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Consider a source which generates one of five possible symbols
S=[a, b, c, d , e]; their probability is
P=[0.2,0.4,0.05,0.1,0.25]](https://image.slidesharecdn.com/presentationppt3-230511111636-0c1f9670/85/Presentation-ppt-3-pptx-14-320.jpg)
Presentation ppt 3.pptx
- 1. INFORMATION NETWOK SECURITYADMINISTRATION INTELLEGENCE TECHNOLOGY EXCELLENCE DIVISION SIGNAL ANALYSIS TEAM NAME TEMESGEN TIZAZU
- 2. Outline Introduction Huffmancoding Convolutional code Interleaving Conclusion MATLAB Implementation
- 3. Introduction Codes areused for data transmission, data compression, cryptography, error- correction Coding is a form of data organization or data presentation There are four types of coding Data compression (source coding) Error correction (channel coding) Cryptographic coding Line coding
- 4. Continued … DataCompression: the process of coding that will effectively reduce the total number of bits needed to represent certain information Source coding:- is used to remove redundancy from source information for efficient transmission Data redundancy is the existence of data that is additional to the actual data Decreasing the level of redundancy is equivalent to lossless compression Because of that source coding is often treated as lossless compression.
- 5. Continued… Lossless: Ifthe compression and decompression processes induce no information loss, otherwise, it is lossy Lossy methods: used when the data are image (JPEG), video (MPEG), audio (MP3) lossless methods: are used when the data are text or programs e.g., Run-length, Huffman, Lempel Ziv….
- 6. Continued… Compression ratio=Bo/B1;where Bo=numberof bits before compression(we need 7 bits to write 128 characters ) B1=number of bits after compression
- 7. Huffman encoding Designthe mapping from source symbols to codewords Entropy Coding (Prefix-free Code):-No codeword is a prefix of another one Lossless compression Symbols that occur more frequently will have shorter Huffman codes than symbols that occur less frequently In an optimum prefix-free code, the two codewords that occur least frequently will have the same length; differing only at the last bit
- 8. Huffman encoding Usedto generate uniquely decodable code:-Good codewords and achieves lower bound L=Average Huffman codeword length: In general: H(S) ≤ L < H(S) + 1; lower bound H(S)= Entropy: Goal: minimizing the average codeword length
- 9. Continued… Huffman algorithm:- Arrange givenmessages in decreasing order of probability Group the least probable message and assign them symbols used for coding Add the probabilities of the grouped message and place them as high as possible rearrange them in decreasing order again Repeat the above step till two probabilities remain Obtain the codeword for each message by tracing the probabilities from left to right Writing the respecting symbols ( 0 or 1) above the path by tracing them from right to left
- 10. Continued… Huffman Coding:Example
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- 13. Continued… They have thesame average length They differ in the variance of the average code length Solution 1 variance =0.16 Solution 2 variance =1.36 So least variance can be taken
- 14. Continued… Consider asource which generates one of five possible symbols S=[a, b, c, d , e]; their probability is P=[0.2,0.4,0.05,0.1,0.25]
- 15. Convolutional code Channel coding:- When digital data is transmitted from one system to another, an unwanted electrical disturbance added to it This can cause an ‘error’ in digital information. That means a 0 can change to 1 or 1 can change to 0. To detect and correct such errors special type of codes capable of detecting and correcting the errors are used.
- 16. Convolutional code Thereare many different types of error control codes like :- polar codes, Reed Solomon codes, Turbo codes, Convolutional codes, cyclic codes and LDPC codes. Block codes and convolutional codes are commonly used in digital communications Different factors affect the choice of a particular coding scheme. Like cost, power, bandwidth, Delay in Decoding, data rate
- 17. Continued… In blockcodes there is a one-to-one correspondence between uncoded blocks of symbols and coded block of symbols Convolutional codes have memory that uses previous bits to encode or decode Each information bits remain in the encoder for up to m+1-time units During each time unit can affect any of the n encoder output (depending on the shift register connection) Encoder output at any given time is dependent on present as well as past inputs
- 18. Continued… Convolution codesare commonly specified by three parameters (n, k, m) A convolutional encoder can be described by shift registers and modulo 2 adders The content of the shift registers determines the state of the encoder The coding rate of convolutional codes is given by R = k/n, Constraint length Lc = m+1 A convolutional encoder has generator sequences (g), given in octal form and each of length Lc.
