1. ADVANCES IN CODING FOR THE FADING CHANNEL
EZIO BIGLIERI
Politecnico di Torino (Italy)
1

2. CODING FOR THE FADING CHANNEL
• Is Euclidean distance
the best criterion?
2

3. MOST OF THE COMMON WISDOM
ON CODE DESIGN
IS BASED ON HIGH-SNR GAUSSIAN CHANNEL:
MAXIMIZE THE MINIMUM EUCLIDEAN DISTANCE
3

4. FOR DIFFERENT CHANNEL MODELS,
DIFFERENT DESIGN CRITERIA MUST BE USED
4

5. FOR EXAMPLE, EVEN ON LOW-SNR
GAUSSIAN CHANNELS
MINIMUM-EUCLIDEAN DISTANCE IS NOT
THE OPTIMUM CRITERION
EXAMPLE: Minimum P(e) for
4-point, one-dimensional constellation:
low SNR
high SNR
5

6. WIRELESS CHANNELS DIFFER
CONSIDERABLY FROM HIGH-SNR
GAUSSIAN CHANNELS:
SNR IS A RANDOM VARIABLE
AVERAGE SNR IS LOW
CHANNEL STATISTICS ARE NOT GAUSSIAN
MODEL MAY NOT BE STABLE
6

7. CODING FOR THE FADING CHANNEL
• Modeling the wireless channel
7

8. COHERENCE BANDWIDTH
DEFINITION:
1
----------------------
DELAY SPREAD
OPERATIONAL MEANING:
Frequency separation at which two frequency
components of TX signal undergo
independent attenuations 8

9. COHERENCE TIME
DEFINITION:
1
---------------------------
DOPPLER SPREAD
OPERATIONAL MEANING:
Time separation at which two time
components of TX signal undergo
independent attenuations 9

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. MOST COMMON MODEL FOR FADING
• channel is frequency-flat
• channel is time-flat (fading is “slow”)
11

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. 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 <T
This is constant over
a symbol interval
13

14. COHERENT DEMODULATION
Received signal:
r (t ) = R x(t ) + n(t ), 0 <t <T
Phase term is estimated
and compensated for
14

15. CHANNEL-STATE INFORMATION
The 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. 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 channel
with coherent demodulation
16

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. MOST COMMON MODEL FOR CODING
Our analysis here is concerned with the
frequency-flat, slow,
FULLY-INTERLEAVED CHANNEL
as the de-interleaving mechanism creates a
fading channel in which the random variables
R in adjacent intervals are independent
18

19. DESIGNING OPTIMUM CODES
Chernoff bound on the pairwise error probability
over 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. DESIGNING OPTIMUM CODES
Design criterion:
Maximize Hamming distance among signa
A consequence:
In trellis-coded modulation, avoid “parallel transitions
as they have Hamming distance = 1.
20

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. DESIGNING OPTIMUM CODES
Problems with optimum fading codes:
• The channel model may be unknown,
or incompletely known
• The channel model may be unstable
22

23. ROBUST CODES
In these conditions, one should look for
robust, rather than optimum,
coding schemes
23

24. CODING FOR THE FADING CHANNEL
• BICM as a robust coding
scheme
24

25. A ROBUST SCHEME: BICM
encoder bit modulator hannel
c demo bit decoder
interleaver d. deinterleav
er
interleaving is done at bit level
demodulation and decoding are separated
25

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 amon
code words at the price of a slight decrease of Euclide
distance ( robust solution if channel model is not stable
26

27. A ROBUST SCHEME: BICM
BICM idea is that Hamming distance
(and hence performance over the fading channel)
can be increased by making it
equal 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. A ROBUST SCHEME: BICM
BICM 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. A ROBUST SCHEME: BICM
The 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. EXAMPLE
: 16QAM, 3bits/2 dimensions
ENCODER BICM TCM
MEMORY dE 2
dH dE 2
dH
2 1.2 3 2 1
3 1.6 4 2.4 2
4 1.6 4 2.8 2
5 2.4 6 3.2 2
6 2.4 6 3.6 3
7 3.2 8 3.6 3
8 3.2 8 4 3
30

31. ANTENNA DIVERSITY & CHANNEL INVERSION
Possible solution to the”robustness problem”:
Turn the fading channel into
a Gaussian channel, and use standard cod
• Antenna diversity
• Channel inversion as a power-allocation
technique
31

