Transcript Introduction

Iterative Equalization and Decoding
John G. Proakis
[email protected]
COMSOC Distinguished Lecture Tour
1
Conventional Equalization
From Receiver
Equalizer
Soft Output
Filter
Decoder
Hard Output
Possible Equalizer Types:
•Linear Equalizer
•Decision Feedback Equalizer (DFE)
•Maximum A posteriori Probability (MAP) Equalizer
•Soft-output Viterbi (MLSE) Equalizer
Possible Decoder Types:
•Maximum A posteriori Probability (MAP) Decoder
•Viterbi (MLSE) Decoder
2
Turbo Principle / Turbo Coding
cs(n)
+
Turbo Encoder:
• Parallel concatenated
recursive systematic
convolutional encoders
P
• Encoders separated by an
interleaver
D
D
+
Encoder 1
+
D
d`(n)
D
+
Encoder 2
cp1(n)
Puncturer and P/S converter
d(n)
b(n)
cp2(n)
Turbo Decoder:
• Two Soft-Input Soft-Output
(SISO) decoders separated
by interleavers
L
• SISO modules can be
• SOVA
• MAP
• Extrinsic information
passed between modules
p1
x
s
x
SISO
Decoder 1
e
12
e
21
P
-1
L
P
p2
x
P
SISO
Decoder 2
Ld
3
Serially Concatenated Systems
Serially Concatenated Coding:
• Serial concatenated
(recursive) convolutional
encoders
d(n)
c(n)
Encoder 1
P
c’(n)
b(n)
Encoder 2
• Encoders separated by an
interleaver
Coded Transmission over
Multipath Channels:
• (recursive) convolutional
encoder
d(n)
c(n)
Encoder
P
c’(n)
Symbol
Mapper
• Interleaved bits mapped to
symbols
x(n)
• Symbols passed through a
multipath channel
Multipath
Channel
r(n)
4
Channel Model / Precoding
Multipath Channel Model:
• Received signal:
L
r (n)   hl (n) x(n - l )  w(n)
x(n)
h0(n)
X
D
D
h1(n)
h2(n)
X
D
hL-1(n)
X
X
l 0
•Rate 1/1 convolutional code
+
w(n)
Precoded System:
• Iteration gain only possible with
recursive inner code (channel) [1], [2]
• Recursive rate 1/1 precoder is
employed before transmission
• Most common precoder: Differential
encoder
r(n)
+
y(n-1)
x(n)
h0(n)

y(n)
X
D
D
h1(n)
h2(n)
X
D
X
hL-1(n)
X
+
w(n)
+
r(n)
5
Iterative Equalization and Decoding (Turbo Equalizer)
• Data bits are convolutionally encoded and interleaved
• M-ary PSK modulated signals transmitted through a multipath channel, which is
treated as an encoder
dn
Convolutional
Encoder
cn
c’n
P
xn
Symbol
Mapper
Multipath
Channel
rn
• Received signals are jointly equalized and decoded in a turbo structure
• First proposed by Douillard, et.al [3], where SOVA modules are employed
• Bauch extended the idea by employing MAP modules [4]
D
D
L e(c’)
Channel
Estimator
P
-1
D
L e(c)
L (c)
+
-
r
MAP
Equalizer
E
L (c’) -
E
+
L e(c’)
E
P
L e(c)
MAP
Decoder
D
L (d)
6
Time-invariant Test Channel
• Proakis C channel [5]
0.688
0.460
0.460
0.227
0.227
t
Impulse Response
Frequency Response
7
Low Complexity Alternative Equalizers: DFE and MLSE
• Performance of DFE and MLSE over the Proakis C channel [5]
Bit error rate performance [5]
8
Iterative Equalization and Decoding Performance
• Iterative equalization and decoding with MAP modules
• Recursive systematic convolutional encoder with R=1/2, K=5,
• Time-invariant 5 tap channel with a spectral null (Proakis C [5])
• Equalizer has perfect knowledge of the channel
• Block length 4096
Bit error rate performance of turbo equalizer [4]
9
Hard Iterative DFE and MAP Decoding
Input from
receiver filter
Forward
Filter
+
Iˆk
P
-1
MAP
Decoder
Output
Data
Symbol
Detector
DFE with
hard input
feedback
Feedback
Filter
~
Ik
P
Hard encoded
symbols
• During the first pass, symbol detector output is passed to the feedback filter
• After the first pass, hard encoded symbol output of the decoder is used in the feedback filter
10
Performance of Hard Iterative DFE and MAP decoder
0
Performance of Hard Iterative DFE(K1=K2=4) in Coded BPSK System
10
First iteration K=7
Second iteration K=7
Third iteration K=7
Fourth iteration K=7
-1
10
-2
P
BER
10
-3
10
-4
10
-5
10
-6
10
12
12.5
13
13.5
14
Eb/N0 (dB)
14.5
15
15.5
• BPSK modulation
• R=1/2, K=7 convolutional coding
• Block length 2048
• Channel Proakis C
11
Soft Iterative DFE and MAP Decoding
Forward
Filter
+
Iˆk
P
Feedback
Filter
DFE with
hard input
feedback
Decision
Device
-1
P
MAP
Decoder
Output
Data
Soft encoded
symbols
Soft decisions of the decoder is combined with the soft outputs of the DFE:
Iˆk

