synchronization techniques for burst-mode continuous phase ... · e. hosseini ph.d. defense:...
TRANSCRIPT
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 1/38
Synchronization Techniques for Burst-Mode Continuous Phase Modulation
Ph.D. Dissertation DefenseEhsan Hosseini
Department of Electrical Engineering & Computer Science University of Kansas
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 2/38
Outline
• Chapter 1: Introduction
• Chapter 2: The Cramér-Rao Bound for Training Sequence Design for Burst-Mode CPM
• Chapter 3: Timing, Carrier and Frame Synchronization for Burst-Mode CPM
• Chapter 4: Applications to SOQPSK
• Chapter 5: Conclusions
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 3/38
External Contributions
• Chapter 1: Introduction
• Chapter 2: The Cramér-Rao Bound for Training Sequence Design for Burst-Mode CPM
• Chapter 3: Timing, Carrier and Frame Synchronization for Burst-Mode CPM
• Chapter 4: Applications to SOQPSK
• Chapter 5: Conclusions
• E. Hosseini and E. Perrins, "The Cramèr-Rao Bound for Training Sequence Design for Burst-Mode CPM," IEEE Transactions on Communications, vol. 61, no. 6, pp. 2396-2407, June 2013.
• E. Hosseini and E. Perrins, "Training Sequence Design for Data-Aided Synchronization of Burst-Mode CPM," in Proceedings of the IEEE Global Communications Conference (GLOBECOM'12), 2012..
• E. Hosseini and E. Perrins, "Maximum Likelihood Synchronization of Burst-Mode CPM," to appear in Proceedings of the IEEE Global Communications Conference (GLOBECOM'13), December 9-13, 2013
• E. Hosseini and E. Perrins, “Timing, Carrier, and Frame Synchronization of Burst-Mode CPM”, IEEE Transactions on Communications, Under Review.
• E. Hosseini and E. Perrins, "The Cramér-Rao Bound for Data-Aided Synchronization of SOQPSK," in Proceedings of the IEEE Military Communications Conference (MILCOM'12), 2012.
• E. Hosseini and E. Perrins, "On Burst-Mode Synchronization of SOQPSK," to appear in Proceedings of the IEEE Military Communications Conference (MILCOM'13), 2013.
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 4/38
Motivation: Aeronautical Telemetry
• integrated Network Enhanced Telemetry (iNET) project Packet-based telemetry in
contrast to streaming telemetry. Benefits: spectrum conservation,
multi-hop, enhanced mobility,.. A major challenge in the RF link
is detection and synchronizationof packets, i.e. bursts
Physical layer waveform: SOQPSK, a subset of CPM
• In this work: 1. Synchronization for general
CPM in burst-mode TX2. Apply the results to SOQPSK,
i.e. iNET standard
GS1
GS2
TA1
TA2
TA3
GS: Ground StationTA: Test Article
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 5/38
What is CPM?
• Continuous phase modulation:
• Frequency pulse:•• Duration of LTs
• Properties:• Constant envelope • Bandwidth efficient• Drawback: complex at Rx
i
sis
s iTtqhTEts )(2exp);( α h: modulation index
q(t): phase response
M-ary data symbols:i
dttdqtg )()(
tLTs
q(t)
21
g(t)
L=1: full-responseL>1: partial-response
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 6/38
Signal Transmission Model
• Burst-Mode Transmission:
• Burst Structure: • Preamble (training sequence): sequence of L0 known symbols• Unique Word (UW): burst identification• Payload: information bits
• Received baseband signal:
• Noise model: AWGN
Preamble UW PayloadGuardInterval
GuardInterval
L0 LUW Lpay
)()()( )2( twtsetr tfj d fd : Frequency offsetθ : Carrier phaseτ : time delay
Synchronization Parameters(Unknown)
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 7/38
Synchronization Problem
