of the bits of a binary word, for which a zero is incorporated into the most significant bit for each bit shift to the right and into the least significant bit for each bit shift to the left.
See Also arithmetic shift, binary point, binary word, bit, most significant bit
mantissa
Part of the numerical representation used to express a floating-point number. Floating-point numbers are typically represented as
See Also exponent, floating-point representation
most significant bit (MSB)
The bit in a binary word that can represent the largest value. The MSB is the leftmost bit in a big-endian-ordered binary word.
See Also big-endian, binary word, bit, least significant bit
nearest (round toward)
A rounding mode that rounds to the closest representable number, with the exact midpoint rounded to the closest representable number in the direction of positive infinity.
See Also ceiling (round toward), floor (round toward), rounding, truncation, zero (round toward)
noncontiguous binary point
A binary point that is understood to fall outside of the word length of a data type. For example, the binary point for the following 4-bit word is understood to occur two bits to the right of the word length
thereby giving the bits of the word the following potential values:
See Also binary point, data type, word length
one's complement representation
A representation of signed fixed-point numbers. Negating a binary number in one's complement requires a bitwise complement. That is, all 0's are flipped to 1's and all 1's are flipped to 0's. In one's complement notation there are two ways to represent zero. A binary word of all 0's represents "positive" zero, while a binary word of all 1's represents "negative" zero.
See Also binary number, binary word, sign/magnitude representation, signed fixed-point, two's complement representation
overflow
The situation that occurs when the magnitude of a calculation result is too large for the range of the data type being used. In many cases you can choose to either saturate or wrap overflows.
padding
Extending the least significant bit of a binary word with one or more zeros.
See Also least significant bit
precision
1. A measure of the smallest numerical interval that a fixed-point data type and scaling can represent, determined by the value of the number's least significant bit. The precision is given by the slope, or the number of fractional bits. The term "resolution" is sometimes used as a synonym for this definition.
2. A measure of the difference between a real-world numerical value and the value of its quantized representation. This is sometimes called quantization error or quantization noise.
See Also data type, fraction, least significant bit, quantization, quantization error, range, slope
Q format
A representation used by Texas Instruments to encode signed two's complement fixed-point data types. This fixed-point notation takes the form
In Q format notation, the most significant bit is assumed to be the sign bit.
See Also binary point, bit, data type, fixed-point representation, fraction, integer, two's complement
quantization
The representation of a value by a data type that has too few bits to represent it exactly.
See Also bit, data type, quantization error
quantization error
The error introduced when a value is represented by a data type that has too few bits to represent it exactly, or when a value is converted from one data type to a shorter data type. Quantization error is also called quantization noise.
See Also bit, data type, quantization
radix point
A symbol in the shape of a period that separates the integer and fractional parts of a number in any base system. Bits to the left of the radix point are integer and/or sign bits, and bits to the right of the radix point are fraction bits.
See Also binary point, bit, fraction, integer, sign bit
range
The span of numbers that a certain data type can represent.
rounding
Limiting the number of bits required to express a number. One or more least significant bit(s) are dropped, resulting in a loss of precision. Rounding is necessary when a value can not be expressed exactly by the number of bits designated to represent it.
See Also bit, ceiling (round toward), floor (round toward), least significant bit, nearest (round toward), precision, truncation, zero (round toward)
saturation
A method of handling numeric overflow that represents positive overflows as the largest positive number in the range of the data type being used, and negative overflows as the largest negative number in the range.
scaling
1. The format used for a fixed-point number of a given word length and signedness. The slope and bias together form the scaling of a fixed-point number.
2. Changing the slope and/or bias of a fixed-point number without changing the stored integer.
See Also bias, fixed-point representation, integer, slope
shift
A movement of the bits of a binary word either towards the most significant bit ("to the left") or towards the least significant bit ("to the right"). Shifts to the right may either be logical, where the spaces emptied at the front of the word with each shift are filled in with zeros, or arithmetic, where the word is sign extended as it is shifted to the right.
See Also arithmetic shift, logical shift, sign extension
sign bit
The bit (or bits) in a signed binary number that indicates whether the number is positive or negative.
sign extension
Adding additional bits that have the value of the most significant bit to the high end of a two's complement number. Sign extension does not change the value of the binary number.
See Also binary number, guard bits, most significant bit, two's complement representation, word
sign/magnitude
A representation of signed fixed-point or floating-point numbers. In sign/magnitude representation, one bit of a binary word is always the dedicated sign bit, while the remaining bits of the word encode the magnitude of the number. Negation using sign/magnitude representation consists of flipping the sign bit from 0 (positive) to 1 (negative), or from 1 to 0.
See Also binary word, bit, fixed-point representation, floating-point representation, one's complement representation, sign bit, signed fixed-point, two's complement representation
signed fixed-point
A fixed-point number or data type that can represent both positive and negative numbers.
See Also data type, fixed-point representation, unsigned fixed-point
slope
Part of the numerical representation used to express a fixed-point number. Along with the bias, the slope forms the scaling of a fixed-point number. Fixed-point numbers can be represented as
where the slope can be expressed as
See Also bias, fixed-point representation, fractional slope, integer, scaling, [Slope Bias]
[Slope Bias]
A representation used by the Fixed-Point Blockset to define the scaling of a fixed-point number.
truncation
A rounding mode that drops one or more least significant bits from a number.
See Also ceiling (round toward), floor (round toward), nearest (round toward), rounding, zero (round toward)
two's complement representation
A common representation of signed fixed-point numbers. Negation using signed two's complement representation consists of a translation into one's complement followed by the binary addition of a one.
See Also binary word, one's complement representation, sign/magnitude representation, signed fixed-point
unsigned fixed-point
A fixed-point number or data type that can only represent numbers greater than or equal to zero.
See Also data type, fixed-point representation, signed fixed-point
word
A fixed-length sequence of binary digits (1's and 0's). In digital hardware, numbers are stored in words. The way hardware components or software functions interpret this sequence of 1's and 0's is described by a data type.
See Also binary word, data type
word length
The number of bits in a binary word or data type.
See Also binary word, bit, data type
wrapping
A method of handling overflow. Wrapping uses modulo arithmetic to cast a number that falls outside of the representable range of the data type being used back into the representable range.
See Also data type, overflow, range, saturation
zero (round toward)
A rounding mode that rounds to the closest representable number in the direction of zero.
See Also ceiling (round toward), floor (round toward), nearest (round toward), rounding, truncation
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