- 19. Continued… A (2,1,3) binaryconvolutional encoder contains linear feed forward shift register In linear feed forward shift register the information sequence u enters the encoder one bit at a time The two-encoder output sequence can be obtained the convolution of the input sequence u with the two-encoder impulse response (generator sequence)
- 20. Continued… So, thetwo-encoder output sequence can be written as 𝒗(𝟏) =(𝒗𝒐 (𝟏) , 𝒗𝟏 (𝟏) ,𝒗𝟐 (𝟏) , … .) and 𝒗(𝟐)=(𝒗𝒐 (𝟐) , 𝒗𝟏 (𝟐) ,𝒗𝟐 (𝟐) , …) The two-encoder impulse response (generator sequence) can be written as 𝒈(𝟏)=(𝒈𝒐 (𝟏) , 𝒈𝟏 (𝟏) ,𝒈𝒎 (𝟏) ) and 𝒈(𝟐)=(𝒈𝒐 (𝟐) , 𝒈𝟏 (𝟐) ,𝒈𝒎 (𝟐) ) The encoding equation cab be written as 𝒗(𝟏)=u*𝒈(𝟏) 𝒗 𝟐 =u*𝒈 𝟐
- 21. Continued… After encoding; thetwo encoder outputs sequences are multiplexed in to a single sequence, called the codeword The codeword is given by V=(𝒗𝒐 (𝟏) 𝒗𝒐 (𝟐) , 𝒗𝟏 (𝟏) 𝒗𝟏 (𝟐) , 𝒗𝟐 (𝟏) 𝒗𝟐 (𝟐) , 𝒗𝟑 (𝟏) 𝒗𝟑 (𝟐) , 𝒗𝟒 (𝟏) 𝒗𝟒 (𝟐) , …) Codeword V has length n(m+L), where L is the length of input sequence and m is memory order
- 22. Continued… We canobtain generator matrix if the generator sequence 𝒈(𝟏) and 𝒈(𝟐) are interlaced and then arranged in matrix
- 23. Continued… Each rowof G is identical to the preceding row but shifted to n places to the right If u has a finite length L, then G has L rows and n*(m+L) columns and V has length n*(m+L) The encoding equation can be written as in matrix form as follows V=u.G Where G is the generator matrix of the code, and u is information sequence
- 24. Continued… In generalcase of an (n,k, m) code, the generator matrix is Where each 𝐺𝑙 is a k× 𝑛 sub matrix as follows
- 25. Continued… Example :-u=(1 0 1 1 1), 𝒈(𝟏)=(1 0 1 1), 𝒈(𝟐)=(1 1 1 1) , Find the transmitted codeword? Solution 𝐯(𝟏)=u*𝐠(𝟏) 𝐯(𝟏)=(1 0 1 1 1)* (1 0 1 1) 𝐯 (𝟏) =(1 0 0 0 0 0 0 1)
- 26. Continued… 𝐯(𝟐)=u*𝐠(𝟐) 𝐯(𝟐)=(10 1 1 1)* (1 1 1 1) 𝐯(𝟐)=(1 1 0 1 1 1 0 1 )
- 27. Continued… So, the transmittedcode word can be obtained by taking the corresponding bit of 𝐯(𝟏) 𝑎𝑛𝑑 𝐯(𝟐) 𝐯(𝟏)= (1 0 0 0 0 0 0 1) 𝐯(𝟐) = (1 1 0 1 1 1 0 1) V= (1 1, 0 1, 0 0 ,0 1,0 1,0 1,0 0 ,1 1)
- 28. Continued… G=generator matrix canbe obtained by
- 29. Continued… So, the transmittedcode word can be obtained by V= U.G= V= (1 1, 0 1, 0 0 ,0 1,0 1,0 1,0 0 ,1 1)
- 30. Interleaving An interleaver takesa given sequence of symbols and permutes their positions, arranging them in a different temporal order To convert error patterns that contain long sequences of serial erroneous data into a more random error pattern Distributing errors among many code vectors For a memory channel with burst errors, rearranging the symbols can cause the bursts of channel errors to be spread in time
- 31. Interleaving To separate thecodeword symbols in time:- This codeword symbol–separation in time effectively transforms a channel with memory into a memoryless one The basic goal of an interleaver is to randomize the data sequence This allows random–error–correcting codes to be useful in burst–noise channel
- 32. Continued… There are threetypes of interleavers that are commonly used in communications: Block interleavers; Convolutional interleavers, Pseudorandom interleavers Block interleavers A block interleaver accepts the coded symbols in blocks from the encoder Shuffles the symbols, and then feeds the rearranged symbols to the modulator The shuffling of the block is accomplished by filling the columns of an M-row–by N - column (M×N) array with the encoded sequence
