1. ChannelCoding Hafiz Malik Dept. of Electrical & Computer Engineering The University of Michigan-Dearborn hafiz@umich.edu http://www-perosnal-engin.umd.umich.edu/~hafiz

2. Channel Coding class of signal transformations designed to improve communication performance by enabling the transmitted signals to better withstand channel distortions such as noise, interference, and fading. Channel coding can be divided into two major classes: 1.Waveform coding by signal design 2.Structured sequences by adding redundancy

3. Waveform Coding – deals with transforming waveform into “better waveform” robust to channel distortion hence improving detector performance. Examples: – Antipodal signaling – Orthogonal signaling – Bi-orthogonal signaling – M-ary signaling – Trellis-coded modulation

4. Structured Sequences – deals with transforming sequences into “better sequences” by adding structured redundancy (or redundant bits). The redundant bits are used to detect and correct errors hence improves overall performance of the communication system. Examples: – Linear codes Hamming codes BCH codes Cyclic codes Reed-Solomon codes – Non-Linear codes Convolution codes Turbo codes

5. Waveform Coding

6. M-ary Communications Send multiple, M, waveforms Choose between one of M symbols instead of 1 or 0. Waveforms differ by phase, amplitude, and/or frequency Advantage: Send more information at a time Disadvantage: Harder to tell the signals apart or more bandwidth needed

7. 4-ary Amplitude Each symbol sends 2 bits Deciding which level is correct gets harder due to fading and noise RCV needs better SNR to achieve accuracy 11 01 10 00

8. Orthogonal Signals Definition This means that the signals are perpendicular to each other in M-dim space For a correlative receiver, this means that each incoming signal can be compared with a model of the signal and the best match is the symbol that was sent

9. Multi-Amplitude Shift Keying (MASK) Send multiple amplitudes to denote different signals Typical signal configuration: – +/- p(t) – +/- 3 p(t) – +/- (M-1) p(t)

10. Multi-Phase Binary Phase Shift Keying (BPSK) 1:   (t)= p(t) cos(  c t) 0:   (t)= p(t)cos(  c t  M-ary PSK Re Im xx Re Im x x x x x x x x

11. Quadrature Amplitude Modulation (QAM) Amplitude-phase shift keying (APK) riri ii

12. Multitone Signaling (MFSK) M symbols transmitted by M orthogonal pulses of frequencies: Receiver is a bank of mixers, one at each frequency Higher M means wider bandwidth needed or tones are closer together

13. M-ary Comments As M increases, it is harder to make good decisions, more power is used But, more information is packed into a symbol so data rates can be increased Generally, higher data rates require more power (shorter distances, better SNR) to get good results

14. How do we compare performance? Symbols have different meanings, so what does the probability of error, P E mean? If a detection error is made, then more than one bit is wrong DCS can be faster at the price of being less sensitive

15. 15 Detect Error On Credit Card

16. 16 Formula for detecting error Let d2, d4, d6, d8, d10, d12, d14, d16 be all the even values in the credit card number. Let d1, d3, d5, d7, d9, d11, d13, d15 be all the odd values in the credit card number. Let n be the number of all the odd digits which have a value that exceeds four Credit card has an error if the following is true: (d1 + d3 + d5 + d7 + d9 + d11 + d13 + d15) x 2 + n + (d2 + d4 + d6 + d8 + d10 + d12 + d14 + d16) 0 mod(10)

17. 17 Detect Error On Credit Card d1 d2d3…d15d16 n = 3

18. 18 Now the test (4 + 4 + 8 + 1 + 3 + 5 + 7 + 9) = 41 (5 + 2 + 1 + 0 + 3 + 4 + 6 + 8) x 2 + 3 = 61 41 + 61 = 102 mod (10) = 2 3

19. 19 Credit Card Summary The test performed on the credit card number is called a parity check equation. The last digit is a function of the other digits in the credit card. This is how credit card numbers are generated by Visa and Mastercard. They start with an account number that is 15 digits long and use the parity check equation to find the value of the 16 th digit. “This method allows computers to detect 100% of single- position errors and about 98% of other common errors” (For All Practical Purposes p. 354).

20. Examples ISBN (international standard book number) 0 – 20 – 1 – 36186 – 8 UPC (universal product codes) – 12-digit sequence – 0 16000 66610 8

21. 21 What is a code? A code is defined as an n-tuple of q elements. Where q is any alphabet. Ex. 1001n=4, q={1,0} Ex. 2389047298738904n=16, q={0,1,2,3,4,5,6,7,8,9} Ex. (a,b,c,d,e)n=5, q={a,b,c,d,e,…,y,z} The most common code is when q={1,0}. This is known as a binary code.

22. 22 The purpose A message can become distorted through a wide range of unpredictable errors. Humans Equipment failure Lighting interference Scratches in a magnetic tape

23. 23 Why error-correcting code? To add redundancy to a message so the original message can be recovered if it has been garbled. e.g. message = 10 code = 1010101010

24. 24 Send a message MessageEncoderChannelDecoderMessage 10101010noise00101010

25. 25 Encoding Naïve approach Hamming codes

26. 26 Take Naïve approach Append the same message multiple times. Then take the value with the highest average. Message:= 1001 Encode:= 1001100110011001 Channel:= 1001100100011001 Decode: = a1 = Average(1,1,0,1) = 1 a2 = Average(0,0,0,0) = 0... (a1,a2,a3,a4) Message:= 1001

27. Naïve Solution: Example Single CheckSum - Truth table: General form: Data=[1 1 1 1] Message=[1 1 1 1 0] Repeats – Data = [1 1 1 1] Message= [1 1 1 1]

28. Why Naïve Solution is Inefficient? Shannon Efficiency Repeat 3 times: This divide W by 3 It divides overall capacity by at least a factor of 3x. Single Checksum: Allows an error to be detected but requires the message to be discarded and resent. Each error reduces the channel capacity by at least a factor of 2 because of the thrown away message.