• Challenge: Acquisition time must be kept minimum in burst-mode communications to save bandwidth, thus:1. Feedforward approach for shorter acquisition time2. Data-aided (DA) algorithms to assist synchronization
• Problems:1. Preamble design: What is the best preamble?2. Timing & carrier recovery: Estimation of , ,3. Frame synchronization: Detection of preamble boundaries
Known Preamble Payload
DA Estimator Synchronizer
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 8/38
Comparison with Related Works
CPM-Frequency-Phase-Symbol Timing
-CPM-Burst-Mode Transmission
CPM-Frequency -Phase-Symbol Timing
Tavares (2007) & Shaw (2010)-PSKDebora (2002)-MSK-Only Symbol Timing
Zhao (2006)-CPM-Phase & Timing (No Freq.)Morelli (2007)-PSK
Choi (2002)-PSK-Continuous Transmission
Best Preamble
Timing & Carrier Recovery
Frame Synchronization
This Work Related Work
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 9/38
The Cramér-Rao Bound for Training Sequence Designfor Burst-Mode CPM
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 10/38
Optimum Training Sequence Design:Cramér-Rao Bound Method• Two approaches can be taken in order to derive the optimum
(best) training sequence for estimation of1. Minimize the estimation error variance:
2. Minimize the CRB, i.e. a theoretical lower bound on estimation error variance of any unbiased estimator:
• CRB is computed via
ααααα
|CRBminarg|CRB|ˆ opt2
iiii uuuuE
αuruIuIα ,;ln where)|(CRB2
,1, p
uuEu
jijiiii
Conditional CRB Fisher Info. Mat. (FIM) Likelihood Function
T,, dfu
MSE
ααα
|ˆminarg 2iiopt uuE
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 11/38
Overview of CRB Calculations
• Closed-form FIM for CPM and joint estimation
• It is shown the optimum training sequence must satisfy these two conditions at the same time1. 0
2. Maximize C
ChhBhAhBTThATT
NE
Ts
s 2220
20
220
30
2
0 828222
823/81)(
uI
0
0
11
11
0'
0'
L
ii
L
ii
B
iA
dutugugtRaTRRC g
L
i
L
iiisgig )()()( where)(2)0(
1
0
2
01
20 0
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 12/38
Proposed Optimum Training Sequence
• Proposed sequence based on our analysis
• It satisfies A’=B’=0,1. Optimum sequence for full-response CPM.2. Only 2 transitions => Near-optimum sequence for partial-response CPM
• We can reduce the approximations for partial-response by using /2 rather than L0 to reduce the edge-effect (EF)
L1/4 L1/2
L0
(M-1)
-(M-1)
00
11
11 0'(2) 0')1(
L
ii
L
ii BiA
1
0
1
01
20 0
)(2)0( Maximize )3(L
i
L
iiisgig aTRRC =0 for full-response CPM
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 13/38
Findings and Properties of the Optimum Sequence1. The same sequence is optimum for frequency, phase, and
timing, which is opposed to linear modulations.2. It follows the same pattern for all CPMs.3. The minimized frequency and phase CRBs are independent of
underlying CPM parameters.4. The minimized symbol timing CRB do depend on CPM
parameters such as frequency pulse.5. It decouples symbol timing estimation from phase and frequency
offset estimates.
*220
20
20
30
2
0
*
800022023/8
1)(Ch
TTTT
NE
Ts
s
uI
timing:phase:frequency:
3
2
1
uuu
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 14/38
More Findings
6. The proposed sequence is asymptotically optimum for partial-response CPMs
7. Computer search confirms the proposed sequence at small L0
8. We also computed CRB for random sequences and compared with the optimum one: The optimum sequence delivers significant gain for symbol timing
estimation of partial-response and/or M-ary CPMs. The frequency and phase estimation are less sensitive to the
selection of training sequence especially for long sequences.
