advanced modulation/synchronization/coding schemes
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Erasmus Summer School 20091
Advanced
Modulation/Synchronization/Coding
Schemes
Parameter Estimation and Synchronization in Digital Receivers
Dipl.Dipl.--IngIng. Dr. . Dr. techntechn. . WilfriedWilfried GappmairGappmair
Institute of Communication Networks and Satellite Communications
Graz University of Technology, Austria
2
Contents
� Introduction– Motivation and background
� Signal model– Modulation schemes
– Distortions
– Transmission parameters
– Pulse shaping and matched filter
� Synchronization in digital receivers– What does synchronization mean
– Basic receiver architecture
– Maximum-likelihood principle
– Estimation criteria
� Feedforward algorithms– Carrier phase estimation
– Carrier frequency estimation
– Symbol timing estimation
� Feedback algorithms– Digital recovery loops
– Carrier phase recovery
– Symbol timing recovery
� Research areas
3
Introduction
� Turbo codes
– Berrou et al. (ICC’93)
– Shannon bound (1948)
– Computational complexity
� Demodulator
– Synchronization with PLL as standard solution
o Continuous vs. burst mode
o Impact of phase noise (small/large bandwidths)
– Challenging operational conditions: very low SNR values, burst mode
– Major parameters to be estimated: carrier frequency/phase, symbol timing
4
Introduction
Basic receiver architecture
NCO
π/2
Decimation + MF
?
I
Q
Antialias filter IF ADC
Decimation + MF
FEC
5
Introduction
Performance of different FEC schemes (BPSK/AWGN)
0 2 4 6 8 10
1.×10−6
0.00001
0.0001
0.001
0.01
0.1
1
BPSK uncoded
ESA/NASA (Rc = ½)
Galileo Code (Rc = ¼)
Turbo Code (Rc = ¼)
Eb/N0 [dB]
BER
6
Signal model
� Modulation schemes
– PSK (spectral efficiency)
– QAM (nonlinear effects: TWT)
– APSK (DVB-S2 standard)
– CPM (MSK, GMSK)
� Distortions
– Additive white Gaussian noise (AWGN)
� Transmission parameters
– Carrier frequency offset: ∆f
– Carrier phase: θ
– Symbol period: T
– Symbol timing: τ
7
Signal model
R2
R1
π/4
π/6
Symbol constellation for 16-APSK
8
� Pulse shaping
– Nyquist criteria (no ISI)
– Root-raised cosines (RRCos): h(t)
– Matched filter: h*(–t)
– Roll-off factor: α
� MF input signal
– Parameter vector: v = (∆f, θ, τ)
– Additive white Gaussian noise (AWGN): w(t)
Signal model
)()()(),(),( )2( twkTthcetwtstrk
k
ftj +
−−=+= ∑+∆ τθπ
νν
9
Signal model
Impulse response h(t) of different RRCos filters
-3 -2 -1 0 1 2 3 4
0
0.25
0.5
0.75
1
1.25
1.5
t/T
α = 0.25 α = 0.50 α = 1.0
10
Signal model
Signal r(t,v) at MF input (α = 0.25)
2 4 6 8 10
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
t/T
Es/N0 = 10 dB
Es/N0 = 20 dB
Es/N0 → ∞
11
Signal model
� MF output signal
– Convolution with MF impulse response → RCos
– Additive non-white Gaussian noise: n(t)
)()()(),(),( )2(* tnkTtgcethtrtxk
k
ftj +
−−=−⊗= ∑+∆ τθπ
νν
)()()(
)2(1
)cos()(sinc)()()(
*
2
*
thtwtn
Tt
TtTtththtg
−⊗=
−=−⊗=
α
απ
12
Signal model
Impulse response g(t) of different RCos filters
-3 -2 -1 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
t/T
α = 0.25 α = 0.50 α = 1.0
13
Signal model
Signal x(t,v) at MF output (α = 0.25)
2 4 6 8 10
-1.5
-1
-0.5
0
0.5
1
1.5
2
t/T
Es/N0 = 10 dB
Es/N0 = 20 dB
Es/N0 → ∞
14
