IKI10201 05cDecoders, Selectors, etc.

1
IKI10201 05c-Decoders, Selectors, etc.

• Bobby Nazief
• Semester-I 2005 - 2006

• The materials on these slides are adopted from
• CS231s Lecture Notes at UIUC, which is derived
  from Howard Huangs work and developed by Jeff
  Carlyle
• Prof. Daniel Gajskis transparency for Principles
  of Digital Design.

2
Decoders

• Next, well look at some commonly used circuits
  decoders and selectors (multiplexers or MUXes).
• They can be used to implement arbitrary
  functions.
• We are introduced to abstraction and modularity
  as hardware design principles.

3
What 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 (ASCII) 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 2k combinations is
  represented
• Produce 2k outputs, only one of which is 1.

4
What a decoder does

• A n-to-2n decoder takes an n-bit input and
  produces 2n outputs. The n inputs represent a
  binary number that determines which of the 2n
  outputs is uniquely true.
• 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.

5
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), based on the
  two inputs (S0-S1).
• In this case theres not much to be simplified.
  Here are the equations

Q0 S1 S0 Q1 S1 S0 Q2 S1 S0 Q3 S1 S0
6
A picture of a 2-to-4 decoder
7
Enable inputs

• Many devices have an additional enable input,
  which is used to activate or deactivate the
  device.
• For a decoder,
• EN1 activates the decoder, so it behaves as
  specified earlier. Exactly one of the outputs
  will be 1.
• EN0 deactivates the decoder. By convention,
  that means all of the decoders outputs are 0.
• We can include this additional input in the
  decoders truth table

8
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
  whats inside.
• It makes diagrams simpler by hiding the internal
  circuitry.
• It simplifies hardware reuse. You dont have to
  keep rebuilding the decoder from scratch every
  time you need it.
• These blocks are like functions in programming!

9
A 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.

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
10
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.

Q0 S2 S1 S0 m0 Q1 S2 S1 S0 m1 Q2
S2 S1 S0 m2 Q3 S2 S1 S0 m3 Q4 S2 S1
S0 m4 Q5 S2 S1 S0 m5 Q6 S2 S1 S0
m6 Q7 S2 S1 S0 m7
11
Decoder expansion

• You can use enable inputs to string decoders
  together. Heres a 3-to-8 decoder constructed
  from two 2-to-4 decoders

12
So 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.

Q0 S1 S0 Q1 S1 S0 Q2 S1 S0 Q3 S1 S0
13
Design example addition

• Lets make a circuit that adds three 1-bit inputs
  X, Y and Z.
• We will need two bits to represent the total
  lets 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.

C(X,Y,Z) ?m(3,5,6,7) S(X,Y,Z) ?m(1,2,4,7)
0 1 1 10
1 1 1 11
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Decoder-based adder

• Here is how a 3-to-8 decoder implements C and S
  as sums of minterms

C(X,Y,Z) ?m(3,5,6,7) S(X,Y,Z) ?m(1,2,4,7)
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Selectors/Multiplexers

• Now well study multiplexers, which are just as
  commonly used as the decoders we presented last
  time. Again,
• Multiplexers can implement arbitrary functions.
• We will put these circuits to use in later weeks,
  as building blocks for more complex designs.

16
Multiplexers

• A 2n-to-1 multiplexer sends one of 2n input lines
  to a single output line.
• A multiplexer has two sets of inputs
• 2n data input lines
• n select lines, to pick one of the 2n data inputs
• The mux output is a single bit, which is one of
  the 2n data 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 S0, then D0 is the output (QD0).
• If S1, then D1 is the output (QD1).

Q S D0 S D1
17
More 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 S0, the output
  QD0, and when S1 the output QD1.
• This is a pretty close match to the equation.

Q S D0 S D1
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A 4-to-1 multiplexer

• Here is a block diagram and abbreviated truth
  table for a 4-to-1 mux.

Q S1 S0 D0 S1 S0 D1 S1 S0 D2 S1 S0 D3
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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 n-to-1 mux
• For each minterm mi of the function, connect 1 to
  mux data input Di. Each data input corresponds to
  one row of the truth table.
• Connect the functions input variables to the mux
  select inputs. These are used to indicate a
  particular input combination.
• For example, lets look at f(x,y,z) ?m(1,2,6,7).

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A more efficient way

• We can actually implement f(x,y,z) ?m(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 xy00, fz
• When xy01, fz
• When xy10, f0
• When xy11, f1
• 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.

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Buses

• Bus drivers have three possible output values
• 0, 1, and Z (high impedance disconnection)

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Priority Encoders

• Encoder is opposite of Decoder, but with priority
  for MSB

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Magnitude Comparators

• Comparator compares two integers A and B and
  sets
• G 1 when A gt B
• L 1 when A lt B
• G L 0 when A B

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Shifter Rotators

• Shifter shifts input bits to the left/right
• Shift left X3X2X1X0 ? X2X1X00
• Shift right X3X2X1X0 ? 0X3X2X1
• Rotator is a Shifter with additional shift from
  one end to the other
• Rotate left X3X2X1X0 ? X2X1X0X3
• Rotate right X3X2X1X0 ? X0X3X2X1

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Read Only Memory (ROM)

• A read-only memory, or ROM, is a special kind of
  memory whose contents cannot be easily modified.
• Data is stored onto a ROM chip using special
  hardware tools.
• ROMs are useful for holding data that never
  changes.
• Arithmetic circuits might use tables to speed up
  computations of logarithms or divisions.
• Many computers use a ROM to store important
  programs that should not be modified, such as the
  system BIOS.
• PDAs, game machines, cell phones, vending
  machines and other electronic devices may also
  contain non-modifiable programs.

