# Chapter 3 Ctd: Combinational Functions and Circuits

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1 Chapter 3 Ctd: Combinational Functions and Circuits 1

2 Value Fixing, Transferring, and Inverting Four different functions are possible as a function of single Boolean variable Transferring Inverting Value Fixing 2

3 Value Fixing for Implementing a Function Y is actually a function of six varibles truth table with 64 rows and 7 clmns Putting I 0 -I 3 in the output column reduces the truth table Y(A,B,I 0,I 1,I 2,I 3 )= A B I 0 + A BI 1 + AB I 2 + ABI 3 3

4 Value Fixing for Implementing a Function 16 different functions can be implemented by setting I 0 -I 3 appropriately 4

5 What is a decoder? 1 A: 0000 B: 0001 C: 0010 : : : A B C Keyboard Decoder 2 16 Decoder: A combinational circuit with an n-bit binary code applied to its inputs and m-bit binary code appearing at the outputs n m 2^n Detect which of the 2^n combinations is represented at the inputs Produce m outputs, only one of which is 1 5

6 1-to-2 (Line) Decoder 6

7 2-to-4 Decoder A 2-to-4 decoder operates according to the following truth table The 2-bit input is called S1S0, and the four outputs are Q0-Q3 If the input is the binary number i, then output Qi is uniquely true S1 S0 Q0 Q1 Q2 Q For instance, if the input S1 S0 = 10 (decimal 2), then output Q2 is true, and Q0, Q1, Q3 are all false This circuit decodes a binary number into a one-of-four code 7

8 How can you build a 2-to-4 decoder? Follow the design procedures from last time! We have a truth table, so we can write equations for each of the four outputs (Q0-Q3) S1 S0 Q0 Q1 Q2 Q In this case there s not much to be simplified. Here are the equations: Q0 = S1 S0 Q1 = S1 S0 Q2 = S1 S0 Q3 = S1 S0 8

9 2-to-4 Decoder S1 S0 Q0 Q1 Q2 Q

10 Enable Inputs Many devices have an additional enable input, which is used to activate or deactivate the device For a decoder, EN=1 activates the decoder, so it behaves as specified earlier. Exactly one of the outputs will be 1 EN=0 deactivates the decoder. By convention, that means all of the decoder s outputs are 0 We can include this additional input in the decoder s truth table: EN S1 S0 Q0 Q1 Q2 Q

11 An aside: abbreviated truth tables In this table, note that whenever EN=0, the outputs are always 0, regardless of inputs S1 and S0. EN S1 S0 Q0 Q1 Q2 Q We can abbreviate the table by writing x s in the input columns for S1 and S0 EN S1 S0 Q0 Q1 Q2 Q3 0 x x

12 Blocks and Abstraction Decoders are common enough that we want to encapsulate them and treat them as an individual entity Block diagrams for 2-to-4 decoders are shown here. The names of the inputs and outputs, not their order, is what matters. A decoder block provides abstraction: You can use the decoder as long as you know its truth table or equations, without knowing exactly what s inside It makes diagrams simpler by hiding the internal circuitry It simplifies hardware reuse. You don t have to keep rebuilding the decoder from scratch every time you need it These blocks are like functions in programming! Q0 = S1 S0 Q1 = S1 S0 Q2 = S1 S0 Q3 = S1 S0 12

13 3-to-8 decoder Larger decoders are similar. Here is a 3-to-8 decoder The block symbol is on the right A truth table (without EN) is below Output equations are at the bottom right Again, only one output is true for any input combination S2 S1 S0 Q0 Q1 Q2 Q3 Q4 Q5 Q6 Q Q0 = S2 S1 S0 Q1 = S2 S1 S0 Q2 = S2 S1 S0 Q3 = S2 S1 S0 Q4 = S2 S1 S0 Q5 = S2 S1 S0 Q6 = S2 S1 S0 Q7 = S2 S1 S0 13

14 So what good is a decoder? Do the truth table and equations look familiar? S1 S0 Q0 Q1 Q2 Q Q0 = S1 S0 Q1 = S1 S0 Q2 = S1 S0 Q3 = S1 S0 Decoders are sometimes called minterm generators For each of the input combinations, exactly one output is true Each output equation contains all of the input variables These properties hold for all sizes of decoders Arbitrary functions can be implemented with decoders. If a sum of minterms equation for a function is given, a decoder (a minterm generator) is used to implement that function 14

15 Design example: Addition Let s make a circuit that adds three 1-bit inputs X, Y and Z We will need two bits to represent the total; let s call them C and S, for carry and sum. Note that C and S are two separate functions of the same inputs X, Y and Z Here are a truth table and sum-of-minterms equations for C and S = 10 X Y Z C S C(X,Y,Z) = Σm(3,5,6,7) S(X,Y,Z) = Σm(1,2,4,7) = 11 15

