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< Reduced-Complexity Joint Frequency, Timing and Phase Recovery for PAM based CPM Receivers>

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Reduced-Complexity Joint Frequency, Timing and Phase Recovery for PAM Based CPM Receivers

Sayak Bose Department of Electrical Engineering and Computer Science University of Kansas, Lawrence, Kansas

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Overview

Introduction to Continuous Phase Modulation (CPM) related work
Motivation of research
Signal models and complexity reduction principle
Joint timing and phase error detector (TED & PED)
Effect of large frequency offsets on TED and PED
Performance analysis metrics and bounds
Simulation results
False lock recovery using reduced complexity detector

configurations

Conclusions and future work

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Introduction to Continuous Phase Modulation (CPM) related work

CPM Receivers CPM Receivers

Conventional CPM Conventional CPM Laurent Laurent Decomposition of Decomposition of CPM CPM PAM PAM-

Based CPM

Based CPM

Symbol Symbol Timing Timing Recovery Recovery D D’ ’Andrea Andrea, , Mengali,Morelli Mengali,Morelli Carrier Phase Carrier Phase Recovery Recovery D D’ ’Andrea Andrea, , Mengali,Morelli Mengali,Morelli Carrier Carrier Frequency Frequency Recovery Recovery D D’ ’Andrea Andrea, , Mengali Mengali Symbol Symbol Timing Timing Recovery Recovery Perrins Perrins, , Bose,Green Bose,Green Carrier Phase Carrier Phase Recovery Recovery Colavope Colavope, , Raheli Raheli Carrier Carrier Frequency Frequency Recovery Recovery D D’ ’Andrea Andrea, , Mengali,Ginesi Mengali,Ginesi Joint Timing & Joint Timing & Phase Recovery Phase Recovery Morelli Morelli, , Vitetta Vitetta Joint Frequency, Joint Frequency, Timing & Phase Timing & Phase Recovery Recovery Joint Timing & Joint Timing & Phase Recovery Phase Recovery Joint Frequency, Joint Frequency, Timing & Phase Timing & Phase Recovery Recovery Conference papers:

E. Perrins, S. Bose, and M. P. Wylie-Green, “Timing Recovery Based on the PAM Representation
f CPM," IEEE Military Communications Conference (MILCOM02008), San Diego, CA, November

2008

E. Perrins, S. Bose, and M. P. Wylie-Green, "PAM-Based Timing Synchronization for ARTM

Modulations," in Proceedings of the International Telemetering Conference, San Diego, CA, October 2008.

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Why CPM?

– Power and bandwidth efficient. – Easy to use with low-cost PAs.

Problems with CPM

– Receiver complexity. – Receiver synchronization. –Motivation for using Pulse Amplitude Modulation (PAM) – Linearize CPM; first proposed for binary CPMs in the well-known paper by Laurent.

– Reduce receiver complexity by discarding low-energy PAM pulses.

– Recover symbol timing using simple algorithms.

Motivation of research

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Conventional CPM signal model

f(t) is the frequency pulse, it has a finite duration of L symbol

times and an area of 1/2.

q(t) is the time-integral of f(t)
h is the modulation index, it is typically a rational number
αn are drawn from an M-ary alphabet

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − =

∑

= n i i

iT t q h j A t s ) ( 2 exp ) ; ( α π α

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Conventional CPM signal model...contd.

The phase can be grouped into two terms since q(t) = 1/2 for t>LT:
Since the modulation indexes are rational numbers, h=k/p, we can

describe the signal with a finite state machine:

4 3 42 1 4 4 4 3 4 4 4 2 1

L n n

L n i i t n L n i i n i i

h iT t q h iT t q h t

−

∑ ∑ ∑

− = + − = =

+ − = − =

φ α φ

α π α π α π α φ

) ; ( 1

) ( 2 ) ( 2 ) ; (

( ) ( )