2
L( Iˆk )
2
+
Hard detected
symbols
Soft APP
from last
iteration
12
Histogram of DFE Output
First Iteration, SNR=12, K=3
First Iteration, SNR=20, K=3
700
2000
D istributation of Equalizer Output Values
Distributation of Equalizer Output Values
1800
600
500
400
300
200
100
1600
1400
1200
1000
800
600
400
200
0
-4
-3
-2
-1
0
1
2
3
4
Equalizer Output values
The histogram of equalizer
estimated output for SNR = 12 dB
0
-5
0
5
Equalizer Output values
The histogram of equalizer
estimated output for SNR = 20 dB
13
Modified Soft Iterative DFE and MAP Decoding
Forward
Filter
+
Iˆk
Conversion
to LLR
P
MAP
Decoder
-1
Output
Data
DFE with
hard input
feedback
Feedback
Filter
Decision
Device
P
+
Soft encoded
symbols
Only extrinsic information is passed to the DFE from the decoder
Variance Estimator
Iˆk

Re
2
2 2
+
Im

Variance Estimator
+
Hard detected
symbols
2
2 2
Soft APP
from last
iteration
14
Performance of Soft Iterative DFE and MAP Decoder
• Recursive systematic convolutional encoder with R=1/2, K=5,
• Time-invariant 5 tap channel with a spectral null (Proakis C [5])
• RLS updates at the DFE
• Block length 4096
• BPSK modulation
15
Performance of Soft Iterative DFE and MAP Decoder
• Recursive systematic convolutional encoder with R=1/2, K=5,
• Time-invariant 5 tap channel with a spectral null (Proakis C [5])
• RLS updates at the DFE
• Block length 4096
• QPSK modulation
16
Iterative Linear MMSE Equalization and Decoding
SISO Linear MMSE Equalizer [6]:
• Known channel:
• Received signal:
hk , k  - M 1 ,- M 1  1,, M 2
zn 
M2
 hk xn-k   k
k  - M1
• Likelihood ratio for MMSE estimator output, xˆ n :
p xˆ n | xn  1
Prxn  1
LMMSE ( xn )  ln
 ln
p xˆ n | xn  -1
Prxn  -1

 


LMMSE
( xn )
e
L ( xn )
0  0 
h-M1 h-M1 1  hM 2
 0
h-M1 h-M1 1  hM 2  0 

• Channel matrix: H  





0

0
h
h
h

M
M

1
M

1
1
2 
• MMSE estimator output:
where,
xˆn  c nH z n - m nz   mnx
c n   2 I N  H cov(x n , x n )H H  cov(x n , xn )
-1
m nz  Ez n   HEx n 
1

mnx  Exn   tanh  L( xn ) 
2

17
Iterative Linear MMSE Equalization and Decoding
Steps to compute symbol estimates with the Linear MMSE equalizer:
1

mnx  tanh  L( xn ) 
2


m nx  mnx M1  N1 mnx M1 N1 -1  mnx-M 2 - N2

T
D n  diag 1 - (mnx M1  N1 ) 2 1 - (mnx M1  N1 -1 ) 2  1 - (mnx-M 2 - N2 ) 2 

s  H 01x ( N1  M1 ) 1 01x ( N2  M 2 )

