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

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1 ELET 7404 Embedded & Real Time Operating Systems Fixed-Point Math Chap. 9, Labrosse Book Fall 2007

2 Fixed-Point Math Most low-end processors, such as embedded processors Do not provide hardware-assisted floating-point 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 Fixed-Point Math (cont.) The big question? Without floating-point math How would you perform the following 12.34V = X 5.4 = /98.7 = FIXED-POINT MATH Integer Math

4 Fixed-Point Math (cont.) Fixed-Point 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 Fixed-Point 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 Fixed-Point 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 2-5 ) 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: 2-5

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 Fixed-Point Math (cont.) It s the programmer s job to keep track of radix placement

9 Fixed-Point 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 16-bit 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 whole-number places

10 When the program performs arithmetic on fixed point numbers, it actually manipulates integers Add Subtract Multiply Divide Fixed-Point Math (cont.) Microprocessors do not provide mechanisms to represent fixed-point numbers

11 Fixed-point number : Fixed-Point 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 Fixed-Point Number Notation Fixed-point number : <mantissa> S <exponent> S means Mantissa needs to be scaled by 2 exponent Example 1: 5S-3 Represents or 5 x 2-3 = 5/2 3 = 5/8 1 x 2-1 = 1 x 1/2 = x 2-2 = 0 x 1/ 2 2 = 1/4 = 0 1 x 2-3 = 0 x 1/ 2 3 = 1/8 = SUM = = Mantissa is a number represented by an integer - Exponent is maintained mentally by the programmer

13 Fixed-point number : <mantissa> S <exponent> S means Mantissa needs to be scaled by 2 exponent Example 2: 31S-8 Fixed-Point Number Notation Represents or 31 x 2-8 = 31/256 1 x 2-4 = 1 x 1/2 4 = 1/16 = x 2-5 = 1 x 1/2 5 = 1/32 = x 2-6 = 1 x 1/2 6 = 1/64 = x 2-7 = 1 x 1/2 7 = 1/128 = x 2-8 = 1 x 1/2 8 = 1/256 = SUM = =

14 Fixed-Point Number Notation Fixed-point number : <mantissa> S <exponent> S means Mantissa needs to be scaled by 2 exponent Example 3: -123S-16 Represents or -123 x 2-16 = -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 16-bit 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 fixed-point math log(65535/1.2345) = log( ) = log(2) = INT(4.725/0.301) = INT(15.697) = 15 Thus, the number is written as S-15

17 Scaling (cont.) To represent in fixed-point 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 fixed-point 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 17-bits 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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