1. Equalization

2. Fig. Digital communication system using an adaptive equaliser at the receiver.

3. Equalization Equalization compensates for or mitigates inter-symbol interference (ISI) created by multipaths in time dispersive channels (frequency selective fading channels). Equalizer must be “adaptive”, since channels are time varying.

4. Zero forcing equalizer Design from frequency domain viewpoint.

5. Zero forcing equalizer ∴ must compensate for the channel distortion. ⇒ Inverse channel filter ⇒ completely eliminates ISI caused by the channel ⇒ Zero Forcing equaliser, ⇒ ZF.

6. Zero forcing equalizer

7. Fig. Pulses having a raised cosine spectrum

8. Zero forcing equalizer

9. Example: A two-path channel with impulse response The transfer function is The inverse channel filter has the transfer function

10. Zero forcing equalizer Since DSP is generally adopted for automatic equalizers ⇒ it is convenient to use discrete time (sampled) representation of signal. Received signal For simplicity, assume say

11. Zero forcing equalizer Denote a T-time delay element by Z − 1, then

12. Zero forcing equalizer The transfer function of the inverse channel filter is This can be realized by a circuit known as the linear transversal filter.

13. Zero forcing equalizer

14. The exact ZF equalizer is of infinite length but usually implemented by a truncated (finite) length approximation. For, a 2-tap version of the ZF equalizer has coefficients

15. Modeling of ISI channels Complex envelope of any modulated signal can be expressed as where h a (t) is the amplitude shaping pulse.

16. Modeling of ISI channels In general, ASK, PSK, and QAM are included, but most FSK waveforms are not. Received complex envelope is where is channel impulse response. Maximum likelihood receiver has impulse response matched to f(t)

17. Modeling of ISI channels Output: where n b (t) is output noise and

18. Least Mean Square Equalizers Fig. A basic equaliser during training

19. Least Mean Square Equalizers Minimization of the mean square error (MSE), ⇒ MMSE. Equalizer input h(t): impulse response of tandem combination of transmit filter, channel and receiver filter. In the absence of noise and ISI The error due to noise and ISI at t=kT is given by The error is

20. Least Mean Square Equalizers The MSE is In order to minimize, we require ……

21. Least Mean Square Equalizers

22. The optimum tap coefficients are obtained as W = R −1 P. But this is solved on the knowledge of x k 's, which are the transmitted pilot data. A given sequence of x k 's called a test signal, reference signal or training signal is transmitted prior to the information signal, (periodically). By detecting the training sequence, the adaptive algorithm in the receiver is able to compute and update the optimum w nk ‘s -- until the next training sequence is sent.

23. Least Mean Square Equalizers Example: Determine the tap coefficients of a 2-tap MMSE for: Now, given that

24. Least Mean Square Equalizers

25. Mean Square Error (MSE) for optimum weights Let:

26. Mean Square Error (MSE) for optimum weights Now, the optimum weight vector was obtained as Substituting this into the MSE formula above, we have

27. Mean Square Error (MSE) for optimum weights Now, apply 3 matrix algebra rules: For any square matrix For any matrix product For any square matrix

28. Mean Square Error (MSE) for optimum weights For the example

29. MSE for zero forcing equalizers Recall for ZF equalizer Assuming the same channel and noise as for the MMSE equalizer for MMSE

30. MSE for zero forcing equalizers The ZF equalizer is an inverse filter; ⇒ it amplifies noise at frequencies where the channel transfer function has high attenuation. The LMS algorithm tends to find optimum tap coefficients compromising between the effects of ISI and noise power increase, while the ZF equalizer design does not take noise into account.

31. Diversity Techniques Mitigates fading effects by using multiple received signals which experienced different fading conditions. Space diversity: With multiple antennas. Polarization diversity: Using differently polarized waves. Frequency diversity: With multiple frequencies. Time diversity: By transmission of the same signal in different times. Angle diversity: Using directive antenna aimed at different directions. Signal combining methods. Maximal Ratio combining.

32. Diversity Techniques Equal gain combining. Selection (switching) combining. Space diversity is classified into micro-diversity and macro-diversity. Micro-diversity: Antennas are spaced closely to the order of a wavelength. Effective for fast fading where signal fades in a distance of the order of a wavelength. Macro (site) diversity: Antennas are spaced wide enough to cope with the topographical conditions ( eg: buildings, roads, terrain). Effective for shadowing, where signal fades due to the topographical obstructions.