Synchronization in digital receivers
� What does synchronization mean
– Synchronization in TDMA systems
– Synchronization as optimization problem
o Multi-dimensional
o Nonlinear
o Stochastic
– Object function
o Minimal Euclidean distance between received / replica signal
o Maximal correlation peak between received / replica signal
– Synchronization stages
o Acquisition (initialization)
o Tracking (controlling)
15
Synchronization in digital receivers
Euclidean distance
Parameter
global minimum
local minima
One-dimensional schematic of the synchronization process
16
Synchronization in digital receivers
� Basic receiver architecture– Demodulator
o IF converter + anti-alias filter
o AD conversion + mixer stage
o MF + decimation
– Synchronization
o Feedforward algorithms
� Used for acquisition (data-aided)
� Specified observation length
� No stability problems
o Feedback algorithms
� Used for tracking (decision-directed, non-data-aided)
� Probabilistic behavior of the settling time
� Stability problems (cycle slips, hang-ups)
17
Synchronization in digital receivers
NCO
π/2
Decimation + MF
Syncrhonization
(Acquisition + Tracking)
I
Q
Antialias-Filter IF ADC
Decimation + MF
Basic receiver architecture
18
Synchronization in digital receivers
� Maximum-likelihood principle (1)
– Replica signal with vector of trial parameters to be estimated
– Normal distribution of the probability function
– Observation length of L symbols
– Maximization procedure
)~(maxargˆ~
ννν
f=
−−= ∫
LT
w
dttstrC
Cf0
2
2
21 )~,(),(
2exp)~( ννν
σ
19
Synchronization in digital receivers
� Maximum-likelihood principle (2)
– Log-likelihood function
– Maximization of the correlation function
=Λ ∫
LT
dttstrf0
* )~,(),(Re~)~(ln)~( νννν
∑ −−= +∆
k
k
tfj kTthcets )~()~,( )~~
2( τθπν
−−=Λ ∑ ∫
+∆−
k
LT
tfj
k dtkTthtrec0
*)~~
2(* )~(),(Re)~( τθπνν
20
Synchronization in digital receivers
� Maximum-likelihood principle (3)
– Data-aided (DA)
– Decision-directed (DD)
– Non-data-aided (NDA)
=Λ
=Λ
=Λ
∑
∑
∑
k
k
k
kk
k
kk
yF
yc
yc
)]~([Re)~(
)~(ˆRe)~(
)~(Re)~(
NDA
*
DD
*
DA
νν
νν
νν
4444 34444 2143421
)~(:)~(
0
*
1«|~
|
)~~
2(
0
*)~~
2(
)~(),(
)~(),()~(
ττ
θπ
θπ
τ
τ
+=
∆
+∆−
+∆−
∫
∫
−−≈
−−=
kTxx
LT
Tf
Tfkj
LT
tfj
k
k
dtkTthtre
dtkTthtrey
ν
νν
21
Synchronization in digital receivers
� Estimator criteria
– Bias (offset)
– Jitter variance (CRLB)
– Consistency property
– Computational complexity
22
Feedforward algorithms
� Carrier phase estimation (1)
– Carrier phase constant (observation length: L symbols)
– No frequency error assumed
– Symbol timing established
– Modified CRLB
02
1MCRLB
NEL s
=
23
Feedforward algorithms
� Carrier phase estimation (2)
– Signal model
– Symbols ck known (DA)
– ML principle applicable
– DA unambiguity: |θ | < π
kk
j
k ncex += θ
=
= ∑∑
−
=
−1
0
*~
*
~ argRemaxargˆL
k
kk
k
j
kk xcexc θ
θθ
24
Feedforward algorithms
� Carrier phase estimation (3)
– Modulation scheme: MPSK
– Symbols ck unknown (NDA)
– Data modulation must be canceled
– Ad-hoc solution: power-law estimator
– NDA unambiguity: |θ | < π / M
= ∑
−
=
1
0
arg1ˆ
L
k
M
kxM
θ
25
Feedforward algorithms
θ
e-j(⋅)
1/M arg(⋅)
L
Phase shift
Power law
xk
NDA feedforward estimator for carrier phase
26
Feedforward algorithms
4 6 8 10 12 14 16
0.0001
