Japan Advanced Institute of Science and Technology
JAIST Repositoryhttps://dspace.jaist.ac.jp/
TitleTurbo Equalization: Fundamentals, Information
Theoretic Considerations, and Extensions
Author(s) Matsumoto, Tad; Anwar, Khoirul; Ahmad, Norulhusna
CitationA Tutorial on the 75th IEEE Vehicular Technology
Conference (VTC-Spring 2012)
Issue Date 2012-05-06
Type Presentation
Text version author
URL http://hdl.handle.net/10119/10523
Rights
Copyright © 2012 Authors. Tad Matsumoto, Khoirul
Anwar and Norulhusna Ahmad, Turbo Equalization:
Fundamentals, Information Theoretic
Considerations, and Extensions, A Tutorial on the
75th IEEE Vehicular Technology Conference (VTC-
Spring 2012), Tutorial Handout ; Place :
Yokohama, Japan ; Date : 6 - 9 May, 2012.
Description
IEEE Vehicular Technology Conference (電気電子学
会移動体技術国際学会)VTC 2012-SpringでのTutorial
Handout
Turbo Equalization: Fundamentals, InformationTheoretic Considerations, and Extensions
A Tutorial on IEEE VTC-Spring 2012
Tad Matsumoto, Khoirul Anwar and Norulhusna Ahmad
Information Theory and Signal Processing Lab., School of Information Science,Japan Advanced Institute of Science and Technology (JAIST),
1-1 Asahidai, Nomi, Ishikawa, 923-1211 JAPAN,E-mail: {matumoto,anwar-k}@jaist.ac.jp
Yokohama, 6 May 2012
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 1 / 40
Part IIChained Turbo Equalization (CHATUE) for Block
Transmission without Guard Interval- Application to Uplink SC-FDMA -
Khoirul Anwar
Japan Advanced Institute of Science and Technology (JAIST)e-mail: [email protected]
1-1 Asahidai, Nomi-shi, Ishikawa, 923-1292, JAPANhttp://www.jaist.ac.jp/is/labs/matsumoto-lab
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 2 / 40
Outline of Presentation
1 Motivations2 Basic Principle
1 System Model2 The Concept of CHATUE Algorithm3 Performance Evaluation
3 Applications1 Uplink SC-FDMA2 Mathematical Formulation3 Performance Evaluation
4 Conclusions
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 3 / 40
General Problem of Wireless Communications
+ noiseRx=
Tx=
+
TimeL
Symbol Duration T
1G
First path
Last Path
2G
3G
<<T No inter-symbol interference (ISI)
T only minor ISI >>T Severe ISI
“0” “1” “1” “0”
“0” “1” “1” “0”
“0” “1” “1” “0”
“0” “1” “1” “0”
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 4 / 40
Motivation
Conventional:
Block 1 Block 2 Block 3GI GI
Proposed
Length of desired
current block
Interference
from the futureInterference
from the past
Block 1 Block 2 Block 3
Saving the Time:
Advantage of CP removal
Normal Guard Interval (GI) (cyclic prefix): 4.69 µs ( Cover 1.4km )
LTE-Advanced SC-FDMA symbol length= 66.7 µs
Data rate loss 4.69/66.7=7.03%
GSM: 3.69 µs
W-CDMA : 0.69 µs
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 5 / 40
The Standard Technique
Fourier
Transform
TimeFrequencyL
Rx:
Tx:
H
Rx:
Tx:
H
CP CP
CP CP
CP
CP
CP
CP
Desired
K
Desired
K
Desired
K
Desired
K
Desired
K
Desired
K
CP: Cyclic Pre x
L L
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 6 / 40
The Benefit of Guard Interval Removal
With Guard Interval:
K KLL
Desired
K
Desired
K
Rate Loss: YESSN
= Eb
N0·R · K
(K+L) ·M
Power Loss: YESR = S
N· N0
Eb· (K+L)
K· 1M
Without Guard Interval:
K K
Desired
K
Desired
K
Rate Loss: NOSN
= Eb
N0·R · K
(K+0) ·M
Power Loss: NOR = S
N· N0
Eb· (K+0)
K· 1M
Notation:SN: Signal-to-Noise Power Ratio, R: Coding rate, Eb
N0: Energy bit per
noise, M : Number of bits per symbol, K: Block length, L: GI length
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 7 / 40
CHATUE Algorithm: The Basic Principle
Khoirul Anwar
E-mail: [email protected]
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 8 / 40
References (Suggested for Further Reading):
1 K. Anwar, H. Zhou, and T. Matsumoto, ”Chained Turbo Equalizationfor Block Transmission without Guard Interval”, 2010 IEEE 71stVehicular Technology Conference (VTC 2010-Spring), pp.1-5, May2010, Taiwan.
