代写代考 ABC00 01 11 10 0

Logical Devices

We are here
Assembly Language Processors

Copyright By PowCoder代写 加微信 powcoder

Arithmetic Logic Units
Devices Circuits
Finite State Machines
Flip-flops
Gates Transistors

Building up from gates…
 Somecommonandmorecomplexstructures:
 Multiplexers(akamux)
 Decoders
 Seven-segment decoders
 Adders(halfandfull)  Subtractors
 Comparators
These are all
combinational circuits

Combinational Circuits
 CombinationalCircuitsareanycircuitswhere the outputs rely strictly on the inputs.
 Everything we’ve done so far and what we’ll do today is all combinational logic.
 Anothercategoryissequentialcircuitsthatwe will learn in the next few weeks.

More Karnaugh Maps

Karnaugh map review
 K-mapsprovideanillustrationofacircuit’s minterms (or maxterms), and a guide to how neighbouring terms may be combined.
Y = A·B·C + A·B·C + A·B·C + A·B·C

Karnaugh map review
 K-mapsprovideanillustrationofacircuit’s minterms (or maxterms), and a guide to how neighbouring terms may be combined.
Y = A·B·C + A·B·C + A·B·C + A·B·C
= B·C + A·C

Reminder on Reducing Circuits
 EliminatingvariablesinK-Mapsbydrawing larger (>1 element) rectangular groupings
results in a circuit with a lower cost function.  Theresultingexpressionisstillinsum-of-
products form.
 But,ifsimplified,itisnolongerinsum-of-minterms
 Note:Itisnotonlythenumberofgatesthat matters when reducing circuits, but also the number of inputs to each gate.

K-Maps – Different Notations A 3-variables map example
Important!
BC Using either notation is fine! 00 01 11 10
Y = B·C + A·C
Important!
_ _ _ _ BC BC BC BC

Helpful Hint
ABC00 01 11 10 0

More Examples w/ K-Maps
ABC00 01 11 10 0
ABC00 01 11 10 0
A BC 00 11
A BC 00 00

More Examples w/ K-Maps
ABC00 01 11 10 0
ABC00 01 11 10 0
A BC 00 11
A BC 00 00

Karnaugh map example
 Createacircuitwith four inputs (A, B, C , D), and two outputs (X, Y):
 The output X is high whenever two or more of the inputs are high.
 The output Y is high when three or more of the inputs are high.

Karnaugh map example
 Createacircuitwith four inputs (A, B, C , D), and two outputs (X, Y):
 The output X is high whenever two or more of the inputs are high.
 The output Y is high when three or more of the inputs are high.

Karnaugh map example
X = A·B + C·D + B·D + B·C + A·D + A·C

Karnaugh map example
X = A·B + C·D + B·D + B·C + A·D + A·C

Karnaugh map example
Y = A·B·D + B·C·D + A·B·C + A·C·D

Alternative for X: Maxterms X:
X = (A+C+D)·(B+C+D)·(A+B+C)·(A+B+D)

Alternative for X: Maxterms X:
X = (A+C+D)·(B+C+D)·(A+B+C)·(A+B+D)

Karnaugh map review
 Note:Therearecaseswherenocombinations are possible. K-maps cannot help in these cases.
 Example:Multi-inputXORgates.
 Outputis1iffoddnumberofinputsis1.
Y = A·B·C + A·B·C + A·B·C + A·B·C

Multiplexers

Logic devices
 Certainstructuresarecommontomany circuits, and have block elements of their own.
 e.g.,Multiplexers(shortform:mux)
 Behaviour:OutputisXifSis0,andYifSis1:
n specifies the number of bits.
 S is the select input; X and Y are the data inputs. S
n Xn0S X0nM n
2-to-1 mux

Multiplexer design
M = Y·S + X·S

Multiplexer uses
 Muxesareveryusefulwheneveryouneedto select from multiple input values.
 Example:surveillance video monitors, digital cable boxes, routers.

Demultiplexers
 Relatedtodecoders:demultiplexers.
 Doesmultiplexeroperation,inreverse.
 Example:modemsreceivingInternetdata.
n0Xn0nW MnM1nX

Mux + Demux
Source: https://upload.wikimedia.org/wikipedia/commons/e/e0/Telephony_multiplexer_ system.gif

 Decodersareessentiallytranslators.
 Translatefromtheoutputofonecircuittothe
input of another.
 Thinkofthemasprovidingamappingbetween2 different encodings!
 Example:Binarysignalsplitter
 Activatesoneoffour
output lines, based on
a two-digit binary number.

7-segment decoder
 Commonandusefuldecoderapplication.  Translatefromabinarynumbertotheseven
segments of a digital display.
 Eachoutputsegmenthasaparticular logic that defines it.
 Example: Decimal number, segment 0 4
 Activate for values: 0, 2, 3, 5, 6, 7, 8, 9.
 In binary: 0000, 0010, 0011, 0101, 0110, 0111, 1000,
 Firststep:BuildthetruthtableandK-map.