0.0005
0.001
0.005
0.01
0.05
DA
NDA
MCRLB
Es/N0 [dB]
Jitter variance [rad2]
Jitter variance for QPSK (L = 32)
27
Feedforward algorithms
� Carrier frequency estimation (1)– Frequency offset due to Doppler shift and oscillator drifts
– Signal power affected at MF output
– Nyquist criterion violated: ISI problems
– Simplified signal model for analysis (|∆fT| « 1 )
– Symbol timing established
k
fTkj
kk necx += +∆ )2( θπ
28
Feedforward algorithms
∆fT
fT
1/2 1 -1/2 -1
signal power
Spectral conditions due to frequency errors
29
Feedforward algorithms
� Carrier frequency estimation (2)
– Maximum-likelihood (ML) solution
– Data-aided estimation
– Modified CRLB
=∆ ∑ +∆−
∆k
Tfkj
kkf
excf )~~
2(*
~,
~Remaxarg)ˆ,ˆ( θπ
θθ
0
22 )1(
1
2
3MCRLB
NELL s−=
π
30
Feedforward algorithms
� Carrier frequency estimation (3)
– ML too complex (searching)
– Application of DFT (FFT) algorithms: Rife-Boorstyn
– Autocorrelation principle: Luise-Reggiannini, Mengali-Morelli
– Main figures of merit
o Operational range
o Jitter variance
o Threshold effect
o Computational complexity
31
Feedforward algorithms
� Estimation of the symbol timing (1)
– Knowledge of carrier frequency/phase irrelevant
– Data symbols ck unknown (NDA)
– ML solution too complex → Oerder-Meyr algorithm
– Modified CRLB
0
2 )0(2
1MCRLB
NEgTL s&&−
=
222
312 8)31()0( ααπ −+=− gT &&
32
Feedback algorithms
� Tracking loops
– Carrier phase recovery
– Symbol timing recovery
– Main figures of merit
o Open-loop (detector) characteristic (S-curve)
o Jitter variance
� Linear range: ~ MCRLB
� Constant range: self-noise (ISI)
� Nonlinear range: threshold effect (DD, NDA)
– Linearization
o Loop filter: order of model
o Slope of S-curve in the stable equilibrium point: detector gain
33
Feedback algorithms
MCRLB
Es/N0 [dB]
log σ
v2
nonlinear
constant
linear
Jitter variance for feedback recovery loops
34
Feedback algorithms
Linearized tracker model
kv
Kd F(z) 1/(z – 1)
nk
vk
Running sum Loop filter Detector gain
∆vk
35
Feedback algorithms
� Digital recovery loops (1)
– First-order loops
o F(z) = KF
o Offset problems if input affected by a drift
o Loop gain K0 = KF Kd
o Transfer function H0(z)
0
01
1
)(
)()(
Kz
z
zv
zvzH
+−
−=
∆=
36
Feedback algorithms
� Digital recovery loops (2)
– Second-order loops
o F(z) = a + b / (z – 1)
o No offset problems if input affected by a drift
o Transfer function H0(z)
dd bKzaKz
zzH
+−+−
−=
)1()1(
)1()(
2
2
0
37
Feedback algorithms
� Digital recovery loops (3)
– Nonlinear effects: hang-ups, cycle slips
– One-sided equivalent noise bandwidth BLT
– Transfer function for loop noise: Hn(z) = 1 – H0(z)
– Jitter variance ~ BLT
∫=21
0
22 )(|)(| fTdeHTB fTj
nL
π
38
Feedback algorithms
� Carrier phase recovery (1)
– Modified CRLB: 2L→ 1 / BLT
– DA signal model
o Symbols ck known
o No carrier frequency offset
o Symbol timing established (no oversampling)
k
j
kk necx += θ
39
Feedback algorithms
� Carrier phase recovery (2)
– ML detector
o DA log-likelihood function
o Error signal uk
o Detector characteristic (S-curve)
=Λ ∑ −
k
j
kk exc θθ~
*
DA Re)~(
]Im[]Im[ *ˆ*
kk
j
kkk ycexcu == − θ
πθθθθ θ <=+=== ||,sin}])(Im[E{}0ˆ|E{)( *
DA k
j
kkk neccuS
40
Feedback algorithms
� Carrier phase recovery (3)
– DD detector: ck → decisions
