June 18, 2002 ©2000-2002 Howard Huang 1Multiplexers• For the rest of the day, we’ll study multiplexers, which are just as commonly used as the decoders we presented last time. Again,– These serve as examples for circuit analysis and modular design.– Multiplexers can implement arbitrary functions.– We will actually put these circuits to good use in later weeks, as building blocks for more complex designs.2/15/2006 Additional Gates and Decoders 2Multiplexers• A 2n-to-1 multiplexer sends one of 2ninput lines to a single output line. – A multiplexer has two sets of inputs:• 2ndata input lines• n select lines, to pick one of the 2ndata inputs– The mux outputis a single bit, which is one of the 2ndata inputs.• The simplest example is a 2-to-1 mux:• 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).Q = S’ D0 + S D12/15/2006 Additional Gates and Decoders 3More truth table abbreviations• Here is a full truth table for this 2-to-1 mux, based on the equation:• 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 D1 D0 Q000 0001 1010 0011 11000101 01101111 1Q = S’ D0 + S D1S Q0D01D12/15/2006 Additional Gates and Decoders 4A 4-to-1 multiplexer• Here is a block diagram and abbreviated truth table for a 4-to-1 mux.• Be careful! In LogicWorks the multiplexer has an active-low EN input signal. When EN’ = 1, the mux always outputs 1.EN’ S1 S0 Q000D0001D1010D2011D31xx1Q = S1’ S0’ D0 + S1’ S0 D1 + S1 S0’ D2 + S1 S0 D32/15/2006 Additional Gates and Decoders 5Implementing functions with multiplexers• Muxes can be used to implement arbitrary functions.• One way to implement a function of nvariables is to use an n-to-1mux:– For each minterm miof 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 f0000001 10101011010001010110111112/15/2006 Additional Gates and Decoders 6A 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• 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.x y z f0000001 10101011010001010110111112/15/2006 Additional Gates and Decoders 7Example: 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 C00000010010001111000101111011111With S1=X and S0=Y, thenQ=X’Y’D0 + X’YD1 + XY’D2 + XYD3Equation for the multiplexer2/15/2006 Additional Gates and Decoders 8Multiplexer-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 C00000010010001111000101111011111C = 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)When XY=00, C=0When XY=01, C=ZWhen XY=10, C=ZWhen XY=11, C=12/15/2006 Additional Gates and Decoders 9Multiplexer-based sum• Here’s the same thing, but for the sum function S(X,Y,Z). X Y Z S00000011010101101001101011001111S = 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)When XY=00, S=ZWhen XY=01, S=Z’When XY=10, S=Z’When XY=11, S=Z2/15/2006 Additional Gates and Decoders 10Dual 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.– In LogicWorks, it’s called a “Mux-4x2 T.S.”2/15/2006 Additional Gates and Decoders 11Dual 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.”)• LogicWorks labels input bits xDy, which means “the xth bit of data input y.”• In the diagram on the right, we’re using S1-S0 to choose one of the following pairsof inputs:– 2D3 1D3, when S1 S0 = 11– 2D2 1D2, when S1 S0 = 10– 2D1 1D1, when S1 S0 = 01– 2D0 1D0, when S1 S0 = 00You can see how 8-way multiplexer (k-to-1) can be used to select from a set of (k) 8-bit numbers2/15/2006 Additional Gates and Decoders 12Summary• A 2n-to-1 multiplexer routes one of 2ninput 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.• Tune in tomorrow as we start to discuss how to build circuits to do
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