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Lecture 3 9/7/11. Lab 202a: Door combo 1 4 2 3 <enter> Do homework problems 2.25, 2.31-2.35 (we will go over in class) for Friday 9/9/11 . Fixed Point Arithmetic. Floating point: overhead too high for many embedded apps – hundreds of operations to do FP ops Fixed point: 2 part representation

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Lecture 3 9/7/11

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Lecture 3 9 7 11 l.jpg

Lecture 3 9/7/11

Lab 202a: Door combo 1 4 2 3 <enter>

  • Do homework problems 2.25, 2.31-2.35 (we will go over in class) for Friday 9/9/11


Fixed point arithmetic l.jpg

Fixed Point Arithmetic

  • Floating point: overhead too high for many embedded apps – hundreds of operations to do FP ops

  • Fixed point: 2 part representation

    • Explicit variable integer I

    • Implicit (i.e. not stored, determined and remembered by programmer!) fixed constant 

    • Represented number F=I

  • Precision: number of distinguishable patterns, governed by number of bits in I

  • Resolution: smallest difference that can be represented,  by definition


Examples l.jpg

Examples

  • Money! Integral pennies: =0.01

    • Decimal-based implicit constant makes sense here – “Decimal fixed-point” though will be represented in binary of course

  • Voltmeter with resolution of 0.001V (i.e. integral mV) also makes sense: =0.001

  • Decimal fixed point useful for managing displays to humans, binary fixed point where =2m is easier to computer with!


Examples4 l.jpg

Examples

  • Voltmeter on 6811/12

    • Built-in 8-bit ADC takes 0-5V input

      • 256 bit patterns to represent the entire range

      • Output N of the ADC is such that

        • Vin = 5*N/255 = 0.019607843*N

      • Smallest change we can detect is about 20mV

        • This one won’t discriminate between 3.005 and 3.008 V, or even 3.01 and 3.02 V

    • Design decisions

      • We now need to represent the voltage in a format convenient for display


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Voltmeter design, cont.

  • Since a decimal display of voltage is the end game, we choose a decimal fixed point display with =0.01V

    • Why not =0.1V?

    • Why not =0.001V?


Voltmeter design cont6 l.jpg

Voltmeter design, cont.

  • Since a decimal display of voltage is the end game, we choose a decimal fixed point display with =0.01V

    • Why not =0.1V?

    • Why not =0.001V?

    • Essential point: this resolution is a bit better than the raw ADC resolution (500 vs 256 patterns): we don’t want to lose resolution as an artifact of display format! Thermometers!!!

    • Complication: doesn’t fit in 8 bits! Need 916


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Fixed point arithmetic

  • Main reason we use it: to add/subtract Fixed point numbers with same , just do integer add/subtract!

  • If numbers have different  values, we have to convert to a common basis first.


Fixed point arithmetic8 l.jpg

Fixed point arithmetic

  • we want z=x+y with

    x=I2n

    y=J2m

    z=K2p

  • algebraic manipulation gives

    K=I2n-p + J2m-p

    • alot like equalizing exponents in FP math


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Fixed point arithmetic

  • multiplication, division more challenging

    • reasons similar to excess-bias problem in FP math

      • if you multiply two numbers representing dollar values stored as pennies (=0.01), straight multiplication gives the answer in terms of 1/10,000 dollars (1/100 of a penny)

        • need to multiply by 100 to get back to pennies representation


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Freescale instruction set

  • Arithmetic, logical operaions

    • full complement of logical bitwise operators

      • AND, OR, NOT, XOR (EOR on Freescale)

      • details of instruction variants will follow in ch3

        • 10 variants of AND instruction, for instance!

    • shift instructions

      • ASR, LSR; ASL, LSL; ROR, ROL

        • difference ASR,LSR?

        • difference ASL,LSL?


Arithmetic operations l.jpg

Arithmetic operations

  • There are integer instructions for + −×÷

  • Challenge: programmer needs to check for anomalies such as overflow explicitly

  • Aid: CCR: Condition Code Register

    • unlike MIPS but like most other processors

    • 4 flags of interest to us now (there are more)

      • N negative: result of last computation was negative

      • Z zero: result of last computation was zero

      • V overflow: result of last computation was signed overflow (i.e. 2’s complement)

      • C carry: result of last computation was unsigned overflow

  • Each operation sets appropriate flag bits, programmer then checks them EVERY TIME!


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Unsigned addition sequence

  • For example, knowing we just did an unsigned add (bcc = Branch if C flag Clear)

    ldaaA8get first input

    addaB8regA = A8 + B8

    bccOK1if C=0 then no error

    so skip to end

    ldaa #255overflow

    OK1 staaR8 R8= (smaller of A8+B8, 255)

  • Strategy here is to round an unsigned overflow back down to largest possible unsigned integer

  • Why not just abort?

  • What if V set?


Unsigned subtraction l.jpg

Unsigned subtraction

  • similar idea: doing R8=A8-B8, set R8 to 0 on unsigned overflow (i.e. negative result)


Signed 2 s comp addition l.jpg

Signed (2’s Comp) addition

  • The -128 – +127 problem complicates life a bit

  • several typos in your book on page 53 (at least in some printings) also complicate understanding

  • nb BMI is Branch if Minus, i.e. N=1.

  • BPL is Branch if Plus, i.e. N=0.

  • BVC is Branch if V flag Clear


Slide16 l.jpg

ldaaA8get first input

addaB8regA = A8 + B8

bvc ok3 if V=0 no error so

skip to end

err3 bmi over3if V=1 and N=1, it

was overflow

ldaa#-128 if V=1 and N=0,

it was underflow

bra ok3

over3 ldaa#127overflow

ok3 staaR8WAS MISSING IN BOOK


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