2 K. Anwar and T. Matsumoto, ”Low Complexity Time-ConcatenatedTurbo Equalization for Block Transmission without Guard Interval:Part 1– The Concept”, Wireless Pers. Commun., Springer, DOI:10.1007/s11277-012-0563-0 (Online: 24 March 2012).
3 K. Kansanen and T. Matsumoto, ”An Analytical Method for MMSEMIMO Turbo Equalizer EXIT Chart Computation”, IEEE Transaction
on Wireless Communications, Vol. 6, No. 1, pp. 59–63.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 9 / 40
System Models
C Mod
+
!"
-
D-ACC-1
P
D-ACC
BCJR
st = [s[0]t , s
[1]t , · · · , s
[k]t · · · s
[K−1]t ]T ∈ C
K×1. (1)
yt = Htst +H′t−1s′t−1 +H′′t+1s
′′t+1 + n ∈ C
(K+L−1)×1, (2)
Block 1 Block 2 Block 3
�me
Length of detected
Block 2
Interference
from the past
Interference
from the future
K KK
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 10 / 40
Channel Model
Ht=
h[0]0 0... h
[1]0
h[0]L−1
.... . .
h[1]L−1
... h[K−1]0
. . ....
0 h[K−1]L−1
∈ C(K+L−1)×K , (3)
H′t−1=
h[K−L+1]L−1 · · · h
[K−1]1
. . ....
h[K−1]L−1
0
, H′′t+1=
0
h[0]0...
. . .
h[0]L−2 · · · h
[L−2]0
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 11 / 40
Avoiding the Confusion on the Channel Models
time
Length of desired
current blockInterference
from the past block
Interference
from the future block
K KK
Ht−1 : A Past channel matrixH′t−1 : A Past channel matrix with the past formH′′t−1 : A Past channel matrix with the future formHt : A Current channel matrixH′t : A Current channel matrix with the past formH′′t : A Current channel matrix with the future form
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 12 / 40
Channel Model: Examples
Given the channel responses of the past, the current, and the futureblocks, respectively, as
ht−1 = [1, 0.7],ht = [1, 0.5],ht+1 = [1, 0.3],
and block length K = 4, write the channels of:
1 H′t−12 H′′t
3 H′′t+1
4 H′t
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 13 / 40
Solution by the CHATUE Algorithm
E1&D1 E2&D2 E3&D3
La,E1 Lp,D1=La,E2
La,E1=Lp,D2 La,E3
Lp,D2=La,E3
La,E2=Lp,D3
EQ’
EQ
EQ’’
C-1’
C-1
C-1’’y’’
y
y’
FD SC/MMSE DecoderBlock 1 Block 2 Block 3
me
Length of detected
Block 2
Interference
from the past
Interference
from the future
K KK
m
e
Chained Equaliza on
With GI Without GI
1 2
1 [1] K. Anwar, Z. Hui, and T. Matsumoto, ”Chained Turbo Equalization for Block Transmission without GuardInterval”, in IEEE VTC-Spring 2010, Taiwan, May 2010.2 [2] K. Anwar and T. Matsumoto, ”Low-complexity Time-concatenated Turbo Equalization for Block Transmission:Part 1 – The Concept”, Wireless Personal Comm., Springer, March 2012 (DOI 10.1007/s11277-012-0563-0).
