Ch.3 Representation and Manipulation of Information

1. Ch.3 Representation and Manipulation of Information From Introduction to Embedded Systems: Interfacing to the Freescale 9s12 by Valvano, published by CENGAGE

2. 3.1 Precision • The number of distinct or different values (page 57). • Alternatives—the total number of possibilities—decimal digits, bytes, or binary bits. • Table 3.1 shows the relationship between bits, bytes and alternatives.

3. Checkpoint • Checkpoint 3.1 How many bytes of memory would it take to store a 50 bit number?

4. Precision (cont.) • Decimal digita are used to specify precision of measurement systems that display results as numerical values. • Table 3.2 (page 58 of text). • (1/2) decimal digit –a digit that can be 0 or 1. • Abbreviations for large numbers (Table 3.3).

5. Checkpoints • Checkpoint 3.2: How many binary bits is equivalent to 3 and (1/2) decimal digits. • Checkpoint 3.3: About how many decimal digits is 64 binary bits? You can this without a calculator, just using the “rule of thumb” (see page 58). • Checkpoint 3.4 A 2 tebibyte storage system can store how many bytes?

6. 3.2 Boolean Information • Two states—logical true and false. • When interfacing to a light, motor, or a heater, the Boolean could mean on and off. • Positive logic—False is all zeros; true is any nonzero value. • Negative logic—absence of a voltage is true and the presence of a voltage is false.

7. 3.3 8-bit numbers • Value of an unsigned number for 8-bits • N = 128*b7 + 64*b6 + 32*b5 + 16*b4 + 8*b3 + 4* b2 + 2*b1 + b0. • Table 3.4 (page 61 of the text)—shows examples of conversions.

8. Conversion and Basis Elements • The basis of a number sysstem is a subset from which linear combinations of the basis elements can be used to construct the entire set. • Algorithm: • Start with MSB. • Do we need the basis element for the number? • If yes, then the bit is a 1—if no, then it is a 0. • Continue to the next basis element. • See Table 3.5, page 61 for an example.

9. Checkpoints • Checkpoint 3.6: Convert the binary number %01101010 to unsigned decimal. • Checkpoint 3.7: Convert the hex number $45 to unsigned decimal. • Checkpoint 3.8: In this conversion algorithm, how can we tell if a basis element is needed? • Checkpoint 3.9 Give the representations of the decimal 45 in 8-bit binary and hexadecimal • Checkpoint 3.10 Give the representations of the decimal 200 in 8-bit binary and hexadecimal

10. Other Number Schemes for Negative Number Representation • One’s complement — complement each bit. • 0001 1001 –the one’s complement is 1110 0110—this is the negative of the number. • Problems: two representations for 0 and no basis elements. • Two’s complement — complement each bit then add 1 to the result. • 0001 1001 when negated becomes 1110 0111. • N = -128*b7 + 64*b6 + 32*b5 + 16 * b4 + 8 * b3 + 4*b2 + 2*b1+ b0.

11. Checkpoints • Checkpoint 3.11 Convert the signed binary number%11101010 to signed decimal. • Checkpoint 3.12 Are the signed and unsigned decimal representations of the 8-bit number $45 the same or different?

12. Other Conversion Techniques • Table 3.7 illustrates conversion of -100 to signed 8-bit binary (page 63). • Other techniques • Convert them into unsigned binary, then do a two’s complement negate. • Add 256 to the number, then conert the unsigned result to binary using the unsigned method.

13. Checkpoints • Checkpoint 3.13: Give the representations of -45 in 8-bit binary and hexadecimal. • Checkpoint 3.14: Why can’t you represent the number 200 using 8-bit signed binary?

14. Other schemes • Sign-Magnitude Representation --if b7 is a 1, then the number is negative. • Problems: • No basis function • Two representations for the number zero. • Different hardware is needed for addition and subtraction (unlike two’s complement). • Binary Coded Decimal (BCD)—easy for humans to read—each decimal digit is represented by a 4-bit binary.

15. Checkpoint • Checkpoint 3.15: What binary values are used to store the number 25 in 8-bit BCD format?

16. 3.4 16-bit Numbers • Word or double byte. • Extension of the 8-bit concept. • See Figure 3.4 (page 64). • See Table 3.9 (page 65). • Two’s Complement—Table 3.10.

