Synchronization of Hamming Codes

Transcription

1 SYCHROIZATIO OF HAMMIG CODES 1 Synchronization of Hamming Codes Aveek Dutta, Pinaki Mukherjee Department of Electronics & Telecommunications, Institute of Engineering and Management Abstract In this report we investigate the symbol and frame synchronization of a (7,4) Hamming code with BPSK modulation and zero mean AWG. A matched filter instead of a product integrator or correlator has been used to establish symbol synchronization by identifying the shape of the received pulses, which is validated with simulations in MATLAB [3]. Frame synchronization has been established by exploiting the syndrome decoding property of block codes. The syndrome is used to identify a valid codeword instead of identifying the error vector as we have assumed that the received codeword is without any bit error. This identification of a correct code block and subsequent frame synchronization has been simulated with a C program. I. ITRODUCTIO Talking about error free communication the first term we think about is Synchronization, i.e., the receiver should be able to track the incoming bits without any prior knowledge of transmission instance. For an encoded data stream it is important to establish both block synchronization and symbol synchronization [1]. The inability to provide acceptable data quality at a given SR due to certain power limitation of the transmitter using a particular form of modulation leaves with the only solution of using error correcting codes which ensures reduction in bit error rate for a particular SR. The redundancy is added to the message are therefore used to correct the error occurring during transmission. This report starts with a brief background on Hamming codes and matched filters. The next section describes the proposed approach of symbol and block synchronization followed by the results obtained from simulations in MATLAB and C. In conclusion we intend to defend our approach against conventional techniques and propose future work in this field. II. HAMMIG CODE AS A CYCLIC CODE Any cyclic code generated by a primitive polynomial is a hamming code of minimum distance 3. For a (7, 4) cyclic code with block length n= 7 we start by factorizing X 7 +1 in to three irreducible polynomials as, X + 1 = ( 1+ X)( 1+ X + X )( 1+ X + X ) By an irreducible polynomial we mean a polynomial that cannot be factored using only polynomials with coefficients from the binary field. An irreducible polynomial of degree m is said to be primitive if the smallest positive integer n for which the polynomial divides X n +1 is n= m+1. This is the fundamental criterion for a Hamming code. Either of the above two primitive irreducible polynomials (1+X (1+X+X 3 ) can be used as a generator polynomial. Let us take the generator polynomial as, 3 g ( x) = (1 + X + X ) and the parity check polynomial is given by, 3 h ( x) = (1 + X )(1 + X + X ) +X 3 ) and 4 = 1+ X + X + X Therefore the following mathematical relation characterizes the entire coding system, 3 X m(x) b(x) = a(x) + g(x) g(x) We may now summarize the steps involved in the encoding procedure for an (n, k) cyclic code assured of a systematic structure. Specifically, we can proceed as follows, 1. Multiply the message polynomial m(x) by X 3.. Divide X 3. m(x) by the generator polynomial g(x), obtaining the remainder b(x) giving the parity polynomial. 3. Add b(x) to X 3 m(x), obtaining the code polynomial c(x). A complete set of codeword for a (7,4) Hamming codes generated using the generator polynomial g(x)=1+x+x is shown below TABLE1 CODEWORDS OF A (7,4) HAMMIG CODE Hamming codes have the property that the minimum distance d min = 3, independent of the values assigned to the number of parity bits m. An important property of the Hamming code is that they satisfy the hamming bound, t (dmin 1)/ with the equality sign, where t is the number of errors the code can correct. This means that Hamming codes can correct single bit errors and therefore can be called as single error correcting binary perfect codes. III. THE MATCHED FILTER The purpose of the Matched Filter is to maximize the signal to noise ratio and to minimize the probability of undetected errors received from a signal. An important fact that is relevant