1
1208
1~)(8~
|CRB|~CRB
000
*
Lsgg
sgsg
TRLRLTR
CTRC
αα
sequence Proposed:sequence alHypothetic :~
*αα
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 15/38
Timing, Carrier, and Frame Synchronization of Burst-Mode CPM
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 16/38
Timing and Carrier Recovery Framework
• Received Signal:• Decompose timing delay:
• For timing and carrier recovery, we assume integer delay is known
• Joint maximum likelihood (ML) estimator:
• fd and ε are embedded inside the above integral → It requires a 2-dimensional grid search
• Objective: decouple timing and frequency in the LLF
)()()( )2( twtsetr tfj d
ss TT
delay fractional :0.50.5-delayinteger :0
where
s
s
d
d
TT
T stfj
df
d dtTtstreftrf
0 )()(Re,,);(maxargˆ,ˆ,ˆ *2
,,
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 17/38
The Trick: Piecewise Linear Approximation
• Similar and consequent symbols cause the phase to increase/decreaseuniformly
• We use linear approximation of the phase in each segment• There is a lag for partial-response CPMs→ Tl
1
-1
0 4 8 12 16
0
t/Ts
Unw
rapp
ed P
hase
GMSK (BTs=0.3, M=2,h=1/2)
-2
2Original phaseApprox. phase
• Phase response of the optimum sequence
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 18/38
ν: normalized freq. offsetε: fractional delay
:baseband signal shifted by ε
Approximated Log-Likelihood Function (LLF)
• Discrete-time LLF:
• Approximated baseband signal:
ll
ll
ll
sl
sl
sl
TTtTTTTtTT
TTtT
TTTthMTTTthM
TTthMt
00
00
0
0
0*
4/34/34/
4/
/)()1(/)2/()1(
/)()1(,
α
1
0
**0
2),(L
isi iTtqht α
00
00
0
0
0
4/34/34/
4/
/1exp2//1exp
/1exp
NLnNLNLnNL
NLnn
LNnhMjLNnhMj
NnhMjns
]Re,,;1
0
*20
NL
n
nj nsnre r
ns
Linear phase approximation
Original phase
will be used in the above LLF
Time shift + sampling + modulation
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 19/38
Maximization of the LLF
• Our approximation decouples timing and frequency offsets:
• The maximization of the LLF becomes straightforward: Step 1: Frequency offset estimation:
Step 2: symbol timing estimation:
Step 3: carrier phase estimation:
2
11
1* Re),,;( hMjhMjj eee r
~)~( maxargˆ 21~
hM
)1(2ˆ)ˆ( argˆ
*21
)}ˆ()ˆ(arg{ˆ2
ˆ)1(1
ˆ)1( hMjhMj ee
Grid search
Functions of r
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 20/38
Implementation of Frequency Estimator
• The received r[n] is pre-processed
• λ1(ν) and λ2(ν) are computed using FFT operations:
• The precision can be increased via:1. Zero padding prior the FFTs → Kf
2. Interpolation of FFT results
ν
True maximumFFT
otherwise4/3
4/0
0][])1(exp[
][][ 00
0
01 NLnNLNLn
nrhLMjnr
nr
otherwise
4/34/0
][]2/)1(exp[][ 000
2
NLnNLnrhLMjnr
1
0
2/)1(11
0
][)(NL
n
njNhnMj eenr
1
0
2/)1(22
0
][)(NL
n
njNhnMj eenr
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 21/38
Joint ML Estimator Block Diagram
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 22/38
Estimator’s Performance:Frequency and Phase Offsets
0 1 2 3 4 5 6 7 8 9 1010-8
10-7
10-6
Es/N0 (dB)
Nor
mal
ized
Fre
quen
cy E
rror V
aria
nce
MSK1RC (M=2,h=1/2)GMSK2RC (M=4,h=1/4)CRB
0 1 2 3 4 5 6 7 8 9 1010-3
10-2
10-1
Es/N0 (dB)
Pha
se E
rror V
aria
nce
MSK1RC (M=2,h=1/2)GMSK2RC (M=4,h=1/4)CRB
• Similar performance for different CPM schemes with L0=64• Within 0.5 dB of CRB at low to medium SNRs
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 23/38
Estimator’s Performance:Timing Offset
• The error variance attains the CRB for various CPMs.• Exception: 1RC due to large deviations from our template signal.
0 1 2 3 4 5 6 7 8 9 1010-4
10-3
10-2
Es/N0 (dB)
Nor
mal
ized
Tim
ing
Erro
r Var
ianc
e
MSKCRB (MSK)1RC (M=2,h=1/2)CRB (1RC)GMSKCRB (GMSK)2RC (M=4,h=1/4)CRB (2RC)
L0 =64
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 24/38
Frame Synchronization:Overview• Prior to Timing and Carrier recovery we must determine preamble
boundaries, i.e. start-of-signal (SoS), in 2 steps:1. SoS detection algorithm: Detects the arrival of a new burst and
tentative location of the SoS
2. SoS estimation algorithm: Estimates the exact location of SoS within an observation window
PreambleIncoming Samples
Noise Only Preamble Random Data
Nw
δ Np
Nw : Observation window lengthNp : Preamble lengthδ : Estimation parameter
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 25/38
SoS Estimation Algorithm
• Apply ML estimation rules to the observation window
• Unwanted parameters, [ ν,θ,αd ] ,are averaged out
• Find δ for which the likelihood function is maximized
1 2)2(2
1
0
222 ][][1exp][1exp
)(1),,,;(
w
w
N
n
nj
nNd ensnrnrp
αr
1 1
1
1*
1**2 ][][)(][][][][2][)(;
w p w
p
pN
n
N
d
dN
Nnss
dN
n
dnrnrdRdnsnsdnrnrnrCp
r
1
1
1*
1**2 ][][)(][][][][2][)( maxargˆ
w w
p
pN
n
D
d
dN
Nnss
dN
ndnrnrdRdnsnsdnrnrnrC
ssimulation viaoptimized :0)()( qNC qw factor Complexity:1 pND
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 26/38
SoS Detection Algorithm
• We apply a simple likelihood ratio test (LRT)
• Using the same techniques as SoS estimator, the LRT becomes
• determines complexity• : test threshold; selected for a given probability of false alarm (Neyman-
Pearson criterion).• ROC: • The LRT is performed using a sliding window.