 Segment0(topsegment)ishighwheneverthe input values are 0000, 0010, 0011, 0101, 0110, 0111, 1000 or 1001, and low whenever input number is 0001 or 0100.
 Thiscreateatruthtableandmaplikethe following….
7-segment decoder
 Thesesegmentsare“active-high”, meaning that setting it high turns it on.
 Example: To display the digits 0-9
 Assumeinputisa4-digitbinarynumber

7-segment decoder
 HEX0 = X3·X2·X0 + X3·X2·X1 + X3·X2·X0 + X3·X1
 Butwait…what about input values 1010 to 1111?

“Don’t care” values
 Inputvaluesthatwillneverhappenorarenot meaningful in a given design, and so their output values do not have to be defined.
 Recorded as ‘X’ in truth-tables and K-Maps.
 IntheK-mapswecanthinkofthese“don’tcare”
values as either 0 or 1 depending on what helps us simplify our circuit.
 Note: you do NOT replace X values with all 0s or 1s, you just include each X in groupings as needed.

“Don’t care” values
 NewequationforHEX0:
HEX0 = X1 + X2·X0 +
Same number of terms, but fewer inputs = smaller gates

Again for segment 1
HEX1 = X1·X0 + X1·X0 + X2

Again for segment 2
HEX2=X2 + X1 + X0

The final 7-seg decoder
 Decodersalllookthe same, except for the inputs and outputs.
 Unlikeotherdevices, the implementation differs from decoder to decoder.
7-seg decoder

Another “don’t care” example
 Climatecontrolfan:
(not related to decoders)
 Thefanshouldturnon(F)ifthetemperatureis hot (H) or if the temperature is cold (C), depending on whether the unit is set to A/C or heating (A).
F = A·H + A·C
0 0 1 0 0 1 X X

Adder circuits

 Alsoknownasbinaryadders.
 Smallcircuitdevicesthataddtwodigitstogether.
 Combinedtogethertocreateiterative combinational circuits.
 Typesofadders:  Halfadders(HA)
 Fulladders(FA)
 RippleCarryAdder

Review of Binary Math
 Eachdigitofadecimalnumberrepresentsa power of 10:
258 = 2×102 + 5×101 + 8×100
 Eachdigitofabinarynumberrepresentsa
power of 2:
011012 = 0x24 + 1×23 + 1×22 + 0x21 + 1×20 = 1310

Decimal to Binary Conversion
 Let’ssayIgiveyounumber11indecimal. How would you represent this in binary?
 Keepdividingby2andwritedownthe remainders!
Use the quotient from previous row.
11 in decimal is 1011 in binary!
Quotient = Number / 2
Remainder = Number % 2
Least Significant Bit
Most Significant Bit

Decimal to Binary Conversion
 Let’ssayIgiveyounumber11indecimal. How would you represent this in binary?
 Keepdividingby2andwritedownthe remainders!
Use the quotient from previous row.
11 in decimal is 1011 in binary!
Quotient = Number / 2
Remainder = Number % 2
Least Significant Bit
Most Significant Bit

Hexadecimal Numbers
 Base16numbers,wherevalidvaluesare:
 0 to 9 as in decimal, and
Hex numbers are typically expressed as 0x____
 Writingabinarynumberinhex(-adecimal):
 0000010111111010 = 0000 0101 1111 1010 = 0x05fa

Unsigned binary addition
27 = 00011011
53 = 00110101 111111

Unsigned binary addition
27 = 00011011
 95+181 01011111
53 = 00110101 111111
With 8 bits we can only represent unsigned numbers 0 to255!

Half Adders
 A2-input,1-bitwidthbinaryadderthatperforms the following computations:
X0011 +Y +0 +1 +0 +1
CS 00 01 01 10
 A half adder adds two bits to produce a two-bit sum.
 Thesumisexpressedasa sumbit SandacarrybitC.

Half Adder Implementation
 Equationsandcircuitsforhalfadderunitsare easy to define (even without Karnaugh maps)
C = X·Y XY
X·Y + X·Y = XY

Half Adder Implementation
 Equationsandcircuitsforhalfadderunitsare easy to define (even without Karnaugh maps)
C = X·Y XY
S = X·Y + X·Y = XY

Full Adders
 Similartohalf-adders,but with another input Z, which
represents a carry-in bit.
 CandZaresometimeslabeledasCoutandCin.
 WhenZis0,theunitbehavesexactlylikea half adder.
 WhenZis1:
+Y +0 +1 +0 +1
+Z +1 +1 +1 +1
CS 01 10 10 11

Full Adder Design
C = X·Y + X·Z + Y·Z S = X  Y  Z

Full Adder Design
 TheCtermcanalsobe rewritten as:
C = X·Y + (X  Y)·Z
 Twotermscomefromthis:  X·Y=carrygenerate(G).
 XY=carrypropagate(P).
 Resultsinthiscircuit

Adder Cin 4
X3Y3 X2Y2 X1Y1 X0Y0
Cout FA C3 FA C2 FA C1 FA Cin
Ripple-Carry Binary Adder
 Fulladderunitsarechainedtogetherinorder to perform operations on signal vectors.

