Advances in coding for the fading channel


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  • For example, in a TDMA or FDMA system, the network resource is shared among users via disjoint frequency and time slots, and this sharing provides a simple abstraction for resource allocation problems at the networking layer .
  • Advances in coding for the fading channel

    2. 2. CODING FOR THE FADING CHANNEL • Is Euclidean distance the best criterion? 2
    7. 7. CODING FOR THE FADING CHANNEL • Modeling the wireless channel 7
    8. 8. COHERENCE BANDWIDTH DEFINITION: 1 ---------------------- DELAY SPREAD OPERATIONAL MEANING:Frequency separation at which two frequencycomponents of TX signal undergoindependent attenuations 8
    9. 9. COHERENCE TIME DEFINITION: 1 --------------------------- DOPPLER SPREAD OPERATIONAL MEANING:Time separation at which two timecomponents of TX signal undergoindependent attenuations 9
    10. 10. FADING-CHANNEL CLASSIFICATION Bx flat selective in in time time and frequency Bc flat flat in in time and frequency frequency Tc Tx 10
    11. 11. MOST COMMON MODEL FOR FADING• channel is frequency-flat• channel is time-flat (fading is “slow”) 11
    12. 12. MOST COMMON MODEL FOR FADING • FREQUENCY-FLAT CHANNEL: Fading affects the received signal as a multiplicative process noise Received signal: r(t ) = R(t )exp jΘ(t ) x(t ) + n(t ) Gaussian process: R Rayleigh or Rice transmitted signal 12
    13. 13. MOST COMMON MODEL FOR FADING • SLOW FADING : Fading is approximately constant during a symbol duration Received signal: r(t ) = R exp jΘ x(t ) + n(t ), 0<t <TThis is constant overa symbol interval 13
    14. 14. COHERENT DEMODULATION Received signal: r (t ) = R x(t ) + n(t ), 0 <t <TPhase term is estimatedand compensated for 14
    15. 15. CHANNEL-STATE INFORMATIONThe value of the fading attenuation is the“channel-state information”This may be:• Unknown to transmitter and receiver• Known to receiver only (through pilot tones, pilot symbols, …)• Known to transmitter and receiver 15
    16. 16. EFFECT OF FADING ON ERROR PROBABILITIES 1 bit error probability, binary antipodal signals 0.1 RAYLEIGH 0.01 FADING 0.001 GAUSSIAN CHANNEL 0.0001 0.00001 0 10 20 30 signal-to-noise ratio (dB)performance of uncoded modulation over the fading channelwith coherent demodulation 16
    17. 17. CODING FOR THE FADING CHANNEL • Optimum codes for the frequency-flat, slow fading channel • Euclid vs. Hamming • How useful is an “optimum code”? 17
    18. 18. MOST COMMON MODEL FOR CODINGOur analysis here is concerned with thefrequency-flat, slow, FULLY-INTERLEAVED CHANNELas the de-interleaving mechanism creates afading channel in which the random variablesR in adjacent intervals are independent 18
    19. 19. DESIGNING OPTIMUM CODESChernoff bound on the pairwise error probabilityover the Rayleigh fading channel with high SNR: Hamming distance Signal-to-noise ratio −dH ( x ,x )  1 Γ 2 P(x → x) ≤ ∏  ≤ δ  Γ 4  k 1+ | xk − xk |2  4 Product distance Most relevant parameter: Hamming 19 distance
    20. 20. DESIGNING OPTIMUM CODESDesign criterion:Maximize Hamming distance among signa A consequence: In trellis-coded modulation, avoid “parallel transitions as they have Hamming distance = 1. 20