holdTest thres :preamble aligned-Fully :
preamble No :
samples ofvector :
1
0
HH
N ppr
:'1 pND 'D
1''0'' |)(Pr1 vs.|)(Pr HrLPHrLP DpDDDpDFA
′ ∗ ∗
1
0
′
1
1≷0
′
;;
≷
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 27/38
Frame Synchronization Performance
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10-4
0.99
0.991
0.992
0.993
0.994
0.995
0.996
0.997
0.998
0.999
1
Probability of False Alarm (PFA)
Pro
babi
lity
of D
etec
tion
(PD
)
D'=2, L0=32
D'=3, L0=32
D'=4, L0=32
D'=2, L0=64
0 1 2 3 410
-5
10-4
10-3
10-2
10-1
100
Es/N0 (dB)
Pro
babi
lity
of F
alse
Loc
k (P
FL)
L0=64, D=63, q=1L0=64, D=8, q=0.5L0=64, D=4, q=0.4L0=64, D=2, q=0.3L0=32, D=31, q=1L0=32, D=8, q=0.5L0=32, D=4, q=0.4L0=32, D=2, q=0.3
SoS Detector Receiver Operating Characteristics (ROCs) SoS Estimator
• Probabilities are computed via simulations• Probability of false lock: Pr
Improvement
SNR = 1 dB
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 28/38
Putting Everything Together:Simulating a Burst-Mode CPM Receiver• Transmit bursts of optimum preamble (L0) + 4096 information bits• Apply AWGN + random frequency, phase and timing offsets• Receiver’s structure:
• Design parameters: N=2, Kf =2, Gaussian interpolator D=D’=4, Nw =2NL0
optimized for each scenario using simulations
SoSDetector
SoSEstimator
Joint MLEstimator MFs Viterbi
Algorithm
Frame Synchronization Timing & Carrier Recovery
CPM Demodulator+PLL
r[n] âi
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 29/38
Bit Error Rate (BER) Performance
GMSK (h=1/2, M=2) 2RC (h=1/4, M=4)
0 2 4 6 8 1010
-5
10-4
10-3
10-2
10-1
100
Eb/N0 (dB)
Bit
Erro
r Rat
e (B
ER
)
L0=32
L0=64
Perfect Synchronization
0 2 4 6 8 1010
-4
10-3
10-2
10-1
100
Eb/N0 (dB)
Bit
Erro
r Rat
e (B
ER
)
L0=32
L0=64
Perfect Synchronization
• A preamble of 64 bits performs within 0.1 dB of the ideal synchronization
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 30/38
Applications to SOQPSK(iNET)
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 31/38
What is SOQPSK?
