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JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 1 -
Iterative Equalization
Speaker:Michael [email protected]
InterleaverDeinterleaver
Decoder
âk
Equalizer / DetectorDemapper Mapper
yk
s(bk)s‘(bk)
s(ck)s‘(ck)
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 2 -
System Configuration and Receiver Structures
Encoder
Interleaver
Mapper
Channel
bk
ck
xk
yk
ak
System Configuration
Receiver A:optimal detector
Receiver B:one-time equalization
and detection
Receiver C:turbo equalization
OptimalDetector
yk
âk
Decoder
Deinterleaver
Demapper
Equalizer / Detector
yk
âk
kb
kc
kx
s(bk)
s(ck)
s(xk)
InterleaverDeinterleaver
Decoder
âk
Equalizer / DetectorDemapper Mapper
yk
s(bk)s‘(bk)
s(ck)s‘(ck)
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 3 -
Interleaver
Example for an interleaver: A 3-random interleaver for 18 code bits
Encoder
Interleaver
Mapper
Channel
bk
ck
xk
yk
ak
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 4 -
Equalization
• Methods to compensate the channel effects
Transmitter Receiver
e.g. Multi-Path Propagation might lead to IntersignalInterference (ISI).
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
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Used Channel Modell
In the following, we will use an AWGN channel with known channel impulse response (CIR). The received signal is given by
L
llklk xhy
0
N 1,2,..., k , n
In matrix form: y = Hx + n
channel coefficient sent signal noise
NNN n
n
n
x
x
x
hhh
hhh
hhh
hh
h
y
y
y
2
1
2
1
012
012
012
01
0
2
1
00
00
00
000
0000
y
As an example, we have alength-three channel with h0=0.407h1=0.815h2=0.407
The noise is Gaussian:2
2
2)(
2
n
enp
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 6 -
The Forward / Backward Algorithm
For Receiver B, the Forward / Backward Algorithm is often used for equalization and decoding.
As this algorithm is a basic building block for our turbo equilization setup, we will discuss it in detail• for equalization• for decoding
We will continue our example to make things clear.
The example uses binary phase shift keying (BPSK)
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
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The Decision Rule
Receiver B:
Decoder
Deinterleaver
Demapper
Equalizer / Detector
yk
âk
kb
kc
kx s(xk)
The decision rule for the equalizer is
0 )|L(c if ,1
0 )|L(c if ,0c
k
kk y
y
with the log-likelihood ratio
)|1()|0(
ln)|(yy
ycPcP
cL
So, we have to calculate L(c|y)
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
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The Trellis Diagram (1)
time k+2state rjstate ri
input xk=xi,j output vk=vi,j
A branch of the trellis is a four-tuple (i, j, xi,j, vi,j)
(1, 1)
(-1, 1)
(1, -1)
(-1, -1)
(1, 1)
(-1, 1)
(1, -1)
(-1, -1)
Skip TrellisOnly FBA Matrix
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 9 -
The Trellis Diagram (2)
If the tapped delay line contains L elements and if we use a binary alphabet {+1, -1}, the channel can be in one of 2L states ri. The set of possible states is
S = {r0,r1,…,r2L-1}
At each time instance k=1,2,…,N the state of the channel is a random variable sk S.
2L = 4
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 10 -
The Trellis Diagram (3)
Using a binary alphabet, a given state sk = ri can only develop into two different states sk+1 = rj depending on the input symbol xk = xi,j = {+1, -1}.
The output symbol vk = vi,j in the noise-free case is easily calculated by
L
llklk xhv
0Hxv
v2,0 = h0x2,0 + h1x3,2 + h2x3,3 = 0.407∙1 + 0.815∙1 + 0.407∙(-1) = 0.815
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 11 -
The Trellis Diagram (4)
xi,j and vi,j are uniquely identified by the index pair (i j). The set of all index pairs (i j) corresponding to valid branches is denoted B.
e.g B = {(00), (01), (12), (13), (20), (21), (33), (32)}
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 12 -
The Joint Distribution p(sk, sk+1, y)
As we are separating the equalization from the decoding task, we assume that the random variables xk are statistically independent (IID), hence
N
kkxPxP
1
)()(
We then have to calculate P(sk=ri, sk+1=rj | y).
This is the probability that the transmitted sequence path in the trellis contains the branch (I, j, xi,j, vi,j) at the time instance k.