33. PDF of SNR for diversity systems Consider an M-branch space diversity system. Signal received at each branch has Rayleigh distribution. All branch signals are independent of one another. Assume the same mean signal and noise power ⇒ the same mean SNR for all branches. Instantaneous

34. PDF of SNR for diversity systems Probability that takes values less than some threshold x is,

35. Selection Diversity

36. Branch selection unit selects the branch that has the largest SNR. Events in which the selector output SNR,, is less than some value, x,is exactly the set of events in which each is simultaneously below x. Since independent fading is assumed in each of the M branches,

37. Selection Diversity

38. Maximal Ratio Combining

39. is complex envelope of signal in the k- th branch. The complex equivalent low-pass signal u(t) containing the information is common to all branches. Assume u(t) normalized to unit mean square envelope such that

40. Maximal Ratio Combining Assume time variation of g k (t) is much slower than that of u(t). Let n k (t) be the complex envelope of the additive Gaussian noise in the k-th receiver (branch). ⇒ usually all k N are equal.

41. Maximal Ratio Combining Now define SNR of k-th branch as Now, Where are the complex combining weight factors. These factors are changed from instant to instant as the branch signals change over the short term fading.

42. Maximal Ratio Combining These factors are changed from instant to instant as the branch signals change over the short term fading. How should be chosen to achieve maximum combiner output SNR at each instant? Assuming n k (t)’s are mutually independent (uncorrelated), we have

43. Maximal Ratio Combining Instantaneous output SNR,,

44. Maximal Ratio Combining Apply the Schwarz Inequality for complex valued numbers. The equality holds if for all k, where K is an arbitrary complex constant. Let

45. Maximal Ratio Combining with equality holding if and only if, for each k. Optimum weight for each branch has magnitude proportional to the signal magnitude and inversely proportional to the branch noise power level, and has a phase, canceling out the signal (channel ) phase. This phase alignment allows coherent addition of branch signals ⇒ “co-phasing”.

46. Maximal Ratio Combining each has a chi-square distribution. is distributed as chi-square with 2M degrees of freedom. Average SNR,, is simply the sum of the individual for each branch, which is Γ,

47. Convolutional Codes Department of Electrical Engineering Wang Jin

48. Overview Background Definition Speciality An Example State Diagram Code Trellis Transfer Function Summary Assignment

49. Background Convolutional code is a kind of code using in digital communication systems Using in additive white Gaussian noise channel To improve the performance of radio and satellite communication systems Include two parts: encoding and decoding

50. Block codes Vs Convolutional Codes Block codes take k input bits and produce n output bits, where k and n are large  There is no data dependency between blocks  Useful for data communications Convolution codes take a small number of input bits and produce a small number of output bits each time period  Data passes through convolutional codes in a continuous stream  Useful for low-latency communication

51. Definition A type of error-correction code in which  each k-bit information symbol (each k-bit string) to be encoded is transformed into an n-bit symbol, where n>k  the transformation is a function of the last M information symbols, where M is the constraint length of the code

52. Speciality k bits are input, n bits are output k and n are very small (usually k=1~3, n=2~6). Frequently, we will see that k=1 Output depends not only on current set of k input bits, but also on past input The “constraint length” M is defined as the number of shifts, over which a single message it can influence the encoder output Frequently, we will see that k=1

53. An Example A simple rate k / n = 1 / 2 convolutional code encoder (M=3) The box represents one element of a serial register + + Code digits Binary information digits Input Output

54. An Example (cont ’ d) The content of the shift registers is shifted from left to right Plus sign represents modulo-2 (XOR) addition Output by encoder are multiplexed into serial binary digits For every binary digit enters the encoder, two code digits are output A generator sequence specifies the connections of a modulo-2 (XOR) adder to the encoder shift register. In this example, there are two generator sequences, g 1 =[1 1 1] and g 2 =[1 0 1]

55. An Example (cont ’ d) t=0 t=1 t=2 + + Code digits Binary information digits Input Output x2x2 x1 x1 x 0 x3x3 x2 x2 x 1 x4x4 x3 x3 x 2 x5x5 x4 x4 x 3 t=3 When t=3, the content of the initial state (x 2, x 1, x 0 ) is missing.

