September 22, 2003 Decoders 1Decoders• Next, we’ll look at some commonly used circuits: decoders and multiplexers.– These serve as examples of the circuit analysis and design techniques from last lecture.– They can be used to implement arbitrary functions.– We are introduced to abstraction and modularity as hardware design principles.• Throughout the semester, we’ll often use decoders and multiplexers as building blocks in designing more complex hardware.September 22, 2003 Decoders 2What is a decoder• In older days, the (good) printers used be like typewriters:– To print “A”, a wheel turned, brought the “A” key up, which then was struck on the paper.• Letters are encoded as 8 bit codes inside the computer.– When the particular combination of bits that encodes “A” is detected, we want to activate the output line corresponding to A– (Not actually how the wheels worked)• How to do this “detection” : decoder• General idea: given a k bit input, – Detect which of the 2^k combinations is represented– Produce 2^k outputs, only one of which is “1”.September 22, 2003 Decoders 3What a decoder does• A n-to-2ndecoder takes an n-bit input and produces 2noutputs. The n inputs represent a binary number that determines which of the 2noutputs is uniquelytrue.• 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.• 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.S1 S0 Q0 Q1 Q2 Q300 1 0 0 00101001000 1 011000 1September 22, 2003 Decoders 4How 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), based on the two inputs (S0-S1).• In this case there’s not much to be simplified. Here are the equations:S1 S0 Q0 Q1 Q2 Q300 1 0 0 00101001000 1 011000 1Q0 = S1’ S0’Q1 = S1’ S0Q2 = S1 S0’Q3 = S1 S0September 22, 2003 Decoders 5A picture of a 2-to-4 decoderS1 S0 Q0 Q1 Q2 Q300 1 0 0 00101001000 1 011000 1September 22, 2003 Decoders 6Enable 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 allof the decoder’s outputs are 0.• We can include this additional input in the decoder’s truth table:EN S1 S0 Q0 Q1 Q2 Q30000 000001 0 0000100000011 00001001000101010011000101110001September 22, 2003 Decoders 7An aside: abbreviated truth tables• In this table, note that whenever EN=0, the outputs are always 0, regardless of inputs S1 and S0.• We can abbreviate the table by writing x’s in the input columns for S1 and S0. EN S1 S0 Q0 Q1 Q2 Q30000 000001 0 000010000001100001001000101010011000101110001EN S1 S0 Q0 Q1 Q2 Q30xx00001001000101010011000101110001September 22, 2003 Decoders 8• 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 namesof 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!Blocks and abstractionQ0 = S1’ S0’Q1 = S1’ S0Q2 = S1 S0’Q3 = S1 S0September 22, 2003 Decoders 9A 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 Q7000100000000010100000001000100000011 0 0 0 1 0 0 0 0100 0 0 0 0 1 0 0 0101 0 0 0 0 0 1 0 01100000001011100000001Q0 = S2’ S1’ S0’Q1 = S2’ S1’ S0Q2 = S2’ S1 S0’Q3 = S2’ S1 S0Q4 = S2 S1’ S0’Q5 = S2 S1’ S0Q6 = S2 S1 S0’Q7 = S2 S1 S0September 22, 2003 Decoders 10So what good is a decoder?• Do the truth table and equations look familiar?• 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.• This means that you can implement arbitrary functions with decoders. If you have a sum of minterms equation for a function, you can easily use a decoder (a minterm generator) to implement that function.S1 S0 Q0 Q1 Q2 Q300 1 0 0 00101001000 1 011000 1Q0 = S1’ S0’Q1 = S1’ S0Q2 = S1 S0’Q3 = S1 S0September 22, 2003 Decoders 11Design 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.X Y Z C S00000001010100101110100011011011010111111 + 1 + 1 = 110 + 1 + 1 = 10C(X,Y,Z) = Σm(3,5,6,7)S(X,Y,Z) = Σm(1,2,4,7)September 22, 2003 Decoders 12• Here, two 3-to-8 decoders implement C and S as sums of minterms.• The “+5V” symbol (“5 volts”) is how you represent a constant 1 or true in LogicWorks. We use it here so the decoders are always active.Decoder-based adderC(X,Y,Z) = Σm(3,5,6,7)S(X,Y,Z) = Σm(1,2,4,7)September 22, 2003 Decoders 13Using just one decoderC(X,Y,Z) = Σm(3,5,6,7)S(X,Y,Z) = Σm(1,2,4,7)• Since the two functions C and S both have the same inputs, we could use just one decoder instead of two.September 22, 2003 Decoders 14Building 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
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