32. CODING FOR THE FADING CHANNEL
• Antenna diversity
32

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. 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. EXPERIMENTAL RESULTS
0
10
-2
10
BR
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. CODING FOR THE FADING CHANNEL
• The block-fading channel
36

37. Most of the analyses are concerned with the
FULLY-INTERLEAVED CHANNEL
as the de-interleaving mechanism creates a
virtually memoryless coding channel.
HOWEVER,
in practical applications such as digital cellular
speech communication, the delay introduced by
long interleaving is intolerable
37

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 kbaud
Consequently, at least
L=20,000 x 0.01 = 200 symbols
are affected approximately by the same fading gain
38

39. FACTS
Consider transmission of a code word of length n.
For each symbol to be affected by an independent
fading gain, interleaving should be used
The actual time spanned by the interleaved code
word becomes at least nL
The delay becomes very large
39

40. FACTS
In some applications, large delays are unacceptable
(real time speech: 100 ms at most)
Thus, an n-symbol code word
is affected by less than n independent fading gains
40

41. BLOCK-FADING CHANNEL MODEL
This model assume that the fading-gain process
is piecewise constant on blocks of N symbols.
It is modeled as a sequence of independent
random variables, each of which is the fading gain
in a block.
A code word of length n is spread over M blocks
of N symbols each, so that n=NM
41

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. 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. 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. 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. System where this model is appropriate:
IS-54 with time-hopping
1 2 1
M=2
46

47. COMPUTING ERROR PROBABILITIES
“Channel use” is now the transmission
of a block of N coded symbols
From Chernoff bound we have, over
Rayleigh 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. COMPUTING ERROR PROBABILITIES
For 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. Relevant parameter for
design
Minimum Hamming block-distance D
between
code words on block basis:
Error probability decreases with
exponent D min
(also called: code diversity )
49

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. 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. 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. Example: Coding in GSM
dfree 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. 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. CODING FOR THE FADING CHANNEL
• Power control
55

56. PROBLEM:
How to encode if CSI is known at
the transmitter (and at the receiver)
56

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 channel
is turned into a Gaussian channel
57

58. Channel inversion is common
n spread-spectrum systems
with near-far imbalance
PROBLEM: For Rayleigh fading channels the avera
transmitted power would be infinite.
SOLUTION: Use average-power constraint.
58

59. CODING FOR THE FADING CHANNEL
• Using multiple antennas
59

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. CHANNEL CAPACITY
RATIONALE: 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 m
th e t tra n sm i t a n ten n a s
Pa 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 ed
b y i n fo rm a ti o n th eo ry ) 61

62. CHANNEL CAPACITY
Assu 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 s
essen 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 e
fra 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. CHANNEL CAPACITY
N o w , th e ch a n n el va ri es w i th ti m e fro m fra m e
to fra m e, so fo r so m e (sm a l l ) p ercen ta g e o f
fra m es d el i veri n g th e d esi red b i t ra te a t th e
d 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 outage
h a s o ccu rred . I n p ra cti ce capacity is a random
variable.
W e a re i n terested i n th e ca p a ci ty th a t ca n b e
a 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. CHANNEL CAPACITY
1 % -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 el
vs. SN R a n d
n u m b er o f
a n ten n a s
N o te: a t 0-d B SN R,
25 b / s/ H z a re
a 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. CHANNEL CAPACITY
1 % -o u ta g e ca p a ci ty
p er d i m en si o n
(u p p er cu rves)
fo r Ra y l ei g h
ch a n n el
vs. SN R a n d
n u m b er o f
a n ten n a s
t= r
65

66. ACHIEVAB R
LE ATES
66

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. 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. 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. SPACE- TIME CODING
t2
=
r1
=
70

71. SPACE- TIME CODING
M RRC=
m a xi m u m -
ra ti o
recei ve
co m b i n i n g 71
SN R (d B)

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. 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. SPACE- TIME CODING
La b el x m ea n s th a t
y
si g n a l x s tra n sm i tted o n a n ten n a 1 , w h i l e
i
si 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
i
2
00 01 02 03
2-sp a ce-ti m e co d e
10 11 12 13 4PSK
4 sta tes
20 21 22 23 2 b i t/ s/ H z
30 31 32 33 74

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. 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. TUR O- CODED MODULATION
B
(Stefa n o v a n d Du m a n , 1 999)
77

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