    21. 21. DESIGNING OPTIMUM CODES If we maximize Hamming distance among signals strange effects occur. For example: if fading acts independently on I and Q parts: 4PSK Effect of a deep fade on Q part (one bit is lost) if fading acts independently on I and Q parts:Rotated 4PSK Effect of a deep fade on(same Euclidean distance) Q part 21 (no bit is lost)
    22. 22. DESIGNING OPTIMUM CODESProblems with optimum fading codes:• The channel model may be unknown, or incompletely known• The channel model may be unstable 22
    23. 23. ROBUST CODESIn these conditions, one should look for robust, rather than optimum, coding schemes 23
    24. 24. CODING FOR THE FADING CHANNEL • BICM as a robust coding scheme 24
    25. 25. A ROBUST SCHEME: BICMencoder bit modulator hannel c demo bit decoder interleaver d. deinterleav er interleaving is done at bit level demodulation and decoding are separated 25
    26. 26. A ROBUST SCHEME: BICM Separating demodulation and decoding is a considerable departure from the “Ungerboeck’s paradigm” , which states that demodulation and decoding should be integrated in a single entity for optimality But this may not be true if the channel is not Gaussian!Bit interleaving may increase Hamming distance amoncode words at the price of a slight decrease of Euclidedistance ( robust solution if channel model is not stable 26
    27. 27. A ROBUST SCHEME: BICMBICM idea is that Hamming distance(and hence performance over the fading channel)can be increased by making itequal to the smallest number of bits(rather than channel symbols)along any error event: 00 00 00     correct path 11 10 11 concurrent path   TCM: Hamming distance is 3 BICM: Hamming distance is 5 27
    28. 28. A ROBUST SCHEME: BICMBICM DECODER USES MODIFIED “BIT METRICS ” With TCM, the metric associated with symbol s is p(r | s) With BICM, the metric associated with bit b is ∑ p( r | s ) s∈Si ( b ) i where S the set of symbols whose label is b in position i is (b ) 01 EXAMPLE: 11 00 S1 (0) 28 10
    29. 29. A ROBUST SCHEME: BICMThe performance of BICM with ideal interleaving depends on the following parameters: • Minimum binary Hamming distance of the code select • Minimum Euclidean distance of the constellation sele so we can combine: • A powerful modulation scheme • A powerful code (turbo codes, …) 29
    30. 30. EXAMPLE : 16QAM, 3bits/2 dimensionsENCODER BICM TCMMEMORY dE 2 dH dE 2 dH2 1.2 3 2 13 1.6 4 2.4 24 1.6 4 2.8 25 2.4 6 3.2 26 2.4 6 3.6 37 3.2 8 3.6 38 3.2 8 4 3 30
    31. 31. ANTENNA DIVERSITY & CHANNEL INVERSION Possible solution to the”robustness problem”:Turn the fading channel intoa Gaussian channel, and use standard cod • Antenna diversity • Channel inversion as a power-allocation technique 31
    32. 32. CODING FOR THE FADING CHANNEL • Antenna diversity 32
    33. 33. ANTENNA DIVERSITY (order M)• The fading channel becomes Gaussian as M → ∞• Codes optimized for the Gaussian channel perform well on the Rayleigh channel if M is large enough• Branch correlation coefficients up to 0.5 achieve uncorrelated performance within 1 dB• The error floor with CCI decreases exponentially with the product of M times the Hamming distance of the code used 33