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Memories and functions

• ROMs are actually combinational devices, not
  sequential ones!
• You cant store arbitrary data into a ROM, so the
  same address will always contain the same data.
• You can think of a ROM as a combinational circuit
  that takes an address as input, and produces some
  data as the output.
• A ROM table is basically just a truth table.
• The table shows what data is stored at each ROM
  address.
• You can generate that data combinationally, using
  the address as the input.
• Data F(A0,A1,A2)

27
Implementing functions with decoders

• We can already convert truth tables to circuits
  easily, with decoders.
• For example, you can think of this old circuit as
  a memory that stores the sum and carry outputs
  from the truth table on the right.

28
Implementing functions with ROM

• ROMs are based on this decoder implementation of
  functions.
• A blank ROM just provides a decoder and several
  OR gates.
• The connections between the decoder and the OR
  gates are programmable, so different functions
  can be implemented.
• To program a ROM, you just make the desired
  connections between the decoder outputs and the
  OR gate inputs.

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Implementing functions with ROM (cont.)

• Here are three functions, V2V1V0, implemented
  with an 8 x 3 ROM.
• Blue crosses (X) indicate connections between
  decoder outputs and OR gates. Otherwise there is
  no connection.

A2 A1 A0
V2 ?m(1,2,3,4)
V1 ?m(2,6,7)
V0 ?m(4,6,7)
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The same example again

• Here is an alternative presentation of the same 8
  x 3 ROM, using abbreviated OR gates to make the
  diagram neater.

V2 ?m(1,2,3,4) V1 ?m(2,6,7) V0 ?m(4,6,7)
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Why is this a memory?

• This combinational circuit can be considered a
  read-only memory.
• It stores eight words of data, each consisting of
  three bits.
• The decoder inputs form an address, which refers
  to one of the eight available words.
• So every input combination corresponds to an
  address, which is read to produce a 3-bit data
  output.

32
ROMs vs. RAMs

• There are some important differences between ROM
  and RAM.
• ROMs are non-volatiledata is preserved even
  without power. On the other hand, RAM contents
  disappear once power is lost.
• ROMs require special (and slower) techniques for
  writing, so theyre considered to be read-only
  devices.
• Some newer types of ROMs do allow for easier
  writing, although the speeds still dont compare
  with regular RAMs.
• MP3 players, digital cameras and other toys use
  CompactFlash, Secure Digital, or MemoryStick
  cards for non-volatile storage.
• Many devices allow you to upgrade programs stored
  in flash ROM.

33
Programmable logic arrays (PLAs)

• A ROM is potentially inefficient because it uses
  a decoder, which generates all possible minterms.
  No circuit minimization is done.
• Using a ROM to implement an n-input function
  requires
• An n-to-2n decoder, with n inverters and 2n
  n-input AND gates.
• An OR gate with up to 2n inputs.
• The number of gates roughly doubles for each
  additional ROM input.
• A programmable logic array, or PLA, makes the
  decoder part of the ROM programmable too.
  Instead of generating all minterms, you can
  choose which products (not necessarily minterms)
  to generate.

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A blank 3 x 4 x 3 PLA

• This is a 3 x 4 x 3 PLA (3 inputs, up to 4
  product terms, and 3 outputs), ready to be
  programmed.
• The left part of the diagram replaces the decoder
  used in a ROM.
• Connections can be made in the AND array to
  produce four arbitrary products, instead of 8
  minterms as with a ROM.
• Those products can then be summed together in the
  OR array.

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Example PLA minimization

• For a PLA, we should minimize the number of
  product terms for all functions together (K-Map
  minimization ? individual function).
• We should express V2, V1 and V0 with no more than
  four total products.

V2 ?m(1,2,3,4) V1 ?m(2,6,7) V0 ?m(4,6,7)
V2 xyz xz xyz V1 xyz xy V0
xyz xy
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Example PLA implementation

• So we can implement these three functions using a
  3 x 4 x 3 PLA

A2 A1 A0
xyz
xy
xz
xyz
V2 V1 V0
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PLA evaluation

• A k x m x n PLA can implement up to n functions
  of k inputs, each of which must be expressible
  with no more than m product terms.
• Unlike ROMs, PLAs allow you to choose which
  products are generated.
• This can significantly reduce the fan-in (number
  of inputs) of gates, as well as the total number
  of gates.
• However, a PLA is less general than a ROM. Not
  all functions may be expressible with the limited
  number of AND gates in a given PLA.
• In terms of memory, a k x m x n PLA has k address
  lines, and each of the 2k addresses references an
  n-bit data value.
• But again, not all possible data values can be
  stored.