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 14 / 40
Two Key Principles
1 Retrieval of Circularity: Matrix J
J =
0(K−L+1)×(L−1) IK×KI(L−1)×(L−1)
∈ CK×(K+L−1) (4)
2 ISI and IBI Removal: Modified FD/SC-MMSE
Example of Matrix J with L = 3,K = 3 and ht = [h0, h1, h2]
J =
0 0 1 0 01 0 0 1 00 1 0 0 1
, JHt =
h2 h1 h0h0 h2 h1h1 h0 h2
(5)
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 15 / 40
Modified SC-MMSE for CHATUE: ISI and IBI Removal
H Est.
wMMSE
LLRH’ Est.H’’ Est.
Channel
Decoder -1
tanh(*/2)tanh(*/2)
input-
+
+
-
La
Future
Block
Past
Block
So Es!ma!on
MMSE Filter
tanh(*/2)
FFT
La’La”
IFFTJ J J
FFT
La;D
Le;D
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 16 / 40
Soft Canceller MMSE for Chained Equalization
J
n
s0; s; s00
H0; H;H00
yr
ø N(0; û2)
Receive signal
yt = Htst +H′t−1s′t−1 +H′′t+1s
′′t+1 + n (6)
Restore the circularity of the channel
rt = Jy,
= JHtst+JH′t−1s
′t−1+JH
′′t+1s
′′t+1+Jn (7)
Soft Replica
rt = JHtst + JH′t−1s′t−1 + JH′′t+1s
′′t+1 (8)
Cancellation rt = rt − rt. To simplify the expression, subscribe notation t may be removed.)
Minimize the error:
w(k) = arg minw(k)H
|w(k)H [r + h(k)s(k)]− s(k)|2 (9)
with h(k) is the k-th column of matrix JH.(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 17 / 40
Modified SC-MMSE for CHATUE: Time Domain
The Wiener-Hopft Solution
E
[
∂|w(k)H [r + h(k)s(k)]− s(k)|2
∂w(k)H
]
= 0 (10)
MMSE Weight
w(k) =(
E[rrH + h(k)|s(k)|2h(k)H)−1
h(k),
=(
JHΛHHJH + JH′Λ′H′HJH + JH′′Λ′′H′′HJH
σ2JJH + h(k)|s(k)|2h(k)H)−1
h(k),
=(
Σ+ h(k)|s(k)|2h(k)H)−1
h(k) (11)
where
Λ = diag(
1− |s(0)|2, 1− |s(1)|2, · · · , 1− |s(K − 1)|2)
, (12)
Λ′ = diag(
0, · · · , 0, 1− |s(L− 1)|2, · · · , 1− |s(−1)|2)
, (13)
Λ′′ = diag(
1− |s(K)|2, · · · , 1− |s(K + L− 1)|2, 0, · · · , 0)
(14)
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 18 / 40
Output of CHATUE SC/MMSE
w(k)H = h(k)H[
Σ+ h(k)|s(k)|2h(k)H]−1
,
(a)=
(
1 + γ(k)|s(k)|2)−1
h(k)HΣ−1 (15)
Therefore, time domain final output:
z(k) =(
1 + γ(k)|s(k)|2)−1
h(k)HΣ−1 (r(k) + h(k)s(k)) (16)
By performing block-wise processing, symbol wise inversion is not required:
z = (IK + ΓS)−1(
Γs+HHJHΣr)
, (17)
S = diag[
|s|2]
, (18)
Γ = diag[
HHJH(
JHΛ(JH)H + JH′Λ′(JH′)H
+JH′′Λ′′(JH′′)H + σ2JJH)−1
JH]
