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1

Coding

with emphasis on

Error Control Coding

A. Marchioro / PH-ESE-ME

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“The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at

another point”

Claude Shannon

A Mathematical Theory of Communication,

The Bell System Technical Journal, vol. 27, pp.379-423, July 1948

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What type of coding?

Four main reasons for using coding:

Adapt to electrical or accidental characteristics

of the channel

Reduce the amount of data to be

transmitted ï Source Coding

Control errors ï Error Coding

Conceal and hide data ï Cryptography

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Motivation: Adapt

Data may need to be coded because of: Electrical reasons

• Clock Recovery

• Line balancing • Transmit statistically same number of 0’s and 1’s

• Aid gain control in AGC amplifiers at receiver’s end

• Facilitate synchronization

• Spectrum limitation and modification

• Line adaptation: • PAM

Logical reasons • Simplification of frame recognition

• E.g. add an easily recognizable start/stop condition

• Protocol reasons • Distinguish data from control information

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Some digital waveforms

NRZ-L

NRZ-M

NRZ-S

Unipolar-RZ

Bipolar-RZ

Bi- -L

Bi- -M

Bi- -S

Delay Mod

Di-code NRZ

Di-code RZ

1 0 1 1 0 0 0 1 1 0 1

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Motivation: Error Control

“Two weekends in a row I came in and found that all my stuff had been

dumped and nothing was done. I was really aroused and annoyed

and I wanted those answers and two weekends had been lost. And

so I said, ‘Damn it, if the machine can detect an error, why can’t it

locate the position of the error and correct it?’”

from an interview with R. Hamming,

February 3-4, 1977, quoted in T. Thompson, p.17

“The purpose of this memorandum is to give some practical codes

which may detect and correct all errors of a given probability of

occurrence, and which detect errors of even a rarer occurrence”. from R. Hamming,

‘Self-Correcting Codes – Case 20878,

Memorandum 1130-RWH-MFW,

Bell Telephone Laboratories, July 27, 1947

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Motivation: Error Control

Hardware is not perfect Errors are introduced by:

• Signal reduction • Transmission from satellite

• Trans-oceanic optical fibers

• Noise and interference • Radio communication

• People • Reading a Credit Card number

• Imperfect parts • Richard Hamming worked first to avoid catastrophic errors in early relay-based

computers

• Radioactive particles causing perturbation in electronic circuits

• Wear-out and aging • e.g. fingerprint and scratches on CD and DVDs

Basic idea: Add redundant information as to make up for lost or garbled information

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Applications

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Applications Voyager 2 Flight System Performance (Dec 2008)

Down transmission speed ~ 7Kbit/sec

Up transmission speed ~50 bits/sec

PROPELLANT/POWER CONSUMABLES STATUS AS OF THIS REPORT

Spacecraft Consumption One Week[*] (gm) 6.29

Propellant Remaining (Kg) 28.63

Output (Watts) 281.5

RANGE, VELOCITY AND ROUND TRIP LIGHT TIME AS OF Nov 7,2008

Distance from the Sun (Km) 13,077,000,000

Distance from the Earth (Km) 13,132,000,000

Total Distance Traveled (Km) 19,409,000,000

Velocity Relative to Sun (Km/sec) 15.524

Velocity Relative to Earth (Km/sec) 41.592

[*] Power Source is Radioisotope Thermoelectric Generator

http://voyager.jpl.nasa.gov/mission/weekly-reports/

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Applications

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ADSL needs FEC

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Parity

In B = {0,1}, start with a message word: S = {s1s2s3s4s5s6s7}

Compute a “Parity” character s8 defined as:

c8 = s1 s2 s3 s4 s5 s6 s7

where is the exclusive-OR (or the sum mod 2).