– NDA detector: nonlinearity (power law)
– Unambiguity: |θ | < π / M
MSScc
ycu
kk
kkk
πθθθ <→→
=
||),()(:ˆ
]ˆIm[
DADD
*
MM
Mnec
MuS
yM
u
M
k
j
kk
M
kk
πθθθθ θ <=+===
=
||),sin(1
}])Im[(E{1
}0ˆ|E{)(
]Im[1
NDA
41
Feedback algorithms
e–j(⋅)
DD/NDA F(z)
kθ
xk
Σ
Decisions
uk
kc yk
Feedback loop for carrier phase recovery
42
Feedback algorithms
-3 -2 -1 1 2 3
-1
-0.5
0.5
1
DA
DD
NDA
θ [rad]
Detector characteristic for QPSK
43
Feedback algorithms
� Symbol timing recovery (1)
– Modified CRLB: 2L→ 1 / BLT
– Interpolation
o Asynchronous (equidistant) sampling
o Polynomial (Lagrange) interpolation
∑−
=+=
1
0
n
i
ikik xy λ
∏∏−
≠=
−
≠= −
−=
−
−=
1
0
1
0
ˆˆ n
ikk
n
ikk ki
ki
ki
k
tt
t ετλ
44
Feedback algorithms
xk
τ
λ3 λ2 λ1 λ0
z-1 z
-1 z
-1
yk
Cubic Lagrange interpolator (n = 4)
45
Feedback algorithms
� Symbol timing recovery (2)
– NDA detector
o ML solution too complex
o NDA and carrier-blind solution
o Gardner (GA) algorithm
o S-curve
])Re[( *
2/11 −−−= kkkk xxxu
)2sin()41(
)2sin()(
2πε
απ
παε
−=S
46
Feedback algorithms
kτ
KF GA Σ
Interpolator xk
uk
yk
Symbol timing recovery with first-order loops
47
Feedback algorithms
kτ
KF GA
e–j(⋅)
Σ
Interpolator
DD F(z)
kθ
xk
Σ
Decisions kc
Joint recovery of carrier phase and symbol timing
48
Research areas
� Satellite monitoring– Detection of modulation schemes
– Estimation of symbol period and roll-off factor
– SNR estimation
� Joint synchronization and decoding– Parameter estimation + data detection (FEC)
– Turbo synchronization
o Maximum-likelihood approach (complexity)
o Expectation-maximization solution (performance/complexity tradeoff )
o Per-survivor processing (low complexity)
� Fading channels– SNR estimation (ACM systems)
– Fading parameters
� PLL models– Nonlinear-stochastic differential equations
– Acquisition behavior
– Phase noise
49
References
1. Mengali, U., D’Andrea, A. N., Synchronization Techniques for Digital Receivers, Plenum Press:
New York, 1997.
2. Meyr, H., Moeneclaey, M., Fechtel, S. A., Digital Communication Receivers, Wiley: New York, 1998.
3. Kay, S. M., Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall: Upper Saddle River, NJ, 1993.
4. Proakis, J. G., Digital Communications, McGraw-Hill: New York, 1989.
5. Berrou, C., Glavieux, A., Thitimajshima, P., “Near Shannon Limit Error-Correcting Coding and Decoding: Turbo Codes”, in Proc. IEEE Int. Conf. Comm. ‘93, Geneva, Switzerland, pp. 1064–1070, May 1993.
6. Viterbi, A. J., Viterbi, A. M., “Nonlinear Estimation of PSK-Modulated Carrier Phase with Application to Burst Digital Transmission”, IEEE Trans. Inf. Theory, vol. 29, pp. 543–551, July 1983.
7. Morelli, M., Mengali, U., “Feedforward Frequency Estimation for PSK: a Tutorial Review”, Euro. Trans. Telecomm., vol. 9, pp. 103 – 116, March/April 1998.
8. Oerder, M., Meyr, H., “Digital Filter and Square Timing Recovery”, IEEE Trans. Comm., vol. 36, pp. 605–612, May 1988.
9. Gardner, F. M., “A BPSK/QPSK Timing-Error Detector for Sampled Receivers”, IEEE Trans.
Comm., vol. 34, pp. 423–429, May 1986.
10. Gardner, F. M., “Interpolation in Digital Modems – Part I: Fundamentals”, IEEE Trans. Comm., vol. 41, pp. 501–507, March 1993.
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