computer science 1000 digital circuits. digital information computers store and process information...
TRANSCRIPT
Computer Science 1000
Digital Circuits
Digital Informationcomputers store and process information
using binaryas we’ve seen, binary affords us similar
advantages as other representationse.g. we saw how we could add binary numbers,
just as we add decimal numbers
Binary Representation recall that the way in which information is represented depends on
the component RAM: capacitor charge hard drive: magnetic fields processor: voltages
for our purposes, the most important point to remember is that there are exactly two states
binary numbers: 1 0 logic: true false gate input/outputs: on off
your text refers to such a binary system as a PandA representation, where information is made up of states that are Present and Absent
PandA from pg. 198 of your text
Processingyour CPU processes informationwhat does that mean?
one possible definition: create a new state (output) from an existing state (input)
by state, I am referring to a binary sequence
Input 1205
Processing Output
Processing – Exampleconsider our addition example from the
previous lecture
1011+0010 1101
Input (Existing State)
Output (New State)
0010
Processing – Example what we require is a device that can:
take two binary numbers as input produces their sum as output
in fact, this is (part of) what the arithmetic logic unit (ALU) in your processor does
the question is: how?
11011011
Binary Adder
0010
1011
Logic Gates at a very basic level, processors are constructed
using gates a logic gate is a very simple device:
has a number of inputs produces a single output
think of each input as a wire that is either on or off on represents a 1 in binary, or a true in logic off represents a 0 in binary, or a false in logic
Logic Gates - Continued the output of a logic gate depends on (and only on)
its inputs in other words, for each possible combination of its
inputs, there is a specific output we represent this using a truth table
the truth table of a gate depends on its type
Not Gate one of the simplest gate types also known as an inverter takes one input, and produces a single output symbol:
Input Output
Not Gate a not-gate simply inverts its input that is
if its input wire is on, its output is off if its input wire is off, its output is on
hence, the truth table is constructed as follows:
Input Output
on off
off on
0 1
1 0
Truth Table Representation in the interest of space, on and off in truth tables
are often represented as 1 and 0 hence, our previous truth would be written as
follows:
unless otherwise stated, this will be the representation that we adopt
Input Output
0 1
1 0
AND Gate an example of a gate with two inputs its output is on only if both of its inputs are on
otherwise, the output is off
Symbol:
A B Output
0 0 0
0 1 0
1 0 0
1 1 1
Truth Table:
Truth Table – Multiple Inputsnote that our truth table now has four rows
one for each possible configuration of inputssuppose we had a gate with three inputs
how many rows would that require?
A B C Output
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
In general, any circuit with n binary inputs will require 2n rows in its truth table.
Digital Circuitsby themselves, gates are quite limited in
what they do there are only so many uses for an AND gate
the real power of gates is when we begin to combine them
that is, the output of one gate becomes the input to another gate
Digital Circuits Example:
suppose we have a NOT gate and an AND gate we could connect them in the following configuration notice that we still have two inputs (A,B) and one output the output of the not gate is attached to the first input of
the AND gate
Digital Circuits Example:
what does the truth table of our circuit look like? to determine this, let’s consider each input state individually
A B Output
0 0
0 1
1 0
1 1
Digital Circuits Example:
when A is 0 and B is 0 the output of the not gate is 1 hence, the input to our AND gate is 1 and 0 the output of our AND gate is 0
A B Output
0 0
0 1
1 0
1 1
0
0
10
0
Digital Circuits Example:
when A is 0 and B is 1 the output of the not gate is 1 hence, the input to our AND gate is 1 and 1 the output of our AND gate is 1
A B Output
0 0
0 1
1 0
1 1
0
1
11
0
1
Digital Circuits Example:
when A is 1 and B is 0 the output of the not gate is 0 hence, the inputs to our AND gate are 0 and 0 the output of our AND gate is 0
A B Output
0 0
0 1
1 0
1 1
1
0
00
0
1
0
Digital Circuits Example:
when A is 1 and B is 1 the output of the not gate is 0 hence, the inputs to our AND gate are 0 and 1 the output of our AND gate is 0
A B Output
0 0
0 1
1 0
1 1
1
1
00
0
1
0
0
Digital Circuit previous example demonstrated a digital circuit
an assembly of logic gates
like logic gates, each digital circuit has: input(s) output(s) a truth table
like logic gates, digital circuits can be used in the construction of other circuits
output from one circuit connected to another digital circuit
Digital Circuit - Rendering circuits are often rendered in an enclosed shape this makes it clear what the inputs and outputs to
the system are
Digital Circuit - Rendering in fact, if we know the inputs, outputs, and truth
table, then we often are not concerned with how the circuit is built
becomes a “black box” handy when this circuit is used as part of a larger circuit
A B Output
0 0 0
0 1 1
1 0 0
1 1 0
Digital Circuit – Example 2suppose I rearrange my previous circuit into
the following configuration
what would the truth table of this circuit be?
Digital Circuit – Example 2Answer (discussed in class):
A B Output
0 0 1
0 1 1
1 0 1
1 1 0
Digital Circuit – Solving our circuits so far are reasonably small, and
therefore easy and quick to derive truth table step through each individual configuration of the inputs
however, as circuits grow, this task becomes more difficult and time consuming
Digital Circuit – Solving as a consequence, it is often helpful to label
intermediate points in the circuit, and solve for those points explicitly in the truth table
usually, these intermediate points are the output from a gate that are being input into another gate
consider our previous example let’s label the output of the NOT gate as point C
C
Digital Circuit – Solving now, when we create our truth table, we will create
an “output” column for C as well although this might initially seem like more work, it
will ultimately make computing the output easier
C A B C Output
0 0
0 1
1 0
1 1
Digital Circuit – Solving compute the values for column C
C A B C Output
0 0
0 1
1 0
1 1
A B C Output
0 0 1
0 1 1
1 0 0
1 1 0
Digital Circuit – Solving now, compute the values for the output column
notice that the output depends only on C and B we need not even consider A the output is 1 if C and B are 1, and 0 otherwise
C A B C Output
0 0
0 1
1 0
1 1
A B C Output
0 0 1
0 1 1
1 0 0
1 1 0
A B C Output
0 0 1 0
0 1 1 1
1 0 0 0
1 1 0 0
OR Gate another common example of a gate with two inputs its output is off only if both of its inputs are off
otherwise, the output is on notice how this contrasts with and
Symbol:
A B Output
0 0 0
0 1 1
1 0 1
1 1 1
Truth Table:
Digital Circuit suppose we replace the AND gate in our previous
circuit with an OR gate compute the resulting truth table for the output
A B Output
0 0
0 1
1 0
1 1
Digital Circuit first, to simplify our computation, let’s add an
intermediate point at the NOT gate call it C C will require a column in the truth table
A B Output
0 0
0 1
1 0
1 1
CA B C Output
0 0
0 1
1 0
1 1
Digital Circuit now, let’s calculate the values for column C
CA B C Output
0 0
0 1
1 0
1 1
A B C Output
0 0 1
0 1 1
1 0 0
1 1 0
Digital Circuit now, we can calculate the values for the output
based on columns B and C recall our definition for the OR gate – the output is off if both of its inputs
are off, and on otherwise
CA B C Output
0 0
0 1
1 0
1 1
A B C Output
0 0 1
0 1 1
1 0 0
1 1 0
A B C Output
0 0 1 1
0 1 1 1
1 0 0 0
1 1 0 1