• An OQSPK-like modulation, which has a CPM representation
• The precoder increases spectral efficiency of SOQPSK compared to a binary CPM by imposing: 1 cannot be followed by 1 and vice versa
• Partial-response SOQPSK-TG is adopted by the iNET standard
Precoder CPMModulator
)(ts 1,0,1n 1,1na
0 1 2 3 4 5 6 7 8-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Time (t/Ts)
Pha
se R
espo
nse
SOQPSK-MIL
SOQPSK-TG
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 32/38
iNET Synchronization Preamble
• Length of L0=128 bits• Periodic and consists of repeating a sequence of 16
bits 8 times as
• One period in terms of ternary symbols
7,,0for 0,0,1,0,1,1,0,10,1,0,1,0,1,0,1
12
2
ka
a
k
k
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
-1
0
1
Time Index
Tern
ary
Sym
bols
( i )
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 33/38
Burst-Mode Synchronization of SOQSPK-TG:iNET Preamble• We apply our CPM techniques to SOQPSK by considering ternary
symbols• iNET preamble phase response
• SOQPSK-TG’s response is approximated by SOQPSK-MIL’s• Derivation of joint ML estimation algorithm is similar to the CPM’s
when h=1/2 and M=2
0 4 8 12 16 20 24 28 32
0
Normalized Time (t/Ts)
Unw
rapp
ed P
hase
(rad
)
SOQPSK-TGSOQPSK-MIL
2
3
4
Tl
iNET
Approximation
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 34/38
Estimation Error Variances for SOQPSK:Optimum vs. iNET preambles
0 1 2 3 4 5 6 7 8 9 1010-9
10-8
10-7
Es/N0 (dB)
Nor
mal
ized
Fre
quen
cy E
rror V
aria
nce
SOQPSK-MIL, OptimumCRB (SOQPSK-MIL, Optimum) SOQPSK-TG, OptimumCRB (SOQPSK-TG, Optimum)SOQPSK-TG, iNETCRB (SOQPSK-TG CRB, iNET)
0 1 2 3 4 5 6 7 8 9 1010-4
10-3
10-2
Es/N0 (dB)N
orm
aliz
ed T
imin
g E
rror V
aria
nce
SOQPSK-MIL, OptimumCRB (SOQPSK-MIL, Optimum) SOQPSK-TG, OptimumCRB (SOQPSK-TG, Optimum)SOQPSK-TG, iNETCRB (SOQPSK-TG, iNET)
• Performance degradation due to transitions in the iNET preamble• Phase error variance has the same behavior as the frequency’s
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 35/38
Overall Performance of SOQPSK:Optimum vs. iNET preambles
• iNET preamble performs very close to the optimum one when L0=128• The optimum preamble with L0 =64 is within 0.2 dB of the iNET’s• False locks for L0 =32 at low SNRs
0 1 2 3 4 5 6 7 8 9 1010
-5
10-4
10-3
10-2
10-1
100
Es/N0 (dB)
Bit
Erro
r Rat
e (B
ER
)
Optimum, L0=32
Optimum, L0=64
Optimum, L0=128
iNET, L0=128
Perfect Synchronizarion
Preamble dB loss @ BER=10-4
iNET 0.4
L0 =128 0.3
L0 = 64 0.6
L0 = 32 1
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 36/38
Conclusions
• Gave an overview of feedforwad synchronization methods for CPM
• Derived the optimum training sequence for CPM in AWGN Minimizes the CRBs for all three synchronization parameters Applicable to the entire CPM family including M-ary and partial response
• Developed a joint ML DA estimation algorithm for timing and carrier recovery Designed based on the optimized training sequence Performs within 0.5 dB of the CRB at SNRs as low as 0 dB for L0 = 64
• Simulated a burst-mode CPM receiver: only 0.1 dB SNR loss with the optimum preamble of L0=64
• Applied CPM results to SOQSPK for iNET standard• Final result: developed a complete feedforward DA
synchronization scheme for general CPM including: best training sequence, timing and carrier recovery, and frame synchronization
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 37/38
Areas of Future Study
• Application to frequency-selective channels: Training sequence design Timing and carrier recovery Frame synchronization
• Joint synchronization and channel estimation• Analysis of frame synchronization algorithm as it can
be applied to other modulations, Probability of false lock (SoS estimator) Probabilities of false alarm and detection (SoS detector)
• Hardware implementation and trade-offs between complexity and performance
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 38/38
Thank You!
Questions?
Please let it be OVER… Interesting
I wonder if this technique would work
for my problem What does that slide say?
Dunno, I’m playing Fruit Ninja
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 39/38
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 40/38
Derivation of CRB
• FIM, for CPM and AWGN, is computed via
• Closed-form FIM for CPM and joint estimation:
dtuu
tstsN
T
jiji ,,),,(Re2)(
0
0
*2
0,
αuαuuI
noise w/oPreamble Received
;2,, ααu tjtfj
s
s eeTEts d
Duration Preamble
00 sTLT
ChhBhAhBTThATT
NE
Ts
s 2220
20
220
30
2
0 828222
823/81)(
uI
PulseFrequency CPM :)(
1
0
1
00
1
00
1
00
0 00
00
00
tg
dtjTtgiTtgC
dtiTtgB
dtiTttgA
L
i
L
j
T
ssji
L
i
T
si
L
i
T
si
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 41/38
Training Sequence for Symbol Timing
• Symbol Timing CRB should be minimized,
• The CRB’s denominator should be maximized, or, simultaneously1. Maximize C2. Minimize the quadratic form: J is positive-definite It is
minimized for
• Assuming C is independent of (A,B):
• In practice,
1. Full-response CPM:
2. Partial-response CPM:
1
21
2221
1211222
28
288)|CRB(
hBhA
hBhACh
IIII
α
0 :form Quadratic T Jvv
0JvvT
0 BA0v0 subject to maxargopt BAC
αα
1
0
1
01
20 0
)(2)0(L
i
L
iiisgig aTRRC
.sufficient is 0const.0)( BACTR sg
.0 AND maximize 0)( 2
0 10
BATR L
i iisg
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 42/38
Proposing the training sequence
• A=B=0 is satisfied when:
• C is maximized when is maximized or sign transitions are minimized in the sequence.