T
c n   2 I N  HD n H H  (mnx ) 2 ssH  s
-1
xˆn  c nH z n - Hm nx  mnx s 
Soft output calculation assuming Gaussian distributed estimates:
 n( 1)  Exˆ n | xn  1  c nH s
 n( -1)  Exˆ n | xn  -1  -c nH s
 n2  c nH s -  n( 1)
MMSE
e
L
( xn ) 
2
2 xˆ n  n( 1)
 n2

2 xˆ n
1 - s H cn
18
Performance of SISO MMSE Linear Iterative Equalizer
• Recursive systematic convolutional encoder with R=1/2, K=5,
• Time-invariant 5 tap channel with a spectral null (Proakis C [5])
• Equalizer has perfect knowledge of the channel
• Block length 4096
19
Comparison of System Performances
• Recursive systematic convolutional encoder with R=1/2, K=5,
• Time-invariant 5 tap channel with a spectral null (Proakis C [5])
• BER results after 6 iterations
• Block length 4096
20
Experimental Study of Iterative Equalizers
Channel
Probe
Dead
Time
Training
Symbols
Information
Symbols
21
Joint MMSE Equalization and Turbo Decoding
P
-1
e
12
L
P
MAP
Decoder 2
Ld
e
12
L
p1
x
s
x
MAP
Decoder 1
p2
x
Lp2
Demapper
&
S/P
Converter
Lp1
P
 (n)
P
2
Adaptive
Algorithm
^
x(n)
e(n)
P
-1
+
-jq(n)
e
y(n)
Forward
Filter
x
+
~
x(n)
Feedback
Filter
x(n)
Decision
Device
Training
Symbols
22
Decision Feedback Equalizer (DFE)
• Soft output of the DFE:
xˆ(n)  wfH (n)y(n) - wbH (n)~
x(n)
•RLS algorithm is used to track channel variation:
k ( n) 
P(n - 1)y (n)
λ  y H (n)P(n - 1)y (n)
P(n)  λ -1 (P(n - 1) - k (n)y H (n)P(n - 1))
ξ (n)  x(n) - (w fH (n - 1)y (n) - w bH (n - 1)~
x (n))
w f (n)  w f (n - 1)  P(n)y (n)ξ  (n)
w (n)  w (n - 1)  P(n)~
x ( n)ξ  ( n )
b
b
• Noise variance estimate:
2
~
σ (n)  λσ (n - 1)  (1 - λ) x(n) - (w (n)y(n) - w (n) x (n))
2
2
H
f
H
b
23
MAP Decoding
• Maximize a posteriori probability:
P(d n  d i | y ), i  0,1,, k - 1
• Decision variable written in the form of log-likelihood ratio:
 P(d n  1 | y ) 

(dn)  log 
 P(d n  -1 | y ) 
p ( sn -1  s ',sn  s ,y ) / p ( y )



S
 log 
 S - p ( sn-1s ',sn s ,y ) / p ( y ) 
24
MAP Decoding (BCJR Algorithm)
• State transition probability:
 n (s, s)  P(dn ) p( yn | xn )
where
y n  ( y ns , y np )
xn  ( xns , xnp )
  Le (d n ) yns  
 ynp xnp 
 n ( s, s)  exp d n 
 2   exp 2 
σ 
 σ 
  2
e




