IMPERIAL COLLEGE of SCIENCE, TECHNOLOGY and MEDICINE, DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING.

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1 IMPERIAL COLLEGE of SCIENCE, TECHNOLOGY and MEDICINE, DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING. COMPACT LECTURE NOTES on COMMUNICATION THEORY. Prof. Athanassios Manikas, version Spring 22 Digital Communications: An Overview of Fundamentals (Continued) 4: Channel Coding

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3 4. ERROR CONTROL CODING (or channel coding) H( f) ^ ^ ^ ^ ^ ^ Digital Communications - An Overview of Fundamentals 8 its per sec =Symbol rate x its per symbol C r b C H+1/m x Redundancy _ l_ H mut r inf H x INF. SOURCE SOURCE ENCODER l opt CHANNEL ENCODER Ideal Case CHANNEL ENCODER CHANNEL A2 1/2 1/2 2 ^ Digital Communications - An Overview of Fundamentals 9

4 AIM: Since practical channels are noisy there is a need to have error detection procedures and then to take some action. This requires the introduction at the transmitter of controlled redundancy which make it possible for the receiver to detect or even correct errors. The coding process described in this section is concerned with the blocks of Discrete Channel Encoder and Channel Decoder and has a totally different meaning from that held in the previous section (Source Encoder). THUS: to reduce the channel noise/interference effects we introduce deliberately some redundancy in the sequence at point and this is what a Discrete Channel Encoder does. This redundancy aids the receiver in decoding the desired sequence; at point 1 ^ : a binary sequence. The channel decoder attempts to reconstruct the sequence at from: ˆ the knowledge of the code used in the channel encoder, and ˆ the redundancy contained in the received data Digital Communications - An Overview of Fundamentals Capacity of a Channel ìthere is a theoretical upper limit to the performance of a specified digital communication system with the upper limit depending on the actual system specified. However, in addition to the specific upper limit associated with each system, there is an overall upper limit to the performance which no digital communication system, and in fact no communication system at all, can exceed. This bound (limit) is important since it provides the performance level against which all other systems can be compared. The closer a system comes, performance wise, to the upper limit the better. ìthe theoretical upper limit was given by Shannon (1948) as an upper bound to the maximum rate at which information can be transmitted over a communication channel. This rate is called channel capacity and is denoted by the symbol G. Digital Communications - An Overview of Fundamentals 121

5 8. Shannon's 2 Coding Theorem (or Shannon's Noisy Coding Theorem) it is related to channel coding or 'error control coding' (i.e. channel encoder & decoder) Theorem: If an information source has an information bit rate channel has capacity G then: r b and a if r b ŸG there exists a CHANNEL coding scheme for which the source output can be transmitted over the channel and recovered with an arbitrary small probability of error. if r b G there is not possible to transmit and recover the information with an arbitrarily small probability of error. Digital Communications - An Overview of Fundamentals 122 CHANNEL Information Source rb E N C O D E R iff r <C b (C,) p= e 0 D E C O D E R p e ^ ^ H( f) Digital Communications - An Overview of Fundamentals 123

6 Two ways to introduce redundancy (i.e. two types of channel coding) ˆ block coding block codes, as the name implies, segment the data stream into blocks or frames of k information symbols, and encode these into codewords Ðframes Ñ of length n Ðwith n 5Ñ. There is no interaction between the individual blocks and each codeword n-tuple Ðframe Ñ is uniquely determined by the message k-tuple. The code rate is simply the ratio k/n, and describes the amount of redundancy in the code. f k-bits at point È n-bits at point 1 Ú f repeat each bit of 8times, 1-bits at point È n-bits at point 1 e.g. Û f Ü Hamming (7,4) 4-bits at point È 7-bits at point 1 ˆ convolutional coding more complicated but more powerful than block coding Digital Communications - An Overview of Fundamentals LINEAR LOCK CODES let =alphabet of H symbols. In a binary case H2. lock codes a set of fixed-length vectors called codewords elements of a codeword if the length of the codeword is 8 and the alphabet is binary 8 then Ö- ß - ß ÞÞÞß - = set of possible codewords (where Q # ) Þ # Q From the above set we select a subset of # 5 codewords ( 5 8Ñto form the set of LOCK CODES Ö- ß - ß ÞÞÞß - # # 5 Digital Communications - An Overview of Fundamentals 125