Parity check can detect all single errors (but can not give the position)

Parity check can not detect double errors

Used:

- often in computer memories

- in serial terminals data transmission

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Two-Dimensional Parity

Pa

rity

X

ParityY

2 Errors

1 0 1 1 1 0 0 0

0 1 0 0 0 1 1 1

1 1 0 0 0 0 1 1

0 1 0 1 1 0 1 0

1 0 0 1 0 1 1 0

0 0 0 1 0 1 0 0

1 1 0 0 1 0 0 1

0 0 1 0 1 1 0

1 0 1 1 1 0 0 0

0 1 0 0 0 1 1 1

1 1 0 0 0 0 1 1

0 1 0 1 0 0 1 1

1 0 0 1 1 1 1 1

0 0 0 1 0 1 0 0

1 1 0 0 1 0 0 1

0 0 1 0 1 1 0

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Two-Dimensional Parity

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Repetition Code

Take each symbol si in S and repeat it n times.

This is an (n, 1) code.

For example the word {s1s2s3} becomes the codeword

{s1s1s1s2s2s2s3s3s3}

Efficiency (= rate) of the code is: 1/n

The minimum distance (see later) is n and the number of errors t that can

be corrected is:

t = (n – 1)

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Repetition anywhere?

Redundancy in mainframes:

http://www.research.ibm.com/journal/rd/435/spainhower.html

http://www.fujitsu.com/downloads/PRMQST/documents/whitepaper/PRIMEQUEST-High-Availability.pdf

In RAID disk arrays:

RAID1 (Mirroring)

Space electronics:

“A Comparison of Fault-Tolerant State Machine Architectures for Space-Borne Electronics”, IEEE Transactions on Reliability, Vol. 45. No. 1. 1996 March

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Triple Module Redundancy

Triple redundancy Three copies of same user logic + state_register

Voting logic decides 2 out of three (majority)

Used in: High reliability electronics

Mainframes

Problems: 300% area and power

corrects only 1 error

can get very wrong with two errors

FSM1

FSM2

FSM3

Voting logic

Input Output

CLK

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ISBN (1)

ISBN: d1d2d3d4d5d6d7d8d9d10

T = (10d1 + 9d2 + 8d3 + 7d4 + 6d5 + 5d6 + 4d7 + 3d8 + 2d9) mod 11

d10 = 11 – T

Example: the check digit in: ISBN 038550420-9 is computed as:

11 – [(10*0 + 9*3 + 8*8 + 7*5 + 6*5 + 5*0 + 4*4 + 3*2 + 2*0) mod 11] =

11 – 2 = 9 q.e.d.

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ISBN (2)

Statement:

All single errors are detected by the ISBN check digit.

Proof:

An error will not be detected if T is changed by a multiple of 11 as the check digit is computed mod 11.

Assume that the ith digit is wrong, i.e. it was di and was changed into d’i with 0 d’i 9.

The difference of the two digits calculated in the check-sum is:

i · di – i · d’i

= i · (di – d’i)

and for the error to be undetected it must be that

gcd(11, i · (di – d’i) ) = 11

which by an elementary theorem in number theory requires that gcd(11, i) = 11 or that gcd(11, (di – d’i) ) = 11. But neither of these statements can be true, as i < 11 and -9 (di – d’i) 9.

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ISBN (3)

Statement:

All exchanges of two consecutive digits can be detected by the ISBN code.

Proof:

Assume that two consecutive digits ni and ni+1 are exchanged in an ISBN

number. In order for the error to be undetected, the T sum must be changed

by a multiple of 11. But if ni is the digit in position i in the sum, the two

terms in the sums are:

i · ni + (i+1) · ni+1

i · ni+1 + (i+1) · ni

and the difference is simply:

ni+1-ni

which for 0 ni 9 is always comprised between -9 and +9, i.e. it is never a

multiple of 11, q.e.d.

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ISBN-13 (4)

The new 13 digit ISBN-13 control code is constructed as:

T = (d1 + 3*d2 + d3 + 3*d4 + d5 + 3*d6 + d7 + 3*d8 + d9 + 3*d10 + d11 + 3*d12 ) mod 10

d13 = 10 – T

Now several digits exchange can NOT be recognized, but more books can be cataloged…

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Families of Error Control Methods

Block Codes: codeword built only on current message-word

Non-block codes: codeword depends on current message word and

of some past words, ex:

Convolutional

Examples of codes:

Hamming

Bose-Chauduri-Hocqueghem (BCH)

Golay

Reed-Solomon (RS)

Reed-Müller

Low Density Parity Check Codes

Turbo Codes

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Theory and practice

Coding and decoding schemes are, especially for

algebraic codes, perfectly well described by mathematical

methods providing the transformations:

The real detection and correction of errors instead require

much more ad-hoc, heuristic methods that allow to

recover the original message from a corrupted received

message

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Hamming (intuitive version)

s1 s2

s3

s4

c5

c6 c7

Definition:

cj = computed to give even parity in the circle

source parity

Notice:

the 16 code words in Hamming(7,4) differ from each other by at least 3 bits.