• We propose the following binary sequence:
• It satisfies A’=B’=0,1. Optimum sequence for full-response CPM.2. Only 2 transitions => Near-optimum sequence for partial-response CPM.
0 0
1 111 0' AND 0'
L
i
L
iii BiA
2
0 10L
i ii
L0/4 L0/2 L0/4
1
-1
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 43/38
Generalization of the Proposed Sequence
• We can reduce the approximations for partial-response by using /2 rather than L0 to reduce the edge-effect (EF)
• M-ary CPM:
• Frequency and Phase CRBs:
Equalities hold, if λ1=λ2=0 or A=B=0 → Proposed sequence for symbol timing.
• Approximations:
onconstelati in the valuemaximum tos'Set 2iiC
233
32|CRB0
20
230
230
0
1
LABLACLACL
L
α
2
3
4434/
4/2
3|CRB 30
22
020
20
20
30
2
2
LABLL
BCL
BCLL
fd
α
Frequency Phase Timing
Full-response Exact Exact ExactPartial-response EF EF C and EF
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 44/38
Proposed Training Sequence and its Approximations• We propose the following sequence for general CPM and all
synchronization parameters:
• EF is reduced as the sequence length is increased• Approximations on C for partial-response CPM and symbol timing:
• The proposed sequence is asymptotically optimum
L1/4 L1/2
L0
(M-1)
-(M-1)
1
1208
1~)(8~
|CRB|~CRB
000
*
Lsgg
sgsg
TRLRLTR
CTRC
αα
sequence Proposed:sequence alHypothetic :~
*αα
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 45/38
PSD of the Optimum Sequence
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 46/38
Computer Search for the Optimum Sequence:Genetic Algorithm (GA)• The ratio of the minimum CRB found via GA to the proposed
sequence CRB:
• The GA is unable to find a better sequence than the proposed one even at short to moderate lengths
20 30 40 50 60 70 80 90 100 110 1200.98
1
1.02
1.04
1.06
1.08
1.1
Sequence Length
CR
BR
atio
1REC (h=1/2, M=2)GMSK (BT = 0.3)2RC (h=1/4, M=4)
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 47/38
Random vs. Optimum Sequences:Symbol Timing Estimation
True CRB vs. Opt. CRB for L=32Comparison of estimator performancefor “unknown and random” with“known and optimum” trainingsequences.
UCRB vs. Opt. CRBComparison of estimator performance for“known but randomly selected” with“known and optimum” training sequences.
-6 -4 -2 0 2 4 610
-3
10-2
10-1
Es/N0 (dB)
CR
B(
)
True CRB (1REC)Opt. CRB (1REC)True CRB (1RC)Opt. CRB (1RC)
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
L0U
CR
B( )
/CR
Bm
in(
) (dB
)
2RC (M=4, h=1/4)
GMSK (BT = 0.3)1REC (M=2, h=1/2)
1RC (M=2, h=1/2)
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 48/38
Random vs. Optimum Sequences:Frequency and Phase Estimation• UCRB vs. Optimum CRB
• Frequency and phase estimation are less sensitive to the selection of training sequence
10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
L0
UC
RB
(f d)/CR
Bm
in(f d) (
dB)
GMSK (BT=0.3)2RC (M=4,h=1/4)1REC (M=2,h=1/2)1RC (M=2,h=1/2)
10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
L0
UC
RB
( )/C
RB
min
( ) (
dB)
GMSK (BT=0.3)2RC (M=4,h=1/4)1REC (M=2,h=1/2)1RC (M=2,h=1/2)
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 49/38
Derivation of CRB
• Assumption: Joint estimation of u = [fd , θ, τ]T in AWGN,
• The FIM elements:
• Observation interval: T0 = L0Ts or L0 symbols• Computation of three partial derivatives:
• Note for CPM:
• Frequency pulse:
tweeTEtr tjfj
s
s d α,2)(
0
0
*
0,
,,Re2 T
jiji dt
uts
uts
NuuuI
uαuuuuu ,;,,,,,,2, tstjtstjststtsjfts
d
1,, * uu tsts Pulse Frequency / where; 1
0
0
ttqtgiTtgt L
isi
α
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 50/38
Derivation of CRBMore Simplifications• A, B, and C relate CRBs to frequency pulse and data symbols.