    34. 34. EXPERIMENTAL RESULTS Performance was evaluated for the following coding schemes: J4: 4-state, rate-2/3 coded 8-PSK optimized for Rayleigh-fading channels U4 & U8: Ungerboeck’s rate-2/3 coded 8-PSK with 4 and 8 states optimized for the Gaussian chan Q64: “Pragmatic” concatenation of the “best” binary rate-1/2 64-state convolutional code (171, 133) mapped onto Gray-encoded 4-PSK 34
    35. 35. EXPERIMENTAL RESULTS 0 10 -2 10BR U4, M=1 E -4 10 J4, M=1 -6 10 U4, M=16 J4, M=16 J4, M=4 U4, M=4 -8 10 5 10 15 20 25 30 35 35 E b /N 0 (dB)
    36. 36. CODING FOR THE FADING CHANNEL • The block-fading channel 36
    37. 37. Most of the analyses are concerned with the FULLY-INTERLEAVED CHANNELas the de-interleaving mechanism creates avirtually memoryless coding channel. HOWEVER,in practical applications such as digital cellularspeech communication, the delay introduced bylong interleaving is intolerable 37
    38. 38. FACTS In many wireless systems:Typical Doppler spreads range from 1 Hz to 100 Hz(hence coherence time ranges from 0.01 to 1 s)Data rates range from 20 to 200 kbaudConsequently, at least L=20,000 x 0.01 = 200 symbolsare affected approximately by the same fading gain 38
    39. 39. FACTSConsider transmission of a code word of length n.For each symbol to be affected by an independentfading gain, interleaving should be usedThe actual time spanned by the interleaved codeword becomes at least nL The delay becomes very large 39
    40. 40. FACTSIn some applications, large delays are unacceptable(real time speech: 100 ms at most)Thus, an n-symbol code wordis affected by less than n independent fading gains 40
    41. 41. BLOCK-FADING CHANNEL MODELThis model assume that the fading-gain processis piecewise constant on blocks of N symbols.It is modeled as a sequence of independentrandom variables, each of which is the fading gainin a block.A code word of length n is spread over M blocksof N symbols each, so that n=NM 41
    42. 42. BLOCK-FADING CHANNEL MODEL 1 M N 2 N 3 N .. N n=NM ..• Each block of length N is affected by the same fading.• The blocks are sent through M independent channels.• Interleaver spreads the code symbols over the M block (McEliece and Stark, 1984 -- Knopp, 1997) 42
    43. 43. BLOCK-FADING CHANNEL MODEL Special cases: M=1 (or N=n) the entire code word is affected by the same fading gain (no interleaving) M=n (or N=1) each symbol is affected by an independent fading gain (ideal interleaving) 43
    44. 44. BLOCK-FADING CHANNEL MODEL The delay constraints determines the maximum M The choice M → ∞ makes the channel ergodic, and allows Shannon’s channel capacity to be defined (more on this later) 44
    45. 45. System where this model is appropriate:GSM with frequency hopping f 4 4 3 3 2 2 2 1 1 1 t M=4 (half-rate GSM) 45
    46. 46. System where this model is appropriate: IS-54 with time-hopping 1 2 1 M=2 46
    47. 47. COMPUTING ERROR PROBABILITIES“Channel use” is now the transmissionof a block of N coded symbolsFrom Chernoff bound we have, overRayleigh block-fading channels: 1 P ( X → X) ≤ ∏ ˆ m∈M 1 + dm / 4N0 2 Set of indices in which Squared Euclidean distance coded symbols differ between coded blocks 47
    48. 48. COMPUTING ERROR PROBABILITIESFor high SNR: Signal-to-noise ratio Hamming block-distance ˆ −d H ( X , X ) 1 Γ 2  P ( X → X) ≤ ∏ ˆ ≤ δ  m∈M 1 + Γ 2 4  dm 4 Product distance 48
