6. FUNDAMENTALS OF CHANNEL CODER

Transcription

1 82 6. FUNDAMENTALS OF CHANNEL CODER 6.1 INTRODUCTION The digital information can be transmitted over the channel using different signaling schemes. The type of the signal scheme chosen mainly depends on the signal to noise ratio of the channel to be used, signal to noise ratio at the input of the receiver and the data rate to be transmitted. In practical systems the maximal signal power and the bandwidth of the channel are restricted and noise power spectral density is also fixed for a particular operating environment. Hence with all these constraints it often becomes difficult to arrive at the signaling scheme which will yield the acceptable probability of error for any given application. The only practical alternative to reduce the probability of the error is to use the channel coding technique [35]. Channel coding [37] is a viable method to reduce information rate through the channel and increase reliability. This goal is achieved by adding redundancy to the information symbol vector. The resulting symbol vector is a long coded vector of symbols that are distinguishable at the output of the channel. This is to improve the capacity of a channel by adding carefully designed redundant information to the data being transmitted through the channel. The process of adding the redundant information is known as channel coding. The channel coding is also known as error control coding where in calculated number of redundancy bits are added to the message bits. These bits doesn t carry any information but are used only for error detection and correction. The channel encoder systematically adds digits to the transmitted message digits and these digits make the channel decoder to detect and correct errors in the information bearing digits. The error detection and correction lowers the over all probability of error.

2 83 By using the speech compression technique the redundancy bits present in the message are removed and this is basically called the source encoder. The channel encoder adds the redundancy bits to lower the overall probability of error to the desired value. The basic block diagram for transmitting the digital data over the channel can be represented as shown in the block schematic below in figure 6.1 Source encoder Channel encoder Modulator Noisy Channel Source decoder Channel decoder Demodulator Fig: 6.1 Basic block diagram for digital communication system In the above blocks the source encoder converts any input data into the binary data and removes any redundancy present in the data. The binary data is given to the channel encoder to add up the redundancy bits. These bits are mainly used for the error detection and correction. The type of error detection and correction technique to be used mainly depends on the type of application, available bandwidth, signal to noise ratio and probability of error. The different types of channel coding techniques are Forward acting error detection: In this technique the decoder checks the received message if the received message is erroneous then it reject the message and request the transmitter to retransmit the message. This procedure repeats until the receiver receives the errorless information. This method requires a reverse channel hence there is wastage of bandwidth, so this technique is not preferred in

3 84 wireless communication because of scarcity of bandwidth. To overcome this disadvantage we choose the next technique. Forward-acting error correction method: In this technique the receiver on receiving the data checks for the error if the received message is erroneous then it automatically does the correction without requesting the transmitter to retransmit the data. In this method no reverse channel is required. Hence is preferred in wireless communication because the required bandwidth is less. The channel coders can be classified into two types, they are Block coder Convolutional coder Block codes are based rigorously on finite field arithmetic and abstract algebra. This coder can be used for only error detection or error detection and correction [38]. The block coder accepts a block of k information bits and produces a block of n coded bits. In case of block codes (n-k) number of redundant bits or check bits will be added. These (n-k) bits are derived from the k information bits. At the receiver these check bits are used to detect and correct errors which may occur in the block of n bits. These coder are commonly referred as (n,k) block coder. The basic block coding techniques are Linear coding technique Cyclic coding technique The Convolutional coders are most widely used channel coders in practical communications systems. These are primarily used for real time error detection and error correction. In this code the check bits are continuously interleaved with information bits. These check bits not only correct a particular coded block but all other blocks as well

4 85 [39]. The Convolutional coders convert the entire data stream into one single codeword. The encoded bits depend not only on the current K input bits but also on the past input bits [37]. In Convolutional codes the Viterbi algorithm provides the basis for the main decoding strategy. The channel coders used in this thesis are Binary cyclic coder and the Convolutional coder. 6.2 BINARY CYCLIC CODES: This forms the subclass of linear block codes. These codes are attraction for two reasons as mentioned below: The encoding and syndrome calculation procedures can be easily implemented The constructional methodology is simple and hence makes it possible to design codes with useful error correcting properties. In cyclic code the theory of Galois field has been used to spot a good code. Cyclic codes are very important because of their underlying Galois field description that leads to encoding and decoding procedures that are algorithmic and computationally efficient Definition of a cyclic code Let c be the codeword in a linear block code of length n. The cyclic code over GF(q) are class of linear codes of block length n if the code obtained after right shift or left shift is also a codeword. Therefore the linear code c is cyclic precisely when it is invariant under all cyclic shifts [38]. These cyclic codes are based on the Galois field and their structure is strongly related to Galois field because of this encoding and decoding algorithms are computationally efficient.

