Combinational Circuits

1
Combinational Circuits

• Objectives
• to study a group of circuits that are used in
  designing processors at the micro-architecture
  level.
• comparators
• decoders/encoders
• multiplexers/demultiplexers
• adders

2
Comparators

• The basic function of a comparator is to compare
  the magnitude of two binary quantities to
  determine the relationship of those quantities.

3
A 1-bit comparator

• Let A B be the inputs and X be the output.
• 1-bit comparator is defined as X 1 if AB.
• X AB AB

4
Decoders

• A k-line-to-nline decoder has k input lines and
  n 2k output lines
• Its behavior is defined by the property that when
  X denotes the binary number i, the corresponding
  line zi with index (subscript) i is active and
  all other lines inactive.
• There is also usually a control line EN, called
  the enable line, which must be active (EN 1)
  for the decoder to operate when EN 0, all the
  output lines are inactive

5
A 3-bit decoder or a 3-line-to-8-line decoder
(k3, n23)

• Simple decoder design

6
3-line-to-8-line decoder

• Simple decoder design
• D0 active ( 1) only when x y z 0 , so D0
  x'y'z'
• D1 x'y'z
• ..
• Di minterm of (x,y,z), with x MSB and z LSB
• As conclusion, we can say that the decoder
  outputs for an active-HIGH(1) decoder are simply
  minterms.

7
Encoders

• An encoder is the inverse of a decoder
• Typically has n 2k input lines and k output
  lines.
• An encoder is intended to identify a single
  active input signal. However there is nothing to
  prevent several of its input lines from being
  active at the same time.
• To deal with this situation, most encoders are
  designed as priority encoders, which have the
  property that when several inputs are active at
  the same time, the output number i that appears
  on Z is the index of the input line xi with the
  highest priority.

8
Multiplexers

• a 2-to-1 multiplexer.

9
Multiplexers

• The main function of a multiplexer M (data
  selector) is to control the transfer of data
  words from one of several sources to a common
  destination.
• It has normally n 2k input data buses
  connecting it to the destination D
• A k-bit input address or "select" bus A
  determines the source to connect through M to D
• A multiplexer with n input data buses, each of
  width m, is called an n-way, m-bit multiplexer.

10
Multiplexers

• A multiplexer that has n data inputs, w0,..,
  wn-1 requires log2n) select inputs.
• Larger multiplexer can also be constructed from
  smaller multiplexers

11
4-to-1 built using three 2-to-1 multiplexers

12
4-1 multiplexer using 2-1 multiplexers
13
Multiplexers

• A general-purpose component to implement
  combinational logic functions.
• Example of a exclusive or gate

14
Multiplexers

• Exclusive or gate

15
De-multiplexers

• A demultiplexer basically reverses the
  multiplexing function. It takes data from one
  line and distributes them to given number of
  output lines. A demultiplexer has N 2k output
  lines, k address selection and one data line. The
  number of the output line active is specified by
  the k bits address.