56. To Determine the Output Codeword There are essentially two ways  State diagram approach  Transform-domain approach Only concentrate on state diagram approach Contents of shift registers make up “state” of code:  Most recent input is most significant bit of state  Oldest input is least significant bit of state  (this convention is sometimes reverse) Arcs connecting states represent allowable transitions  Arcs are labeled with output bits transmitted during transition

57. To Determine the Output Code Word ---State Diagram Rate k / n = 1 / 2 convolutional code encoder (M=3) State is defined by the most (M-1) message bits moves into the encoder D0D0 D1D1 D2D2 + + Code digits Binary information digits State (recent M-1 digits)

58. State Diagram (cont ’ d) There are four states [00], [01], [10], [11] corresponding to the (M-1) bits Generally, assuming the encoder starts in the all-zero [00] state

59. State Diagram (cont ’ d) Easiest way to determine the state diagram is to first determine the state table as shown below

60. State Diagram (cont ’ d) 1/01 means (for example), that the input binary digit to the encoder was 1 and the corresponding codeword output is 01 10 00 11 01 1/10 1/01 0/11 0/01 0/00 1/11 1/00 0/10

61. Trellis Representation of Convolutional Code State diagram is “unfolded” a function of time Time indicated by movement towards right

62. Code Trellis It is simply another way of drawing the state diagram Code trellis for rate k / n = 1 / 2,M=3 convolutional code shown below 00 11 10 01 00 11 10 0/11 0/01 1/01 0/00 1/11 1/00 0/10 1/10 Start state Final state

63. Encoding Example Using Trellis Diagram Trellis diagram, similar to state diagram, also shows the evolution in time of the state encoder Consider the r=1/2, M=3 convolutional code

64. Encoding Example Using Trellis Diagram 00 01 10 11 01 10 11 00 0/00 0/11 1/11 1/00 0/10 1/010/01 1/10 State Input data 0 1 0 0 1 Output 00 11 10 11 11

65. Distance Structure of a Convolutional code The Hamming distance between any two distinct code sequences is the number of bits in which they differ: The minimum free Hamming distance of a convolutional code is the smallest Hamming distance separating any two distinct code sequences:

66. The Transfer Function This is also known as the generating function or the complete path enumerator. Consider the r=1/2, M=3 convolutional code example and redraw the state diagram. a0a0 b a1a1 c d JND 2 JN JDJD 2 JDJND

67. The Transfer Function (Con ’ d) State “a” has been split into an initial state “a 0 ”and a final state “a 1 ” We are interested in the number of paths that diverge from the all aero path at state “a” at some point in time and remerges with the all-zero path. Each branch transition is labeled with a term, where are all integers such that:  -----corresponds to the length of the branch  -----Hamming weigh of the input zero for a “0” input and one for a “1” input  -----Hamming weight of the encoder output for that branch

68. The Transfer Function (Con ’ d) Assuming a unity input, we can write the set of equations By solving these equations, From the transfer function, there is one path at a Hamming distance of 5 from the all-zero path. This path is of length 3 branches and corresponds to a difference of one input information bit from the all zero path. Other terms can be interpreted similarly. The minimum distance is thus 5.

69. Search for Good Codes We would like convolutional codes with large free distance  Must avoid “catastrophic codes” Generators for best convolutional codes are generally found via computer search  Search is constrained to codes with regular structure  Search is simplified because any permutation of identical generators is equivalent  Search is simplified because of linearity

70. Best Rate ½ Codes MGenerators (in Octal) d free 3575 415176 523357 653758 713317110 824737110 956175312

71. Best Rate 1/3 Codes MGenerators (in Octal) d free 35778 413151710 525333712 647537513 713314517115 822533136716 955766371118

72. Best Rate 2/3 Codes MGenerators (in Octal) d free 21716154 32775726 42361553377

73. Summary What is convolutional code The transformation of a convolutional code We can represent convolutional codes as generators, block diagrams, state diagrams and trellis diagrams Convolutional codes are useful for real-time applications because they can be continuously encoded and decoded

74. Assignment Question: Construct the state table and state diagram for the encoder below. + + Code digits Binary information digits Input (k=1)Output (n=3) +

75. THANK YOU