(19)
Note: (a). γ(k) = h(k)HΣ−1h(k)(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 19 / 40
Output of Block-Wise Processing CHATUE
Σ = JHΛHHJH + JH′Λ′H′HJH + JH′′Λ′′H′′HJH , (20)
Γ = diag[
HHJHΣ−1JH]
, (21)
S = diag(|s|2) (22)
Lemma: JH = FHΦF→ HHJH = FHΦF
Σ = FHΦFΛFHΦHF+ JH′Λ′H′HJH + JH′′Λ′′H′′HJH , (23)
Γ = diag[
FHΦHFΣ−1FHΦF] (a)= diag
[
FHΦHX−1ΦF]
(24)
Finally, output of z:
z = [IK + Γs]−1[
Γs+HHJHΣ−1r]
,
(b)= [IK + Γs]−1
[
Γs+ FHΦHX−1Fr]
(25)
Note: (a). X = FΣFH , (b). X−1 still require simplification.(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 20 / 40
Approximation to Diagonal
IJJIJHJHIJHJHXH
K
HH
K
HH
K
H )tr()''''''tr()'''tr( 2111 σ+Λ+Λ+ΦΛΦ≈
HHHHHHHHHHFJJFFJHFJHFJHFJHFFX
2''''''''' σ+Λ+Λ+ΦΛΦ=
Index of P ast S ymbols
Ind
ex
of
Pa
st
Sym
bo
ls
10 20 30 40 50 60
10
20
30
40
50
60
Index of F uture S ymbols
Ind
ex
of
Fu
ture
Sym
bo
ls
10 20 30 40 50 60
10
20
30
40
50
60
Index of noise samples
Ind
ex
of
no
ise
sa
mp
les
10 20 30 40 50 60
10
20
30
40
50
60
Index of C urrent S ymbols
Ind
ex
of
Cu
rre
nt
Sym
bo
ls
10 20 30 40 50 60
10
20
30
40
50
60
Index of P ast S ymbols
Ind
ex
of
Pa
st
Sym
bo
ls
10 20 30 40 50 60
10
20
30
40
50
60
Index of F uture S ymbols
Ind
ex
of
Fu
ture
Sym
bo
ls
10 20 30 40 50 60
10
20
30
40
50
60
Index of noise samples
Ind
ex
of
no
ise
sa
mp
les
10 20 30 40 50 60
10
20
30
40
50
60
Index of C urrent S ymbols
Ind
ex
of
Cu
rre
nt
Sym
bo
ls
10 20 30 40 50 60
10
20
30
40
50
60
Note: The mutual information (MI) of the past, the current, and thefuture blocks are I ′a,Et
= Ia,Et= I ′′a,Et
= 0.5.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 21 / 40
Extrinsic LLR Formulation
zt can be expressed as being equivalent to a Gaussian channel output as,
zt = µtst + vt ∈ CK×1 (26)
µt = E[zt · s∗t ] =
1
Ktr{Γ(IK + ΓSt)
−1}, (27)
where vt is the equivalent noise vector with variance beingσ2t = µt(1− µt). In (27), we used the approximation,
St = diag{|st|2} ≈
1
K
K∑
k=1
|s[k]t |
2 · IK ∈ CK×K . (28)
Finally, the extrinsic LLR of the transmitted binary symbol is
Le,Et[s
[k]t ] = ln
Pr(z[k]t |s
[k]t = +1)
Pr(z[k]t |s
[k]t = −1)
=4ℜ(z
[k]t )
1− µt
, (29)
with z[k]t being the k-th component of zt and ℜ(z
[k]t ) denoting the real
part of the complex z[k]t .