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Hamming (2)

r1 r2

r3

r4

r5

r6 r7

During transmission the message word

s1s2s3s4c5c6c7

is (potentially) modified by an error

in (the unknown)

position j and is received as:

r1r2r3r4r5r6r7

for example, for j=2:

r1r2r3r4r5r6r7 = s1s2s3s4c5c6c7 0100000

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Hamming (3)

1 1*

0

0

1

0 1

Example:

for an original word: 1000101

assume that e=0100000 occurred,

resulting in r=1100101

Circles with odd (=wrong) parity

are now marked

Decoding and correcting trick:

can we find a single bit (assuming that

there was just one error) that lies inside

all the marked circles and outside

of the unmarked one?

0

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Hamming Codes (1)

Another simple construction of Hamming Code:

Given the four data bits (a0,a1,a2,a3), construct

three parity bits as follows:

p0 = a0 + a1 + a2

p1 = a1 + a2 + a3

p2 = a0 + a1 + a3

(here “+” is modulo 2 addition) and send the

codeword: (a0, a1, a2, a3, p0, p1, p2).

The valid codewords are therefore given in the table

on the right:

Notice that we use 27 code-words to represent 24

possible message-words

0 0 0 0 0 0 0

0 0 0 1 0 1 1

0 0 1 0 1 1 0

0 0 1 1 1 0 1

0 1 0 0 1 1 1

0 1 0 1 1 0 0

0 1 1 0 0 0 1

0 1 1 1 0 1 0

1 0 0 0 1 0 1

1 0 0 1 1 1 0

1 0 1 0 0 1 1

1 0 1 1 0 0 0

1 1 0 0 0 1 0

1 1 0 1 0 0 1

1 1 1 0 1 0 0

1 1 1 1 1 1 1

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Hamming Codes (2)

The decoder receives: (a’0, a’1, a’2, a’3, p’0, p’1, p’2) and computes:

s0 = p’0 + a’0 + a’1 + a’2

s1 = p’1 + a’1 + a’2 + a’3

s2 = p’2 + a’0 + a’1 + a’3

called the “syndromes”. If there has been no error, these are all zero, if there has been one

error, one or more of them may be non-zero. The syndromes depend only on the error

pattern, as in the table below:

0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 1

0 1 0 0 0 0 0 0 1 0

0 1 1 0 0 0 1 0 0 0

1 0 0 0 0 0 0 1 0 0

1 0 1 1 0 0 0 0 0 0

1 1 0 0 0 1 0 0 0 0

1 1 1 0 1 0 0 0 0 0

Syndrome Error

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Hamming Codes (3)

a0

a1

a2

a3

a0

a1

a2

a3

p0

p1

p2

Hardware for encoder

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Hamming Codes (4)

a0

a1

a2

a3

a’0

a’1

a’2

a’3

p’0

p’1

p’2

Hardware for decoder

Correction Logic

+

+

+

+

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Hamming Codes (5)

A compact description of the encoding operation and of the syndrome

computations may be given by using matrix notation such as:

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Hamming in use

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Block coding as linear maps

Error Coding always assumes an increase in the amount of

information to be sent or stored, as the ‘extra’ information has to take

into account some means to recover what gets corrupted or lost

during transmission or storage (retrieval) or data.

Block coding performs this on finite blocks of data, without

reference to previous blocks, and with all redundant information

contained in the block itself.

Unlike block coding, convolutional coding performs encoding based

on the current set of data to be coded and on the history of previous

blocks, i.e., a given data set is mapped on a number of different data

sets, depending on the content of the previously coded sets.