29. 29 Hamming [7,4] Code The seven is the number of digits that make the code. E.g. 0100101 The four is the number of information digits in the code. E.g. 0100101

30. 30 Hamming [7,4] Encoding Encoded with a generator matrix. All codes can be formed from row operations on matrix. The code generator matrix for this presentation is the following:

31. 31 Hamming [7,4] Codes 1000011 0100101 0010110 0001111 1100110 1010101 1001100 0110011 0101010 0011001 1101001 1001010 1111111 0111100 0011001 0000000 Codes Possible codes

32. Multi-dimensional Codes Code Space: 2-dimensional 5 element states Circle packing makes more efficient use of the code-space

33. Cannon Balls http://wikisource.org/wiki/Cannonball_stacking http://mathworld.wolfram.com/SpherePacking.html Efficient Circle packing is the same as efficient 2-d code spacing Efficient Sphere packing is the same as efficient 3-d code spacing Efficient n-dimensional sphere packing is the same as n-code spacing

34. 34 Minimum Weight Theorem Definitions Proof of Theorem

35. 35 Definitions The weight of a code is the number of nonzero components it contains. e.g. wt(0010110) = 3 The minimum weight of Hamming codes is the weight of the smallest nonzero vector in the code. e.g. d(G)= 3

36. 36 Definitions The distance between two codes u and v is the number of positions which differ e.g. u=(1,0,0,0,0,1,1) v=(0,1,0,0,1,0,1) dist(u,v) = 4 Another definition of distance is wt(u – v) = dist(u,v).

37. 37 Definitions For any u, v, and w in a space V, the following three conditions hold:

38. 38 Definitions The sphere of radius r about a vector u is defined as: e.g. u=(1,0,0,0,0,1,1) (0,0,0,0,0,1,1) (1,1,0,0,0,1,1) (1,0,0,0,0,0,1) (1,0,1,0,0,1,1) (1,0,0,1,0,1,1) (1,0,0,0,1,1,1)

39. 39 Minimum Weight Theorem If d is the minimum weight of a code C, then C can correct t = [(d – 1)/2] or fewer errors, and conversely.

40. 40 Proof Want to prove that spheres of radius t = [(d – 1)/2] about codes are disjoint. Suppose for contradiction that they are not. Let u and w be distinct vectors in C, and assume that uwv

41. 41 Proof By triangle inequality uw v

42. Since spheres of radius t = [(d – 1)/2] so and this gives But since We have a contradiction. Showing the sphere of radius t about codes are disjoint. 42 Proof

43. 43 Result of Theorem Since d(G) = 3 then for t = [(3 – 1)/2] = 1 or fewer errors, the received code is in a disjoint sphere about a unique code word.

44. Parity-Check Matrix The parity check matrix is found by solving the generator matrix for For each generator matrix G, there exists an (n – k) x n matrix H, such that the rows of G are orthogonal to the rows of G; i.e., where H T is the transpose of H, and 0 is an k x (n – k) all zeros matrix. The matrix H is called the parity-check matrix, that can be used to decode the received code words.

45. 45 Channel Decoding 1.Brut force Approach: list all possible messages 2.Syndrome Decoding

46. 46 Brute Force Approach: List all messages This is done by generating a list of all the possible messages. For something small like the Hamming [7,4] codes the task is feasible, but for codes of greater length it is not. An example of a list is as follows: Code words 100001101001010010110 … 000001100001010000110 Other100000101001110010100 Received 001001100011011010110 Words110001111001010110110 ………

47. 47 Brute Force Approach: List all messages For example, if the received code was 0001101 then it would be decoded to 0100101 from the list. Code words 100001101001010010110 … 000001100001010000110 Other100000101001110010100 Received 001001100011011010110 Words110001111001010110110 ………

48. 48 Syndrome Decoding Consider a transmitted code c m and y is the received sequence, y can be expressed as, where e denotes binary error vector. The decoder calculate product (n – k)–dimensional vector s is called the syndrome of the error pattern. Or in other words, s contains the pattern of failure in parity checks.

49. Standard Array Let us arrange the 2 n n -tuples that represent all possible received sequences in an array, such that first row contains all the codewords (2 k ) and first column contains all correctable error patterns. The standard array format of ( k,n ) code is as follows:

50. Standard Array Note that codeword c0 is trivial codeword, that is, all- zeros codeword. In addition, it is also error pattern of all-zeros. Each row in the standard array is called a coset. An error pattern in the first column, called coset leader. A coset leader is the vector with the minimum weight in the coset. Each coset consists of 2 k n -tuples  there are 2 n –k cosets.

51. 51 Syndrome Decoding The first step is to create a list of syndromes corresponding the coset leaders. The syndrome of each vector y is found by When a code is received, the syndrome is computed and compared to the list of syndromes. Let the coset leader to the syndrome by e. Finally the code is decoded to x = y – e.

52. 52 Syndrome example Note that G=(I |P) and H = ( | I).

53. 53 Syndrome example Let x:= 1001100 be the original message MessageEncoderChannelDecoderMessage 10011001100noise1000100? Compute the syndrome of the received code

54. 54 Conclusion A code of minimum weight d is called perfect if all the vectors in V are contained in the sphere of radius t = [(d – 1)/2] about the code-word. The Hamming [7,4] code has eight vectors of sphere of radius one about each code-word, times sixteen unique codes. Therefore, the Hamming [7,4] code with minimum weight 3 is perfect since all the vectors (128) are contained in the sphere of radius 1.