4 4 4 4 3 4 4 4 4 2 1 L

states 1 1 1 2 3

1

, ,..., , , , , ,

−

− + − − − − −

= ↔ =

L

pM n n L n L n n n n n n n

α α α φ σ α α α α σ α n α n−1 α n−2 α n−L +1

…

k

mod p

1 + −L n

φ

L n−

φ

L −1 elements

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Pulse Amplitude Modulation (PAM) based CPM model

M-ary Single-h
PAM complexity reduction principle

Number of PAM Components and

Subset the largest amplitude pulses to reduce the number of matched

filters (MF)

Reduce number of trellis states in the detector

∑∑

− =

1 ,

) ( ) ; (

N k s k i i k s s

iT t g b T E t s α

) 1 ( 2

) 1 (

− =

−

M N

L P

∑ ∑

− =

−

1

) ( ) ( ) ; (

N k s k i k i j s s

iT t g c b e T E t s

L i

φ

α

{ }

1 , , 1 , − ⊆ N L κ

M P

2

log =

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PAM based joint timing and phase error detector

Received signal model in AWGN channel
Coherent detection
Symbol detection using the Viterbi algorithm (VA)
Decision-directed timing recovery
Decision-directed phase recovery
Noncoherent detection
Symbol detection using the Viterbi algorithm
Decision-directed timing recovery without explicit phase

information

Note:

Timing and phase recovery uses decisions from the receiver.
Symbol detection and signal recovery are based on maximum

likelihood principle.

( )

∑∑

− =

+ − − =

1 ,

) ( ) (

N k i s k i k s s j

t w iT t g b T E e t r τ

θ

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PAM based joint timing and phase error detector

Metric increment in VA for sequence detection
PAM-based TED is given by

where the TED increment

PAM-based PED is given by

where the PED increment MF bank filter output and

s

T L t ≤ ≤

{ }

) , ~ , ~ ( Re

1

'

=

∑

− = − − L i j L i i i

e c y

θ

τ φ

∫

+ + + −

− − ≅

s k s

T D i iT s k j i k

dt iT t g e t r x

) ( ,

) ( ) ( ) (

τ τ θ

τ τ

∑

∈ − −

=

TED

k k i k i k j L i i i

x b e c y ) ~ ( ) ~ , , (

, , *

τ τ φ

θ

& &

∑

− = − −

=

1

} ) ~ , , ( Re{

L i j L i i i

e c y

θ

τ φ &

∑

∈ − −

=

PED

k k i k i k j L i i i

x b e c z ) ( ) , , (

, , * ~

τ τ φ

θ

∑

− = − −

=

1 ~

} ) , , ( Im{

L i j L i i i

e c z

θ

τ φ

1 1 + ≤ ≤ L Dk

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Error detector implementation

Error signal from the TED Error signal from the PED Note:

Matched filters estimate data symbols through VA implementation.
Derivative matched filters generate TED Error hence timing estimate.

A discrete-time differentiator approximates the derivative.

κ ∈

−

k k

t g )} ( { MF bank ] [m r

} ˆ {

n

α

D n

c − ˆ

D L n − −

ϕ ˆ ] [ D n e −

τ

] [ ˆ D n− τ

VA TED Timing PLL Interpolator Phase PLL PED

] [ D n e −

θ

] [ ˆ D n j

e

− − θ

{ }

] [ ˆ

]) [ ˆ , ˆ , ˆ ( Re ] [

D n j D L n D n D n

e D n c y D n e

− − − − − −

− = −

θ τ

τ φ &

{ }

] [ ˆ

]) [ ˆ , ˆ , ˆ ( Im ] [

D n j D L n D n D n

e D n c z D n e

− − − − − −

− = −

θ θ

τ φ

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Effect of large frequency offsets on TED and PED

Frequency offset on the order of the symbol rate 1/Ts
A non-data-aided (NDA) frequency recovery is done before attempting symbol

sequence, timing and phase recovery. A Frequency Difference Detector (FDD) is employed for this purpose.

Timing recovery without the explicit recovery of phase (noncoherent) is more

suitable as phase recovery is still difficult due to the average residual frequency jitter .