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Coding as a map FkØ Fn

Fk

Fn

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Error Detection in Fn

Degradation due to

Transmission or storage

(retrieval)

Recoverable

Undetected Error

Confused, unrecoverable

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Linear Codes: Structure

Definition: Linear Block Code

A linear code C is a subspace of dimension k of the vector space Bn

for simple binary vectors, or in general of GF(q)n.

In other words, C is a non-empty set of n-tuples over GF(q)n, called

codewords, respecting the structure of a vector space, that is the

addition of two codewords is always a codeword, and the

multiplication of a codeword by a scalar element of the field is also

always a codeword. From the definition, it follows that the zero word

must always be part of a linear codeword, as

" c œ C , (c ) + (-c) = 0 œ C

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Cyclic Codes: Basic definitions

Let c = (c0, c1,…, cn-2, cn-1) œ GF(q)n.

The vector:

c’= (cn-1, c0, c1, …, cn-2)

is called a right cyclic shift of c.

Definition: Cyclic code

An (n, k) block code C is said to be a cyclic code if it is linear and if for

every c its right shift c’ is also a codeword.

Polynomials can be considered as ‘support’ for codewords:

One can conveniently represent a codeword c = (c0, c1, …, cn-2, cn-1) as follows:

c(x) = c0 + c1x + c2x2 + cn-1x

n-1

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Cyclic Codes: Simple Example

The code:

c0 : 0000000 c1 : 1011100

c2 : 0101110 c3 : 1110010

c4 : 0010111 c5 : 1001011

c6 : 0111001 c7 : 1100101

is cyclic, in fact it can be noticed that using shift and linearity, starting with

cg=(1011100):

c0 : 0000000 c1 : cg

c2 : cg>>1 c3 : c2+c4

c4 : cg>>2 c5 : c1+c4

c6 : c2+c4 c7 : c1+c2+c3

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Reed-Solomon: very

elementary introduction

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RS coding, very simplified version

Let’s imagine we want to transmit just two numbers and that we are ready

to add some redundancy to ‘repair’ faulty data.

One method could be:

Compute the straight line that goes through the two numbers intended as y

values in a plot at fixed x=1 and x=2 coordinates, i.e. the line going though

the points (1, y1) and (2, y2).

Instead of transmitting just the two numbers, now transmit the five y numbers

on the same straight line corresponding to x-coordinates x=1,2, and 3,4,5

Now assume that one of the yi gets corrupted

Clearly the other 4 numbers are sufficient to establish a straight line and the y

value corresponding to the corrupted one can be reconstructed from the other

points

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RS coding, simplified version(2)

If two numbers are corrupted during transmission, then there are still

three on a straight line (well…, in the real worst case there might be two

straight lines, can you find out why?)

Once the straight line is found again among the “good” three y-points,

the exact location of the two corrupted y-points will be reconstructed.

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RS coding, simplified version(3)

To be capable of correcting three numbers, a parabola is built first :

Let y1, y2, y3 be the three numbers to be transmitted

Let x1 = 1, x2 = 2, x3 = 3 be the corresponding x-coordinate of a parabola

on (x1, y1), (x2, y2) and (x3, y3)

Introduce some new x=points, say at x4 = 4, x5 = 5, x6 = 6 and then

compute the extra y-coordinates on the parabola y4, y5, y6.

Now repeat the method used for the straight line to correct one or two

corrupted y-points.

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RS coding, simplified version(4)

The original points are

(1,3), (2,1), (3,2)

to these we add:

(4,6), (5,13), (6,23)

and we actually transmit:

(3,1,2,6,13,23)

(The polynomial in this

case is 8 - 6.5 x + 1.5 x2 )

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RS coding, simplified version(5)

So, the trick is:

Let the number of points y={y1, y2, …, yk} to be transmitted be k

Determine a polynomial of degree k – 1 fitting them all at standard ‘x’

coordinates x1, x2, …, xk

Calculate some redundant points on the same polynomial at coordinates

xk+1, xk+2,…,xk+r

When the y data are received see if they all fit on a polynomial of degree

k – 1

• If yes, no error has occurred

• If not

• Try to see if one point is not on the curve and recover it

• If the search for one is not ok, try to see if two are not on the

curve and recover them

• …

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Example

1. Music or Data CD ROM Coding

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CD recording block diagram

A/D EFM CIRC

Encoder

Control and

Meta info Synchronization

44.1KHz/ch

1.41Mbps

1.88Mbps

1.94Mbps 4.32Mbps

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Cross-Interleaver

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Encoding block in CD

RS

(28,24)

RS

(32,28) …

2

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Din{24x8} Dout{32x8}

C2 Encoder C1 Encoder

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C1,2 Encoders

Both work in GF(28) and are derived from

(255, 251) codes.