- 33. Continued… After the arrayis filled, the symbols are then fed to the modulator one row at a time and transmitted over the channel Input to the interleaver is column–wise, while output is row–wise At the receiver, the deinterleaver performs the inverse operation, the symbols are entered by rows and removed one column at a time
- 34. Continued… Example: we havethe following data sequence before interleaving The given data sequence is the input to a 4 ×4 block interleaver The out put of 4 ×4 block interleaver is as follows
- 35. Continued… We have thefollowing data sequence before deinterleaving The given data sequence is the input to a 4 ×4 block deinterleaver The out put of 4 ×4 block deinterleaver is as follows
- 36. Continued… The following sequenceis the input to a 4 ×6 block interleaver 1101 0101 1111 0000 1010 1011 Let there be a noise burst of five symbol times, the colored symbols shown below experience errors in transmission Input the bit stream column by column The output of the interleaver is then read row by row 101011 111000 001011 111001
- 37. Continued… The output of the interleaver (101011 111000 001011 111001) is the input to deinterleaver as follows Then input the bit stream row by row The output of the deinterleaver is then read column by column 1101 0101 1111 0000 1010 1011
- 38. Continued… Convolutional Interleavers:- The code symbols are sequentially shifted into the bank of N registers Each successive register provides J symbols more storage than did the preceding one The zeroth register provides no storage the symbol is transmitted immediately Each new code symbol, the commutator switches to a new register The new code symbol is shifted while the oldest code symbol in that register is shifted out to the modulator
- 39. Continued… After the (𝑁− 1)𝑡ℎ register, the commutator returns to the zeroth register and starts again The deinterleaver performs the inverse operation The input and output commutators for both interleaving and deinterleaving must be synchronized The synchronized deinterleaver is shown simultaneously feeding the interleaved symbols to the decoder
- 40. Continued… Convolutional Interleavers:- Thecode symbols are sequentially shifted into the bank of N registers
- 41. Continued… Example :-we have the following data sequence before interleaving Convolutional four–register (J= 1) interleaver being loaded by a sequence of code symbols Symbols 1 to 4 are being loaded; the x represent unknown states
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- 44. Continued….. Find theinterleaving matrix, entering the bits column–wise We then add columns of xs to the left and right of the columns containing the bit The output of the interleaver is then read diagonally from top–left of the message bits to bottom–left of the xs
- 45. Continued….. 1 x xx 5 2 x x 9 6 3 x 13 10 7 4 x 14 11 8 x x 15 12 x x x 16
- 46. Application Area Convolutional codesare used extensively in numerous applications in order to achieve reliable data transfer Including digital video, radio, mobile communication, and satellite communication These codes are often implemented in concatenation with a hard-decision code, particularly Reed Solomon,turbocodes
- 47. Continued….. This interleaving techniqueprovides a high degree of robustness to variability in the burst parameters Most block or convolutional codes are designed to combat random independent errors Interleaver one of the most important component that construct Turbo code To enhance overall Turbo code performance against burst errors
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