16 Decoder-based Adder Here, two 3-to-8 decoders implement C and S as sums of minterms. 16

17 Using just one decoder Since the two functions C and S both have the same inputs, we could use just one decoder instead of two. 17

18 Building a 3-to-8 decoder You could build a 3-to-8 decoder directly from the truth table and equations below, just like how we built the 2-to-4 decoder Another way to design a decoder is to break it into smaller pieces Notice some patterns in the table below: When S2 = 0, outputs Q0-Q3 are generated as in a 2-to-4 decoder When S2 = 1, outputs Q4-Q7 are generated as in a 2-to-4 decoder S2 S1 S0 Q0 Q1 Q2 Q3 Q4 Q5 Q6 Q Q0 = S2 S1 S0 = m 0 Q1 = S2 S1 S0 = m 1 Q2 = S2 S1 S0 = m 2 Q3 = S2 S1 S0 = m 3 Q4 = S2 S1 S0 = m 4 Q5 = S2 S1 S0 = m 5 Q6 = S2 S1 S0 = m 6 Q7 = S2 S1 S0 = m 7 18

19 Decoder Expansion You can use enable inputs to string decoders together. Here s a 3-to-8 decoder constructed from two 2-to-4 decoders: S2 S1 S0 Q0 Q1 Q2 Q3 Q4 Q5 Q6 Q

20 Modularity Be careful not to confuse the inner inputs and outputs of the 2-to-4 decoders with the outer inputs and outputs of the 3-to-8 decoder (which are in boldface) This is similar to having several functions in a program which all use a formal parameter x 20

21 A variation of the standard decoder The decoders we ve seen so far are active-high decoders EN S1 S0 Q0 Q1 Q2 Q3 0 x x An active-low decoder is the same thing, but with an inverted EN input and inverted outputs EN S1 S0 Q0 Q1 Q2 Q x x

22 Separated at birth? Active-high decoders generate minterms, as we ve already seen Q3 = S1 S0 Q2 = S1 S0 Q1 = S1 S0 Q0 = S1 S0 The output equations for an active-low decoder are mysteriously similar, yet somehow different Q3 = (S1 S0) = S1 + S0 Q2 = (S1 S0 ) = S1 + S0 Q1 = (S1 S0) = S1 + S0 Q0 = (S1 S0 ) = S1 + S0 It turns out that active-low decoders generate maxterms 22

23 Active-low Decoder So we can use active-low decoders to implement arbitrary functions too, but as a product of maxterms For example, here is an implementation of the function f(x,y,z) = ΠM(4,5,7) using an active-low decoder The ground symbol connected to EN represents logical 0, so this decoder is always enabled 23

24 Summary of Decoders A n-to-2 n decoder generates the minterms of an n-variable function As such, decoders can be used to implement arbitrary functions Later on we ll see other uses for decoders too Some variations of the basic decoder include: Adding an enable input Using active-low inputs and outputs to generate maxterms We also talked about: Applying our circuit analysis and design techniques to understand and work with decoders Using block symbols to encapsulate common circuits like decoders Building larger decoders from smaller ones 24

25 Encoder An encoder is a digital function that performs the inverse operation of a decoder Octal-to-Binary Encoder: This encoder has eight inputs, one for each of the octal digits, and three outputs that generate the corresponding binary number Only one input can have the value of 1 at any given time D7 D6 D5 D4 D3 D2 D1 D0 Encoder A2 A1 A0 25

26 Octal-to-Binary Encoder A0= D1 + D3 + D5 + D7, A1= D2 + D3 + D6 + D7 A2= D4 + D5 + D6 + D7 26

27 Priority Encoder A priority encoder is a combinational circuit that implements a priority function Octal-to-Binary: If more than one input is active simultaneously, the output produces an incorrect combination Priority Encoder: If two or more inputs are active simultaneously, the input having the highest priority takes precedence 27

28 4-Input Priority Encoder 28

29 4-Input Priority Encoder 29

30 Multiplexers Multiplexers, or muxes, are used to choose between resources A real-life example: in the old days before networking, several computers could share one printer through the use of a switch. 30

31 Multiplexers A 2 n -to-1 multiplexer sends one of 2 n input lines to a single output line A multiplexer has two sets of inputs: 2 n data input lines n select lines, to pick one of the 2 n data inputs The mux output is a single bit, which is one of the 2 n data inputs The simplest example is a 2-to-1 mux: Q = S D0 + S D1 The select bit S controls which of the data bits D0-D1 is chosen: If S=0, then D0 is the output (Q=D0). If S=1, then D1 is the output (Q=D1). 31

