Transcript Ch. 10
Ch. 10 Numerical Calculations From Valvano’s text Introduction to Embedded Systems 10.1 Fixed Point Numbers • Fixed point numbers are used to express noninteger values. • Two parts: – Variable integer (I). – Fixed Constant (Δ). • Values is product of the two parts. – Example: • Dec. Fixed Point = I x 10m for some constant m. • Binary Fixed Point = Ix 2m for some constant m 10.1 Cont. • Precision— of a number system is the total number of distinguishable values. • Resolution—of a number is the smallest difference that can be represented which is Δ for fixed point numbers. (Sometimes this is represented by units –eg mv when .001 is the resolution.) Checkpoint 10.1 • Give an approximation of π using the decimal fixed-point (Δ = .001) format. • ANSWER: Let π be about 3.14159, then with m = 3 we have decimal fixed-point as 3141 (3.14159 x 1000). Checkpoint 10.2 • Give an approximation of π using the binary fixed point ( Δ= 2-8). • ANSWER: Again, π is about 3.14159 so if we are using binary fixed point we have the approximation, with m = 8, 3.14159 x 256 = 804. 2477, so the variable part is 804. Example: 9S12 ADC • The ADC is 10 bits and the range of voltages is 0 to 5 volts. • Let Vin be the input voltage and N be the digital ADC output. • Then Vin = 5 x N/1023 = 0.0048876 x N. • The resolution is about 5/1023 which is about 5 mv (note that 2-10 is about 10-3). • See Table 10.1 10.1 (cont.) • Errors to Avoid and Order of Operations – Overflow—the result exceeds the range of the number system. • Solutions: promotion and ceiling/floor – See page 90 (Chapter 3). – Drop-out—occurs after a right shift or a divide. • Can be avoided by dividing last when performing multiple integer calculations. 10.2 Extended Precision Calculations • Program 10.2 (page 372)—A 32-bit addition operation (P = N + M). This starts with the least significant byte. • Program 10.3 is a 32-bit subtraction algorithm. 10.2.5 Table Lookup and Interpolation • Example 10.4 – Design a fixed-point sin() function 10.3 Expression Evaluations • The precision of the expression can be increased using subroutine calls. – See Program 10.14 and 10.15 (page 382) 10.4 IEEE Floating-Point Numbers • IEEE Standard for Binary Floating-Point Arithmetic—ANSI/IEEE STD 754-1985. • Most widely used format for floating point numbers.