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# A few more things about binary numbers - PowerPoint PPT Presentation

A few more things about binary numbers. Errors. Recall that using a binary representation maximizes our tolerance of fluctuations without loss of the information represented. Still errors occur. They have fall into two categories: Errors that cannot be interpreted as a 1 or 0

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### A few more things about binary numbers

• Recall that using a binary representation maximizes our tolerance of fluctuations without loss of the information represented.

• Still errors occur. They have fall into two categories:

• Errors that cannot be interpreted as a 1 or 0

• Flipped bits: interpreting a 1 where a 0 should be and vice versa.

• One may worry about errors wherever data is held temporarily, stored semi-permanently or transmitted.

• The discovery that a bit or bits were flipped is known as error detection.

• The restoration of the correct data is known as error correction.

• Data being transmitted by a modem is chunked into "blocks" of a certain byte size and sent to the destination modem.

• The destination modem checks each block for errors and returns

• ACK if no errors are found

• NAK if errors are found, which leads to a retransmission.

• The kind of checking (checksum or cyclic redundancy checking) varies from protocol to protocol.

• The bit error rate (BER) is the ratio of bits having errors (opposite of the value they are supposed to have 01 or 10) to the total number of bits sent

• It is often expressed as ten to a negative power.

• For example, a BER of 10-6, means that on average, for every million bits sent, one error occurs.

• The BER indicates how often a packet has to be resent.

• Sometimes increasing the data transmission rate also increases the BER, making it actually beneficial to reduce the transmission rate so that fewer packets need to be resent.

• A BERT (bit error rate test or tester) is a procedure or device that measures the BER.

• Error detection schemes involve transmitting additional information which the receiver can use to validate the data.

• Error detection can inform the receiver of the presence of an error but cannot fix the error, the packet must be resent.

• Error correction schemes (which are distinct) attempt to pinpoint the flipped bit.

• Parity is a simple error detection scheme.

• One chooses ahead of time on the type of parity to be used (even or odd). Let us assume even.

• Then one takes a string of binary data and calculates whether the number of 1’s in the string is even or odd.

• The transmitter includes an extra bit

• 0 if the number of data-bit 1’s is even

• 1 if the number of data-bits 1’s is odd

• The string consisting of the data plus parity bit is now guaranteed to have an even number of 1’s.

• When the data is going to be used or received in the case of transmission, one checks that the data plus parity bit have even parity (an even number of 1’s).

• If the data now has odd parity, some error must have occurred.

• But we do not know which bit was flipped (detection not correction).

D – data P – parity

• (You may read that the number of 1’s is counted, this is not true and is unnecessarily complicated.)

• A multi-input XOR (excluded OR) gate will determine the parity bit, this circuitry is much simpler than what is required for counting the number of 1’s.

• Parity can be used when data is moved around within computers as well as when data is sent from computer to computer.

• Modems use parity. Users may also select no parity, meaning that the modems will not include the parity bit. Sending fewer bits speeds up transmission.

Choose the Advanced Tab Preferences button

Modem Parity Choices Preferences button

Checksum Preferences button

• A checksum is another error-detection scheme often used in transmission.

• The transmission string of data bits is broken up into units, for instance consisting of 16 bits each.

• These 16-bit numbers are then added.

• The sum is sent along as part of the frame’s trailer field.

• The receiver repeats the calculation. If the sums match, it's assumed that no transmission error occurred.

Checksum (Cont.) Preferences button

• Actually instead of including the sum in the trailer, one can transmit instead its negative.

• The receiver then repeats the sum over the data and includes in that summation the negative of the sum sent in the trailer.

• The result should be zero. It is easier to test for a result of all zeros (an multi-input OR gate).

Catching errors Preferences button

• Since the 16-bit string might end up in any of 216 (65536) possible states, you might assume that the checksum would catch errors 65535/65536  99.9985% of the time.

• But this would assume that all possible errors are equally likely, which is not true.

IP Datagram Protocol Preferences button

TCP Segment Protocol Preferences button

Burst errors Preferences button

• For multi-bit errors, it is more likely that a group of consecutive bits are affected as opposed to randomly selected bits, such an error is known as a burst error.

• It is also common for periodic errors to occur (e.g. the first two bits in every byte).

CRC Preferences button

• Cyclic Redundancy Check is better at catching burst errors than checksums.

• The idea is the same, perform some mathematical operation on the data, send the result, have the receiver do the same calculation and check that the same answer is obtained.

CRC (Cont.) Preferences button

• CRC is somewhat like division

• If one thinks of the data as a large number, one can divide it by another number N giving a whole number answer and a remainder

• The remainder could be any number between 0 and N-1.

• The problem here is that division is a fairly difficult computation (more circuitry and a slow process).

CRC (Cont.) Preferences button

• CRC uses a variation on division that while mathematically abstract is very simple to build a circuit for.

• The circuitry needs only a shift register and some (2-input) XOR gates.

Shift Register Preferences button

• A register is a small piece of memory that holds values.

• In addition to holding values, a shift register performs a simple operation on the values; it moves them to the left to to the right.