5 Polynomial Description of cyclic code A linear code C over GF(q) is called a cyclic code if c=(c 0,c 1,c 2,c 3,c 4,...c n-1 ) is a code in C then the cyclic shifted value of c be c =(c n-1,c 0,c 1,c 2,c 3,...c n-2 ) is also a code in C. Every cyclic code is a special kind of subspace because it has cyclic property. Each vector in Galois field is represented as a polynomial in x of degree of less than or equal to (n-1). This set of polynomials have ring structure and are defined as GF(q)[x]/[x n -1]. Hence if the code words of a code are denoted by a polynomial and the code is a subset of GF(q)[x]/[x n -1]. Then code is a cyclic code if x.c(x) is a codeword polynomial whenever c(x) is a codeword polynomial [40]. In a (n, k) cyclic code there exist a generator polynomial of degree (n-k). This is a unique polynomial from which all the code word polynomials can be generated. For an (n, k) binary cyclic code the generator polynomial can be written as g(x)=g n-k x n-k +g n-k-1 x n-k g 1 x 1 +g In the above equation the coefficients g n-k, g n-k-1...g 1 and g 0 can take the values either 0 or 1. The term x n-k is always present in the generator polynomial and there are no higher order terms, hence the degree of the generator polynomial is always n-k. The degree of g(x) is equal to the number of parity check bits of the code [41] Encoding cyclic codes The cyclic codes can be encoded in two different ways Nonsystematic encoding: This is the easiest way of encoding. In this technique the message polynomial i(x) is multiplied with the generator polynomial g(x) to get the codeword polynomial c(x).

6 87 c( x) i( x) g( x) 6.2 Systematic encoding: The check bits are generated by dividing the message polynomial i(x) by the generator polynomial g(x). The remainder obtained will be used as check bits. i( x) x rx ( ) gx ( ) nk 6.3 c( x) i( x) r( x) 6.4 The encoder circuit for the cyclic code can be designed using the set of shift registers. The number of registers or flip-flops required will be equal to the degree of the generator polynomial. The division circuit using the flip-flops is designed as shown below in the figure 6.2 for the generator polynomial g(x) g(x)=g n-k x n-k +g n-k-1 x n-k g 1 x 1 +g ON g g g g r 0 r r 2 r n-k Check bit Input message bits To channel Fig: 6.2 Binary Cyclic encoder circuit

7 88 The encoding operation consists of two steps [31] 1. Initially the channel switch will be in position 1 where the message bits are shifted into the register and also into the communication channel simultaneously. As soon the k information bits are shifted into the register, the register contains parity check bits from r 0, r 1, r 2,.r n-k Once all the k message bits are transferred the AND gate is turned off and the switch is moved to the position 2, this in turn shifts the contents of the encoder circuit into the channel. The design of this encoder is much easier than that of linear block encoder where the portion of the G or H matrix have to be stored Decoding cyclic codes Decoding can be done using the linear feedback shift registers [41]. Both systematic and non-systematic codes have got the generator polynomial g(x) as a factor. Let the code vector received over the noisy channel be v(x). The received polynomial may or may not be similar to the transmitted polynomial c(x). The function of the decoder is to check for the error present in the received polynomial by calculating the syndrome. A zero syndrome indicates errorless transmission and a nonzero syndrome indicates the erroneous transmission. If the received polynomial v(x) is similar to that of the transmitted codeword polynomial c(x) in a (n, k) linear cyclic code then v(x) is divisible by the generator polynomial g(x).i.e. the remainder r(x) will be equal to zero. The error detection at the receiver is done by checking whether the received polynomial is divisible by g(x), if yes then v(x) is equal to c(x) else v(x) is not equal to c(x). The

8 89 nonzero remainder obtained indicates the presence of error and is called the syndrome polynomial represented by s(x). vx ( ) sx ( ) 6.6 gx ( ) Where s(x) is the remainder obtained by dividing v(x) and g(x). The degree of the syndrome polynomial is (n-k-1) or less. The corrected codeword c(x) can be obtained from the received polynomial v(x) using the equation given below c( x) v( x) e( x) 6.7 Where e(x) is the error polynomial and is related to the syndrome polynomial s(x) as mentioned in the equation below e( x) [ q( x) m( x)] g( x) s( x) 6.8 G11 ON g 0 g 1 g g + s 0 s 1 s 2 s n-k-1 G2 + + Input received bits Syndrome bits + Output Fig: 6.3 Binary cyclic syndrome calculation circuit for an (n, k) cyclic code Where e(x) is the error polynomial, g(x) generator polynomial, m(x) is the message polynomial and s(x) is the syndrome polynomial. The term q(x) is the quotient obtained by dividing the received polynomial by the generator polynomial. The s(x) contains the information about the error pattern. Hence the syndrome calculation circuit can be accomplished using divider circuit shown in the figure 6.3 The syndrome calculations are carried out as follows [35]:

9 90 1. The register is first initialized by turning the gate 2 ON and the gate 1 OFF then the received vector is entered into the shift register. 2. After the entire vector is shifted into the register, the contents of the register will be the syndrome. Now the gate 2 is turned off and gate 1 is turned ON to shift out the content of the register. Once the contents are shifted the syndrome circuit is ready for processing the next received vector. The implementation of the cyclic decoder is easy compared to the non-cyclic linear binary coder. To perform the correction the decoder has to determine a correctable error pattern e(x) from the syndrome using the following operations performed in the decoder block given in the figure 6.4 The error correction procedure consists of the following steps: 1. The received vector is shifted into the buffer register and the syndrome register. 2. The syndrome of the received vector is calculated and is placed in the syndrome register. This in turn is read into the detector. The detector is the combinational circuit designed to output 1 if the syndrome obtained corresponds to the correctable error pattern with the error at the highest position x n-k. 3. If the detector output is 0 then the received vector is assumed to be correct. 4. The first received digit is shifted out of the buffer and at the same time the syndrome register is shifted right once. If the first received digit is erroneous then the detector output will be 1 and will be used to correct the first received digit. The output of the detector is fed to syndrome register to modify the syndrome. This result in a new syndrome corresponding to the received vector shifted to the right by one place.

10 91 5. The obtained new syndrome is used to check whether the second received digit which is at right most stage of the buffer register is erroneous. The above 3 rd and 4 th steps are repeated. 6. The decoder operates on the received vector digit by digit until the entire received vector is shifted out of the buffer. + + Feedback Syndrome connection register Input Error pattern detector S out s in Buffer register Received vector input s in s out Corrected vector Fig: 6.4 Decoder for cyclic codes + This above decoder can be used for decoding any cyclic code. This is less expensive and less complex. The cyclic coder is basically used in this thesis to correct single or double error pattern. These cyclic coders are very efficient to correct single and double error pattern but not preferred to correct multiple error corrections as complexity of the algorithm increases. As the complexity of the algorithm increases the processing time required will also increase. Hence to overcome this drawback we use cyclic coding technique for less erroneous channels or for errorless channels.

11 CONVOLUTIONAL CODING In case of block coder the block of n digits generated by the encoder mainly depends only on the k message within the particular time unit. In case of Convolutional coder the n code bits generated mainly depends not only on the k message bit with in particular time slot but also on the previous blocks of message bits. These codes can be designed to detect and correct the errors. Encoding in case of Convolutional coder can be accomplished using simple shift register and decoding can be done using many practical procedures. Recent studies have shown that the Convolutional coders perform better than that of block coders in many error control applications [35]. The Convolutional coders are mainly used to achieve reliable data transfer in different applications like digital video, mobile communication and satellite communication. The Convolutional codes can be either systematic codes or non systematic codes. Systematic codes are those where in the message bits are in a separate sub block and the check bit are in a separate sub block. The systematic Convolutional codes are preferred as the implementation is easier than that of non-systematic codes. The very important advantage of systematic code is that it prevents the propagation of error. The error correction property of both the coding methodology is the same Convolutional encoder A Convolutional encoder takes sequence of message as inputs and generates sequence of code digits as outputs. At any time unit k message bits are fed into the encoder and the encoder generates a code block consisting of n code digits. The code generated by this encoder is called an (n, k) Convolutional code of constraint length nn digits, where n is the size of the codeword generated and N is the total number of message blocks. The rate efficiency of the code can be written as k/n. The convolutional

12 93 encoders are mainly designed using shift registers and modulo-2 adders. The working of the encoder is as given below. Initially the shift registers are cleared. Then the first bit is entered into the flipflop F1, during this period the commutator samples the modulo-2 adder output c1, c2, c3. Thus a single input bit yield to three output bits. When the next message sequence enters into the D1 flip-flop, the information bit present previously in D1 gets shifted into the next flip-flop. The new bit in turn adds three more output bits. + Input message bits D1 D2 D3 D4 Coded output data + Fig 6.5 Convolutional Encoder This above process repeats until the last message bits are shifted into the last flipflop. The convolutional encoder operates on the message stream in a continuous manner, hence the convolutional coder requires less buffering compared to the block codes. The selection of which bits are to be added mainly depends on the generator polynomial g(x). Each and every polynomial will have unique error correction property. The convolutional

