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Chapter 8

Reliability and

Channel Coding

The Three Main Sources of Transmission Errors •  Noise

–  Thermal / white noise

•  Interference –  electromagnetic radiation emitted from devices such as electric motors –  background cosmic radiation can cause noise that can disturb radio transmissions and

signals traveling across wires •  Cosmit radiation consists of energetic charged particles, such as protons and Helium ions, moving

through space •  When these particles enter the Earth's atmosphere they collide with, and disrupt, atoms in our

atmosphere, producing secondary, less intense, radiation

•  Distortion –  All physical systems distort signals –  As a pulse travels along an optical fiber, the pulse disperses –  Wires have properties of capacitance and inductance

•  that block signals at some frequencies while admitting signals at other frequencies –  Placing a wire near a large metal object can change the set of frequencies that can

pass through the wire •  Metal objects can block some frequencies of radio waves, while passing others

•  Attenuation –  As a signal passes across a medium, the signal becomes weaker

•  signals on wires or optical fibers become weaker over long distances, just as a radio signal becomes weaker with distance

Ways to Improve Noise

•  Increase the signal-to-noise ratio (either by increasing the signal or lowering noise if possible) –  Physical transmission system is always susceptible to errors –  It may not be possible to change the signal-to-noise ratio

•  Mechanisms like shielded wiring can help lower noise •  Increase capacity by reducing BER

–  Reducing noise improves SNR and thus BER

Types of Error •  An error occurs when a bit is altered between

transmission and reception •  Single bit errors

–  only one bit altered –  caused by white noise

•  Burst errors –  contiguous sequence of B bits in which first last and any

number of intermediate bits in error –  caused by impulse noise or by fading in wireless –  effect greater at higher data rates

•  Erasure •  Sometimes a signal is neither clearly 1 nor clearly 0, but falls in an

ambiguous region, which is known as an erasure

If Data rate is 10 Mbps and the signal is lost for only 1usec, how many bit errors occur?

1usec / 0.1 usec = 10 bits!

Higher data rate à more errors

Types of Error

Bursts of errors due to impulse noise:

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Effect of Transmission Errors on Data

•  For a burst error, the burst size, or length, is defined as the number of bits from the start of the corruption to the end of the corruption

Two Strategies for Handling Channel Errors

•  A variety of mathematical techniques have been developed that overcome errors during transmission and increase reliability –  Known collectively as channel coding

•  The techniques can be divided into two broad categories: –  Forward Error Correction (FEC) mechanisms –  Automatic Repeat reQuest (ARQ) mechanism

•  In either case we are adding overhead –  There is always a tradeoff - adding redundancy vs. error detection

•  What is the impact of channel error?

ARQ Mechanism •  Not appropriate for wireless applications

–  bit error rate is high causing lots of retransmissions –  when propagation delay long (satellite) compared with frame

transmission time, resulting in retransmission of frame in error plus many subsequent frames

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Block and Convolutional Error Codes •  The two types of FEC techniques satisfy separate needs: •  Block Error Codes

–  It divides the data to be sent into a set of blocks –  It attaches extra information known as redundancy to each block –  The encoding for a given block of bits depends only on the bits

themselves, not on bits that were sent earlier –  They are memoryless in the sense that the encoding mechanism

does not carry state information from one block of data to the next

•  Convolutional Error Codes –  It treats data as a series of bits, and computes a code over a

continuous series –  Thus, the code computed for a set of bits depends on the current

input and some of the previous bits in the stream –  Convolutional codes are said to be codes with memory

Error Correction Motivation •  Errors can be detected and corrected

–  Error correction is more complex •  Correction of detected errors usually requires data

block to be retransmitted •  Instead need to correct errors on basis of bits received

Error Correction Basic Idea •  Adds redundancy to transmitted message •  Can deduce original despite some errors

–  Errors are detected using error-detecting code –  Error-detecting code added by transmitter –  Error-detecting code are recalculated and checked by receiver

–  map k bit input onto an n bit codeword –  each distinctly different –  When error occurs the receiver tries to guess which codeword

sent was (e.g., teh à the)

Error Detection

Error Correction with Row and Column

(RAC) Parity

Redundancy Check 1- Vertical Redundancy Check (VRC)

- Parity Check 2- Longitudinal Redundancy Check (LRC) 3- Cyclic Redundancy Check

Error Detection – Parity Check

•  Basic idea –  Errors are detected using error-detecting code –  Error-detecting code added by transmitter –  error-detecting code are recalculated and checked by receiver

