ELET 7404 Embedded & Real Time Operating Systems. FixedPoint Math. Chap. 9, Labrosse Book. Fall 2007


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1 ELET 7404 Embedded & Real Time Operating Systems FixedPoint Math Chap. 9, Labrosse Book Fall 2007
2 FixedPoint Math Most lowend processors, such as embedded processors Do not provide hardwareassisted floatingpoint math The PIC18F452 only supports a 8x8 hardware multiply Look at Table 1: Performance Comparison Without Hardware Math support With Hardware Math support Table 1. Performance Comparison for Processors With and Without Hardware Math Support
3 As an embedded system programmer confronted with the task of writing code that is: Fast Small in bytes FixedPoint Math (cont.) The big question? Without floatingpoint math How would you perform the following 12.34V = X 5.4 = /98.7 = FIXEDPOINT MATH Integer Math
4 FixedPoint Math (cont.) FixedPoint Math is integer math that allows fractions. The idea: Trick the computer into thinking you are talking about an integer when in fact you, as the programmer, know that you are dealing with a number that has a fractional component.
5 FixedPoint Math (cont.) The computer only thinks in bits, e.g or 2 4, or 16 However, this assumes that the right most bit represents 2 0 Does it have to be?
6 FixedPoint Math (cont.) Implied decimal point (radix) normally rightmost bit Why can t I put it someplace else? I can Programmer decides to place radix between the 5 th and 6 th bit position (rightmost bit weight is 25 ) is no longer 16 now it is 0.5 In other words, 16 has been scaled by = 1/2 5 = 1/32 = i.e. 16 * = 0.5
7 Radix between the 5th and 6th bit position This means that the rightmost bit weight is: 25
8 So as far as the computer knows Its dealing with plain old 16 But the programmer (THAT S YOU!) Know otherwise So, important point FixedPoint Math (cont.) It s the programmer s job to keep track of radix placement
9 FixedPoint Math (cont.) By manipulating the radix position the programmer (YOU!) can scale integers into fractional values. Location of the radix defines a convention for how the program will interpret a 16bit string As the radix point moves to the left The factional portion of string increases The fraction becomes more precise However, the overall range of numbers diminishes Because there are fewer wholenumber places
10 When the program performs arithmetic on fixed point numbers, it actually manipulates integers Add Subtract Multiply Divide FixedPoint Math (cont.) Microprocessors do not provide mechanisms to represent fixedpoint numbers
11 Fixedpoint number : FixedPoint Math (cont.) <mantissa> S <exponent> Where S means: That the Mantissa needs to be Scaled by 2 exponent  Exponent is also called the scale factor  Mantissa is always an integer number
12 FixedPoint Number Notation Fixedpoint number : <mantissa> S <exponent> S means Mantissa needs to be scaled by 2 exponent Example 1: 5S3 Represents or 5 x 23 = 5/2 3 = 5/8 1 x 21 = 1 x 1/2 = x 22 = 0 x 1/ 2 2 = 1/4 = 0 1 x 23 = 0 x 1/ 2 3 = 1/8 = SUM = = Mantissa is a number represented by an integer  Exponent is maintained mentally by the programmer
13 Fixedpoint number : <mantissa> S <exponent> S means Mantissa needs to be scaled by 2 exponent Example 2: 31S8 FixedPoint Number Notation Represents or 31 x 28 = 31/256 1 x 24 = 1 x 1/2 4 = 1/16 = x 25 = 1 x 1/2 5 = 1/32 = x 26 = 1 x 1/2 6 = 1/64 = x 27 = 1 x 1/2 7 = 1/128 = x 28 = 1 x 1/2 8 = 1/256 = SUM = =
14 FixedPoint Number Notation Fixedpoint number : <mantissa> S <exponent> S means Mantissa needs to be scaled by 2 exponent Example 3: 123S16 Represents or 123 x 216 = 123/2 16 = 123/ Mantissa is a number represented by an integer  Exponent is maintained mentally by the programmer
15 Scaling Scaling is done to allow almost any number to be represented using a 16bit number. Various equations have been developed to make this representation possible. INT( ) means take the integer part of the result. Results of INT must be truncated. Meaning, everything after the decimal point is removed. No rounding occurs
16 This truncation, INT( ), does cause us to lose a little accuracy but it is worth the gain in processor resources. Example 4: Scaling (cont.) To represent the number in fixedpoint math log(65535/1.2345) = log( ) = log(2) = INT(4.725/0.301) = INT(15.697) = 15 Thus, the number is written as S15
17 Scaling (cont.) To represent in fixedpoint math, with 16 bits, numbers greater than 2 16 = , the formula is: However, this will cause loss in resolution, and we would need more than 16 bits to represent large numbers without loss in resolution.
18 Scaling (cont.) Example 5: To represent the number in fixedpoint math log(107573/65535) = log( ) = log(2) = INT( /0.301) =INT(0.715) = 0 Thus, the number is written as S1 However, loss in resolution, we would need 17bits to represent this number
19 Scaling (cont.) To represent any signed number between to To represent a signed number that is less than and greater than
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