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Adders Used to perform addition, subtraction, multiplication, and division (sometimes) Half-adder adds rightmost (least significant) bit Full-adder adds all other bits, since a 1 may be carried into it. Use carry-out from one adder as the carry-in for the next adder Combinational circuit (no memory)

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**Inputs two 1-bit values, X and Y **

Half-Adder A Half Adder (HA) is a 2-input, 2-output combinational circuit that adds the inputs and produces a Sum and a Carry The Boolean expressions and the circuit for the Sum S and the carry C are given below: Half adder Inputs two 1-bit values, X and Y Outputs their 2-bit sum as bits C and S C is carry S is the sum

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**Half-Adder Only this circuit is a Half-Adder Result Carry**

Other circuits may be equivalent, however Result Carry

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Half-Adder

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**Half-Adder How to know a circuit is equivalent to half-adder ?**

Circuit is equivalent to a half-adder if both truth tables have identical output Cannot be used if there could be a carry-in bit So used to add least significant (rightmost) bit. Very important circuit in computers. Addition is very common Half-adder is found in every computer, calculator, digital watch…

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Binary Full Adder A Full Adder (FA) is a 3-input, 2-output combinational circuit that adds the inputs and produces a Sum and a Carry The third input can be perceived as the carry from a previous addition The Boolean expressions and for the Sum S and the carry C, obtained from their K-Maps, are given below:

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Multiplexer A multiplexer (MUX) is a device that accepts data from one of many input sources for transmission over a common shared line. To achieve this the MUX has several data lines and a single output along with data-select inputs, which permit digital data on any one of the inputs to be switched to the output line. The logic symbol for a 1-of-4 data selector/ multiplexer is shown below, along with its associated table. Logic symbol for 1-of-4 multiplexer

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Active high MUX

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Active Low MUX

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**Note that if a binary zero appears on the data-select lines then **

Data Select Inputs Input Selected S1 S0 D0 1 D1 D2 D3 Table of Operation Note that if a binary zero appears on the data-select lines then data on input line D0 will appear on the output. Thus, data output Y is equal to D0 if and only if S1=0 and S0=0

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**Similarly, the data output is equal to D1, D2 and D3 for**

, and , respectively. Thus the total multiplexer logic expression, formed from ORing terms is

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**Larger MUX Technique: use hierarchies of smaller components**

Example: creating 4x1 mux from 2x1 mux Will create a 2-level mux tree First level takes the initial inputs The results from the first level are fed into the second level

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**Hierarchy Approach Technique**

Divide the truth table into equal sections Number of sections given by type of second-level MUX If the second-level MUX is 2x1 then need 2 sections in the TT S1 S0 F Input 0 1 Input 1 Input 2 Input 3 Section 1 Section 2

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**Hierarchy Approach Technique S1 S0 F**

Connect the outputs corresponding to individual sections of the TT to the data lines of the individual first-level MUXs Input 0 S1 S0 F Input 0 1 Input 1 Input 2 Input 3 Outputs from Section 1 of Truth Table 2x1 Mux Input 1 Outputs from Section 2 of Truth Table 2x1 Mux Input 2 Input 3

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**Hierarchy Approach Technique**

Connect the outputs of the first-level MUXs to the data lines of the second-level MUX following the order of the sections Input 0 2x1 Mux Input 1 2x1 Mux 2x1 Mux Input 2 Input 3

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**Hierarchy Approach Technique**

Connect the outputs of the first-level MUXs to the data lines of the second-level MUX following the order of the sections Input 0 2x1 Mux Input 1 2x1 Mux 2x1 Mux Input 2 Input 3

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**4-to-1 MUX made from 2-to-1 MUXs**

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Decoders A decoder: Accepts a Boolean value (number) and activates the corresponding output line All other lines are deactivated For n inputs there are 2n output lines Each possible input value corresponds to an output line

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**COMMUNICATION CHANNEL**

Decoders Motivation (one example) CPU needs to select channel => assign address to each device Need a way to activate device. MEMORY PRINTER COMMUNICATION CHANNEL CPU MOUSE DISPLAY

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Decoders ACTIVATION LINE Motivation CPU needs to select channel => assign address to each device Need a way to activate device. MEMORY ADDRESS CODE PRINTER ADDRESS DECODER CPU MOUSE DISPLAY

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**Decoder An n input decoder has 2n outputs.**

B 1, iff A,B is 01 1, iff A,B is 10 1, iff A,B is 11 i = 0 i = 1 i = 2 i = 3 An n input decoder has 2n outputs. Outputi is 1 iff the binary value of the n-bit input is i. At any time, exactly one output is 1, all others are 0. 1, iff A,B is 00

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Decoder with Enable In general, attach m-enabling circuits to the outputs See truth table below for function Note use of X’s to denote both 0 and 1 Combination containing two X’s represent four binary combinations Alternatively, can be viewed as distributing value of signal EN to 1 of 4 outputs In this case, called a demultiplexer

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**2-to-4 decoder with active high enable**

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**2-to-4 decoder with active low enable**

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**Encoders An encoder: For 2n inputs there are n output lines**

Outputs the Boolean value corresponding to the input line number There is a special output line V that indicates whether any input lines are active.

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Encoder

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Priority Encoder

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