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ECE 3110: Introduction to Digital Systems

ECE 3110: Introduction to Digital Systems. Signed Addition/Subtraction unsigned Multiplication/Division. Previous class Summary. Signed Conversions. signed addition/subtraction. Two’s-complement Addition rules Subtraction rules Overflow

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ECE 3110: Introduction to Digital Systems

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  1. ECE 3110: Introduction to Digital Systems Signed Addition/Subtraction unsigned Multiplication/Division

  2. Previous class Summary • Signed Conversions

  3. signed addition/subtraction • Two’s-complement • Addition rules • Subtraction rules • Overflow • An operation produces a result that exceeds the range of the number system.

  4. Two’s Complement Overflow Consider two 8-bit 2’s complement numbers. I can represent the signed integers -128 to +127 using this representation.What if I do (+1) + (+127) = +128. The number +128 is OUT of the RANGE that I can represent with 8 bits. What happens when I do the binary addition? +127 = 7F16 + +1 = 0116------------------- 128 /= 8016 (this is actually -128 as a twos complement number!!! - the wrong answer!!!)

  5. Detecting Two’s Complement Overflow Two’s complement overflow occurs is: Add two POSITIVE numbers and get a NEGATIVE result Add two NEGATIVE numbers and get a POSITIVE resultWe CANNOT get two’s complement overflow if I add a NEGATIVE and a POSITIVE number together. The Carry out of the Most Significant Bit means nothing if the numbers are two’s complement numbers.

  6. Some Examples All hex numbers represent signed decimal in two’s complement format. FF16 = -1 + 0116 = + 1-------- 0016 = 0Note there is a carry out, but the answer is correct. Can’t have 2’s complement overflow when adding positive and negative number. FF16 = -1 + 8016 = -128-------- 7F16 = +127 (incorrect!!) Added two negative numbers, got a positive number. Twos Complement overflow.

  7. Adding Precision (unsigned) What if we want to take an unsigned number and add more bits to it? Just add zeros to the left. 128 = 8016 (8 bits) = 008016 (16 bits) = 0000008016 (32 bits)

  8. Adding Precision (two’s complement) What if we want to take a twos complement number and add more bits to it? Take whatever the SIGN BIT is, and extend it to the left. -128 = 8016 = 100000002 (8 bits) = FF8016 = 11111111100000002 (16 bits) = FFFFFF8016 (32 bits) + 127 = 7F16 = 011111112 (8 bits) = 007F16 = 00000000011111112 ( 16 bits) = 0000007F16 (32 bits)This is called SIGN EXTENSION. Extending the MSB to the left works for two’s complement numbers and unsigned numbers.

  9. Signed 2’s-complement and unsigned binary numbers • Basic binary addition and subtraction algorithms are same • Interpretation may be different. 1101+1110

  10. Unsigned multiplication/division • Multiplication: shift-and-add: partial product • Division: shift-and-subtract • Overflow: • divisor is 0 or • Quotient would take more than m bits

  11. Unsigned binary multiplication: example • 1011 x 1101=? • Add each shifted multiplicand as it is created to a partial product.

  12. What do you need to know? • Addition, subtraction of binary, hex numbers • Detecting unsigned overflow • Number ranges for unsigned, SM, 1s complement, 2s complement • Overflow in 2s complement • Sign extension in 2s complement

  13. Next… [Chapter 2.10--2.16] • BCD, GRAY, and other codes

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