2 SYCHROIZATIO OF HAMMIG CODES in discussing matched filter is that the receiver must know shape it is looking for. A matched filter aims at acheiving two objectives: 1. Maximize signal power, i.e. power of g(t) at t = T.. Minimize noise, i.e. power of n(t). Therefore the combined design criterion is max η, where η is peak pulse SR Derivation of matched filter [5][]: For the incoming signal we can write, G ( f ) = f ) g ( t) = g The objective is to find h(t) that maximizes pulse peak SR η = ( T ) = j π f t j π f T j π f T For the additive noise, S ( f ) = E{ n ( t)} = S ( f ) = with this idea in mind an expression for the Matched filter can be derived which is based on Schwartz s Inequality. Schwartz s inequality for functions can is as follows: Therefore based on the above equations and discussions we can derive an expression for optimum impulse response of a matched filter. j π f T Let φ ( f ) = H ( f ) and φ ( f ) = G( - η = η H g(t) η = E{n (t)} max opt 1 - = Hence, h ( f ) = k G opt G( f ) * = j π f T ( ( t) = k g Instantaneous Power AveragePower * j π f T ( T t) j π f T 1 φ ( x ) - - φ ( x ) φ ( x ) dx φ 1 ( x ) dx - is true iff * φ 1 ( x) = k φ ( x), which by k k R occurs G( f ) when Schwartz G( f ) 's dx inequality Therefore we can conclude that for a given transmitter pulse shape g(t) of duration T, matched filter has the following properties, Duration and shape of impulse response of the optimal filter is determined by pulse shape g(t). h opt (t) is scaled, time-reversed, and shifted version of g(t) IV. BLOCK AD SYMBOL SYCHROIZATIO MODEL This section aims at serving the process of symbol and block synchronization of hamming codes. The first part involves the use of matched filter for symbol synchronization and bit decision, which leads to the construction of Hamming coded data vector block. Second part consists of the simulated model of block synchronization based on syndrome decoding. A. Symbol synchronization[6][7] The objective behind this job is to synchronize the receiver without any prior knowledge of the transmission time under the assumption that the signal shape, its duration and kind of modulation is known but not the transmission instance. So it is up to the receiver to extract the bit timing and tune its local clock according to the incoming modulated bits. Once the bit time or the clock is extracted, the receiver tracks the pulses and adjusts itself with any kind of pulse broadening. The proposed communication model is given in Fig 1. Fig. 1: Block diagram of the symbol synchronizer The main steps for simulating the problem are as follows. a) First we defined a binary Hamming coded data stream by zeros and ones, which simulate the transmitter section and added a zero mean white Gaussian noise. b) ext we choose an arbitrary sampling instance, i.e., any sample point in the above array and select the next sampling duration as T b, which is the actual, bit time. c) This sampled data is passed through a matched filter. d) The filter used here is matched to a High as well as to a Low pulse as we have used a non-return to zero level coding scheme. e) Therefore an ideal high or low pulse having a duration T b should peak at t=t b. This theory has been exploited in our work. f) Any deviation from the actual bit time T b will result in an error signal and accordingly the sample duration or rather the sample time is to be either advanced or delayed. g) In this work we have tried to trace the high pulse and not the low pulse. This objective could have also been fulfilled it the low pulse was tracked.