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 22 / 40
EXIT Chart Analysis
Eb/N0 = 4 dB
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
A
B
64-path Rayleigh Fading
BPSK, FFT: 512, GI: 0
Interleaver: 512
Eb/N0=4dB, Without ACC
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
64-path Rayleigh Fading
BPSK, FFT: 512, GI: 0
Interleaver: 512
Eb/N0 = 4dB, With ACC P = 1:8
Note: EXIT analysis of the CHATUE Algorithm without and with dopedaccumulator (D-ACC) with P = 8.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 23 / 40
The Advantage of GI/CP Removal
0 1 2 3 4 5 6 7 8 9 1010
-5
10-4
10- 3
10-2
10-1
100
Eb/N0(dB)
Avera
ge B
ER
R=2/3,
CP:25%
R=1/2,
CHATUE
64-path Rayleigh Fading
BPSK, K=256
Interleaver: 256, CC-3[7,5]
Truncation Length = 1 block
Note: The total power (over all path) is∑L−1
ℓ=0 |hℓ|2 = 1. The block
length is kept by KCHATUE = KSCCP + L with advantage ofN
RCHATUE= NRSCCP
+ 1.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 24 / 40
BER Performances
1 2 3 4 5 6 7 8 9 1010
-5
10-4
10-3
10-2
10-1
100
Eb /N0[dB]
Avera
ge B
ER
64-path Rayleigh Fading
BPSK, K = 512, GI: 0
Interleaver: 512, CC-3[7,5]
Lower Bound:
Coded, AWGN,
K=512, GI=0
Truncation Length = 3 Blocks,
1 2 3 4 5 6 7 8 9 1010
-5
10-4
10-3
10-2
10-1
100
Eb/N0 (dB)
Avera
ge B
ER
64-path Rayleigh Fading
BPSK, K = 512, GI: 0
ACC P=8, Interleaver: 512, CC-3[7,5]
Truncation: 1 Block
Truncation Length = 3 Blocks,
Note: BER performances of the CHATUE Algorithm without and withdoped accumulator (D-ACC) with P = 8.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 25 / 40
How to Solve the Channel Variation?
hö`
H
hℓ =1K
∑K−1k=0 h
[k]ℓ ,
Ht =
h[0]0 0... h
[1]0
h[0]L−1
.... . .
h[1]L−1
... h[K−1]0
. . ....
0 h[K−1]L−1
t
≈
h0 0... h0
hL−1...
. . .
hL−1... h0. . .
...0 hL−1
t
= Ht ∈ C(K+L−1)×K
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 26 / 40
Performances Evaluation
10-5
10-4
10-3
10-2
10-1
100
Av
erage
BE
R
w/o ACC (Lower Bound)
w/o ACC (Truncation)
w/ ACC (Lower Bound)
w/ ACC (Truncation)
10.10.010.0010.00010.000010.000001
Eb/N0= 6 dB
Eb/N0= 4 dB
Truncation Length = 3 Blocks
Lower Bound:
fdTs
Block length, K = 512
Ia;E = I0
a;E= I
00
a;E= 1
Normalized Doppler Spread ( ) over single carrier symbol duration
In general, broadband communication (e.g. 4G) is sensitive toDoppler shift.
Highway speed: 100 km/h at 3.5 GHz (WiMAX)→ fdTs = 0.00162/512 = 0.0000031
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 27 / 40
CHATUE Algorithm: Application to
SC-FDMA Systems
Khoirul Anwar
E-mail: [email protected]
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 28 / 40
References (Suggested for Further Reading):
1 H. Zhou, K. Anwar, and T. Matsumoto, ”Chained Turbo Equalizationfor SC-FDMA Systems without Cyclic Prefix”, IEEE Globecom 2010
Workshop on Broadband Single Carrier and Frequency Domain
Communications, pp.1318-1322, Dec. 2010, USA.
2 H. Zhou, K. Anwar, and T. Matsumoto, ”Low ComplexityTime-Concatenated Turbo Equalization for Block Transmissionwithout Guard Interval: Part 2 – Application to SC-FDMA,” Wireless
Personal Communications, Springer, Sept. 2011 (DOI:10.1007/s11277-011-0409-1).
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 29 / 40
Application to (4G) Uplink SC-FDMA without GI
Ref.: H. Zhou, K. Anwar and T. Matsumoto, ”Low Complexity
Time-Concatenated Turbo Equalization for Block Transmission without Guard
Interval: Part 2. Application to SC-FDMA”, Wireless Personal Commun.,
Springer, DOI 10.1007/s11277-011-0409-1, Pub. online: 25 Sept. 2011.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 30 / 40
SC-FDMA: A Review
K-D
FT
Source S/PSubcarrier
Mapping P/S ChannelHi
M-ID
FT
i-th User
User 1
User 2
User i Base Sta�on
Subcarrier Alloca�on:
Localized Distributed: User 1
: User 2
: User 3
SM
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 31 / 40
SC-FDMA: System Model
Subcar.De-Mapping/
CHATUE SC-FDMA J
Channel
H
Past LLR
Future LLR
Sub
Map.Mod
+-
+
--
Sub.