κ ∈

−

k k

t g )} ( {

MF bank

] [m r

] [ ˆ n ν

VCO DMF bank

κ ∈

−

k k

t g )} ( {&

FED Loop Filter

] [n e

κ ∈

−

k k

t g )} ( {

MF bank

} ˆ {

n

α

D n

c − ˆ

D L n − −

ϕ ˆ ] [ D n e −

τ

] [ ˆ D n − τ

VA TED Timing PLL Interpolator Phase PLL PED

] [ D n e −

θ ] [ ˆ D n j

e

− − θ κ ∈ k n k

x } {

,

] [m r

] [ ˆ 2 m

avg

e

ν π −

1

S

2

S

avg

v ˆ

FDD

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Performance analysis metrics and bounds

We use modified Cramer-Rao bound (MCRB) to establish a lower bound on the

degree of accuracy to which , and can be estimated.

Normalized MCRB - timing

where for uncorrelated data symbols special cases : 1) LREC: 2) LRC:

Normalized timing error variance
MCRB – phase
Phase error variance
MCRB – frequency
Normalized frequency error variance

τ θ

ν

2 2 2

/ 1 8 1 ) ( 1 N E L C C h MCRB T

s f s

× = ×

α

π τ

3 / ) 1 ( } {

2 2

− = ≅ M E C

n

α

) 4 /( 1 L C C

LREC f

≅ = ) 8 /( 3 L C C

LRC f

≅ =

} ] [ ˆ { 1 1

2 2 2

τ τ σ τ − × ≅ × n Var T T

s s

/ 1 2 1 ) ( N E L MCRB

s

× = θ

} ] [ ˆ {

2

θ θ σ θ − ≅ n Var

3 2 2

/ 1 2 3 ) ( N E L MCRB T

s s

× = × π ν

} ] [ ˆ {

2 2 2

ν ν σν − × ≅ × n Var T T

s s

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Performance analysis metrics and bounds…contd.

Phase Locked Loop (PLL) Considerations
Error detector outputs are refined into suitable offset estimates
The loop bandwidth is an important parameter determining the

performance of the synchronizers.

Timing PLL
First order timing PLL implementation refines the TED output

after every

, . PLL step size is

Phase PLL
The new phase estimate from the PLL is obtained as
First order PLL with no carrier frequency offset and
Second order PLL with residual carrier frequency offset

K1 and K2 are proportional and integration constants respectively

Frequency PLL
First order frequency PLL refines FDD output after every as

. PLL step size is

] [ ] 1 [ ˆ ] [ ˆ n e n n

τ τ

γ τ τ + − ≅

τ τ τ

γ

p s

k T B 4 ≅

] [n eτ

] [ ] 1 [ ˆ ] [ ˆ n n n ξ γ θ θ

θ

+ − ≅ ] 1 [ 2 ] [ ) 2 1 ( ] 1 [ ] [ − − + + − = n e K n e K K n n

p p θ θ θ θ

κ κ ξ ξ

] [ ] [ n e n

θ

ξ =

ν ν ν

γ

p s

k T B 4 ≅

] [ ] 1 [ ˆ ] [ ˆ n e n n

ν ν

γ ν ν + − ≅

s

BT

s

T

s

T

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Performance analysis metrics and bounds…contd.

S-Curve identifies the stable lock points
These are the zero-crossing positive slope points on the curve.
We want the such a point at zero error, otherwise it is a false lock point
Decision directed M-ary TED and PED have false lock points
FDD is NDA, therefore, free of false lock points.
S-curve for TED
where

is timing offset

{ }

τ τ τ

δ δ | ] [ . / ) ( n e E T E S

s s

≅

τ τ δτ ˆ − ≅

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Performance analysis metrics and bounds…contd.

