The Basics of digital logic

• Two terms
  • Combinational logic is a system with no memory/ feedback.
  • A full adder is an example of combinational logic circuit.
  • Sequential logic is a system with memory/feedback.
  • A register is an example of a sequential logic circuit.
• I care that you can represent digital logic items three ways
  • As a truth table.
  • In my experience we design the truth table first.
  • As a boolean expression.
  • In my experience, we use boolean expressions to simplify the logic produced by the truth table.
  • As a circuit.
  • This is the ultimate goal of this section.
• Do this for and, or and not.
  • A+B for or
  • juxtaposition (AB)or · (A·B) for and
  • A for not
• Some rules you need to know
  • A · 0 = 0, A·1 = A
  • A + 1 = 1, A + 0 = A
  • A + A = 1, A · A = 0
• A multiplexer is a standard circuit which
  • has 2ⁿ input lines
  • has 1 output line
  • has n control lines.
  • The data on the n control lines select the single input line which will be transmitted on the output line.
  • The simplest case is a 2 input (I₀, I₁), 1 control (C) and one output (O) mux.

  • C	O
    0	I0
    1	I1

  • Using the above boolean expressions, we can and each input line with the control line to produce something that either passes the data through or a 0.
  • Then we can or all of the results together, and only the data will be passed through.
  • 
  • And this really becomes O = (C · I₁) + (C · I₀)
• In general, we can add input ports to and and or gates.
  • A·B·C = (A·B)·C = A·(B·C)
  • There is a small problem eventually with power being diminished
  • But we are not worried about that.
  • Just "right click, add port"
• So I can build a 4 input, 2 control line, one output mux
  • 
  • source code
• The other thing we do in TKgate is build a "wide" version of the gates.
  • Just select instance properties and then edit the port widths.
  • 
  • source code
• Muxes
  • Are things you can buy on a chip.
  • Components→ MSI → select correct mux.
  • 
• We will use muxes in our construction of the MIPS machine, see for example the future at the top of page 263.
• Decoders
  • A decoder has n input lines and 2ⁿ output lines.
  • A decoder will take an address, on the input, and select a correspondign output.
  • For example a 1-4 decoder has 2 inputs and 4 outputs.

    I1	I0	O3	O2	O1	O0
    0	0	0	0	0	1
    0	1	0	0	1	0
    1	0	0	1	0	0
    1	1	1	0	0	0

  • The control for this is much like the control for the MUX
  • 
  • source code
  • In TKgate the decoder has an extra control line.
    • This is a enable/disable line
    • This can be set to high
    • 
  • We will use decoders in the construction of the register file.