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

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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.

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

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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).

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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?

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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.

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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.

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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.

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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

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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.

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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?

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Other Conversion Techniques Table 3.7 illustrates conversion of -100 to signed 8-bit binary (page 63). Other techniques 1.Convert them into unsigned binary, then do a two’s complement negate. 2.Add 256 to the number, then conert the unsigned result to binary using the unsigned method.

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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?

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Other schemes Sign-Magnitude Representation --if b7 is a 1, then the number is negative. –Problems: 1.No basis function 2.Two representations for the number zero. 3.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.

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

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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.

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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.

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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.

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

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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.

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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.

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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.

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

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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.

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

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

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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.

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

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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?

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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.

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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.

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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).

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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.

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

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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.

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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

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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.

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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.

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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.

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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).

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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.

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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

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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)

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

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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)

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3.10 Character Information ASCII—American Standard Code for Information Interchange. Usually 7 bits with 8 th bit (MSB) set to 0. See Table 3.21 (page 98). ISO/IEC 8859 uses the 8 th 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.

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3.11 Conversions C-examples illustrate the conversion process.

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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.

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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.

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