    49. 49. Relevant parameter fordesign Minimum Hamming block-distance D between code words on block basis: Error probability decreases with exponent D min (also called: code diversity ) 49
    50. 50. EXAMPLE (N=4) Block #1 Block #2 00 00 00 00 11 11 11 Dmin=2 00 10 10 01 01 11 4 binary symbols 4 binary symbols 50
    51. 51. Bound on Dmin With S-ary modulation, Singleton bound holds for a rate-R code:   R  Dmin ≤ 1 +  M 1 −  log S      2  51
    52. 52. Example: Coding in GSM + +Rate-1/2 convolutional code (0.5 bits/dimension)used in GSM with M=8. It has dfree=7 52
    53. 53. Example: Coding in GSMdfree path is: {0...011010011110...0}Symbols in each one of the 8 blocks: 1: 0...0110...0 2: 0...0110...0 Dmin=5 3: 0...0000...0 4: 0...0100...0 5: 0...0000...0 6: 0...0000...0 7: 0...0100...0 8: 0...0100...0 53
    54. 54. This code is optimum!With full-rate GSM, R=0.5 bits/dim, M=8, S=2. Hence: Dmin ≤ 5 achieved by the code. (With S=4 the upper bound would increase to 7). 54
    55. 55. CODING FOR THE FADING CHANNEL • Power control 55
    56. 56. PROBLEM:How to encode if CSI is known atthe transmitter (and at the receiver) 56
    57. 57. We have:r (t ) = R x(t ) + n(t ) Assume R is known to transmitter and receiver γ If: x (t ) = s (t ) R( channel inversion) then the fading channelis turned into a Gaussian channel 57
    58. 58. Channel inversion is commonn spread-spectrum systemswith near-far imbalancePROBLEM: For Rayleigh fading channels the avera transmitted power would be infinite.SOLUTION: Use average-power constraint. 58
    59. 59. CODING FOR THE FADING CHANNEL • Using multiple antennas 59
    60. 60. MULTIPLE- ANTENNA MODEL (Si n g l e-u ser) ch a n n el w i th t tra n sm i t a n d r recei ve a n ten n a s: t r H 60
    61. 61. CHANNEL CAPACITYRATIONALE: U se sp a ce to i n crea se d i versi ty (Freq u en cy a n d ti m e co st to o m u ch )Ea ch recei ver sees th e si g n a l s ra d i a ted fro mth e t tra n sm i t a n ten n a sPa ra m eter u sed to a ssess sy stem q u a l i ty :CHANNEL CAP ACITY(Th i s i s a limit to error- f ree bit rate, p ro vi d edb y i n fo rm a ti o n th eo ry ) 61
    62. 62. CHANNEL CAPACITYAssu m e th a t tra n sm i ssi o n o ccu rs i n f rames:th ese a re sh o rt en o u g h th a t th e ch a n n el i sessen ti a l l y u n ch a n g ed d u ri n g a fra m e,a l th o u g h i t m i g h t ch a n g e co n si d era b l y fro m o n efra m e to th e n ext (“ quasi- stationary” vi ew p o i n t)W e a ssu m e th e ch a n n el to b e u n k n o w n to th e tra n sm i tter, b u t k n o w n to th e recei ver H o w ever, th e tra n sm i tter h a s a p a rti a l k n o w l ed g e o f th e ch a n n el q u a l i ty , so th a t i t ca n ch o o se th e tra n sm i ssi o n ra te 62
    63. 63. CHANNEL CAPACITYN o w , th e ch a n n el va ri es w i th ti m e fro m fra m eto fra m e, so fo r so m e (sm a l l ) p ercen ta g e o ffra m es d el i veri n g th e d esi red b i t ra te a t th ed esi red BER m a y b e i m p o ssi b l e.W h en th i s h a p p en s, w e sa y th a t a channel outageh a s o ccu rred . I n p ra cti ce capacity is a randomvariable.W e a re i n terested i n th e ca p a ci ty th a t ca n b ea ch i eved i n n ea rl y a l l tra n sm i ssi o n s (e.g ., 99% ). 63