17. Checkpoints • Checkpoint: 3.16: Convert the 16-bit binary number %0010 0000 0110 1010 to unsigned decimal. • Checkpoint 3.17: Convert the 16-bit hex number $1234 to unsigned decimal. • Checkpoint 3.18: Convert the unsigned decimal number 1234 to 16-bit hexadecimal. • Checkpoint 3.19: Convert the unsigned decimal number 10000 to 16-bit binary. • Checkpoint 3.20: Convert the 16-bit hex number $1234 to signed decimal. • Checkpoint 3.21: Convert the 16-bit hex number $ABCD to signed decimal. • Checkpoint 3.22: Convert the signed decimal number 1234 to 16-bit hexadecimal. • Checkpoint 3.23: Convert the signed decimal number -100000 to 16-bit binary.

18. 3.5 Extended Precision Numbers • Unsigned numbers with n bits (see page 66). • Two’s complement n-bit numbers. • Binary Coded Decimal n-bit numbers.

19. Checkpoint • Checkpoint 3.24: What hexadecimal values are used to store the number 3456 in 16-bit BCD format?

20. 3.6 Logical Operations • Unary Operations—produces its result given a single input parameter—negate, increment, decrement. • Logical Not Operation—Figure 3.5. • Binary Operations—produce a single result given two inputs—AND(&), OR(|), and exclusive OR(^)—Table 3.12, Figure 3.6.

21. Logical Operations and the 9S12 • Operations are performed in a bit-wise fashion. • N bit will be set if the result is negative. • Z bit will be set if the result is zero. • Logical operations at bottom of page 68 will clear the V bit (signed overflow) and leave the C bit unchanged.

22. Examples (pages 69-74) • 3.1 Write software to set bit 4 and clear bits 1 and 0 of an 8-bit variable N. (page 69). • 3.2 Write software that sets a global variable to true if a switch is pressed. • 3.3 Write software that make PT4 and PT5 outputs and clears both outputs without affecting the other bits of PTT. • 3.4 Write software that togles the PT 3 output without affecting the other bits of PTT. • 3.5 Generate two out-of-phase square waves a shown in Figure 3.8 (page 72). • 3.6 The goal is develop a means for the microcontroller to turn on and turn off an AC-powered appliance. The interface will use a solid-state relay with a control parameters of 2 V and 10 ma. Write necessary subroutines to operate the system.

23. Digital Storage Elements • Figure 3.11— (page 74) • Table 3.14—D flip-flops

24. 3.7 Shift Operations • In assembly language, the shift is a unary operation and is for one bit. • lsr – logical shift right • asr– arithmetic shift right • lsl– logical shift left • asl – arithmetic shift left • C will contain the carry out. • Figure 3.13 – 3.16 illustrates the operations. • Roll (ror, rol) — operations can be used to create multiple-byte shift functions (Fig. 3.7). • See page 77 for a list with related registers.

25. Example 3.7 • Write assembly code to implement M=N>>2, where M and N are 16-bit unsigned variables. • Solution— • ldd N • lsrd • lsrd • std M

26. Checkpoint • Checkpoint 3.31: Let N and M be 8-bit signed locations. Write assembly code to implement M = 4*N.

27. Example 3.8 • Take two 4-bit nibbles and combine them into one 8-bit value. • Solution—Use the shift operation to move the bits into position, then use the or operation to combine the two parts into one number.—See page 78.

28. 3.8 Arithmetic operations: Addition and Subtractions. • Operations are performed using hardware. • Overflows occur and have to be checked.

29. Checkpoints • Checkpoint 3.32: How many bit does it take to store the result of two unsigned 8-bit numbers added together? • Checkpoint 3.33: How many bits does it take to store the result of two signed 8-bit numbers added together? • Checkpoint 3.34: How many bits does it take to store the result of two unsigned 8-bit numbers multiplied together? • Checkpoint 3.35: How many bit does it take to store the result of two signed 8-bit numbers multiplied together?

30. 3.8 Arithmetic Operations (cont.) • Four of the condition code bits stored in the Condition Code Register (CCR) are used in Addition/Subtraction. • See Table 3.16, page 79. • The adda and addb instructions work for both signed and unsigned data. • N, Z, V (signed overflow), and C (unsigned overflow) are set as shown (page 79.

31. Example 3.9 (page 79) • Write assembly code to implement M = N+10, where M and N are 8-bit variables. • Solution: • Perform an 8-bit read to get N into RegA. • 10 is added to Reg A. • Result is stored in M. • C and V bits are set when overflows occur on unsigned and signed operations.

32. Arithmetic Operations –16 bit numbers • The addd instruction can be used to add 16 bit numbers as discussed at the top of page 80. • N, Z, V, and C are set as needed (see page 80).