32 More truth table abbreviations Here is a full truth table for this 2-to-1 mux, based on the equation: Q = S D0 + S D1 S D1 D0 Q Here is another kind of abbreviated truth table Input variables appear in the output column This table implies that when S=0, the output Q=D0, and when S=1 the output Q=D1 This is a pretty close match to the equation S Q 0 D0 1 D1 32

33 2-to-1 Mux 33

34 4-to-1 Mux Y= (S1 S0 )I0 + (S1 S0)I1 + (S1S0 )I2 + (S1S0)I3 Decoder 34

35 4-to-1 Mux 35

36 4-to-1 Mux with Enable Input EN S1 S0 Q D D D D3 1 x x 1 Q = EN S1 S0 D0 + EN S1 S0 D1 + EN S1 S0 D2 + EN S1 S0 D3 + EN 36

38 Implementing functions with multiplexers Muxes can be used to implement arbitrary functions One way to implement a function of n variables is to use an 2^n-to-1 mux: For each minterm m i of the function, connect 1 to mux data input Di. Each data input corresponds to one row of the truth table Connect the function s input variables to the mux select inputs. These are used to indicate a particular input combination For example, let s look at f(x,y,z) = Σm(1,2,6,7). x y z f

39 A more efficient way We can actually implement f(x,y,z) = Σm(1,2,6,7) with just a 4-to-1 mux, instead of an 8-to-1 Step 1: Find the truth table for the function, and group the rows into pairs. Within each pair of rows, x and y are the same, so f is a function of z only. When xy=00, f=z When xy=01, f=z When xy=10, f=0 When xy=11, f=1 x y z f Step 2: Connect the first two input variables of the truth table (here, x and y) to the select bits S1 S0 of the 4-to-1 mux. Step 3: Connect the equations above for f(z) to the data inputs D0-D3. 39

40 Example: multiplexer-based adder Let s implement the adder carry function, C(X,Y,Z), with muxes There are three inputs, so we ll need a 4-to-1 mux The basic setup is to connect two of the input variables (usually the first two in the truth table) to the mux select inputs X Y Z C With S1=X and S0=Y, then Q=X Y D0 + X YD1 + XY D2 + XYD3 40

41 Multiplexer-based carry We can set the multiplexer data inputs D0-D3, by fixing X and Y and finding equations for C in terms of just Z. X Y Z C When XY=00, C=0 When XY=01, C=Z When XY=10, C=Z When XY=11, C=1 C = X Y D0 + X Y D1 + X Y D2 + X Y D3 = X Y 0 + X Y Z + X Y Z + X Y 1 = X Y Z + X Y Z + XY = Σm(3,5,6,7) 41

42 Multiplexer-based sum Here s the same thing, but for the sum function S(X,Y,Z) X Y Z S When XY=00, S=Z When XY=01, S=Z When XY=10, S=Z When XY=11, S=Z S = X Y D0 + X Y D1 + X Y D2 + X Y D3 = X Y Z + X Y Z + X Y Z + X Y Z = Σm(1,2,4,7) 42

43 Dual multiplexer-based full adder We need two separate 4-to-1 muxes: one for C and one for S But sometimes it s convenient to think about the adder output as being a single 2-bit number, instead of as two separate functions A dual 4-to-1 mux gives the illusion of 2-bit data inputs and outputs It s really just two 4-to-1 muxes connected together 43

44 Dual muxes in more detail You can make a dual 4-to-1 mux by connecting two 4-to-1 muxes. ( Dual means two-bit values. ) In the diagram on the right, we re using S1-S0 to choose one of the following pairs of inputs: 2D3 1D3, when S1 S0 = 11 2D2 1D2, when S1 S0 = 10 2D1 1D1, when S1 S0 = 01 2D0 1D0, when S1 S0 = 00 44

45 Selecting based on 3-state buffers 45

46 Selecting based on TGs 46

47 Summary of Muxes A 2 n -to-1 multiplexer routes one of 2 n input lines to a single output line Just like decoders, Muxes are common enough to be supplied as stand-alone devices for use in modular designs. Muxes can implement arbitrary functions We saw some variations of the standard multiplexer: Smaller muxes can be combined to produce larger ones We can add active-low or active-high enable inputs As always, we use truth tables and Boolean algebra to analyze things 47

Review: Additional Boolean operations Operation: NAND (NOT-AND) NOR (NOT-OR) XOR (exclusive OR) Expressions: (xy) = x + y (x + y) = x y x y = x y + xy Truth table: x y (xy) x y (x+y) x y x y 0 0 1 0 1

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