•  Parity bit –  Odd (odd parity)

•  If it had an even number of ones, the parity bit is set to a one, otherwise it is set to a zero

•  (P=0 if odd ones)à always odd number of ones in the frame •  Asynchronous applications and Standard in PC memory

–  Even (even parity) •  Synchronous applications

F(1110001)à odd parity 1 111 000 1 Parity Bit + Data Block

Error Detection – Parity Check An Example Block Error Code:

Single Parity Checking

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If even number of 1s à Even parity =0

Error Detection Basic Mechanism

•  for block of k bits transmitter •  Represented by (n,k) encoding scheme

–  k dataword length –  n codeword –  r added bits

Example: 8-bit data + single parity bità 2^9 (512) possibilities / only 2^8

(255=256-1) valid code words (excluding all-zero)

What is the minimum number of bits we should add?

Redundancy Check

•  Longitudinal Redundancy Check (LRC) –  Organize data into a table and create a parity for each column

11100111 11011101 00111001 10101001

11100111 11011101 00111001 10101001 10101010

11100111 11011101 00111001 10101001 10101010 Original Data LRC

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Hamming Distance: A Measure of a Code's Strength

•  No channel coding scheme is ideal! –  changing enough bits will always transform to a valid codeword

•  What is the minimum number of bits of a valid codeword that must be changed to produce another valid codeword? –  To answer the question, engineers use a measure known as the

Hamming distance –  Given two strings of n bits each, the Hamming distance is defined as

the number of differences

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Hamming Distance: A Measure of a Code's Strength

•  Hardware implementation –  One way to compute the Hamming distance consists of taking the

exclusive or (xor) between two strings –  Then, counting the number of 1 bits in the answer

•  For example, consider the Hamming distance between strings 110 and 011 –  The xor of the two strings is:

which contains two 1s –  Therefore, the Hamming distance between 011 and 101 is 2

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The Tradeoff Between Error Detection and Overhead

•  A large value of dmin is desirable –  because the code is immune to more bit errors, if fewer than dmin bits

are changed, the code can detect that error(s) occurred

•  The maximum number of bit errors that can be detected:

•  A code with a higher value of dmin sends more redundant information than an error code with a lower value of dmin

•  Code rate that gives the ratio of a dataword size to the codeword size

Error Detection and Correction

•  Relation between Hamming Distance and Error –  When a codeword is corrupted

during transmission, the Hamming distance between the sent and received codewords is the number of bits affected by the error

–  Ex : if the codeword 00000 is sent

and 01101 is received, 3 bits are in error and the Hamming distance between the two is d (00000, 01101) = 3

•  To guarantee the detection of up to e errors in all cases, the minimum Hamming distance in a block code must be dmin = e + 1 à e= dmin -1

•  To guarantee the maximum t correctable errors in all cases

⎥⎦

⎥⎢⎣

⎢ −=

21mindt

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Cyclic Redundancy Codes (CRC)

•  Term cyclic is derived from a property of the codewords: –  A circular shift of the bits of any codeword produces another one

•  A (n=7, k=4) CRC by Hamming

CRC generator and checker

Refer to your notes for examples!

transmits n bits which is exactly divisible by some number (predetermined divisor) receiver divides frame by that number

n-k bit

n-k+1 bit

n-k bit

k + (n-k) bits predetermined divisor

Cyclic Redundancy Codes (CRC)

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CRC is the remainder!

CRC generator and checker

•  Example : Division in CRC Encoder

predetermined divisor

CRC generator and checker

•  At the Receiver: –  Example : Division in CRC Decoder

Known divisor

Another CRC Example

http://www.macs.hw.ac.uk/~pjbk/nets/crc/

Message: 1010001101 Pattern (divisor): 110101 Transmit: 1010001101 + 01110

Step through to calculate the remainder!