This APP (a posteriori probability) can be computed efficiently with the forward / backward algorithm, based on a suitable decomposition of the joint distribution
p(sk, sk+1, y) = p(y) ∙ P(sk, sk+1 | y)
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
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The Decomposition
We can write the joint distribution as
p(sk, sk+1, y) = p(sk, sk+1, (y1,…,yk-1), yk, (yk+1,…,yN))
and decompose it to
probability that contains all paths through the Trellis to come to state sk
probability that contains all possible pathes from state sk+1 to sN
probability for the transition from sk to sk+1 with symbol yk
sNso s1 sN-1
yk… …sk
)(
11
),(
1
)(
111
111
)|,...,()|,(),...,,(),,(
kkkkkkk s
kNk
ss
kkk
s
kkkk syypsyspyyspssp
y
sk+1
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 14 -
The Transition Probability γ
We can further decompose the transition probability into
k(sk, sk+1) = P(sk+1|sk) ∙ p(yk|sk, sk+1)
Using the index pair (i j) and the set B we get
Bj) (i if 0
B j) (i if )|()(),( ,, jikkjikjik
vvypxxPrr
k(r0, r3) = 0 as (03) B
k(r0, r0) = P(xk=+1)∙p(yk|vk=1.63)
From the channel law yk=vk+nk and the Gaussian distribution we know that
2
2
)(
2)|(
2
2
kk vy
kk
evyp
Skip Probabilities
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 15 -
The Probability (Forward)
The term k(s) can be computed via the recursion
Ss
kkk ssss ),()()( 11
with the initial value 0(s) = P(s0=s).
so s1
1(s1) 1(s1,s2)
s2
2(s2)
ri rj
4
0112 ),()()(
ijiij rrrr
…
Note: k contains all possible paths leading to sk.
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 16 -
The Probability β (Backward)
Analogous, the term βk(s) can be computed via the recursion
Ss
kkk ssss ),()()( 1
with the initial value βN(s) = 1 for all s S.
sN
2(s2,s3)
yk
β3(s3)
ri rj …
β2(s2)
… s3s2
4
0232 ),()()(
ijijj rrrr
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 17 -
The Formula For The LLRNow, we know the APP P(xk = x|y). All we need to accomplish this task is to sum the branch APPs P(sk, sk+1|y) over all branches that correspond to an input symbol xk=x
xxBij
jkjikikxxBij
jkikk
jiji
rrrrrsrsPxxP,, :)(
1:)(
1 )(),()()|,()|( yy
To compute the APP P(xk=+1|y) the branch APPs of the indexpairs (00),(12),(20) and (32) have to be summed over
-1
1:)(1
1:)(1
,
,
)(),()(
)(),()(
ln)|1()|1(
ln)|1()|0(
ln)|(
ji
ji
xBijjkjikik
xBijjkjikik
k
k
k
kk rrrr
rrrr
xPxP
cPcP
cL
yy
yy
y
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 18 -
The FBA in Matrix FormFor convenience, the forward/backward algorithm may also be expressed in matrix-form. We need to create two matrices.
Pk |S|x|S| with {Pk}I,j = k(ri, rj)
A(x) |S|x|S| with
A third matrix is created by elementwise multiplication:
B(x) = A(x)∙Pk |S|x|S|
otherwise 0
x withbranch a is j) (i if 1)( ji,
,
xx jiA
0100
0001
0100
0001
)1(A
1000
0010
1000
0010
)1(A
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 19 -
The Algorithm
Input: Matrices Pk and Bk(x)
We calculate vectors fk |S|x1 and bk |S|x1
Initialize with f0 = 1 and bN = 1
For k = 1 to N step 1 (forward)
Output the LLRs: 1
1
)1()1(
ln)|(
kk
Tk
kkTk
kcL bBfbBf
y
fk = Pk-1fk-1
bk=PkTbk+1
For k = N to 1 step -1 (backward)
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 20 -
Soft Processing
Decoder
Deinterleaver
Demapper
Equalizer / Detector
yk
âk
kb
kc
kx
s(bk)
s(ck)
s(xk)
A natural choice for the soft information s(xk) are the APPs or similarly the LLRs L(ck|y), which are a “side product” of the maximum a-posteriori probability (MAP) symbol detector.
Also, the Viterbi equalizer may produce approximations of L(ck|y).