C2 Shortened to (28, 24)

C1 Shortened to (32, 28)

Remember that for RS codes

dmin= n – k + 1

therefore each can correct two errors.

C1,2 Encoders (2)

C1 could correct for two errors, but actually

when 1 error or more is detected, the decoder

signals a failure to the logic and inserts all

zeros (i.e. it replaces errors with erasures, i.e.

zeros). The de-interleaver spreads then the

zeros, and the C2 decoder fixes those errors.

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EFM (8 to 14 Modulation)

CD ROM Error Correction Summary

Longest completely

correctable burst

~ 4,000 data bits (2.5 mm track)

Longest interpolatable

burst

~12,300 bits (7.7 mm track)

Undetected Error Rate < 1 every 750 hours @

BER=10-3

Negligible @ BER<10-4

Storage necessary 2048 Byte RAM

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Conclusion

Error Control Coding is a heavily mathematical theory which

has been proved to be fundamental for modern

communication technologies

Error correction can be traded off for transmission power when the

coding scheme can make up for losses generated by small S/N ratios

In the case of the GBT project (see seminar in three weeks),

the objectives were:

To be able to correct burst of errors

• Generated by SEU in the pin-diode

• Generated by hits in the analog or digital circuitry of the transceiver

• To introduce FEC with minimal latency

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Bibliography

Good books on Coding:

R. Blahut, Algebraic Codes for Data Transmission, Cambridge U.P., 2003

O. Pretzel, Error Correcting Codes and Finite Fields, Oxford U.P. 1992

S. Wicker, Error Control Systems, Prentice Hall, 1995

The Mathematics underneath:

J. A. Gallian, Contemporary Abstract Algebra, Houghton Mifflin, 2006

McEliece, Finite Fields for Scientists and Engineers, Kluwer, 1986

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Extra material

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CRC (1)

CRC codes are a class of binary codes built on cyclic codes, i.e. with codewords

built as multiples of a generator polynomial g(x) with coefficients in GF[2].

CRC codes allow error detection (see below), but no error correction and are used

extensively in network protocols which allow for retransmission.

Let us denote as:

Rg(x)[·]

the operation of computing the remainder of the argument after dividing by g(x).

If m(x) is the original message polynomial, a CRC encoding operation can be

written as:

c(x) = xr m(x) + Rg(x)[xr m(x) ]

notice that deg( c(x) ) = r + deg( m(x) )

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CRC (2)

If an error occurs during transmission, the received word is now:

r(x) = c(x) + e(x)

where e(x) is the polynomial representing the error pattern.

To find out if an error has occurred we now compute the syndrome:

s(x) = Rg(x)[ r(x) ]

that is:

s(x) = Rg(x)[ c(x) + e(x) ] = Rg(x)[ c(x) ] + Rg(x)[ e(x) ]

but by construction of the cyclic code Rg(x)[ c(x) ] = 0 (see Encoding (systematic)) , therefore:

s(x) = Rg(x)[ e(x) ]

Notice that if e(x) is a code polynomial, say e(x) = c1(x), then s(x)=0 and the error

passes undetected.

