Binary Representation and Logic Design for Computing
Explore the basics of binary representation, digital technology, number systems, signed numbers, and twos complement in computer logic design. Learn about the fundamental concepts, terminology, and methodologies used in processing binary values for computing applications.
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Number Representation and Logic Design CS 3220 Fall 2015 A C Hadi Esmaeilzadeh hadi@cc.gatech.edu Georgia Institute of Technology T Some slides adopted from Prof. Milos Prvulovic Alternative,Computing,Technologies
Computing with digital technology Physical Layer Low voltage (0 V) High voltage (5 V, then 3.3 V, 1.1 V, and now is 0.9 V or lower) Abstraction (we do not deal with voltage levels) 0 1 Groups of binary values construct words, numbers, pixels, audio signals, 2
What does this binary value represent? 01001000011000010110010001101001 = 0100_1000 0110_0001 0110_0100 0110_1001 3
What does this binary value represent? 01001000011000010110010001101001 = 0100_1000 0110_0001 0110_0100 0110_1001 Hadi 1214342249 230801.64 4
Terminology Physical: Bit: one binary digit Abstract: Nibble: four binary digits Byte: eight binary digits = two nibbles Word: Usually 32 binary digits = 4 bytes 5
Number Representation Positional notation Same as base-10 but now it s base 2: In base 10, we have 9807 = 9*103 + 8*102 + 0*101 + 7*100 In base 2, we have 1011 = 6
Signed numbers Easy: one bit for sign, then absolute value E.g. 1011 (- 011) is actually -3 How do we add two such numbers? First check the sign bits If both are 1 or both 0, add the absolute values and retain the same sign bit If one is 1 and one is 0, compare the two absolute values, then subtract the smaller from the larger and use the sign of the larger number Lots of circuitry needed for all this! Also we have to representation for zero: (-0, +0) We need a better way! 7
Twos complement OK, let s say we want 4-bit signed numbers but Want to just add the numbers as if they were unsigned Want to quickly tell if number is positive or negative So if we add 1 to -1 we should get 0 0 is 0000, 1 is 0001, so -1 has to be 1111 Now, if we add 1 to -2, we should get -1 So -2 has to be 1110 We can still tell if positive or negative But add, subtract, etc. is much simpler now 8
Twos complement What is the range? 9
Twos complement Quickly negate a 2 s number: 1011 Invert(all bits) + 1 Start from right, copy until the first 1 , then invert the remaining Sign extension: Store 1011 in a byte 10
Note on Number Representation Digital logic still operates on binary signals 2 s complement vs. sign-and-value is all about how we choose to represent numbers using the underlying binary signals If four wires have values of 1, 0, 1, 1, then If sign-and-value, it represents -3 If 2 s complement, it is -5 If unsigned number, it is 11 (eleven) May not even be a number! 11
Hexadecimal Notation Writing binary numbers is inconvenient More than 3 times as many digit as decimal notation So we also use hexadecimal (base 16) notation Fewer digits needed than in binary (or even decimal) Each hex digit represents exactly 4 binary digits, so it is easy to convert back-and-forth Example: Hexadecimal E04C is in binary: Note: no actual hexadecimal hardware Hardware still operates in binary Hexadecimal notation is only for our convenience 12
Digital Logic Implemented using MOS transistors Source Drain Gate - - - - - - P-type substrate N-type Channel 13
MOS transistor +V +V +V 14
Inverter (NOT gate) +V 15
NOR +V 16
NAND +V 17
XOR? 19
1-bit add A B Carry OUT OUT 21
Full Adder A B Carry OUT Carry IN OUT 22
Full adder A B S (Ouptut) Cin Cout A B 23
3-bit add? A2 B2 A1 B1 A0 B0 1-bit Full Adder 1-bit Full Adder 1-bit Full Adder Cout Cin Cout Cin Cout Cin S2 S1 S0 Data dependence serializes the additions! 26
Keeping State We use latches and flip-flops Here is an SR latch 27
D latch When E is 1 When D=1, make the S signal be 0 (OUT -> 1) When D=0, make the R signal be 0 (OUT -> 0) D(ata) S OUT E(nable) OUT R D (inverted D input) When E is 0, both S and R are 1 (OUT unchanged) 28
Flip-Flop? Essentially two latches in series: Latch 1 has CLK connected to its Enable Keeps latching changes in input value while clock is 0 When clock becomes 0, it keeps what it had Latch 2 has CLK connected to its enable Keep latching the output of Latch 1 while clock is 1 Keeps same output value when clock is 0 While clock is 0, Latch 2 outputs the stored bit When clock becomes 1, Latch 2 outputs 29
How the flip-flop works While clock is 0 Output of Latch 1 follows the input But Latch 2 outputs the stored bit When clock changes from 0 to 1 Latch 1 stops following the input Latch 2 now outputs what Latch 1 is outputting Result: FF output == FF input when clock went 0->1 When clock changes back to 0 Latch 1 starts following the input again But Latch 2 now keeps what it had Result: FF output unchanged until clock goes 0->1 30