For filter-based equalizers extracting s(xk) is more difficult. A common approach is to assume that the estimation error is Gaussian distributed with PDF p(ek)…
kkk xxe ˆ
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 21 -
Decoding - Basics
• Convert the LLR L(ck|y) back to probabilities:
• Deinterleave P(ck|y) to P(bk|y)
• is the input set of probabilities to the decoder
•With the forward/backward algorithm we may again calculate the LLR L(ak|p)
1,01
)|( )|(
)|(
ce
eccP
k
k
cL
cLc
k y
y
y
)|(
)|(
)|(
2
1
y
y
y
p
NbP
bP
bP
For the example:Encoder of a convolutional code, where each incoming data bit ak yields two code bits (b2k-1, b2k) via
b2k-1 = ak ak-2
b2k = ak ak-1 ak-2
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 22 -
Decoding - Trellis
The convolutional code yields to a new trellis with branches denoted by the tuple (i, j, ai,j, b1,i,j, b2,i,j). Set B remains {(00),(01),(12),(13),(20),(21),(33),(32)}
state rjstate ri input ak=ai,joutput (b2k-1, b2k)=(b1,i,j, b2,i,j)
(0,0)
(1,0)
(0,1)
(1,1)
(0,0)
(1,0)
(0,1)
(1,1)
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 23 -
Decoding – Formulas (1)To apply the forward/backward algorithm, we have to adjust the way Pk and A(x) are formed. For Pk we have to redefine the transition probability
B j) (i if 0
B j) (i if)|()|()P(a),( ,,22,,112,k yy jikjikjijik
bbPbbParr
equalizer from
2k1-2k
IID of because 21
00k )|0P(b)|0P(b)0(),( g. e. yy
kaPrr .
{Pk}I,j = k(ri, rj)
otherwise 0
a withbranch a is j) (i if 1)( ji,
,
xx jiaA
Ba(x) = Aa(x)∙Pk
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 24 -
Decoding – Formulas (2)
So, we calculate L(ak|p) using the forward / backward algorithm. By changing A(x) we can also calculate L(b2k-1|p) and L(b2k|p) which will later serve as a priori information for the equalizer.
L(b2k-1|p):
otherwise 0
b withbranch a is j) (i if 1)( j1,i,
,1
xx jibA
otherwise 0
b withbranch a is j) (i if 1)( j2,i,
,2
xx jibAL(b2k|p):
with the set of probabilities:
)|(
)|(
)|(
2
1
y
y
y
p
NbP
bP
bP
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 25 -
Decoding - Example
otherwise 0
a withbranch a is j) (i if 1)( ji,
,
xx jiaA
L(b2k-1|p):
otherwise 0
b withbranch a is j) (i if 1)( j1,i,
,1
xx jibA
otherwise 0
b withbranch a is j) (i if 1)( j2,i,
,2
xx jibAL(b2k|p):
L(ak|p):
1000
0010
1000
0010
)1(aA
0100
0001
0100
0010
)1(1bA
0100
0001
1000
0010
)1(2bA
0100
0001
0100
0001
)0(aA
0100
0010
1000
0001
)0(1bA
1000
0010
0100
0001
)0(2bA
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 26 -
Decoding - AlgorithmFor decoding, we may use the same forward/backward algorithm with different initialization, as the encoder has to terminate at the zero state at time steps k=0, k=K. Change Bk(x) to output L(b2k-1|p) or L(b2k|p).
Input: Matrices Pk and Bk(x)
Initialize with f0 = [1 0…0]T |S|x1 and bN = [1 0…0]T |S|x1
For k = 1 to N step 1 (forward)
Output the LLRs: 1
1
)1()0(
ln)|(
kk
Tk
kkTk
k paLbBfbBf
fk = Pk-1fk-1
bk=PkTbk+1
For k = N to 1 step -1 (backward)
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 27 -
Bit Error Rate (BER)
Performance of separate equalization and decoding with hard estimates (dashed lines) or soft information (solid lines). The System transmits K=512 data bits and uses a 16-random interleaver to scramble N=1024 code bits.
With soft information, we may gain 2dB, but it is still a long way to -1.6 dB.
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
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Block Diagram - Separated Concept
EqualizerForward/BackwardAlgorithm
Observations
Prior Probabilities
a PosterioriProbabilities
L(ck|y)
y
Deinterleaver
DecoderForward/BackwardAlgorithm
L(bk|y)
Prior Prob.