They are approximated as,
• Edge-effect: for partial-response CPM• Correlation function of g(t):
dutugugtRTjiRC
BduuugTiA
L
i
L
jgsgji
L
ii
L
i
LTsi
s
1
0
1
0
1
0
1
00
0 0
00
where
21 where
2
2/01 LLL
1
0
1
01
20 0
)(2)0(1for 0L
i
L
iiisgigsg aTRRCnnTR
Pulse Shape 2REC 3REC 2RC 3RC GMSK0.125 0.0833 0.1875 0.125 0.13270.0625 0.0556 0.0312 0.0589 0.0551
0 0.0278 0 0.0036 0.0036
)0(gR
)( sg TR
)2( sg TR ≈ 0
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 51/38
Exhaustive Computer Search for CPM
• Exact computer exhaustive search vs. approximated proposed sequence:
CPM Exhaustive Search Proposed Sequence ΔCRB
1REC 0
2REC 0.6%
3REC 0
1RC 0
2RC 0.7%
3RC 0
Length = 16, Binary CPM
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 52/38
CRB for Random Sequences
• Comparison of the optimum sequence and a random sequence• True CRB:
1. The likelihood function is averaged over α2. Only feasible for MSK-type CPMs, i.e. binary, h=1/2, and full-response
3. The above expectation is computed via simulations
• Unconditional CRB (UCRB):1. Conditional CRB is averaged over α.
2. A closed-form expression was found for symbol timing
1
0
1
0 00
2
0,
0 0
ˆ2tanhˆ2tanhˆˆ2 L
i
L
jj
si
s
l
j
k
islk a
NEa
NE
ua
uaE
NEuI
)(UCRB|CRBˆ|CRBˆ 2over Averaging2| αα αr
ααr EEE
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 53/38
CRB for Random Data Sequence:True CRB• NDA estimators with random (unknown) data
sequence:• Computation of true CRB, as a nuisance parameter.• Quite complex; may not be feasible for general CPM.
• Special case: binary, h=1/2 and full-response:1. PAM representation:
2. pseudo symbols: independent Gaussian RVs.3. Averaging LF over + partial derivatives→ FIM elements:
4. Monte-Carlo simulations → CRBs.
odd)(Im
even)(Re~ Receiver)(
20
22
02
0 ndtnTtctre
ndtnTtctreaiTtcats
Tss
tfjT
stfj
ni
sid
s
d
1
0
1
0 00
2
0,
0 0 ~2tanh~2tanh~~2 L
i
L
jj
si
s
l
j
k
islk a
NEa
NE
ua
uaE
NEuI
s'ias'ia
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 54/38
CRB for Random Data Sequence:Unconditional CRB (UCRB)• Averaging the conditional CRB w.r.t the training sequence results
in unconditional lower bound on the MSE:
• Symbol timing CRB can be expressed as:
• Where:
• Thus:
)(UCRB|CRBˆ|CRBˆ 2over Averaging2| αα αr
ααr EEE
Simplify
Numerical1/
)|(UCRB0 Qαα
α α Ts
s ENE
T
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 55/38
Computing Timing UCRB
• Define:• Approximation via Taylor series:
• Eigen-decomposition of Q:
• Expected value of Z
• Using the same approach and employing central limit theorem:
QααTZ
3
2
00 /1
/UCRB
ZEZE
NET
ZE
NET
s
s
s
s
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 56/38
UCRB:Computation of E{Z2}• Recall , the second moment of Z is written as
• Where and• requires computation of and . • Approximation:
• Xi`s are uncorrelated and hence independent.• Finally
0
1
2L
iiXZ
31,0~ remlimit theo Central
2MNXi
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 57/38
Maximization of LLF(Details 1/2)1. Discrete-time LLF:
2. Approximated CPM signal:
3. Approximated LLF
]Re,,;1
0
*20
NL
n
nj nsnre r
00
00
0
0
0
4/34/34/
4/
/1exp2//1exp
/1exp
NLnNLNLnNL
NLnn
LNnhMjLNnhMj
NnhMjns
1
4/3
)/()1(2
14/3
4/
)2//()1(214/
0