33. Example 3.10 • Write assembly code to implement M = N + 1000, where M and N are 16-bit variables. • Solution • Need to use 16-bit register (D). • 16 bit read to get N (use ldd N). • Add 1000 (addd #1000). • Store the result in M ( std M). • Check C or V (whichever is appropriate) for possible overflows.

34. Checkpoint • Checkpoint 3.36: Wrie assembly code that adds a constant 100 to Register X.

35. Subtraction and Compare • The instructions at the bottom of page 80 show that compare instructions subtract a value from memory. • The other instructions (subtraction and test) all subtract values as shown. • 16-bit instructions are shown on page 81. • Note that the programmer keeps track of the values being signed or unsigned, since the computer sets both C and V.

36. Example 3.11 • Write assembly code to implement M = N-10, where M and N are 8-bit variables. • Solution (page 81) • ldaa N • suba #10 • staa M

37. Example 3.12 • Write assembly code to implement M = N-1000, where M and N ar 16-bit variables. • Solution: (page 81) • ldd N • subd #1000 • std M • Object and source code are illustrated on page 82.

38. Adder/Subtractor Hardware • Figure 3.18 shows a binary full adder. • Figure 3.19 shows an 8-bit adder using 8 full-adders. • Consider the operation adda #64 • The contents of RegA and constant binary 64 are placed at the inputs of the hardware. • The result is placed in Reg A. • Condition codes are set. • Figure 3.20 Shows a number wheel description. • Figure 3.21 Shows how a two’s complement approach to subtraction can use full-adders. • Figure 3.22 Shows a number wheel for subtraction. • More number wheels—Fig. 3.23, Fig. 3.24.

39. Checkpoints • Checkpoint 3.37 Assume Register A is initially -100. After executing the instruction adda #64 what is the value in Register A, and the NZVC bits. • Checkpoint 3.38 Assume Register A is initially -100. After executing the instruction adda #64 what is the value in Register A, and the NZVC bits. • Checkpoint 3.39 Assume Register A is initially 200. After executing the instruction suba #-64 what is the value in Register A, and the NZVC bits? • Checkpoint 3.40 Assume Register A is initially 200. After executing the instruction suba #64 what is the value in Register A, and the NZVC bits.

40. Error Handling • Usually only have to deal with C or V. • Promotion involves the increasing of the precision of the input numbers. (Page 88) • Ceiling and floor —establishing upper and lower bounds for the result of an operation. (Page 90).

41. 3.9 Arithmetic Operations: Multiplication and Divide • As an embedded programmer, it is important to understand the strengths and weaknesses of the computers used. • Many embedded computers have a limited ability for mathematical operations (page 92). • If precision is not supported, then a different processor is needed (for speed) or develop software algorithms for extended precision (slower). • A combination of shifts and additions can be used.

42. Example 3.13 • Write assembly code to implement unsigned M = 5*N + 25 where M is 16 bits and N is 8 bits. • Solution: • ldaa N ; 0 to 255 • ldab #5 • mul ; RegD = 5*N, 0 to 1275 • addd #25 ;RegD+5*N+25, 25 to 1300 • std M

43. Example 3.14 • Write assembly code to implement M = 2.3*(N + 5.5), where M is 16 bits and N is 8 bits. • Solution • (use integer operations)—page 96. • Use the idiv instruction (bottom of page 95)

44. Example 3.15 • Write assembly code to scale an unsigned 8-bit integer into a number from 0 to 500. • Solution • (see page 96)

45. Example 3.16 • Write assembly code to implement M = 12.34*N, where M and N are unsigned 16 bits. • Solution • Page 97 • fdiv instruction is used--(performs a 16-bit by 16-bit unsigned divide)

46. 3.10 Character Information • ASCII—American Standard Code for Information Interchange. • Usually 7 bits with 8th bit (MSB) set to 0. • See Table 3.21 (page 98). • ISO/IEC 8859 uses the 8th bit to define additional characters. • Unicode Standard—handles some ambiguities, but is more complex. • Figure 3.33 illustrates the concept of “null-termination for storage of a string of ASCII.

47. 3.11 Conversions • C-examples illustrate the conversion process.

48. 3.12 Debugging Monitor Using a LED • Monitors are used in real-time systems as a debugging tool (page 102). • An LED attached to a port is an example of a “Boolean” monitor.

49. Checkpoints • Checkpoint 3.45: How is the character 0 represented in ASCII? • Checkpoint 3.46: Assume Register A contains an ASCII code 0 to 9. Write assembly code that converts the ASCII code into the corresponding decimal number.