Map.
User i
-
J
P
Doped Accumulator
n
The received composite signal is
rt =I
∑
i=1
ri,t + Jn, (30)
where
ri,t=JHi,tFHMDiFKsi,t+JH
′
i,t−1FHMDiFKs
′
i,t−1+JH′′
i,t+1FHMDiFKs
′′
i,t+1
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 32 / 40
Matrix D for Subcarrier Allocations: Example
Di is a M ×K matrix for the i-th user, by which, the κ-th sub-carriercomponent of the K-point Discrete Fourier Transform (DFT) is mappedto the m-th sub-carrier of the M -point DFT, where 0 ≤ κ ≤ K − 1, and0 ≤ m ≤M − 1.For localized sub-carrier mapping,
Di =
{
1, m = Ru ·K + κ
0, otherwise(31)
and for distributed sub-carrier mapping,
Di =
{
1, m = Ru + MK· κ
0, otherwise(32)
with Ru indicating the resource unit allocation, which is subjected to0 ≤ Ru ≤
MK− 1.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 33 / 40
Soft Cancellation in CHATUE-SC-FDMA
J
n Soft Cancellation rt = rt − rtSoft Replica
rt =I
∑
i=1
JHi,tFHMDiFK si,t +
I∑
i=1
JH′i,t−1FHMDiFK s
′i,t−1
+I
∑
i=1
JH′′i,t+1FHMDiFK s
′′i,t+1 (33)
Soft Symbol Estimates
si,t(k) = E[si,t(k)|Le,C−1i
] = tanh{Le,C−1
i[si,t(k)]/2}, (34)
s′i,t−1(k) = E[s′i,t−1(k)|L′p,C−1
i,t−1
] = tanh{L′p,C−1
i,t−1
[s′i,t−1(k)]/2}, (35)
s′′i,t+1(k) = E[s′′i,t+1(k)|L′′p,C−1
i,t+1
] = tanh{L′′p,C−1
i,t+1
[s′′i,t+1(k)]/2}. (36)
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 34 / 40
CHATUE-SC-FDMA Output
zi,t = (Ik + Γi,tSi,t)−1[Γi,tsi,t + FH
KΦHi,tFKΣ
−1i,t ri,t]
= (Ik + Γi,tSi,t)−1[Γi,tsi,t + FH
KΦHi,tX
−1FK ri,t] ∈ CK×1, (37)
where the Γi,t can be expressed as
Γi,t = diag[HHi,tΣi,t
−1Hi,t]
= diag[FHKΦ
Hi,tFKΣi,t
−1FHKΦi,tFK ]
= diag[FHKΦ
Hi,tX
−1Φi,tFK ] ∈ CK×K (38)
with X being the frequency domain covariance matrices given by
X = FKΣi,tFKH = Φi,tFKΛi,tF
HKΦ
Hi,t+FKσ2
iDTi JJ
HDiFHK
+FKH′i,t−1Λ
′i,t−1H
′Hi,t−1F
HK
+FKH′′i,t+1Λ
′′i,t+1H
′′Hi,t+1F
HK ∈ C
K×K (39)
and σ2i = K
Mσ2n for the i-th user.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 35 / 40
Approximation for Computational Complexity Reduction
Based on FΛFH ≈ 1Ktr[Λ]IK , we can perform:
X ≈ Φi,tΛi,tΦHi,t+ diag(FKσ2
iDTi JJ
HDiFHK + FKH
′
i,t−1Λ′
i,t−1H′Hi,t−1F
HK
+FKH′′
i,t+1Λ′′
i,t+1H′′Hi,t+1F
HK)
≈ Φi,tΛi,tΦHi,t+
1
Ktr
[
σ2iD
Ti JJ
HDi+H′
i,t−1Λ′
i,t−1H′Hi,t−1+H
′′
i,t+1Λ′′
i,t+1H′′Hi,t+1
]
IK
(40)Residual IBI from past and future NoiseICI
50 100 150 200 250
50
100
150
200
250
0
050 100 150 200 250
50
100
150
200
250
0
050 100 150 200 250
50
100
150
200
250
0
0
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 36 / 40
EXIT Chart Analysis for CHATUE-SC-FDMA
Eb/N0 = 5 dB
Q
P
IA-E, IE-D
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decoder (conv. constraint length 3, rate ½ code)
Equalizer, when MI from Past/Future = 0