3 SYCHROIZATIO OF HAMMIG CODES 3 h) After a few iterations of delay and/or advance is made the actual starting point of a high pulse is detected and once the bit time is extracted the sampler is excited at every T b interval. This is how the symbol time is extracted and symbol synchronization is achieved. i) This approach is also beneficial if there is any kind of pulse broadening or change in the bit time. In such a case the receiver has to track the pulse duration and change its matched filter characteristic according to the incoming pulse. The error signal calculation is realized by the following examples In fig. and fig.3 where we consider an incoming pulse train without any random noise. For simplicity a return to zero level coded data with bit time of T b has been considered. CASE: 1 Fig : The Input is sampled in the middle of a low pulse We first compute the two maxims at t = T 1 and at t = T b. Then compute their difference it time scale. So we get a high pulse of duration T 1 followed by a low pulse of duration (T b -T 1 ), and the matched filter output is shown which does not peak at t=t b. ow if we advance the sampling time by (T b -T 1 ), i.e., if we add this difference with the initial estimate then we will certainly arrive at the start of the high pulse. Once the starting position is identified then the sampler will sample further at intervals of T b and the matched filter output will peak at t=t b, and hence the symbol will synchronize. CASE: Fig 3: The input is sampled in the middle of a high pulse In the above sampling situation we compute the two maxims at t = T b and at t = T b -T 1. Then compute their difference it time scale. So we first get a low pulse of duration (T b -T 1 ) and then a high pulse of duration T 1. The output of the matched filter obtained by convoluting the input with the time-reversed image of the actual pulse shows that sampled input does not have the shape of a correct pulse. If we delay the sampling time by (T b -T 1 ), i.e., to subtract the difference from the original estimate and we will arrive at the start of the high pulse. Once the starting position is identified then the sampler will sample at intervals of T b and the matched filter will peak at t=t b. CASE: 3 Fig 4: The input is sampled at the start of a high pulse If the input is sampled at t=t b as shown in fig. 4 the output peaks at t=t b and we can conclude that the input sample has the shape of the correct pulse and the symbol time is extracted. This process is equally beneficial in synchronization in presence of noise, RZ coding and digital modulation. B. Block Synchronization After symbol synchronization is achieved the receiver has the estimates of whether the received binary data is a 1 or a. This information is useful to demodulate the signal but not decode it. We used a syndrome decoder to decode the incoming Hamming coded data. For decoding we need to have a block of 7 bits (as we have used (7,4) Hamming encoding scheme). At the receiver section a block of 7 bit is collected but it is possible that the blocks so obtained may not be a valid code word. The communication model involving a transmitter, syndrome calculator and synchronizer is simulated using C language with the objective of identifying the start and end position of a valid Hamming code frame of 7 bits. For example, let the matched filter output is, Received Bits: When these blocks are fed into a syndrome decoder an error pattern will be generated. ow if we correct the block using the error pattern, it will not be the exact data that was sent from the receiver, because the received block is in the middle of a code word and not one of the 16 possible codewords for a (7,4) Hamming coded data. The correct code word that should have been received is, Correct Bits: Therefore for correct decoding of the code sent by the transmitter we need to identify the exact block of codeword. We exploited syndrome decoding considering the fact that the bits from the decision device prior to the syndrome calculator is not erroneous, so the error pattern obtained from a non zero syndrome is not essentially used to denote that it is not a valid code word and not to correct a channel induced bit error. 1) The Transmitter Section [1]: The transmitter section starts with the generation of the generator matrix. The columns of the coefficient matrix is randomly chosen such that at least one element in each column is zero and no two columns have a zero in the same row. This enables us to achieve two objectives, i) the parity equations are unique, i.e., the parity bits are formed by linear combination of (n-1) data bits and, ii) the rows of the generator matrix are linearly independent. This 4 3 parity matrix is then attached with a 4 4 identity matrix to form a 4 7 generator matrix. A different generator matrix is created every time the program runs, which gives some kind of dynamic nature to the system. The next step involves the random generation of 1 4 data vectors. The

4 SYCHROIZATIO OF HAMMIG CODES 4 transmitter generates 16 random data vectors, but this number can be increased to any limit and are multiplied with the generator matrix performing modulo addition. These 16 codewords are collected as a serial datastream, which emulates the transmitting buffer. ) The Receiver Section []: At the receiver section, first the 3 7 parity check matrix (H) is generated from the prior knowledge of the generator matrix and the transpose of this matrix is obtained. After the generation of H-Transpose a random sampling position in the transmitted datastream is selected and next seven code bits are collected in an array recdata that emulates the receiver buffer and the syndrome is calculated. A non-zero syndrome proves that the received code is not the actual frame transmitted from the transmitter. But it may also happen that a set of 7 code bits, which is not a valid frame, may yield a zero syndrome. Thus to get rid of this problem successive 1 codeword are checked for zero syndrome. During this process if any set of code vectors yield a non-zero syndrome then it is ascertained that the first vector is not one of the 16 transmitted vectors. In such a case the original codeword is shifted by one bit position allowing the next bit from the transmitter to enter the receiver buffer and the above procedure is repeated. However if all the ten iterations yield a zero syndrome then we can safely conclude that the all the frames are correct frames and synchronization is achieved. The proposed block diagram of frame synchronization is shown in fig. 5 the matched filter shows that it peaks at t=.5sec, which identifies the starting position of a high pulse. Thereafter the sampler is excited at every.5sec and the matched filter detects a valid 1 or a. It is noteworthy how the peak value of the two filters varies according to the sampled input. The program also takes care of consecutive ones and zeros as shown in fig. 8. The critical part is that if the receiver only tracks the high bit then for consecutive ones and zeros there is a chance that the receiver reaches a deadlock or falls in an infinite loop. The software takes care of such deadlocks. Fig 6. Transmitted datastream and the Matched filter in time scale Fig. 5 Block diagram of synchronization of coded blocks V. RESULTS The results of the simulation are divided into two parts symbol synchronization and block synchronization. 1) Symbol Synchronization: A hamming coded datastream with zero mean AWG and the matched filter characteristics are shown in fig. 6. This simulates the transmitter buffer. The simulation starts with the receiver selecting a random sampling instance of.8sec, and samples the data for duration of.5sec which is the bit duration T b. The algorithm in section IV was simulated for twelve iterations and fig. 7, 8 and 9. The synchronization starts by analyzing the output of the matched filter as shown in fig. 7. The time T of the peak output of the matched filer is offset from the actual bit time T b =.5sec. The software calculates the difference between T b and T and the next sampling instance is advanced by that difference amount in time scale. During the next iteration the output of Fig 7. Output of the matched filter with sampling at T=..8 sec Fig 8. Output of the matched filter for consecutive high pulses

