- An important advantage
of CRC codes is:
- The "divide" algorithm
used in the CRC can be
easily implemented with
cheap hardware:
- Shift-register (actually: D-flipflops) and
- XOR circuits
- The "divide" algorithm
used in the CRC can be
easily implemented with
cheap hardware:
- Shift-register circuit:
- 1 bit shift register = a circuit with that copies the input value to its output during the time that the clock signal changes from 0 ⇒ 1 (or from 1 ⇒ 0)
- You can see the behavior
of a serie of shift-register
in this demo:
/home/cs455001/demo/Logic-Sim/shift-register
- XOR circuit:
- XOR circuit =
a circuit with 2 inputs
and one output
- The XOR function is as follows:
0 XOR 0 = 0 0 XOR 1 = 1 1 XOR 0 = 1 1 XOR 1 = 0
- XOR circuit =
a circuit with 2 inputs
and one output
- You can see the behavior
of the XOR circuit
in this demo:
/home/cs455001/demo/Logic-Sim/xor
- How to create
a circuit
that compute the CRC code
for a given
generator polynomial:
- Let
- N = length of the generator polynomial
- Use (N−1) D-flipflops (= shift registers)
to store
the remainder of the division.
- Start by
- Putting (N−1) Dffs
in a row.
- Connect the output
of the ith Dff to
the input
of the i+1th Dff
Example: (N = 4 or N−1 = 3)
- Putting (N−1) Dffs
in a row.
- The input of the
first Dff will
receive the
message
one bit at a time:
- We must modify the
circuit as follows:
- Read the CRC generator polynomial
in the reverse order.
- In this step, you will only use N−1 bits of the CRC generator polynomial
- For bit i in the
(reversed ordered)
CRC generator polynomial do:
- if bit i = 1, then:
- Change the
direct connection
to the input
of the Dff (shift register)
to
an XOR gate
- I.e.: insert an XOR circuit before the ith Dff
- Connect the output
of the last Dff
as the second input of
the new
XOR circuit
- I.e.: the output of the last Dff is fed back into the ith Dff)
- Change the
direct connection
to the input
of the Dff (shift register)
to
an XOR gate
- if bit i = 1, then:
- Read the CRC generator polynomial
in the reverse order.
It's much easier to see and example (I'll do it in class)
- Let
- Example:
compute circuit
for generator polynomial
1001
- N = 4
Put the 3 Dffs in a row.
Connect the output of the ith Dff to the input of the i+1th Dff:
- Add the feed backs:
- N = 4
- Example:
compute circuit
for generator polynomial
1101
- N = 4
Put the 3 Dffs in a row.
Connect the output of the ith Dff to the input of the i+1th Dff:
- Add the feed backs:
- N = 4
- How to compute
the CRC for
an input message:
- Reset
all the Dffs (memory elements)
to ZERO
- Toggle key 'a' to
ONE
to
clear the
shift registers
- Toggle key 'a' back to ZERO (blank cell) so you can run the circuit
- Toggle key 'a' to
ONE
to
clear the
shift registers
- Present
the first bit of the message
- Use key 0 to
set it to the
value of the
first bit
(0 or
1)
- The input bits are
presented from
left to right:
Message: 101010 Present the bits as follows: 1 0 1 0 1 0
- Use key 0 to
set it to the
value of the
first bit
(0 or
1)
- Give the
compute circuit
a clock signal
- You must toggle the key 1 twice
- Present the second bit of the message (see above)
- Give the compute circuit
another clock signal
(see above)
- And so on, until all bits in the message are
processed
Note:
- Remember that the sender must
append
N−1
ZERO (0) bits to the
input message
- These N−1 ZERO (0) bits must be processed also !!
- Remember that the sender must
append
N−1
ZERO (0) bits to the
input message
- After processing all bits:
- Sender:
- Remainder of the "division" = values of the shift registers read in the reverse order
- Receiver:
- If remainder of the
"division" = all 0's,
then accept the
message
- Otherwise, reject the message
- If remainder of the
"division" = all 0's,
then accept the
message
- Sender:
- Reset
all the Dffs (memory elements)
to ZERO
- Example Program:
(Circuit for generator polynomial
1101)
- Prog file: click here
How to use it:
- Run: /home/cs455001/demo/Logic-Sim/CRC1
- Make a = 0 (blank)
to run circuit
- Toggle 0:Input to the next bit in
input
- Press 1:Clk to give circuit a clock signal (2 toggles per input bit).
This is a logic-sim program from my CS355 class...
Example:
00011 ---------------- 1101 / 10001000 1101 ---- (XOR) 1011 1101 ---- (XOR) 1100 1101 ---- (XOR) 000100 <---- remainder = 100Do this:
- Toggle 2 to run circuit
- Toggle 0 (input) to 1, then toggle 1 (clock twice)
- Toggle 0 (input) to 0, then toggle 1 (clock twice)
- Toggle 0 (input) to 0, then toggle 1 (clock twice)
- Toggle 0 (input) to 0, then toggle 1 (clock twice)
- Toggle 0 (input) to 1, then toggle 1 (clock twice)
- Toggle 0 (input) to 0, then toggle 1 (clock twice)
- Toggle 0 (input) to 0, then toggle 1 (clock twice)
- Toggle 0 (input) to 0,
then toggle 1 (clock twice)
- Read the bits backwards: 100