Functional Block: Decoders

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1 University of Wisconsin - Madison EE/omp Sci 352 Digital Systems Fundamentals harles R. Kime Section 2 Fall 2 hapter 3 ombinational Logic Design Part 2 Tom Kaminski & harles R. Kime harles Kime & Thomas Kaminski Functional lock: Decoders Decoder converts n binary bits to a maximum of 2n unique output lines. n m-to-n line decoder, where m < 2 n, can be used to: Generate 2 n (or fewer) minterms, Select one-of 2 n items Decoders are sometimes known as demultiplexers when enabled with a separate data-in line. hapter 3-Part 2 2

2 2-to-4 Line Decoder This device takes: n=2 input lines and decodes minterms for: m=2 n = 4 output lines. hapter 3-Part to-4 Line Demultiplexer This device takes: n=2 input lines and decodes minterms for: m=22 =4 output lines where each output is: NDed with an input,. If is viewed as an Enable, all outputs are for = and one output is for =. If is viewed as Data, then this data is sent to one or the outputs. hapter 3-Part 2 4 2

3 Example: 74F38 Demultiplexer 74F38 truth table: Enables E E2 E3 Inputs x y z Outputs D D D2 D3 D4 D5 D6 D7 x x x x x x x x x x x x x x x Note: This "Truth Table" uses the x (or - )to mean "this could be either or ". Thus, it "compacts" some of 2 6 = 64 lines. hapter 3-Part 2 5 Implementing Logic with Decoders Decoders provide minterms directly. Simply "OR" the appropriate minterm outputs to make any logic function desired. ctive low decoders behave as the first NND gate in a NND-NND, Sum of Products implementation. ctive high decoders behave as first stage ND gates in a ND-OR Sum of Products implementation. Two or more active high decoders driven from different bits of a binary code can be used to form minterms by "NDING" their outputs. Similarly, active low decoders can be used to form minterms by "ORING" their outputs. hapter 3-Part 2 6 3

4 Example : F(,) = m(,3) For this we use a 2-to-4 line decoder and sum minterms and 3 with an OR gate: m3 F m hapter 3-Part 2 7 Example : F(,Y,Z) = m(,3,5,6) hapter 3-Part 2 8 4

5 Implementing Larger Minterms Minterm m5 is formed by "NDING" the D3 outputs of each decoder. Similarly m is formed by "NDING" the D outputs of each decoder. D S S S S D3 D2 D D D3 D2 D D What minterm is formed by "NDING" D (upper) and D2 (lower) outputs? This works best with widely scattered, sparse minterms. m5 m?? m hapter 3-Part 2 9 Functional lock: Encoders Encoders perform the "inverse" operation of decoders, taking a code in one format and encoding it into another format. Many encoders consist of just OR gates. For example an 8- to-3 binary encoder consists of three 4-input OR gates, OR2,OR and OR. Input I i, i =,,7 is connected to an input on OR j if the binary representation of i has a in position j. priority encoder is used to generate a code for the "most significant" bit set in a string of bits. This can be used to find the first one in a word, or to select external events in priority order. n example of a MSI priority encoder is the 74F48, 8 line to 3 line priority encoder. It can be cascaded to encode higher numbers of bits. hapter 3-Part 2 5

6 Encoder Example Encode 4 lines,, 2, 3 into the corresponding binary codes. hapter 3-Part 2 Review: Decoders and Encoders Decoder converts n binary bits to a maximum of 2 n unique output lines. Decoders are sometimes know as demultiplexers when enabled with a separate data-in line. Decoders implement minterms directly. Use a decoder and an OR gate to form Sum-of- Minterms directly. Encoders perform the "inverse" operation of decoders, taking a code in one format and encoding it into another format. hapter 3-Part 2 2 6