    64. 64. CHANNEL CAPACITY1 % -o u ta g e ca p a ci ty(u p p er cu rves)fo r Ra y l ei g h ch a n n elvs. SN R a n dn u m b er o fa n ten n a sN o te: a t 0-d B SN R,25 b / s/ H z a rea va i l a b l e w i th t= r= 32! t= r (SN R i s P/ N a t ea ch recei ve a n ten n a ) 64
    65. 65. CHANNEL CAPACITY1 % -o u ta g e ca p a ci typ er d i m en si o n(u p p er cu rves)fo r Ra y l ei g hch a n n elvs. SN R a n dn u m b er o fa n ten n a s t= r 65
    66. 66. ACHIEVAB R LE ATES 66
    67. 67. SPACE- TIME CODING(Al a m o u ti , 1 998) Co n si d er t 2 a n d r 1 . = = Den o te s0 th e si g n a l fro m a n ten n a 0 a n d s1 th e si g n a l fro m a n ten n a 1 Du ri n g th e n ext sy m b o l p eri o d -s1 * i s tra n sm i tted b y a n ten n a 0 s0* i s tra n sm i tted b y a n ten n a 1 67
    68. 68. SPACE- TIME CODING Th e si g n a l s recei ved i n tw o a d j a cen t ti m e sl o ts a re r0 = r (t ) = h0 s0 + h1s1 + n0 ∗ ∗ r1 = r (t + T ) = − h s + h1s0 + n1 0 1 Th e co m b i n er y i el d s ~ = h ∗r + h r ∗ s0 0 0 1 1 ~ = h ∗r − h r ∗ s 1 1 0 0 1 68
    69. 69. SPACE- TIME CODING So th a t: ~ = h 2 + h 2 s + noise s0 0 1 0 ~ = h 2 + h 2 s + noise s1 0 1 1 A m a xi m u m -l i k el i h o o d d etecto r m a k es a d eci si o n o n s0 a n d s1 . Th i s sch em e h a s th e sa m e p erfo rm a n ce a s a sch em e w i th t 1 , r 2 a n d = = m a xi m a l -ra ti o co m b i n i n g . 69
    70. 70. SPACE- TIME CODINGt2=r1= 70
    71. 71. SPACE- TIME CODINGM RRC=m a xi m u m -ra ti orecei veco m b i n i n g 71 SN R (d B)
    72. 72. SPACE- TIME CODING Th e p erfo rm a n ce o f th i s sy stem w i th t 2 = a n d r 1 i s 3-d B w o rse th a n w i th t 1 a n d r 2 = = = p l u s M RRC. Th i s p en a l ty i s i n cu rred b eca u se th e cu rves a re d eri ved u n d er th e a ssu m p ti o n th a t each TX antenna radiates half the energy as the single transmit antenna w i th M RRC. 72
    73. 73. SPACE- TIME CODING(Ta ro k h , Sesh a d ri , Ca l d erb a n k , et a l .) Co n si d er tw o tra n sm i t a n ten n a s Exa m p l e: Sp a ce-ti m e co d e a ch i evi n g d i versi ty 2 w i th o n e recei ve a n ten n a (“ 2-sp a ce-ti m e co d e” ), a n d d i versi ty 4 w i th tw o recei ve a n ten n a s 73
    74. 74. SPACE- TIME CODINGLa b el x m ea n s th a t ysi g n a l x s tra n sm i tted o n a n ten n a 1 , w h i l e isi g n a l y s (si m u l ta n eo u sl y ) tra n sm i tted o n a n ten n a i200 01 02 03 2-sp a ce-ti m e co d e10 11 12 13 4PSK 4 sta tes20 21 22 23 2 b i t/ s/ H z30 31 32 33 74
    75. 75. SPACE- TIME CODING• I f y j n d en o tes th e si g n a l recei ved a t a n ten n a j a t ti m e n , th e b ra n ch m etri c fo r a tra n si ti o n l a b el ed qq… qi s 1 2 t r t 2 ∑j =1 y n − ∑ hi , j qi j i =1 (n o te th a t ch a n n el -sta te i n fo rm a ti o n i s n eed ed to g en era te th i s m etri c) 75
    76. 76. SPACE- TIME CODING Fo r w i rel ess sy stem s w i th a sm a l l n u m b er o f a n ten n a s, th e sp a ce-ti m e co d es o f Ta ro k h , Sesh a d ri , a n d Ca l d erb a n k p ro vi d e both coding gain and diversity Using a 64- state decoder these come within 2—3 dB of outage capacity 76
    77. 77. TUR O- CODED MODULATION B(Stefa n o v a n d Du m a n , 1 999) 77
    78. 78. TUR O- CODED MODULATION B BER fo r severa l tu rb o co d es a n d a 1 6-sta te sp a ce-ti m e co d e 78