13 94 coder is as shown in the figure 6.5. The Convolutional codes can be generated by the following techniques as mentioned below Time domain approach Transfer domain approach Time Domain approach In time domain approach a binary convolutional encoder can be defined by a set of n impulse response. The impulse response will characterize its behavior in time domain. This method is as explained below: Let generator sequence be denoted as g 1, g 2, g 3,......g m+1. These also denote the impulse response. In an encoder number of generator sequence will be equal to the number of modulo-2 adders. Let the message sequence be represented as d 1, d 2,...d L. These are given as input to the encoder one bit at a time. Then the encoder generates the output sequence denoted as c 1, c The number of output sequence will be equal to the number of modulo-2 adders or number of output nodes. The output sequence can be written as c d g 1 1 c d g and so on From the definition of discrete convolution we can find c using the equation given l below: m c d g l l1 i1 i0 6.11

14 95 In the above equation m is the total number of flipflops. The l varies from 1 to L+m, where L is the number of message sequence. After finding different values of c then these output sequences are multiplexed into a single sequence called codeword [38]. The codeword at the output of the convolutional encoder is given by the equation c l Transfer domain approach: The linear filter operation in time domain is replaced by the multiplication of Fourier transforms in the frequency domain. The advantage of transfer domain over the time domain is due to its less complex computation. In transfer domain the impulse response of each path is replaced by polynomial whose coefficients are represented by the elements of the impulse response. If the convolutional encoder has m number of flip-flops with j number of output terminals then its generator polynomial and its code sequence for each output terminal can be written as given below g x g x g x g x... g x j j 0 j 1 j 2 j m m The corresponding output sequence for the adder is given by j c ( x) d( x) g ( x) j 6.13 After getting the polynomials at the output of each adder, the final encoder polynomials can be obtained from the equation 6.14 c x c x xc x x c x x c x 1 n 2 n 2 3 n n1 n n ( ) ( ) ( ) ( )... ( )

15 96 The indeterminate x can be regarded as a unit-delay operator and the power of x defining the number of time units by which the associated bit is delayed with respect to the initial bit in the sequence[42] : Structural properties of Convolutional codes: The Convolutional codes can be viewed as a finite state machine and in fact this is a sequential circuit. The structural properties of a Convolutional encoder can be portrayed in graphical manner using any of the three equivalent diagrams like State diagram, Code tree and Trellis The state of the encoder is defined as its shift register contents. For an (n, k, m) code where n is the number of output terminals, k is the number of input terminals and m is the number of flip-flops. With k > 1, the i th shift register contains k i previous information bits. The number of different states an encoder can be mainly depends on the number of flip-flops. Hence total number of states for single input encoder can be written as 2 m and with k inputs causes transition to a new state. These transitions can be graphically represented by a state diagram. The procedure to draw the state diagram consists of three parts [42]. 1. Identify the different possible states. 2. Draw state transition table clearly showing the present state, next state, input bits and the corresponding output bits. 3. The state diagram is drawn showing all the states of the flip-flops. Using the state diagram we can draw the code tree and also the Trellis

16 97 The procedure to draw Code tree and the Trellis: Using the state diagram the code tree is drawn by considering the upward path if the input is 0 and if input is 1 the lower path is followed. Each state in the code tree is represented by a node and the transition from one state to the other is represented by the lines called branches. The output codeword for each input is shown on the branches. The length of the tree to be constructed must be equal to n(l+ m). Hence to obtain the complete code sequence corresponding to an information sequence of length kl the tree must be extended by n(m-1) time units and this extended part is called the tail f the tree and the 2 kl is the right most node and is called the terminal codes of the tree. Once the tree is drawn we can see that the tree becomes repetitive after first three branches, hence we may collapse the tree graph into a new form called a Trellis. This is so called because it is a tree like structure with remerging branches. The Trellis diagram contains (L+m+1) levels and is also called the depth of the Trellis and are labeled as 0 to (L+m). The convention followed while drawing the trellis is that a code branch produced by an input 0 is shown by a solid line while that produced by 1 as dotted lines. This is widely preferred because the buffer space required to store the Trellis is comparatively very less Free distance and error distribution: The minimum free distance is denoted as d free. The d free is the minimum Hamming distance between any obtained codes. The error detecting and correcting capability of a code mainly depends on the minimum free distance. The error detecting capability can be calculated as