Cyclic Redundancy Codes (CRC) Mathematical Representation

•  Let M(x) be the message polynomial •  Let P(x) be the generator polynomial (divisor)

–  P(x) is fixed for a given CRC scheme –  P(x) is known both by sender and receiver

•  Create a block polynomial F(x) based on M(x) and P(x) such that F(x) is divisible by P(x)

)(0)(

)()(

xPxQ

xPxF

+=

Example of CRC

•  Send –  M(x) = 110011 à x5+x4+x+1 (6 bits) –  P(x) = 11001 à x4+x3+1 (5 bits, n = 4)

à 4 bits of redundancy –  Form xnM(x) à 110011 0000

à x9+x8+x5+x4 –  Divide xnM(x) by P(x) to find C(x)

100001

1001 11001 10000

11001110011000011001

Send the block 110011 1001

•  Receive

00000 11001 11001

11001110011100111001

No remainder à Accept

à C(x)

)(0)(

)()(

xPxQ

xPxF

+=

n-k bit

n-k+1 bit

n-k bit

k + (n-k) bits

CRC Example

1010001101

110101

01110

1010001101 + 01110

1010001101 + 01110

110101

00000

Message: 1010001101 Pattern (divisor): 110101 Transmit: 1010001101 + 01110

Block Code – Error Detection and Correction

•  Hamming Distance –  The Hamming distance between two words is the number of differences

between corresponding bits. –  Easily found by applying the XOR operation on the two words and count the

number of 1s in the result. –  Ex : Hamming distance d (000,011) = 2 –  Ex : Hamming distance d (10101, 11110) = 3

•  Minimum Hamming Distance

–  The minimum Hamming distance is the smallest Hamming distance between all possible pairs in a set of words

–  Ex : Find the minimum Hamming distance of the coding scheme in table 1

Sol: = 2

Table1

•  Minimum Hamming Distance –  Ex : Find the minimum Hamming distance of the coding scheme in

the table.

–  3 parameters to define the coding scheme •  Codeword size n •  Dataword size k •  The minimum Hamming distance dmin

•  We can call this (Table 2) coding scheme C(5,2) with dmin =3 •  Note : the coding scheme for (Table 1) is C(3,2) with dmin =2

Table2

Block Code – Error Detection and Correction

Error Detection and Correction

•  Example: Give the above coding scheme we experience 1000 bit errors when transmitting 1Gigbit bits –  Calculate the bit error rate. –  Calculate the code rate.

–  What is the coding scheme: C(n,k),What is dmin? –  How many errors can be detected using the given encoding technique? –  How many errors can be corrected using the given encoding technique? –  Assume the received codeword is 00101. What will be the likely original

dataword?

Error Detection and Correction

•  Example: Give the above coding scheme we experience 1000 bit errors when transmitting 1Gigbit bits –  Calculate the bit error rate. –  Calculate the code rate.

–  What is the coding scheme: C(n,k),What is dmin? –  How many errors can be detected using the given encoding technique? –  How many errors can be corrected using the given encoding technique? –  Assume the received codeword is 00101. What will be the likely original

dataword? e = dmin -1=3-1=2

t= 1

BER: 10^-6

Code rate = k/n = 2/5

⎥⎦

⎥⎢⎣

⎢ −=

21mindt

C(5,2) with dmin =3

Error Detection and Correction •  The larger the dmin the better •  The code should be relatively easy to encode/decode •  We like n-k to be small à reduce bandwidth •  We like n-k to be large à reduce error rate

System Performance

•  Assume n=4, k=2 à Code rate ½

•  Given BER, coding can improve Eb/No –  Lower Eb/No is required –  Code gain is the reduction in

dB in Eb/No for a given BER •  E.g., for BER=10^-6 à

code gain is 2.77 dB •  Energy per coded bit (Eb) = ½ data bit (Eb) –  Hence, BER will be 3dB less –  This is because Ebit=2xEdata

•  For very high BER, adding coding requires higher Eb –  Not due to overhead

10^-6

Channel bit error rate

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Error Correction with Row and Column (RAC) Parity

•  Imagine an array of 3-rows and 4-columns, with a parity bit added for each row and for each column –  this arrangement is known as a Row and Column (RAC) code

•  Example RAC has n= 20, which means that it is a (20, 12) code •  Total of 20 bits; 12 data bits

In cases where two or three bits are changed an RAC encoding will be able to detect an odd number of errors

Even Parity (P=0 if even 1s)

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Error Correction with Row and Column (RAC) Parity

•  How error correction works? Assume that one of the bits in the figure below is changed during transmission: –  When the receiver arranges the bits into an array and parity bits are

recalculated •  two of the calculations will disagree with the parity bits received •  a single bit error will cause two calculated parity bits to disagree with the

parity bit received

RAC can only correct single-bit errors