Observationsa Posteriori
Probabilities
L(bk|p)
DecisionRule
L(ak|p)
âk
Let’s look again at the transition propability:
)()|(),( ,, jikjikkjik xxPvvyprr .
Prior informationlocal evidence aboutwhich branch in thetrellis was transversed.
So far:
•The equalizer does not have any prior knowledge available, so the formation of entries in Pk relies solely on the observation y.
•The decoder forms the corresponding entries in Pk without any local observations but entirely based on bitwise probabilities P(bk|y) provided by the equalizer.
Forward/BackwardAlgorithm
Observations a PosterioriProbabilities
Block diagram of the f/b algorithm
Prior Probabilities
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 29 -
Block Diagram - Turbo Equalization
EqualizerForward/BackwardAlgorithm
Observations
Prior Probabilities
a PosterioriProbabilities
Lext(ck|p)L(ck|y)
y
Deinterleaver
Interleaver
Extrinsic InformationLext(bk|p)
Extrinsic InformationLext(ck|y)
+
_
+_
DecoderForward/BackwardAlgorithm
Lext(bk|y)
Prior Prob.
Observationsa Posteriori
Probabilities
L(bk|p)
DecisionRule
L(ak|p)
âk
EqualizerForward/BackwardAlgorithm
Observations
Prior Probabilities
a PosterioriProbabilities
L(ck|y)
y
Deinterleaver
DecoderForward/BackwardAlgorithm
Prior Prob.
Observationsa Posteriori
Probabilities
L(bk|p)
DecisionRule
L(ak|p)
âk
Let’s look again at the transition propability:
)()|(),( ,, jikjikkjik xxPvvyprr .
Prior informationlocal evidence aboutwhich branch in thetrellis was transversed
So far:
•The equalizer does not have any prior knowledge available, so the formation of entries in Pk relies solely on the observation y
•The decoder forms the corresponding entries in Pk without any local oobservations but entirely based on bitwise probabilities P(bk|y) provided by the equalizer.
Turbo Equalization
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 30 -
Block Diagram - Comparison
EqualizerForward/BackwardAlgorithm
Observations
Prior Probabilities
a PosterioriProbabilities
L(ck|y)
y
Deinterleaver
DecoderForward/BackwardAlgorithm
L(bk|y)
Prior Prob.
Observationsa Posteriori
Probabilities
L(bk|p)
DecisionRule
L(ak|p)
âk
EqualizerForward/BackwardAlgorithm
Observations
Prior Probabilities
a PosterioriProbabilities
Lext(ck|p)L(ck|y)
y
Deinterleaver
Interleaver
Extrinsic InformationLext(bk|p)
Extrinsic InformationLext(ck|y)
+
_
+_
DecoderForward/BackwardAlgorithm
Lext(bk|y)
Prior Prob.
Observationsa Posteriori
Probabilities
L(bk|p)
DecisionRule
L(ak|p)
âk
EqualizerForward/BackwardAlgorithm
Observations
Prior Probabilities
a PosterioriProbabilities
L(ck|y)
y
Deinterleaver
DecoderForward/BackwardAlgorithm
Prior Prob.
Observationsa Posteriori
Probabilities
L(bk|p)
DecisionRule
L(ak|p)
âk
Receiver B:separated equalization
and detection
Receiver C:Turbo Equalization
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 31 -
Turbo Equalization - Calculation
EqualizerForward/BackwardAlgorithm
Observations
Prior Probabilities
a PosterioriProbabilities
Lext(ck|p)L(ck|y)
y
Deinterleaver
Interleaver
Extrinsic InformationLext(bk|p)
Extrinsic InformationLext(ck|y)
+
_
+_
DecoderForward/BackwardAlgorithm
Lext(bk|y)
Prior Prob.
Observationsa Posteriori
Probabilities
L(bk|p)
DecisionRule
L(ak|p)
âk
EqualizerForward/BackwardAlgorithm
Observations
Prior Probabilities
a PosterioriProbabilities
L(ck|y)
y
Deinterleaver
DecoderForward/BackwardAlgorithm
Prior Prob.
Observationsa Posteriori
Probabilities
L(bk|p)
DecisionRule
L(ak|p)
âk
Caution: We have to split L(ck|y)=Lext(ck|y) + L(ck)as only extrinsic information is fed back. Lext(ck|y) does not depend on L(ck). L(ck) would create direct positive feedback converging usually far from the globally optimal solution.