Shift Register Preferences button

time

Output

Shift register

Input

How shift registers are used? Preferences button

• Multiplication

• Converting Parallel Data (the form inside the computer) to Serial (the form sent over transmission lines)

• Cyclic Redundancy check (CRC)

CRC (Cont.) Preferences button

• Again the 16-bit string might end up in any of 216 (65536) possible states, so you might expect that CRC would catch errors 65535/65536  99.9985% of the time same as a 16-bit checksum.

• But CRC is better at detecting burst errors which are more likely than purely random errors.

• The positions of the XORs are important in determining what kinds of burst errors are detected.

CRC: Transmission Only Preferences button

• The data must be serialized for a Cyclic Redundancy Check.

• This is fine for transmission error checking since the data was serialized for transmission.

• However, serializing the data would waste a lot of time if the data were in a parallel form (as it is inside the computer).

CRC = Shift register + XORs Preferences button

Basically one has a shift register with a few excluded OR gates inserted in strategic positions.

Ethernet Frame Protocol Preferences button

Ethernet uses CRC because at that low level the data is serialized. The higher levels in the stack (e.g. IP and TCP) used checksums.

Preferences button

0

0

0

0

11000001010

0

0

0

1

1000001010

1

0

0

1

000001010

1

0

1

0

00001010

Preferences button

0001010

0

1

1

0

0

0

0

1

001010

1

0

0

0

01010

0

0

1

0

1010

Preferences button

0

1

0

1

010

1

1

0

1

10

1

1

1

0

0

0

0

1

1

Hamming Code Preferences button

• Hamming code extends the idea of parity. It can be used as either as

• An extended error detection scheme

• Parity will discover if an odd number of bits have been flipped. Hamming code can be used to detect two-bit errors.

• Or an error correction scheme

• If a single bit error is assumed, Hamming code can locate the offending bit.

Hamming code (Cont.) Preferences button

• Hamming code breaks the bit string into a few overlapping groups.

• By overlapping here we mean that a given bit can belong to more than one group. But each bit should belong to a unique set of groups. That’s how the bit is located.

• One generates a parity bit for each group.

• On checking for errors, one identifies which groups violate parity. From these one can locate the bit in error.

Hamming code (Cont.) Preferences button

• The set of data and parity bits are numbered such that the parity bits correspond to numbers which are powers of 2: 1, 2, 4, 8 etc.

• Recall in binary the powers of 2 consist of one 1 and the rest 0’s.

• 1: 0001

• 2: 0010

• 4: 0100

• 8: 1000

Hamming code (Cont.) Preferences button

• The groups:

• Any bit whose count in binary has a 1 in the 1’s position belongs to the first group.

• Any bit whose count in binary has a 1 in the 2’s position belongs to the second group.

• Any bit whose count in binary has a 1 in the 4’s position belongs to the third group.

• Etc.

Groups for Hamming code Preferences button

Parity bits

Data bits

Groups for Hamming code Preferences button

Locating the offending bit Preferences button

• Let us assume that in a case having four data bits and three parity bits that parity errors were found in check groups 1 and 3.

• We look for the bit that belongs to groups 1 and 3 and does not belong to group 2.

• If it belongs to group 1, it has a 1 in the 1’s position.

• If it does not belong to group 2, it has a 0 in the 2’s position.

• If it belongs to group 3, it has a 1 in the 4’s position.

• It must be 5.

Locating the offending bit Preferences button

101  5

Now reverse the situation! Preferences button

Neither row violates even parity for the first group.

Both rows have parity violations for group 2. Preferences button

Neither row has a parity violation for group 3.

Both rows have parity violations in group 4. Preferences button

It just so happens in this example that the two rows have the same parity violation. That’s a coincidence. It doesn’t have to be like that.

If there are parity violations in groups two and four then the binary number identifying the offending bit is 1010 (for both rows, unfortunate example) which is the 10th bit.

Big-endian/Little-endian Preferences button

• In memory, one addresses bytes. Often the data being written to memory involves more than one byte.

• Big-endian and Little-endian refer to the two different ways in which bytes are placed in memory when the word being written consists of more than one byte.

Big-endian/Little-endian Preferences button

• In big-endian architectures, the more significant bytes are placed in the locations with the lower addresses.

• “Big end first”

• Many mainframe computers, particularly IBM

• In little-endian architectures, the less significant bytes are placed in the locations with the higher addresses.

• “Little end first”

• Most modern computers, including PCs

• The PowerPC is bi-endian because it can understands both ways.

12345678 in Big-endian Preferences button

“Big end first”

12345678 in Little-endian Preferences button

“Little end first”

Byte versus bit Preferences button

• Note that the previous example shows only big- and little-endian byte orders.

• One can also consider the bit ordering within each byte. It can also be either big- or little-endian.

• Some architectures use a mix: big-endian ordering for bits and little-endian ordering for bytes, or vice versa.

Origin of the term Preferences button

• “The terms big-endian and little-endian are derived from the Lilliputians of Gulliver's Travels, whose major political issue was whether soft-boiled eggs should be opened on the big side or the little side. Likewise, the big-/little-endian computer debate has much more to do with political issues than technological merits.”

• www.webopedia.com

Other references Preferences button

• http://www.webopedia.com

• http://www.whatis.com

• Understanding Data Communications & Networks, William Shay

• http://www.cs.utsa.edu/~wagner/laws/hamming.html