Equalizer, when MI from Past/Future = 1
Trajectory
Block Length, M=512
Path Length, L=64
Eb=N0 = 5 dB
IE-E
, IA
-D
IA-E,IE-D
Decoder (R=3/5 (punctured) , const. length =3)
Equalizer, when MI from Past/Future = 0
Equalizer, when MI from Past/Future = 1
Trajectory
Q
P
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Block Length, M=512
Multi-path Length, L=64
Eb=N0 = 5 dB
IE-E
,IA
-D
Note: EXIT analysis of the CHATUE Algorithm without and with dopedaccumulator (D-ACC) of foping rate P = 8.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 37 / 40
BER Performances of CHATUE-SC-FDMA: User 1
Block Length, M = 512
CP Length = Multi-path L = 64
Conv. Code: Memory 2
G = [7, 5]8
0 1 2 3 4 5 6 7 8 9
Iter.1
AWGN (Coded)
Iter.5, Conv. CP.
Iter.2
Iter.5, CHATUE
Av
erag
e B
ER
Iter.0
10-5
10-4
10-3
10-2
10-1
100
Eb/N0
(dB) for user 1 Eb/N0 (dB) for user 1
Av
erag
e B
ER
Iter.1
MI past/future = 1
Iter.2
0 1 2 3 4 5 6 7 8 9 10
Block Length, M=512
CP Length = Multi-path L=64
Conv. Code: Memory 2,
G=[7, 5]8
Iter.5, Conv. CP-Trans.
Iter.5, CHATUE (DA)
10-5
10-4
10-3
10-2
10-1
100
Iter.0
Note: BER performances of the CHATUE Algorithm without and withdoped accumulator (D-ACC) with P = 8.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 38 / 40
CHATUE-SC-FDMA: Performance of Multiple Users
0 10 20 30 40 50 6010
-6
10-5
10-4
10-3
10-2
10-1
100
Number of Users
CHATUE-SC-FDMA (Orthogonal sub-carrier
allocation in one frame)
Over-Frame CHATUE-SC-FDMA
Block Length, M = 512
Multi-path L = 64
Conv. Code: Memory 2
Iteration = 5
G = [7, 5]8
Eb=N0 = 6 dB
Av
erag
e B
ER
/U
ser
It is found that the BER performance is not significantly affected, even when the
512 sub-carriers are shared by 64 users.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 39 / 40
Conclusions
GI causes power loss or total rate-loss with a factor of K/(K + L).
CHATUE Algorithm can excellently improve the performance oftransmission without GI with low computational complexity (it is alsoapplicable for system with insufficient GI).
Better performance (ICI, ISI, IBI Removal) can be achieved asdemonstrated by: BER performance & Trajectory Analysis of theEXIT chart
Further Advantages: (1) Lower Code Rate, (2) Turbo Cliff, (3)Multi-User Systems, (4) Uplink 4G SC-FDMA
Using the CHATUE Algorithm for SC-FDMA, the next generation 4Gsystems is possible without cyclic prefix/guard interval.
A comparison of CHATUE SC-FDMA with the conventionalSC-FDMA with GI/CP-Transmission verifies that the performance isalmost similar, but CHATUE SC-FDMA has a better spectralefficiency by eliminating the necessity of the GI/CP transmission.
(IEEE VTC-Spring 2012) Tutorial on Turbo Equalizations 40 / 40