17 98 t d f ree The correcting capability can be written as t d f ree In case convolutional codes the processing is done in a continuous manner instead on a block, hence the value of t applies to a quantity of errors located relatively near to each other. Free distance is also interpreted as the minimal length of an erroneous burst at the output of the convolutional decoder. The convolutional coders are mainly designed to correct multiple error patterns. There are different methodologies in convolutional codes to correct the multiple or burst error patterns [43] Convolutional decoder There are several different techniques to decode the convolutional codes and these are grouped into Sequential decoding Maximum likely hood decoding The Sequential decoding algorithm allows both forward and backward movement in the Trellis. The decoder keeps the track of its decision and each time and each time it makes the ambiguous decision it compares with the threshold value stored. If the computed value is greater than the threshold then that particular path is given-up by the decoder and comes to the previous path where the computed value is less than the threshold value. In the sequential decoding the number steps will very much greater than (L+m). The number of computation increases with the level of noise present in the channel, hence is not preferred for the real time application. If the E b /N is greater than the

18 99 decoder threshold then decoding will be done at a faster rate else the delay in decoding increases. The different sequential decoding algorithms [41] are Fano decoding algorithm Stack decoding algorithm The maximum likely hood decoding algorithm computes the metric value for 2 k number of paths and selects the path with highest metric value. The same computation is followed until the number of levels are less than (L+m). The number of computation does not depend on the noise level present in the channel, hence the computation time remains constant irrespective of the noise present in the channel. Due to this advantage this is used for real time implementation. One of different maximum likelyhood decoding algorithms is Viterbi decoding algorithm. There are many different decoding algorithms to decode Convolutional codes.one such decoding algorithm is the Viterbi algorithm. This is widely used algorithm as it is very easy to implement and also provides maximum likely hood performance. This algorithm provides the basis for the main decoding strategy of Convolutional codes. The Viterbi decoding algorithm mainly is a graphical portrayal of Convolutional codes and uses the maximum likely-hood decoding technique to decode the received data. This decoder logically explores every possible user data sequence to decode hence for this reason this is also known as maximum likelihood decoder [44]. The procedure used by the Viterbi algorithm to decode is as given below: Let the information sequence U=(u 1,u 2,u 3,.u L ) be of length kl and let this sequence be encoded into a codeword V=(v 1,v 2,v 3, v L+m ) of length N, where N=n(L+m). Let the received sequence at the receiver be R=(r 1,r 2,r 3,r r L+m ). By maximum

19 100 likelihood technique we need to find the estimated ˆ V from the received vector R at the receiver.the decoding rule for choosing the estimate value ˆ V given the received vector R is optimum when the probability of decoding error is minimum. For equiprobable messages the probability of error will be minimum if the estimated value will maximizes the log-likelihood function. Hence the basic rule is to choose the estimate ˆ V which maximizes the ln PR V becomes maximum when d free becomes minimum. value. Hence we can conclude that the log-likely hood function The procedure to decode using the Viterbi decoding algorithm is as follows [42]: Start with the initial level. Compute the partial metric for the single path entering each node. Store the path and its metric for each state. Increment the present level by one and compute the partial metric for all the paths entering a state by adding the branch metric to the metric of the connecting survivor at the preceding unit. Store the path with the largest metric. If the number of levels are less than (L+m) repeat the above step. Note that although we use the tree graph for the decoding procedure the number of nodes at any level of the Trellis does not continue to grow as the number of incoming message bits but remains constant at 2 m.the decoding procedure is very advantageous for smaller message bits hence is widely preferred. The Viterbi algorithm is widely used in many commercial satellite systems. For shorter codes this performs better against noise. The main advantage of the Viterbi algorithm is its constant executing speed or processing speed regardless of the received signal-to- noise ratio. Hence this makes the algorithm attractive for real time, delay limited application. This algorithm is chosen in this thesis because of the following reasons

20 101 This is more tolerant to the channel errors than sequential decoders. These are easy to implement. The processing delay is very less compared to the sequential decoders under any condition hence is preferred for real time applications. Constructional feature of Viterbi is very simple as its uses only simple operator like addition, compare and select 6.4. SUMMARY In this chapter we study about the importance of channel coding techniques for mobile communication along with the different types of channel coding techniques. In this thesis we are designing the adaptive channel coding technique using cyclic coder and the convolutional coder. This chapter comments on choosing these above said coders.