Floating Point Representation · 2018-10-29 · EE 3610 Digital Systems Suketu Naik . Fixed Point...
Transcript of Floating Point Representation · 2018-10-29 · EE 3610 Digital Systems Suketu Naik . Fixed Point...
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EE 3610 Digital Systems Suketu Naik
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Floating PointRepresentation
EE 3610: Digital Systems
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EE 3610 Digital Systems Suketu Naik
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Floating boat
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EE 3610 Digital Systems Suketu Naik
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Floating numbers
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EE 3610 Digital Systems Suketu Naik
4Fixed Point Numbers
Generic Binary RepresentationExample: 1101012 1 x 25 + 1 x 24 + 0 x 23 + 1 x 22 + 0 x 21 + 1 x 20
= 32 + 16 + 4 + 1 = 5310Example: 1011.10112 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 1 x 2-1 + 0 x 2-2 + 1 x 2-3 + 1 x 2-4
= 11. 6875
N-bit fixed point, 2’s complement integer representation X = -bN-1 2N-1 + bN-2 2N-2 + … + b020
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EE 3610 Digital Systems Suketu Naik
5Fixed Point Numbers‘Fixed Point’ number has a fixed number of digits after and before the decimal point
Qm.n notation m bits for integer portion n bits for fractional portionTotal number of bits N = m + n + 1, for signednumbersExample: 16-bit number (N=16) and Q2.13 format 2 bits for integer portion 13 bits for fractional portion 1 signed bit (MSB)
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EE 3610 Digital Systems Suketu Naik
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Examples:1110 in Integer Representation Q3.0-23 + 22 + 21 = -211.10 Fractional Q1.2 Representation-21 + 20 + 2-1 = -2 + 1 + 0.5 = -0.5
1.110 Fractional Q0.3 Representation-20 + 2-1 + 2-2 = -1 + 0.5 + 0.25 = -0.25
Difficult to use due to possible overflow In a 16-bit processor, the dynamic range is -32,768 to 32,767.Example: 200 × 350 = 70000: an overflow!
Fixed Point Numbers: Qm.n notation
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EE 3610 Digital Systems Suketu Naik
7Fixed Point Numbers: Qm.n notationDynamic Range and Precision of 16-Bit Numbers for Different Q Formats
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EE 3610 Digital Systems Suketu Naik
8Fixed Point Numbers: Adv. And Disadv. Disadvantage of fixed point: dynamic range is small
Advantages of fixed point: logic circuits simpler, less memory,less processor speed: smaller, faster, cheaper, lower power consumption
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EE 3610 Digital Systems Suketu Naik
9Application of Fixed Point DSPADSP Blackfin: multi-format audio, video, voice and image
processing Example: Undersea pipline monitoring in Orman Natural
Gas Field Hydro-acoustics and non-contact sensors together with ADSP Blackfin and LabVIEW Embedded Module for sub sea state monitoring and analysis tasks
ADI Blackfin DSP Orman Natural Gas Field (Norway)
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EE 3610 Digital Systems Suketu Naik
10Why Floating Point?