S-curve for PED

where

is the phase offset

S-curve for FDD
where

is the frequency offset

{ }

θ θ θ

δ δ | ] [ . / ) ( n e E T E S

s s

≅

{ }

ν ν ν

δ δ | ] [ . / ) ( n e E T E S

s s

≅

θ θ δθ ˆ − ≅

ν ν δν ˆ − ≅

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Simulation results

Binary GMSK system with Gaussian pulses

(Gaussian Minimum Shift Keying) M=2, 4G, h=1/2

Optimal PAM based detector
Trellis state = 16
8 single-h MFs/Pulses
Reduced complexity detectors chosen for this example
4 state detector with L’ = 2
MFs/pulses.
MFs and pulse.

2 | | | | = = =

PED TED

κ κ κ 2 | | = κ

1 | | = =

PED TED

κ κ

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Simulation results (Binary GMSK…)

Timing error variance with no carrier frequency offset
Timing error variance with a large carrier frequency offset

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Simulation results (Binary GMSK)

Phase error variance with no carrier frequency offset
Frequency error variance with a large carrier frequency
ffset

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Simulation results (Binary GMSK)

BER with no carrier frequency offset
BER with a large carrier frequency offset

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Simulation results

M-ary CPM system with partial response

M=4, 2RC, h=1/4

Optimal PAM based detector
Trellis state = 16
12 single-h MFs/Pulses

Reduced complexity detectors chosen for this example

4 state detector with L’ = 1
MFs/pulses.
MFs and pulse.

2 | | = κ 2 | | | | = = =

PED TED

κ κ κ

1 | | = =

PED TED

κ κ

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Simulation results (M-ary CPM…)

Timing error variance with no carrier frequency offset
Timing error variance with a large carrier frequency offset

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Simulation results (M-ary CPM…)

Phase error variance with no carrier frequency offset
Frequency error variance with a large carrier frequency
ffset

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Simulation results (M-ary CPM…)

BER with no carrier frequency offset
BER with a large carrier frequency offset

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Summary of simulation results

PAM based reduced-complexity CPM detectors provide very good tracking

characteristics under no carrier frequency offset.

Coherent and noncoherent detection can be done based on PAM based
detectors. The noncoherent detectors are worse by about 2-3 dB in BER under

all practical requirements and under no frequency offset condition.

With a frequency offsets on the order of of the symbol rate, the performance
f PAM based detectors does not suffer deterioration in terms of tracking

accuracy and BER.

With the carrier frequency offset on the order of the symbol rate, noncoherent

detection outperforms coherent detection in terms of tracking accuracy and BER.

Noncoherent detection allows further simplification of the receiver structure by

alleviating the need for a second stage of frequency recovery.

4

10−

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False lock recovery using reduced complexity detector configurations

M-ary partial response CPM systems suffer from false lock problems during

signal acquisition.

Under false lock, the synchronizers settle at incorrect timing and phasing

instants rendering poor BER, timing and phase error variance.

NDA auxiliary lock detectors

remove false locks but has a longer acquisition time. CPM (M=4, 3RC, h=1/2) Due to the variable lengths of the PAM filter components, PAM based configurations can deal with this problem more effectively than its conventional CPM counterpart.

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False lock recovery … contd.

S-curves for M-ary CPM (M=4, 3RC, h=1/2)

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False lock recovery … contd.

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False lock recovery … contd.

Observations
CPM (M=4, 3RC, h=1/2), conventional and PAM based with 1 pulse

noncoherent TED with during initial acquisitions.

3

10 5

−

× =

s

T Bτ

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Conclusions and future work

Conclusion
Synchronizers provide a comparable performance against

conventional CPM receivers.

Under a large carrier frequency offset, a PAM based receiver in

noncoherent mode offer similar performance as its CPM counterpart

A novel method of decision-directed false lock recovery for PAM based

CPM receivers.

Future work
Joint phase and timing recovery in wireless fading channels.
Possibility of different PAM based error detector configurations for

acquisition and tracking stages.

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Acknowledgements

I would like to thank

My Advisor, Dr. Erik Perrins
Committee members Dr. Sam Shanmugan and Dr. Shannon Blunt
Nokia-Siemens Networks and The University of Kansas General

Research Funds for their support throughout this research

Communication theory and systems research group at KU
Family and friends

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Thank you!

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