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Computer Architecture Lecture 3 Combinational Circuits

NYU. Computer Architecture Lecture 3 Combinational Circuits. Ralph Grishman September 2015. Time and Frequency. time = 1 / frequency frequency = 1 / time units of time millisecond = 10 -3 second microsecond = 10 -6 second nanosecond = 10 -9 second picosecond = 10 -12 second

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Computer Architecture Lecture 3 Combinational Circuits

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  1. NYU Computer ArchitectureLecture 3Combinational Circuits Ralph Grishman September 2015

  2. Time and Frequency • time = 1 / frequency • frequency = 1 / time • units of time • millisecond = 10-3 second • microsecond = 10-6 second • nanosecond = 10-9 second • picosecond = 10-12 second • units of frequency • kiloHertz (kHz) = 103 cycles / second • megaHertz (MHz) = 106 cycles / second • gigaHertz (GHz) = 109 cycles / second Computer Architecture lecture 3

  3. Today’s Problem • A typical clock frequency for current PCs is 2 GHz. What is the corresponding clock period? (a) 200 ps (b) 500 ps (c) 2 ns (d) 5 ns Computer Architecture lecture 3

  4. Solution • Frequency = 2 GHz = 2 * 109 Hz • Period = 1 / frequency = 1 / (2 * 109) sec = (1 / 2) * (1 / 109) sec  = 0.5 * 10-9 sec = 0.5 ns = 500 * 10-6 sec = 500 ps Computer Architecture lecture 3

  5. Assignment #1 • various short questions about combinational circuits Computer Architecture lecture 3

  6. Design tools • see lecture outline Computer Architecture lecture 3

  7. Propagation Delay • delay of individual transistor -- how fast it can switch -- determined by physical factors (e.g., size) • speed of transistor determines speed of gate voltage in out time Computer Architecture lecture 3

  8. Propagation Delay • the propagation delay (speed) of a combinatorial circuit is the length of time from the moment when all input signals are stable until the moment when all outputs have stabilized Computer Architecture lecture 3

  9. Propagation Delay • propagation delay of a combinatorial circuit can be determined as longest path (in number of gates) from any input to any output delay=2 Computer Architecture lecture 3

  10. A Very Rough Estimate • After transistor switches, it has to charge output wires • this may be a large part of total delay • so assuming all gate delays are the same produces a very rough estimate of circuit delays • but is good enough for understanding principles of circuit design • so we will make that assumption in this course Computer Architecture lecture 3

  11. Fan-in • sum-of-products form suggests any combinatorial function can be computed in 3 gate delays (one delay for inverters, one for ANDs, one for OR) Computer Architecture lecture 3

  12. Fan-in • but gates are limited in their fan-in (number of inputs a gate has) Computer Architecture lecture 3

  13. Fan-in • for example, if fan-in is f, it takes log (base f) n gate delaysto OR or AND together ninputs log2 8 = 3 gate delays Computer Architecture lecture 3

  14. Adders • The simplest case: adding two one-bit numbers • Sum = A xor B • Carry = A and B Computer Architecture lecture 3

  15. n-bit Adder • adding multi-bit numbers: • have to keep track of a carry out of one bit position and into the next position to the left 0 0 1 1 + 0 0 0 1 0 1 0 0 Computer Architecture lecture 3

  16. n-bit Adder • Do this with full adders, which have 3 inputs: A, B, and Cin, and 2 outputs, Sum and Cout. Computer Architecture lecture 3

  17. Full Adder • We will show the connections of the full adder as follows: A B Cout Cin Sum Computer Architecture lecture 3

  18. n-bit Adder • Then we can draw a 3-bit adder like so: A2 A1 A0 B2 B1 B0 Cin Cin Cout Cout Sum2 Sum0 Sum1 Computer Architecture lecture 3

  19. n-bit adder: delay • ripple-carry adder: carry ripples from bit 0 to high-order bit • total delay (for large n) = n * delay(CinCout) Computer Architecture lecture 3

  20. Signed Numbers • So far we assumed the bits represent positivve numbers: Computer Architecture lecture 3

  21. Signed Numbers • We could use some of the bit patterns to represent negative numbers, like so: sign and magnitude Computer Architecture lecture 3

  22. Signed Numbers • Or like so: two’s complement Computer Architecture lecture 3

  23. Signed Numbers • Or even like so: Computer Architecture lecture 3

  24. Why do we prefer two’s complement? Computer Architecture lecture 3

  25. Why do we prefer two’s complement? • Can use same logic as for unsigned addition Computer Architecture lecture 3

  26. Computing two’s complement • Given representation of v, how to compute representation of –v ? Computer Architecture lecture 3

  27. Computing two’s complement • Given representation of v, how to compute representation of –v: • flip every bit in representation of v • add 1 Computer Architecture lecture 3

  28. Computing two’s complement A2 A0 A1 0 0 1 Cin Cin Cout Cout Acomp2 Acomp0 Acomp1 Computer Architecture lecture 3

  29. Subtracting B – A = B + (-A) A2 A0 A1 0 0 1 Cin Cin Cout Cout B1 B2 B0 Acomp2 Acomp0 Acomp1 Computer Architecture lecture 3

  30. Can we simplify this? Computer Architecture lecture 3

  31. Subtracting: B – A = B + (-A) A2 A0 A1 B2 B1 B0 Cin Cin Cout Cout 1 (A-B)2 (A-B)0 (A-B)1 Computer Architecture lecture 3

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