7 Multiplexers Multiplexer (MU) is another common functional block. Multiplexer uses n binary select bits to choose from a maximum of 2 n unique input lines. Like a decoder, it decodes minterms internally. Unlike a decoder, it has only one output line. The decoded minterms are used to select data from one of up to 2 n unique data input lines. The output of the multiplexer is the data input whose index is specified by the n bit code. hapter 3-Part 2 3 Example: 4-to- multiplexer The 4-to- line Multiplexer uses the same minterm decoder core. It is like a demultiplexer with individual data input lines (instead of just one) and an output OR gate. I3 I2 I I S S hapter 3-Part 2 4 7

8 Multiplexer Versus Decoder I3 I2 I S I S Note how similar the two are internally. hapter 3-Part 2 5 Functions with Multiplexers It is possible to implement any oolean function of n variables with a 2 n input multiplexer. Simply tie each input to the "" or "" line as desired. It is also possible to implement any n+ variable function with a 2 n multiplexer. Simply use the (n+)st variable in true or complement form depending upon what the truth table requires. oolean function of more than n variables can be partitioned into several easily implemented sub-functions defined on a subset of the variables. The multiplexer will then select among these sub-functions. hapter 3-Part 2 6 8

9 Example: Gray to inary ode The Gray code has adjacent elements separated by only one bit change. We wish to convert a 3-bit Gray code to a binary code. The function table on the right documents the required conversion. Gray inary x y z The Gray to inary ode onverter requires us to implement three separate, three-input oolean functions. hapter 3-Part 2 7 Gray to inary (ontinued) First step: Let's get the function table into a logical order by reordering the input Gray code values in binary sequence: y inspection: x = F(,,) = m(, 3, 5, 7) y = G(,,) = m(, 2, 5, 6) z = H(,,) = m(, 2, 4, 7) Gray inary x y z hapter 3-Part 2 8 9

10 Gray to inary (ontinued) The K-Maps Z Y Note: x(,,) =, is an easy function to implement. (No logic gates needed!) Function y(,,) = ' + ' is a bit harder to implement. Function z(,,) looks familiar. What is it? hapter 3-Part 2 9 Gray to inary (ontinued) We know that 2n to Multiplexers can be used to implement arbitrary functions of n bits. We simply connect the inputs to "" or "" as needed. Use two eight-input multiplexers to implement functions for y and z: D7 D6 D5 D4 D3 D2 D D S2 S S Out 8-to- MU Z D7 D6 D5 D4 D3 D2 D D S2 S S Out 8-to- MU Y In this case, the MU elements are acting like a "Read Only Memory" (ROM). hapter 3-Part 2 2

11 Other MU Implementations We can also use two 4-to- MU blocks and implement y and z. Suppose we factor out and use and as the select inputs to the multiplexers Y Z D D D 3 D 2 D D D 3 D 2 hapter 3-Part 2 2 MU Implementations (ont.) Factoring out variable leads to the following implementation with two, 4-to- Multiplexers: ' ' D3 D2 D D Out Z D3 D2 D D Out Y S S 4-to- MU S S 4-to- MU s before, x =. hapter 3-Part 2 22

12 MU: (ont.) Factoring Out We could have factored out other variables. s in the book, we will factor out and apply to the select inputs: Gray inary x y z D = D = D = D = D = ' D = ' D2 = D2 = D2 = ' D3 = D3 = ' D3 = This is slightly larger than selecting to factor out. hapter 3-Part 2 23 MU: (ont.) Factoring out Gray inary x y z x = "" y = z = x = "" y = ' z = ' x = "" y = z = ' x = "" y = ' z = Note: We re-arranged the table (fixing and and varying from to in each cell) to simplify this procedure. It still looks like factoring was better. hapter 3-Part

13 MU: (ont.) Factoring out Gray inary x y z D = D = D = D = D = D = ' D2 = D2 = D2 = ' D3 = D3 = D3 = Note: We re-arranged the table (fixing and and varying from to in each cell) to simplify this procedure. Factoring is best! Note also that x = holds. hapter 3-Part 2 25 Summary Know the functions performed by the following functional blocks: Decoders Demultiplexers Encoders Multiplexers Know how to implement oolean functions using: Multiplexers Decoders hapter 3-Part

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