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Implementation Considerations-old

# Implementation Considerations-old - Microcomputer Systems 1...

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Microcomputer Systems 1 Implementation Considerations

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Data Representations &  Arithmetic Fixed-Point Numbers and Arithmetic
February 11, 2012 Veton Këpuska 3 Fixed-Point There are several different binary number systems.  Most notable: 1. Sign Magnitude 2. One’s Complement 3. Two’s Complement Example of 4-bit signed numbers in three different  formats

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February 11, 2012 Veton Këpuska 4 Binary Representations of 4-bit Signed  Numbers Decimal Value Sign Magnitude One’s Complement Two’s Complement +7 0 111 0 111 0 111 +6 0 110 0 110 0 110 +5 0 101 0 101 0 101 +4 0 100 0 100 0 100 +3 0 011 0 011 0 011 +2 0 010 0 010 0 010 +1 0 001 0 001 0 001 +0 0 000 0 000 0 000 -0 1 000 1 111 - -1 1 001 1 110 1 111 -2 1 010 1 101 1 110 -3 1 011 1 100 1 101 -4 1 100 1 011 1 100 -5 1 101 1 010 1 011 -6 1 110 1 001 1 010 -7 1 111 1 000 1 001 -8 - - 1 000
February 11, 2012 Veton Këpuska 5 Fixed-Point Representations Integers vs. Fractional Numbers Representations Notation: Qm.n Format: m – Number of bits to the left of the radix point n – number of bits to the right of the radix  point Let N – total number of bits N=m+n+1 Signed, and N-bit signed number in Qm.n format with MSB as sign  bit (b N-1 ) N=m+n Unsigned

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February 11, 2012 Veton Këpuska 6 Examples Q16.0 Format is Full unsigned integer number representation Q15.0 Format is Full signed integer number representation Q15.1 Format represents unsigned 16 bit integer value Q14.1 Format represents signed 15 bit integer value Q0.16 (or Q.16 or simply Q16) is a 16 bit format that for unsigned number that  uses 16 bits for the fractional value. Q0.15(or Q.15 or simply Q15) is a 15 bit format that for signed number that uses  16 bits for the fractional value. Fractional Representations (e.g., Q1.15) have the advantage over the Full format  representations that results of the multiplication are always smaller than either of  the numbers QX.0 or QY.1 Formats one must check for overflow and handle it Q0.X or Q1.Y Formats may lead to underflow but no special handling is required.
February 11, 2012 Veton Këpuska 7 Fixed-Point Representations Integers vs. Fractional Numbers Representations Numbers represented as 16/32 bits: 2 16 =65,536 or 2 32 =4,294,967,296 bit patterns. 1. Unsigned Integer Format Stored Value:  16-bit: 0..65,536 or  32-bit: 0..4,294,967,296 1. Signed Integer Format Stored Value: 16-bit: -32,768..32,767 or 32-bit: -2,147,483,648..2,147,483647 1. Unsigned Fractional Format Stored Value:  16-bit: 0..1 (65,536 levels) or  32-bit: 0..1 (4,294,967,296 levels) 1. Signed Fractional Format Stored Value:  16-bit: -1..1 (65,536 levels) or  32-bit: -1..1 (4,294,967,296 levels)

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February 11, 2012 Veton Këpuska 8 Blackfin Fixed-Point Representation  ADI DSP’s including Blackfin support Fractional Format  Representation; 16-bit Example in Q1.15 Format: -2 0 2 -1 2 -2 2 -3 2 -4 2 -5 2 -6 2 -7 2 -8 2 -9 2 -10 2 -11 2 -12 2 -13 2 -14 2 -15 MSB LSB HEX BINARY DECIMAL 7FFF 0111 1111 1111 1111 +0.999969 0001 0000 0000 0000 0001 +0.000031 0000 0000 0000 0000 0000 +0.000000 FFFF 1111 1111 1111 1111 -0.000031 8000 1000 0000 0000 0000 -1.000000
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