)/()1(2
0
0
0
0
0
00
][
][][Re),,;r(
NL
NLn
LNnhMjnj
NL
NLn
LNnhMjnjNL
n
NnhMjnjj
enre
enreenree
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 58/38
Maximization of LLF(Details 2/2)4. Rearrange LLF as
5. If , the LLF is maximized when phase is
6. The LLF becomes , which is maximized by maximizing
7. That is,
2
11
1 Re),,;( hMjhMjj eee r
)()(, 2)1(
1)1( hMjhMj ee
),(arg~
)~,~(
)~()~(Re2)~()~()~,~( *21
~1222
21
2 hMje
hM
)1(2)~()~(arg~
*21
~)~( maxargˆ 21~
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 59/38
Linear Phase Approximation:Other CPM Examples
0 4 8 12 16
0
1RC (M=2,h=1/2)
t/Ts
Unw
rapp
ed P
hase
0 4 8 12 16
0
t/Ts
2RC (M=4,h=1/4)
Unw
rapp
ed P
hase
2
-2
-2
2
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 60/38
FFT and Interpolation Precision
0 1 2 3 4 510-7
10-6
10-5
10-4
Es/N0 (dB)
Nor
mal
ized
Fre
quen
cy E
rror V
aria
nce
No Interpolation, Kf=1No Interpolation, Kf=2No Interpolation, Kf=4Parabolic, Kf=1Parabolic, Kf=2Parabolic, Kf=4Gaussian, Kf=1Gaussian, Kf=2Gaussian, Kf=4CRB
• Frequency MSE for different interpolators and zero padding factors.• Both interpolation and zero padding must be present based on simulations.
No zero padding
No interpolation
Our choice:Kf =2, Gaussian
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 61/38
SoS Estimation Algorithm(Details 1/3)1. Likelihood function of all unknown parameters
2. Drop constant factors
3. Approximate the exponential function using 2nd degree Taylor series
1 2)2(2
1
0
222 ][][1exp][1exp
)(1),,,;(
w
w
N
n
nj
nNd ensnrnrp
αr
1
)2(*22 ][][Re2expexp),,,;(
wN
n
vnjwd ensnrNp
αr
1 1)2)(2(*
4
1 1)2)(2(**
4
12*
2
][][][][Re1
][][][][Re1
][][Re21)(),,,;(
w w
w w
w
N
n
N
m
mnj
N
n
N
m
nmj
N
n
njd
emsnsmrnr
emsnsmrnr
ensnrCp
αr
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 62/38
SoS Estimation Algorithm(Details 2/3)4. Average the likelihood function w.r.t θ
5. Rearrange the above likelihood function
6. Take the expectation w.r.t data symbols
1 1)(2**
4 ][][][][Re1)(
),,,;(21),,;(
w wN
n
N
m
mnj
dd
emsnsmrnrC
drp
αr
1
1
1**2
12 ][][][][Re2|][|)(),,;(
w ww N
d
dN
n
djN
nd dnsnsdnrnrenrCp αr
1
1
1*
1**2
12 ][][)(][][][][Re2|][|)(),;(
p w
p
pw N
d
dN
Nnss
dN
n
djN
ndnrnrdRdnsnsdnrnrenrCp
r
Preamble Random Data
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 63/38
SoS Estimation Algorithm(Details 3/3)7. A frequency estimate can be derived from each term in the LLF
8. If we use each estimate in its corresponding term, the LLF becomes independent of frequency offset
9. Since the LLF is now only a function of δ, it can be used for estimation of SoS
1*
1** ][][)(][][][][arg
21ˆ
dN
Nn
dN
nssd
w
p
p
dnrnrdRdnsnsdnrnrd
1
1
1*
1**
12 ][][)(][][][][2|][|)();(
p w
p
pw N
d
dN
Nnss
dN
n
N
ndnrnrdRdnsnsdnrnrnrCp
r
1
1
1*
1**
12 ][][)(][][][][2|][|)(maxargˆ
p w
p
pw N
d
dN
Nnss
dN
n
N
ndnrnrdRdnsnsdnrnrnrC
qwNC )()(
E. Hosseini Ph.D. Defense: Synchronization Techniques for Burst-Mode CPM 64/38
Optimization of C(δ)
• PFL is computed via simulations• C(δ) is more important for larger values of D
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210
-3
10-2
10-1
100
q
Pro
babi
lity
of F
alse
Loc
k (P
FL)
D=2D=4D = 8D=63
qwNC )()( GMSK
Np= 64Es /N0=1 dB