Floating point numbers are used to represent fractions and allow more precision
e.g. 35.261, 3.14159, 1.25
Arithmetic units for floating-point numbers are more complex than fixed-point numbers
IEEE 754 Floating-Point Formats
Single precision (32-bit) Double precision (64-bit) Extended precision (128-bit)
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EE 3610 Digital Systems Suketu Naik
11Basics
N= F x BE
N = a floating point (real) number F = fraction B = base E = exponentBase can be 2, 10, 16 or other
Here we consider B=2
Fraction F and Exponent E can be represented in a number of ways
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EE 3610 Digital Systems Suketu Naik
12Basics
Binary numbers, like decimal numbers, can be represented in scientific notation
Example: 923.5210 can be represented as 9.2352 x 102
Similarly, 101011.1012 = 43.62510 can be represented as 1.01011101 x 25
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EE 3610 Digital Systems Suketu Naik
13Basics: Sign Field
43.62510 101011.1012 1.01011101 x 25
0 xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxx
Sign field is 1-bit long: 0 (positive) or 1 (negative)
N= (-1)S x (1 + F) x 2E
S = Sign fieldE = Exponent fieldF = Mantissa (Fractional) Field
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EE 3610 Digital Systems Suketu Naik
14Basics: Exponent Field
43.62510 101011.1012 1.01011101 x 25
0 10000100 xxxx xxxx xxxx xxxx xxxx xxx
Exponent field is 8-bits long: range 0 to 255Add 127 to the exponent in scientific notationHere 5 + 127 = 132: 10000100 (If the exponent is less than 127 then it is negative)
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EE 3610 Digital Systems Suketu Naik
15Basics: Mantissa (Fraction) Field
43.62510 101011.1012 1.01011101 x 25
0 10000100 01011101000000000000000
Mantissa field is 23-bits long: throw away 1 to the leftof binary point and pad it with zeros at the end1.01011101 01011101000000000000000
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EE 3610 Digital Systems Suketu Naik
16Single Precision Representation
43.62510 101011.1012 1.01011101 x 25
Sign Exponent Fraction
0 10000100 01011101000000000000000
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EE 3610 Digital Systems Suketu Naik
17Formats: Single PrecisionSingle Precision
Example: 13.4510 : 1101.01 1100 1100 1100 ... 2: 1.10101 1100 1100 ... x 23
(Note: 0.45 produces repeating binary fractions)Sign: positive: 0Exponent: 3 + 127 = 130: 10000010Mantissa (Fraction): 10101 1100 1100 1100 1100 11:
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EE 3610 Digital Systems Suketu Naik
18Formats: Double PrecisionDouble Precision
Example: 13.4510 : 1101.01 1100 1100 1100 ... 21.10101 1100 1100 ... x 23
(Note: 0.45 produces repeating binary fractions)Sign: positive: 0Exponent: 3 + 1023 = 1026: 10000000010 Mantissa (Fraction): 10101 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 110
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EE 3610 Digital Systems Suketu Naik
19IEEE 754 Floating Point: Range
Type Size Exponent Mantissa Range
Single 32-bit 8-bit 23-bit 2x10+/-38
Double 64-bit 11-bit 52-bit 2x10+/-308
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EE 3610 Digital Systems Suketu Naik
20Floating Point (FP) to DecimalExample: HEX 0xC0B40000, assume signal precision FP
Step 1: Convert HEX into Binary
HEX C 0 B 4 0 0 0
Binary 1100 0000 1011 0100 0000 0000 0000
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EE 3610 Digital Systems Suketu Naik
21Floating Point (FP) to DecimalStep 2: Reorganize into packets of 1, 8, and 23 bits long
Sign Exponent Fraction
1 10000001 01101000000000000000000
HEX C 0 B 4 0 0 0
Binary 1100 0000 1011 0100 0000 0000 0000
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EE 3610 Digital Systems Suketu Naik
22Floating Point (FP) to DecimalStep 3: Conversion
Sign Exponent Fraction
1 10000001 01101000000000000000000
Sign=1 (negative)Exponent= 10000001 = 1 x 27+ 1 x 20 = 129 => 129-127 = 2Mantissa= 1.01101
N= -1.01101 x 22
= -101.101= -(1 x 22 + 1 x 20 + 1 x 2-1 + 1 x 2-3)= -(4 + 1 + 0.5 + 0.125 ) = -5.625
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EE 3610 Digital Systems Suketu Naik
23Floating Point (FP): VHDL
use ieee.float_pkg.all; variable x, y, z : float (5 downto -10); begin y := to_float (3.1415, y); -- Uses “y” for the sizing only. z := “0011101010101010”; –- 1/3 x := z + y;
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EE 3610 Digital Systems Suketu Naik
24Application of Floating Point DSPADSP SHARC: most computationally intensive, real-time signal-processing applications Examples: High Definition Audio, Precision Motor/Tool Control, Precision Sensors
ADI SHARC DSP
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EE 3610 Digital Systems Suketu Naik
25ReferencesFloating Point Converter http://babbage.cs.qc.cuny.edu/IEEE-754.old/Decimal.html
32-bit single precision floating point adderhttp://upcommons.upc.edu/pfc/bitstream/2099.1/15467/4/32BitFloatingPointAdder.pdf
ADI SHARC Processors http://www.analog.com/en/processors-dsp/sharc/products/index.html