7 model: ( 8ß 5) with 8 5 Channel Encoder (Linear lock Coder) where x 3 = cx3 ß x 3# ß... ß x3k d = a vector of binary digits c 3 = cc3 ß c 3# ß... ß... ß c3n d Digital Communications - An Overview of Fundamentals EXAMPLES of lock Codes: 1) Triple Repetition Code: This is a single-error corrected code, then use majority logic system to decide at the receiver, i.e. Source Encoder (Linear lock Coder) Source Decoder (Linear lock Coder) Errors occur only when there are 2 or 3 erroneous digits in a word e.g. 0 encoded as 0, then transmitted and finally received as 1, will be decoded as 1. Digital Communications - An Overview of Fundamentals 127

8 example IF the probability of a bit in error is : / =0.2 THEN the probability of a codeword in error (if the above repetitive code is used) can be found as follows: Pr( error ) Pr( 2bits or 3 bits in errors in a 3bit sequence) Pr( 2bits in error in a 3bit sequence) Pr( 3bits in error in a 3bit sequence)= 3 # $ $ 2 : / Ð :/ Ñ $ :/ Ð :/ Ñ =.... = for <0.5 the above coding improves the reliability. : / Digital Communications - An Overview of Fundamentals 128 The triple repetition code also works for double-error detection e.g. only correct received sequences are 0 and 1 so triple errors will cause an undetected error with probability: Pr( undetected error) Pr ( $bits in error in a 3bit sequence) $ $ / $.:. Ð : / Ñ = Digital Communications - An Overview of Fundamentals 129

9 2) An Example of HAMMING CODE The Hamming Ð7,4 Ñ code is constructed as follows: the information sequence [ u,u,u,u # $ %] is encoded into the codeword [ u,u,u,u # $ %,u Šu# Šu %, u Šu$ Šu %, u Šu# Šu$ Ñ : : : x 3 Òuuuu # $ Ó - 3 Òuuuu::: Ó # $ # $ % & ' ( ) * # $ % & ' point A3 u u# u$ u % point A4 u u# u$ u% p p# p$ - -# -$ -% -& -' -( -) -* - - -# -$ -% -& - ' Digital Communications - An Overview of Fundamentals Implementation of lock Codes using shift registers and modulo-two adders e.g. Hamming Ð7,4Ñ Shift register u1 u2 u3 u4 Mod-2 adders u1 u2 u3 u4 p1 p2 p3 Shift register Digital Communications - An Overview of Fundamentals 131

10 Generator Matrix & Parity bits Matrix : - Ô p p #... pð8 5Ñ p# p ##... p#ð8 5Ñ where -= Ö Ù= c Õ p p... p Ø is the generator matrix for the 5 5# 5Ð8 5Ñ (n,k)-code. ˆk ß d Digital Communications - An Overview of Fundamentals 132 Parity Check matrix. ˆ Associated with any linear (n,k)-code is the DUAL code. This is defined as the linear ( n,n-k) code with # n k codewords (vectors) ˆ Generator matrix for the DUAL-code :. c T, ˆ d= c T, ˆ d n-k n-k ˆ This matrix has the following properties: c 3. Ṭ T. -. Digital Communications - An Overview of Fundamentals 133

11 ˆ The generator matrix. of the Dual-code is called parity-check-matrix since it is used by the receiver to check that the received word r satisfies the equation r. Ṭ = 0 Digital Communications - An Overview of Fundamentals 134 Implementation of Hamming Ð7,4Ñ Digital Communications - An Overview of Fundamentals 135