The role of Cin
 Whycan’twejusthaveahalf-adderforthe smallest (right-most) bit?
 Wecould,ifwewereonlyinterestedin addition. But the last bit allows us to do subtraction as well!
 Timeforalittlefunwithsubtraction!

What is 11111111?
 Assumeyouhavean8-bitbinarynumber.
 Howdoyourepresentnegativenumbers?  Forinstance,thenumber-1?
 Add1to11111111andseewhathappens. 11111111
100000000 00000000
 Therefore,11111111mustbe-1!

Subtractors
 Subtractorsareanextensionofadders.
 Basically,performadditiononanegativenumber.
 Beforewecandosubtraction,needto understand negative binary numbers.
 Twotypesofnumbers:
 Unsigned=allnumbersarepositive.
 Still use signed representation to perform subtraction
 Signed=allbitsareusedtostorea2’scomplement negative number.
 More common, and what we use for this course.

Two’s complement
 Firststep:getting1’scomplement:
 Given number X with n bits, take (2n-1)-X
 Negates each individual bit (bitwise NOT).
01001101  10110010 11111111  00000000
 2’scomplement=(1’scomplement+1)
 Note:Addinga2’scomplementnumbertothe original number produces a result of zero.
01001101  10110011 11111111  00000001

Signed subtraction
 Negativenumbersaregenerallystoredin2’s complement notation.
 Reminder:1’scomplementbitsarethebitwise NOT of the equivalent positive value.
 2’scomplementonemorethan1’scomplement value; results in zero when added to equivalent positive value.
 Subtraction can then be performed by using the binary adder circuit with negative numbers.

Signed 3-Digit Numbers

Rules about signed numbers
 When thinking of signed binary numbers, there are a few useful rules to remember:
 The largest positive binary number is a zero followed by all ones.
 The binary value for -1 has ones in all the digits.
 The most negative binary number is a one followed by all zeroes.
 There are 2n possible values that can be stored in an n-digit binary number.
 2n-1 are negative, 2n-1-1 are positive, and one is zero.
 For example, given an 8-bit binary number:
 Thereare256possiblevalues
 Oneofthosevaluesiszero
 128arenegativevalues(11111111to10000000)  127arepositivevalues(00000001to01111111)
-1 to -128

Practicing 2’s complement
 Assume4-bitsignednumbers,writethe following decimal numbers in binary:
2 => 0010
 -1 0 8
 Whatismaxpositivenumber?  Whatisminnegativenumber?
=> Notpossibletorepresentin4digits! => 1000
=> 7 (or 24-1 -1) => -8 (or -24-1)

At the core of subtraction
 Subtractionofanumberissimplythe addition of its negative value.
 Where the negative value is found using the 2’s complement process.
 7–3 = 7+(-3)  -3–2 = -3+(-2)

Signed Subtraction example
 7 – 3 0111
0100 = 410
 -3 – 2 1101
1011 = -510

What about bigger numbers?
 53 – 27 00110101
00110101 discarded +11100101 00011010
00011010 = 2610
 27 – 53 00011011
11100110 = -2610

Subtraction circuit
 4-bitsubtractor:X–Y
 Xplus2’scomplementofY
 Xplus1’scomplementofYplus1
Use NOT gates to get the 1’s complement of Y.
Y3 Y2 Y1 X3 X2 X1
FA C3 FA C2 FA C1 S3 S2 S1
Feed 1 as carry-in in the least significant adder.

Addition/Subtraction circuit
Y3 Y2 Y1 X3 X2 X1
FA C3 FA C2 FA C1 S3 S2 S1
If Sub is 0 => addition.
If Sub is 1 => subtraction.
 The full adder circuit can be expanded to incorporate the subtraction operation.
 Remember: 2’s complement = 1’s complement + 1  WeneedSubtoprovidethevalueforCin

Food for Thought
 Whathappensifweaddthesetwopositivesigned binary numbers 0110 + 0011 (i.e., 6 + 3) ?
 The result is 1001.
 But that is a negative number (-7)! 
 Whathappensifweaddthetwonegativenumbers 1000 + 1111 (i.e., -8 + (-1))?
 The result is 0111 with a carry-out. 
 Weneedtoknowwhentheresultmightbewrong.
 This is usually indicated in hardware by the Overflow flag!
 More about this when we’ll talk about processors.

Sign & Magnitude Representation
 Instead of signed numbers, some (older) processors use sign and magnitude representation.
 The sign part: one bit is designated as the sign (+/-).  0forpositivenumbers
 1fornegativenumbers
 The magnitude part: remaining bits store the positive (i.e., unsigned) version of the number.
 Example: 4-bit binary numbers:
 0110 is 6 while 1110 is -6 (most significant bit is the sign)
 What about 0000 and 1000? => zero (two ways)
 Sign-magnitude computation is more complicated.
 2’s complement is what today’s systems use!

 Acircuitthattakesin two input vectors, and determines if the first is greater than, less than or equal to the second.
 Howdoesonemake that in a circuit?

Basic Comparators
 B,whereAandBareonebitlong.
 Thecircuitsforthiswouldbe:
A=B Comparator A>B AB:
A·B + A·B A·B