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Lo’ai Tawalbeh Lecture #2

Lo’ai Tawalbeh Lecture #2. Digital Logic Review. Standard combinational modules: decoders, encoders and Multiplexers. 1/3/2005. Decoders. General decoder structure Typically n inputs, 2 n outputs 2-to-4, 3-to-8, 4-to-16, etc. Binary 2-to-4 decoder. Note: “x” = (don’t care) cases.

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Lo’ai Tawalbeh Lecture #2

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  1. Lo’ai TawalbehLecture #2 Digital Logic Review Standard combinational modules: decoders, encoders and Multiplexers 1/3/2005 cpe 252: Computer Organization

  2. Decoders • General decoder structure • Typically n inputs, 2n outputs • 2-to-4, 3-to-8, 4-to-16, etc. cpe 252: Computer Organization

  3. Binary 2-to-4 decoder Note: “x” = (don’t care) cases. cpe 252: Computer Organization

  4. Decoder Use – Operation Decoding • Microprocessor instruction decoding opcode field instruction other fields 4-input binary decoder 1 En 0 1 2 ………. 15 jump load decoded instructions store add cpe 252: Computer Organization

  5. 2-to-4-decoder logic diagram cpe 252: Computer Organization

  6. E x2 x1 x0 x y7 y6 y5 y4 y3 y2 y1 y0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 2 0 0 0 0 0 1 0 0 1 0 1 1 3 0 0 0 0 1 0 0 0 1 1 0 0 4 0 0 0 1 0 0 0 0 1 1 0 1 5 0 0 1 0 0 0 0 0 1 1 1 0 6 0 1 0 0 0 0 0 0 1 1 1 1 7 1 0 0 0 0 0 0 0 0 - - - - 0 0 0 0 0 0 0 0 3-input Binary Decoder Inputs: x = (x2, x1, x0), with xi in {0,1} and E in {0,1} Outputs: y = (y7,y6,y5,…,y1,y0) with yi in {0,1} Function: yi = E. mi(x), i = 0,1,…,7 cpe 252: Computer Organization

  7. z2 x2 x1 x0 z2 z1 z0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 1 2 3 4 5 6 7 x2 x1 x0 2 1 0 z1 Binary Decoder z0 E 1 Implementing functions using a Binary Decoder and OR Gates Function Remember that any function can be represented as a sum of minterms cpe 252: Computer Organization

  8. Binary Encoders • Only one input Ij has value 1 at any given time • Output Y corresponds to the binary code of j when Ij = 1 cpe 252: Computer Organization

  9. 8-to-3 Binary Encoders Y0 = I1 + I3 + I5 + I7 (odd input indices) Y1 = I2 + I3 + I6 + I7 Y2 = I4 + I5 + I6 + I7 (indices > 3) cpe 252: Computer Organization

  10. 1 0 1 1 0 1 1 0 1 0 0 0 1 1 0 Multiplexers data inputs 0 1 2 3 MUX 1 0 selection inputs cpe 252: Computer Organization

  11. 4-input Multiplexer cpe 252: Computer Organization

  12. Typical Multiplexer use selection between multiple paths to a functional unit cpe 252: Computer Organization

  13. Multiplexers as universal modules • Universal module: using only this module you are able to implement ANY logic function. • NAND and NOR gates for example are universal gates. • Question: how do you assign inputs for the multiplexer in order to implement a given function? cpe 252: Computer Organization

  14. Exercise Implement the following function using: a) 8-input multiplexer. b) 4-input multiplexer. F=x,y,z(1,2,6,7) cpe 252: Computer Organization

  15. Lo’ai TawalbehLecture #2 Digital Logic Review Signed and Unsigned Numbers cpe 252: Computer Organization

  16. 4-bit Unsigned Numbers • Range of values for n-bit vector is: • 0 ≤ x ≤ (2^n-1) cpe 252: Computer Organization

  17. Representation of Signed Integers and Basic Operations • Two common representations • Sign and Magnitude (SM) • True and Complement (TC) • In both cases there is a mapping from signed values to positive values. cpe 252: Computer Organization

  18. Sign and Magnitude • x represented by a pair (s,m) where • s is the sign: 0 for positive and 1 for negative • m is the magnitude • example: (-23)10 = -(10111) = (1,10111) • Range of values for n-bit vector (n-1 bits for m) • - (2n-1 – 1) ≤ x ≤ (2n-1 – 1) • Two representations for zero cpe 252: Computer Organization

  19. 2’s complement • No separation between sign and magnitude • Signed integer x represented by positive integer xR such that: Example: n=4, 2^4=16. To represent x = -7; xR = 9 • Range of values for n-bit vector ( 2’s comlement) • - (2n-1 ) ≤ x ≤ (2n-1 – 1) cpe 252: Computer Organization

  20. 4-bit Two’s Complement Numbers cpe 252: Computer Organization

  21. Change of Sign • complement each bit of x • add 1 Example: n = 4 x = (0011)2= 3 x’ = 1100 + 1 1101  representation of -3 cpe 252: Computer Organization

  22. Positive integer addition/subtraction cpe 252: Computer Organization

  23. Two’s complement addition/subtraction • Addition: same as positive integer addition • just discard carry out • Subtraction: x - y • step 1: change the sign of y to obtain -y • step 2: add x and -y • Example: x = 8, y = 5, 5-bit vectors y = 00101 -y = 11011 <<< change sign x - y = 01000 11011 + 00011 << carry out discarded cpe 252: Computer Organization

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