12 Important Parameters of lock Codes name: The code is called Ðn,kÑ-code rate of the code: is defined as R = < 1 c k n weight of a codeword c 3 : A 3 No. of 's that this codeword c 3 contains Hamming distance between two codewords c3 & c4:. 34 No. of elements (or positions) in which the codewords c3 & c4 differ. 0 Ÿ. 34 Ÿ8 minimum distance of the code: d min = min d 34 e 34 f N..: let c = c0ß 0 ß... ß 0 dþ Consider now another codeword c3 with weight w. 3 You can see that d 3=w 3 and d min = min ew 3f 3 3Á Digital Communications - An Overview of Fundamentals 136 for a given code with min. dist. d 738 it can be proven that: 1. we can detect up to. 738 errors 2. by moving to the nearest acceptable word we can correct. # # 738 errors if. 738 even # errors if. 738 odd Digital Communications - An Overview of Fundamentals 137

13 4.5. CONVOLUTIONAL CODES Notation: GG( 85O ß ß Ñ where Ú 8 À Û 5 À Ü O À output bits number of bits per shift register constraint length Convolutional coding retains these two parameters k and n Ðand therefore the same definition of code rate Ñ, but also requires a third integer O which is the constraint length of the code. This value represents the number of k-bit stages in the linear finite-state shift register of the convolutional encoder Ðsee figure below Ñ. Digital Communications - An Overview of Fundamentals 138 The data is shifted in to the register k bits at a time, while the corresponding n-tuple codeword is taken from the outputs of n linear algebraic function generators in turn. i/p 1st Shift register 2nd Shift register K-th Shift register k k k Mod-2 adders... 1st 2nd n-th Multiplexer o/p Digital Communications - An Overview of Fundamentals 139

14 A codeword n-tuple is therefore not uniquely determined by the information k-tuple present at the input to the encoder, but also by the previous ÐO- Ñ information k -tuples before that as well: the main difference between a block and a convolutional encoder is the memory of past information words. Thus for the same code rate, convolutional codes have additional errorcorrecting capability, but conversely require a more complex decoder. For more information see: R. Steele Mobile Radio Communications pp Channel Decoder for a convolutional channel encoder: Viterbi algorithm (too complex to be discussed in this course) Digital Communications - An Overview of Fundamentals 140 EXAMPLE (see Appendix) GGÐ#ß ß $Ñà 8 #à 5 à O $ st + + 2nd Digital Communications - An Overview of Fundamentals 141

15 ðóóñóóò O$ # ðóóñóóò H H ÐßßÑ 1st + + 2nd # ïh ÐßßÑ Digital Communications - An Overview of Fundamentals 142 GG convolution of impulse response of encoder with i/p sequence i/p :... o/p:,,,,,,... GGÐ#ß ß $Ñà 8 #à 5 à O $ st 2nd - Ô ß ß ß ß ß ß ÞÞÞ ß ß ß ß ß ß ÞÞÞ Ö Ù ß ß ß ß ß ß ÞÞÞ Õ ÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞÞ Ø semi-infinite matrix Digital Communications - An Overview of Fundamentals 143

16 STATE TRANSITION DIAGRAM of GG current state State a State b State c o/p i/p=1 i/p=0 next state State a State b State c State d State d Digital Communications - An Overview of Fundamentals 144 STATE DIAGRAM of GG State d State b i/p=1 State a i/p=0 o/p State c 2 ÐO Ñ5 %=>+>/= Digital Communications - An Overview of Fundamentals 145

17 X VIPPMW HMEKVEQ (VITERI ALGORITHM) State a o/p i/p=0 i/p=1 State b State c State d Digital Communications - An Overview of Fundamentals 146 Digital Communications - An Overview of Fundamentals 147

18 4.6. Interleaving Most well known codes have been designed for AWGN ÐAdditive White Gaussian Noise Ñ channels i.e. the errors caused by the channel are statistically independent. Thus when the channel is AWGN-channel these codes increase the reliability in the transmission of information. However, if the channel exhibits bursty error characteristics then these error clusters are not usually corrected by these codes. How we define a burst of errors: a sequence of bit errors the 1st and last of which are 1s. Digital Communications - An Overview of Fundamentals 148 solution: use an interleaver Its objective is to interleave the coded data in such a way that the bursty channel is transformed into a channel having independent error and thus a code designed for independent channel errors Ðshort burst Ñ is used. Interleaver of degree m: reorders the output of the channel encoder as follows: Digital Communications - An Overview of Fundamentals 149

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