α
(
s
)
γ

2

n -1
n ( s , s )β n ( s )
e
S


Λ(d n )  2 yn  L (d n )  log
e
  - α n -1 ( s) γ n ( s, s)β n ( s) 
σ
 S

channel
value
a priori
information
extrinsic
information
25
MAP Equalizer
• γen (s, s) depends on the channel trellis defined by hl(n)
with 2(L-1) states
L -1
 1

e
l
γ n ( s, s)  exp - 2 yn -  h (n) xn-l 
l 0
 2σ

•If xn-l for J<l<L-1 is known
J -1
L -1

1
γ en ( s, s)  exp - 2 yn -  hl (n) xn -l -  hl (n) xˆn-l
l 0
lJ
 2σ



Number of states is reduced to 2(J-1)
26
Per-Survivor Processing
Survivor Paths
Discarded Paths
Yn0 , n0
0
00
0
0
0
01
0
Yn1 , n1
1
1
1
1
1
0
10
0
0
1
0
1
11
1
1
n-4
n-3
n-2
Survivor path:
S
Yn0  [1,1,0,0]
Yn2  [1,1,0,1]
1
n-1
   α S
Path metric: n m  P Yn m 
S
Yn2 , n2
0
n
Sm
|
Y
m
n
n
Yn3 , n3

