basic_gates.ppt
- 2. 2 Review of Boolean algebra • Just like Boolean logic • Variables can only be 1 or 0 – Instead of true / false
- 3. 3 Review of Boolean algebra • Not is a horizontal bar above the number – 0 = 1 – 1 = 0 • Or is a plus – 0+0 = 0 – 0+1 = 1 – 1+0 = 1 – 1+1 = 1 • And is multiplication – 0*0 = 0 – 0*1 = 0 – 1*0 = 0 – 1*1 = 1 _ _
- 4. 4 Review of Boolean algebra • Example: translate (x+y+z)(xyz) to a Boolean logic expression – (xyz)(xyz) • We can define a Boolean function: – F(x,y) = (xy)(xy) • And then write a “truth table” for it: _ _ _ x y F(x,y) 1 1 0 1 0 0 0 1 0 0 0 0
- 5. 5 Basic logic gates • Not • And • Or • Nand • Nor • Xor x x x y xy x y xyz z x+y x y x y x+y+z z x y xy x+y x y xÅy x y
- 6. 6 Converting between circuits and equations • Find the output of the following circuit • Answer: (x+y)y – Or (xy)y x y x+y y (x+y)y __ x y y
- 7. 7 x y Converting between circuits and equations • Find the output of the following circuit • Answer: xy – Or (xy) ≡ xy x y x y x y _ _ ___
- 8. 8 Converting between circuits and equations • Write the circuits for the following Boolean algebraic expressions a) x+y x y x y x y __ x x+y
- 9. 9 x y x y x y Converting between circuits and equations • Write the circuits for the following Boolean algebraic expressions b) (x+y)x _______ x y x+y x+y (x+y)x
- 10. 10 Writing xor using and/or/not • x Å y (x + y)(xy) x y xÅy 1 1 0 1 0 1 0 1 1 0 0 0 x y x+y xy xy (x+y)(xy) ____
- 11. 11 Converting decimal numbers to binary • 53 = 32 + 16 + 4 + 1 = 25 + 24 + 22 + 20 = 1*25 + 1*24 + 0*23 + 1*22 + 0*21 + 1*20 = 110101 in binary = 00110101 as a full byte in binary • 211= 128 + 64 + 16 + 2 + 1 = 27 + 26 + 24 + 21 + 20 = 1*27 + 1*26 + 0*25 + 1*24 + 0*23 + 0*22 + 1*21 + 1*20 = 11010011 in binary
- 12. 12 Converting binary numbers to decimal • What is 10011010 in decimal? 10011010 = 1*27 + 0*26 + 0*25 + 1*24 + 1*23 + 0*22 + 1*21 + 0*20 = 27 + 24 + 23 + 21 = 128 + 16 + 8 + 2 = 154 • What is 00101001 in decimal? 00101001 = 0*27 + 0*26 + 1*25 + 0*24 + 1*23 + 0*22 + 0*21 + 1*20 = 25 + 23 + 20 = 32 + 8 + 1 = 41
- 13. 13 A note on binary numbers • In this slide set we are only dealing with non-negative numbers • The book (section 1.5) talks about two’s- complement binary numbers – Positive (and zero) two’s-complement binary numbers is what was presented here – We won’t be getting into negative two’s- complmeent numbers
- 14. 15 How to add binary numbers • Consider adding two 1-bit binary numbers x and y – 0+0 = 0 – 0+1 = 1 – 1+0 = 1 – 1+1 = 10 • Carry is x AND y • Sum is x XOR y • The circuit to compute this is called a half-adder x y Carry Sum 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0
- 15. 16 The half-adder • Sum = x XOR y • Carry = x AND y x y Sum Carry x y Sum Carry x y Carry Sum 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 0
- 16. 17 Using half adders • We can then use a half-adder to compute the sum of two Boolean numbers 1 1 0 0 + 1 1 1 0 0 1 0 ? 0 0 1
- 17. 18 How to fix this • We need to create an adder that can take a carry bit as an additional input – Inputs: x, y, carry in – Outputs: sum, carry out • This is called a full adder – Will add x and y with a half-adder – Will add the sum of that to the carry in • What about the carry out? – It’s 1 if either (or both): – x+y = 10 – x+y = 01 and carry in = 1 x y c carry sum 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 0
- 18. 19 HA X Y S C HA X Y S C x y c c s HA X Y S C HA X Y S C x y c The full adder • The “HA” boxes are half-adders x y c s1 c1 carry sum 1 1 1 0 1 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 s1 c1
- 19. 20 The full adder • The full circuitry of the full adder x y s c c
- 20. 21 Adding bigger binary numbers • Just chain full adders together HA X Y S C FA C Y X S C FA C Y X S C FA C Y X S C x1 y1 x2 y2 x3 y3 x0 y0 s0 s1 s2 s3 c . . .
- 21. 22 Adding bigger binary numbers • A half adder has 4 logic gates • A full adder has two half adders plus a OR gate – Total of 9 logic gates • To add n bit binary numbers, you need 1 HA and n-1 FAs • To add 32 bit binary numbers, you need 1 HA and 31 FAs – Total of 4+9*31 = 283 logic gates • To add 64 bit binary numbers, you need 1 HA and 63 FAs – Total of 4+9*63 = 571 logic gates
- 22. 23 More about logic gates • To implement a logic gate in hardware, you use a transistor • Transistors are all enclosed in an “IC”, or integrated circuit • The current Intel Pentium IV processors have 55 million transistors!
- 23. 24 Flip-flops • Consider the following circuit: • What does it do?
- 24. 25 Memory • A flip-flop holds a single bit of memory – The bit “flip-flops” between the two NAND gates • In reality, flip-flops are a bit more complicated – Have 5 (or so) logic gates (transistors) per flip- flop • Consider a 1 Gb memory chip – 1 Gb = 8,589,934,592 bits of memory – That’s about 43 million transistors! • In reality, those transistors are split into 9 ICs of about 5 million transistors each
- 25. 26 Hexadecimal • A numerical range from 0-15 – Where A is 10, B is 11, … and F is 15 • Often written with a ‘0x’ prefix • So 0x10 is 10 hex, or 16 – 0x100 is 100 hex, or 256 • Binary numbers easily translate:
- 28. 29 DEADBEEF • Many IBM machines would fill allocated (but uninitialized) memory with the hexa- decimal pattern 0xDEADBEEF – Decimal -21524111 – See http://www.jargon.net/jargonfile/d/DEADBEEF.html • Makes it easier to spot in a debugger
- 29. 30 • 0xDEAD = 57005 • Now add one to that...