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CRC (3)

CRC: an example

g(x) = x16 + x15 + x2 + 1

m = [0,1,1,0,1,1,0,1,0,0,1,0,0,1,1,1] ñ x14 + x13 + x11 + x10 + x8 + x5 + x2 + x + 1

We get: x16 m(x) + Rg(x)[x16 m(x) ] =

x30 + x29 + x27 + x26 + x24 + x21 + x18 + x17 + x16 + x14 + x13 + x11 + x10 + x9 + x7

+ x6 + x4 + x2

or in bit format:

c=[0,1,1,0,1,1,0,1,0,0,1,0,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,1,0,0]

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CRC (4)

Table of commonly used CRC generator polynomials

CRC-4 g(x) = x4+x3+x2+x+1

CRC-7 g(x) = x7+x6+x4+1

CRC-8 g(x) = x8+x7+x6+x4+x2+1

CRC-12 g(x) = x12+x11+x3+x2+x+1

CRC-ANSI g(x) = x16+x15+x2+1

CRC-CCITT g(x) = x16+x12+x5+1

CRC-24 g(x) = x24+x23+x14+x12+x8+1

CRC-32b g(x) = x32+x26+x23+x22+x16+x12+x11+x10+x8+x7+x5+

x4+x2+x+1

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CRC (5)

Statement

All single bit errors can be detected by CRC

Proof:

The generator polynomial must have more than one non-zero term,

otherwise the code would be an (n, n) code (this can be seen by shifting and

linearly combining all codewords), and the minimum distance between

codewords would be 1 (i.e. a meaningless code).

Now, if the codeword polynomial have more than one non-zero terms, then it

can not divide a polynomial with a single xi term evenly, so there is always a

non-zero remainder, i.e. all single bit errors can be detected.

For an error to be undetected, it must itself be a codeword (seen above); but a

valid codeword is generated with a generator codeword that has more than

one xi term, so it can not be a polynomial with a single xi term set.

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Fundamental questions on codes

1. How is the code described and represented

2. How is encoding accomplished

3. How is decoding accomplished

4. How are these two operations achieved in a computationally

tractable time

5. What is the performance of the code

1. What are the properties: # of codewords, weight of codewords

6. Are there other families of codes that can provide better coding gain

7. How are these codes found and described

8. Are there constraints on the allowable values of n, k and dmin

9. Is there any limit on the amount of coding gain possible

10. For a given available SNR, is there a lower limit on the probability

of error than can be achieved

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RS combined with interleaving UDP packets in TCP/IP protocol do not have guaranteed delivery

RS is used to replace lost packets (“erasures”)

Data stream is framed into blocks of 249 bytes and encoded in RS(255,249) blocks, this has dmin = 7 and can correct 6 erasures

Messages are interleaved in blocks of 255xN

Blocks are send from columns

If a packet is lost, it is replaced by a “0” column

The receiver knows that packet “j” is lost because it is missing in the sequence

The RS code (organized in N rows) can recover up to 6 missing columns

c1,1 c1,2 c1,3 … c1,255

c2,1 c2,2 c2,3 … c2,255

… … … … …

cN,1 cN,2 cN,3 … cN,255

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EFM Coding

Data for EFM coding are considered in bytes. Each byte is translated according to a lookup table into a corresponding 14-bit codeword.

The 14-bit words are chosen such that binary ones are always separated by a minimum of two and a maximum of ten binary zeroes. This is because bits are encoded with NRZI encoding, or modulo-2 integration, so that a binary one is stored on the CD surface as a change from a zero to a one or a one to a zero, while a binary zero is indicated by no change. A sequence 0011 would be changed into 1101 or its inverse 1101 depending on the previous one written. If there are 2 zeros between 2 consecutive ones, then the written sequence will have 3 consecutive zeros (or ones), for example, 010010 will translate into 1100011 (or 0011100). The EFM sequence 100100010010000100 will translate into 111000011100000111 (or its inverse).

Because EFM ensures there are at least 2 zeroes between every 2 ones, it thus ensures that every one and zero is at least three bit clock cycles long. This property is very useful since it reduces the demands on the optical pickup used in the playback mechanism. The ten consecutive-zero maximum ensures worst-case clock recovery in the player.

EFM requires 3 merging bits between adjacent 14-bit codewords to ensure that consecutive codewords can be cascaded without violating the specified minimum and maximum runlength constraint. The 3 merging bits are also used to shape the spectrum of the encoded sequence. Thus, in the final analysis, 17 bits of disc space are needed to encode 8 bits of data.