n
27
Channel Estimator
xˆ (n)
hl (n)
y(n)
- +
e(n)
Adaptive
Algorithm
• Each survivor path has a separate channel estimator
• The input to the channel estimator, xˆ (n) , is the estimates
within the survivors
• RLS algorithm is employed
28
Channel Estimator
• Initial channel estimate is based on the correlation of the
preamble
• RLS algorithm is employed to track the channel
P(n - 1)xˆ (n)
k ( n) 
λ  xˆ H (n)P(n - 1)xˆ (n)
P(n)  λ -1 (P(n - 1) - k (n)xˆ H (n)P(n - 1))
ξ (n)  y (n) - (w H (n - 1)xˆ (n))
w (n)  w (n - 1)  P(n)xˆ (n)ξ  (n)
• Noise variance estimate:
σ 2 (n)  λσ 2 ( n - 1)  (1 - λ) ξ ( n)
2
29
Experimental Results
Channel impulse response
estimate for transducer seven
obtained using the channel probe
DFE results for transducer 7.
Eye Pattern - Filter coefficients
PLL phase estimate - Bit error distribution
30
Experimental Results
Channel impulse response
estimate for transducer seven
obtained using adaptive channel
estimator
Comparison of received signal with the
estimated received signal based on the
channel estimate
31
Experimental Results
• Sparse Channel with multipath delay in the order of 200 symbols
• Length of the DFE or channel estimator filters cannot cover the channel
• Sparse processing is needed
32
Results of DFE Turbo Decoder
Event Modulation DFE Iter
1
0
QPSK
334 25
BPSK
70
0
1
QPSK
488 80
BPSK
22
0
2
QPSK
392 43
BPSK
64
4
3
QPSK
374 27
BPSK
51
0
4
QPSK
290 11
BPSK
38
0
5
QPSK
121 2
BPSK
20
0
6
QPSK
165 1
BPSK
7
0
7
QPSK
247 6
BPSK
1
0
8
QPSK
207 7
BPSK
17
1
Iter Iter Iter Iter
2
3
4
5
4
3
3
3
56
49
47
49
18
2
17
15
2
12
15
15
11
11
7
7
7
7
2
2
2
2
2
2
2
2
0
1
0
33
Results of Iterative DFE Turbo Decoder
Event
0
1
2
3
4
5
6
7
8
Mod.
QPSK
QPSK
QPSK
QPSK
QPSK
QPSK
QPSK
QPSK
QPSK
DFE
334
488
392
374
290
121
165
247
207
Iter1
25
80
43
27
11
2
1
6
7
Iter2
4
56
18
17
7
0
2
2
1
DFE
140
203
186
165
139
Iter1
1
16
7
5
5
Iter2
2
9
7
6
5
DFE
133
179
177
148
131
Iter1
1
8
6
2
3
Iter2
1
5
6
0
3
73
104
89
1
0
0
1
75
2
0
DFE
138
171
173
Iter1
1
7
6
Iter2
1
4
6
DFE
139
172
173
Iter1
1
5
6
Iter2
1
4
6
129
3
3
129
3
3
34
Results of Iterative DFE MAP Decoder
Event
Mod.
0
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
1
2
3
4
5
6
7
8
DFE1 MAP
1
512
136
46
0
307
0
40
0
522
167
42
0
288
40
47
0
238
5
36
0
153
0
10
0
208
9
6
0
91
0
7
0
282
15
16
0
DFE2 MAP
2
268
49
DFE3 MAP
3
215
31
265
13
210
0
155
5
144
5
119
0
81
0
109
5
101
0
DFE4 MAP
4
212
31
146
5
DFE5 MAP
5
210
31
146
5
DFE6 MAP
6
210
31/16
146
5/2
35
Results for Iterative Map Equalizer Turbo Decoder
PK
#
MOD.
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
BPSK
QPSK
Equ
Map
1
21
447
7
682
8
427
14
392
16
313
0
34
2
81
0
52
4
130
Tur
Iter
1
0
73
0
241
0
64
0
59
0
14
Tur
Iter
2
Equ
Map
2
Tur
Iter
1
Tur
Iter
2
Equ
Map
3
Tur
Iter
1
50
80
3
3
8
0
222
444
154
115
197
38
45
76
1
3
6
0
35
62
2
1
2
0
6
10
0
1
0
1
0
1
3
0
2
6
1
1
4
1
0
0
2
Tur
Iter
2
Equ
Map
4
Tur
Iter
1
Tur
Iter
2
Equ
Map
5
Tur
Iter
1
Tur
Iter
2
25
33
4
1
3
3
0
1
4
1
1
4
1
1
36
Results for Iterative MAP Equalizer MAP Decoder
PK
#
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
MOD. Equ
MAP 1
BPSK 13
QPSK 600
BPSK 10
QPSK 407
BPSK 7
QPSK 643
BPSK 20
QPSK 294
BPSK 18
QPSK 327
BPSK 1
QPSK 85
BPSK 8
QPSK 113
BPSK 2
QPSK 77
BPSK 2
QPSK 277
Dec
MAP 1
0
333
0
88
0
415
0
9
0
34
0
0
0
0
0
0
0
14
Equ
Dec
Equ
Dec
MAP 2 MAP 2 MAP 3 MAP 3
212
38
33
0
313
95
6
0
19
0
1
0
29
0
32
0
37
Conclusions
• Due to error propagation in the DFE, turbo decoder cannot provide
performance improvement beyond the second iteration  Error Floor
• Joint DFE and turbo decoding adds an additional loop to the system
and lowers the error floor
• Joint channel estimator and iterative equalizer is able to decode
packets with low SNR, which cannot be decoded with the DFE
• Tail cancellation is an effective way to reduce the computational
complexity of the MAP equalizer
• If the channel is sparse, although the DFE filter lengths are short, the
DFE is able to provide enough information to the turbo decoder
• A sparse DFE can be used to improve the performance of the
DFE/MAP Decoder and the DFE/Turbo Decoder
38
References
[1] S. Benedetto, et.al., “Serial concatenation of interleaved codes: Design and performance
analysis,” IEEE Trans. Info. Theory, vol. 42, pp. 409-429, April 1998
[2] I. Lee, “The effect of a precoder on serially concatenated coding systems with ISI channel,” IEEE
Trans. Commun., pp. 1168-1175, July 2001
[3] C. Douilard, et.al., “Iterative correction of intersymbol interference: Turbo-equalization,” European
Transactions on Telecommunications, vol. 6, pp. 507-511, Sep.-Oct. 1995
[4] G. Bauch, H. Khorram, and J. Hagenauer, “Iterative equalization and decoding in mobile
communications systems,” in Proc. European Personal Mobile Commun. Conf., pp. 307-312
[5] J. Proakis, Digital Communications, McGraw-Hill Inc., 2001
[6] M. Tuchler, A. Singer, and R. Koetter, “Minimum mean squared error equalization using a priori
information,” IEEE Trans. Signal Proc., vol. 50, pp. 673-683, March 2002
39