5 SYCHROIZATIO OF HAMMIG CODES 5 fig. 7 and fig. 8 shows how the matched filter at the receiver tracks the code bits and at the same time maintains a one to one synchronism with the transmitter Furthermore the algorithm followed here also suggest that the receiver is adaptive in nature and if it looses synchronization in the middle it can track the code and lock it in a couple of iterations as shown in fig. 9. position of sampling is defined which is equal to 9. It signifies that the receiver has no prior knowledge of when the transmitter has transmitted the bits. So when the receiver is turned on, it starts to collect blocks of data, each of 7 bits starting at bit number 9. The Parity-Check Matrix For A (7,4) Hamming Code : The Transpose Of Parity-Check Matrix For A (7,4) Hamming Code Is: The random starting position of sampling : 9 Fig 9. Output of the matched filter for changes in polarity of input signal ) Block Synchronization: The simulation starts at the transmitter end with the generation of a generator matrix, random data matrix, and 16 set of Hamming codeword by using matrix multiplication and modulo- arithmetic. These codewords are arranged in an array to simulate the transmitter buffer. The Generator Matrix For A (7,4) Hamming Code Is: DATA VECTORS CODE VECTORS Datastream : At the receiver end the parity check matrix is calculated from the generator matrix. A software generated random starting Syndrome is calculated for each of the blocks received. A nonzero syndrome definitely signifies that the received codeword is not a valid Hamming codeword. As the receiver has no prior idea of the transmitted block, so in order find the starting position of a Hamming code block the received bits are right shifted to accommodate a new symbol from the transmitter end. The Shift in the output defines a one bit shift or rather the receiver starts buffering from the next bit. Thus after six sequential shifts a zero syndrome vector is obtained. Once a zero syndrome vector is received in this case next consecutive 1 blocks are checked for zero syndrome. After 1 iterations we can conclude that the system is synchronized for Hamming coded data frame CODE VECTORS SYDROMES shift shift shift shift shift shift Zero Zero Zero Zero Zero Zero Zero Zero Zero Zero THE SYSTEM IS SYCHROISED... V. COCLUSIO We have investigated the symbol and block synchronization using the fundamental properties of signal processing in form of a Matched Filter and Syndrome Decoding. Although in this method, single bit error has been assumed to be zero, what can be measured by this method is the block error instead of BER. The synchronizer proposed starts at a random instance and

6 SYCHROIZATIO OF HAMMIG CODES 6 eventually synchronizes with the receiver. But in reality the system often moves out of synchronization and the process of synchronization is thus an iterative process. Lastly we would like to conclude that this report is a small step towards the vast magnitude of digital communication and some aspects of digital signal processing involved carrier synchronization [4], Matched Filters and complex modulations are left out, which are to be investigated in future attempts. VI. REFERECES [1] Simon Haykin, Communication Systems, John Wiley and sons. [] John G. Proakis & Dimitri G. Manolakis, Digital Signal Processing, Prentice-Hall of India Pvt. Ltd. [3] MATLAB website: [4] Victor Vilnrotter and Marvin Simon - Information-Reduced Carrier Synchronization for Coded PSK Operation at Low-SR [Online] Available [5] Behnaam Aazhang, Matched filters [Online] Available [6] A.S.J. Helberg, W.A. Clarke, H.C. Ferreira and A.J. Han Vinck "A Class of DC-Free, Synchronization Error Correcting Codes" IEEE Trans. on Magnetics, Vol. MAG-9, o. 5, pp , September [7] J. B. Carruthers, D. D. Falconer, H. M. Sandler, L. Strawczynski, Bit Synchronization In The Presence Of Co-Channel Interference, Canadian Conference on Electrical and Computer Engineering, Onawa. Ontario" Canada. September 3-6 September 199