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CC (1)

CC: d1d2d3d4 d5d6d7d8 d9d10d11d12 d13d14d15d16

T = (t(d1) + d2 + t(d3) + d4 + t(d5) + d6 + t(d7) + … + d14 + t(d15) ) mod 10

where

t(di) = add digits of (2*di),

i.e.

t(di) = (2 · di) for (2 · di) < 10

= ((2 · di) -10) + 1 for (2 · di) 10

d16 = 10 – T

Example: the check digit in: CC 1234 5678 9012 345 is d16 = 2.

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CC (2)

Statement:

All single digit errors are detected by the CC check digit.

Proof:

A change of digit di into digit d’i can change the sum T by

di – d’i

or by

t(di – d’i).

For the error to be undetected, the check digit has to change by a

multiple of 10.

But, neither of these terms can be a multiple of 10, the first is in fact

-10 < (di – d’i) < 10

and the second is also (try the function t(x) on all x from 0 to 9) in the

same range, so they are not a multiple of 10, q.e.d.

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CC (3)

Statement:

Almost all exchanges of adjacent digits can be detected by the CC check digit system.

Proof:

An exchange of two adjacent digits di and di+1 can modify the check sum by:

d = ( t(di) + di+1 ) - ( di + t(di+1 ) )

Well, convince yourself just by looking at the table of possible results for d:

… continue

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67

CC (4)

Proof (cntd):

The values on the diagonal are irrelevant, as the digits are exchanged with

themselves, the only case where d=0 is for 9ñ0 and for 0ñ9. But these are

only 2 cases out of 90, i.e. the efficiency for detecting exchanges of adjacent

CC digits is 2/90 97.8%.

Why 44.1 KHz

Explanation of 44.1 kHz CD sampling rateThe CD sampling rate has to be larger than about 40 kHz to fulfill the

Nyquist criterion that requires sampling at twice the maximum analog frequency, which is about 20 kHz for audio.

The sampling frequency is chosen somewhat higher than the Nyquist rate since practical filters neede to prevent

aliasing have a finite slope. Digital audio tapes (DATs) use a sampling rate of 48 kHz. It has been claimed that thier

sampling rate differs from that of CDs to make digital copying from one to the other more difficult. 48 kHz is, in

principle, a better rate since it is a multiple of the other standard sampling rates, namely 8 and 16 kHz for telephone-

quality audio. Sampling rate conversion is simplified if rates are integer multiples of each other.From John

Watkinson, The Art of Digital Audio, 2nd edition, pg. 104:In the early days of digital audio research, the necessary

bandwidth of about 1 Mbps per audio channel was difficult to store. Disk drives had the bandwidth but not the capacity for

long recording time, so attention turned to video recorders. These were adapted to store audio samples by creating a

pseudo-video waveform which would convey binary as black and white levels. The sampling rate of such a system is

constrained to relate simply to the field rate and field structure of the television standard used, so that an integer number

of samples can be stored on each usable TV line in the field. Such a recording can be made on a monochrome recorder,

and these recording are made in two standards, 525 lines at 60 Hz and 625 lines at 50 Hz. Thus it is possible to find a

frequency which is a common multiple of the two and is also suitable for use as a sampling rate.The allowable sampling

rates in a pseudo-video system can be deduced by multiplying the field rate by the number of active lines in a field

(blanking lines cannot be used) and again by the number of samples in a line. By careful choice of parameters it is

possible to use either 525/60 or 625/50 video with a sampling rate of 44.1KHz.In 60 Hz video, there are 35 blanked lines,

leaving 490 lines per frame or 245 lines per field, so the sampling rate is given by :60 X 245 X 3 = 44.1 KHzIn 50 Hz video,

there are 37 lines of blanking, leaving 588 active lines per frame, or 294 per field, so the same sampling rate is given by50

X 294 X3 = 44.1 Khz.The sampling rate of 44.1 KHz came to be that of the Compact Disc. Even though CD has no video

circuitry, the equipment used to make CD masters is video based and determines the sampling rate.(Reference kindly

provided by Kavitha Parthasarathy.)

from H. Schulzrinne, Professor and Chair in the Dept. of Computer Science; also with the Dept. of Electrical Engineering at Columbia University

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