The interleavers are included into the iterative update loop to further disperse the direct feedback effect. The forward/backward algorithm creates locally highly correlated output. These correlations between neighboring symbols are largely suppressed by the interleaver.
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 32 -
Turbo Equalization - Algorithm
Input: Observation sequence y
Channel coefficients hl for l=0,1,…,L
Initialize:Predetermine the number of iterations T
Initialize the sequence of LLRs Lext(c|p) to 0
Output: Compute data bit estimates âk from L(ak|y)
L(c|y) = Forward/Backward(Lext(c|p))Lext(c|y) = L(c|y) – Lext(c|p)L(b|p) = Forward/Backward(Lext(b|y))Lext(b|p) = L(b|p) – Lext(b|y)
Compute recursively for T iterations
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 33 -
Turbo Equalization - BER
A
B C
The system transmits K=512 data bits and uses a 16-random interleaver to scramble N=1024 code bits. Figure A uses separate equalization and detection.
Figure B uses turbo MMSE Equalization with 0, 1, 2, 10 iterations. Figure C uses turbo MAP equalization after the same iterations. The line marked with “x” is the performance with K = 25000 and 40-random interleaving after 20 iterations.
Turbo MMSE Turbo MAP
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 34 -
Turbo Equalization – Exit Charts [2]
Receiver EXIT chart at 4 dB ES/N0 Receiver EXIT chart at 0.8 dB ES/N0
No MMSE
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 35 -
Linear Equalization
The computational effort is so far determined by the number of trellis states. An 8-ary alphabet gives 8L states in the trellis. Linear filter-based approaches perform only simple operations on the received symbols, which are usually applied sequentially to a subset of M observed symbols yk.e.g. yk=(yk-5 yk-4 … yk+5)T M=11
A channel of length L can be expressed as with M x (M+L).Any type of linear processing of yk to compute can be expressed as .
The channel law immediately suggests , the zero-forcing approach. With noise present, an estimate is obtained. This approach suffers from “noise enhancement”, which can be severe if is ill conditioned.
This effect can be avoided using linear minimum mean square error (MMSE) estimation minimizing . The equation used is
It is also possible to nonlinearly process previous estimates to find the besides the linear processing of yk (decision-feedback equalization (DFE).
kkk nxHy ~H~
kx kkTkkx byf ˆ
Hf ~Tk
kTkkk nfxx ˆ
H~
]|ˆ[| 2kkE xx
kx
uHHHIfyfx ~~~ˆ12
Hkk
Tk
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 36 -
Complexity [2]
Approach Real Multiplications Real Additions
MAP Equalizer 3 ∙ 2mM + 2m ∙ 2m(M-1) 3 ∙ 2mM + 2(m-1) ∙ 2m(M-
1)
exact MMSE LE16N2 + 4M2 + 10M – 4N
- 48N2 + 2M2 - 10N + 2M
+ 4
approx. MMSE LE (I) 4N + 8M 4N + 4M - 4
approx. MMSE LE (II)
10M 10M - 2
MMSE DFE16N2 + 4M2 + 10M – 4N
- 48N2 + 2M2 - 10N + 2M
+ 4M: Channel impulse response lengthN: Equalizer filter length2m: Alphabet length of the signal constellation
DFE: Decision Feedback Equalization
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 37 -
Comparison• The MMSE approaches have reduced complexity.
•The MMSE approaches perform as well as the BER-optimal MAP approach, only requiring a few more iterations.
•However, the MAP equalizer may handle SNR ranges where all other approaches fail.
Ideas
• treat scenarios with unknown channel characteristics, e. g. combined channel estimation and equalization using a-priori information
• Switch between MAP and MMSE algorithms depending on fed back soft information
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 38 -
Thank you for your attention!
Questions & Comments ?
JASS 05 St. Petersburg - Michael Meyer - Iterative Equalization
- 39 -
References
[1] Koetter, R.; Singer, A.; Tüchler, M.: Turbo EqualizationIEEE Signal Processing Magazine, vol. 21, no. 1, pp 67-80, Jan 2004
[2] Tüchler, M.; Koetter, R.; Singer, A.: Turbo Equalization: Principles and New ResultsIEEE Trans. Commun., vol. 50, pp. 754-767, May 2002
[3] Tüchler, M.; Singer, A.; Koetter, R.:Minimum Mean Squared Error Equalization Using A-priori InformationIEEE Trans. Signal Processing, vol. 50, pp. 673-683, March 2002