Mult lticarrie ier Underw rwater Acoustic Communication Gilad - - PowerPoint PPT Presentation

Andrew and Erna Viterbi Faculty

• f Electrical Engineering

Tim ime Vary rying Carrie ier Frequency Offset Estimation in in Mult lticarrie ier Underw rwater Acoustic Communication

Gilad Avrashi

Supervised by Prof. Israel Cohen and Dr. Alon Amar

Contents

• Introduction
• Signal Space Estimation
• Pilot Design Optimization
• Time-Varying CFO Estimation
• Conclusions

2/

Introduction

Why underwater communications?

Autonomous Underwater Vehicles Manned Vehicles Mine Detection Pipeline Inspection Submarines Divers

Challenges of f Underw rwater Communications

• EM signals are attenuated quickly in the UW medium → pressure

waves (sound) have been chosen for long range communications

• Sound waves characteristics:
• Propagation speed: 1500 m/s

(times 200,000 slower than EM waves!)

• Frequency dependent losses
• Frequency related ambient noise

Orthogonal Frequency Div ivision Mult ltiplexing

• The comm. bandwidth is divided into sub-carriers
• Each subcarrier is modulated to carry a digital communication symbol
• Pros:
• Easy to implement using FFT operations
• Robustness to frequency selective channels
• Simple channel equalizer
• Cons:
• Very sensitive to frequency shifts
• High peak-to-average power ratio (PAPR)

2.095 2.1 2.105 2.11 x 10

4

• 0.5

0.5 1 1.5 frequency [Hz] Magnitude No Doppler scaling

𝐿Δ𝑔 = 𝑋 Orthogonality is achieved by Δ𝑔 = 1

𝑈

OFDM modulation

Information bits QPSK Mapping IFFT Zero Padding

Upsampling & Modulation

𝐭 ∈ ℂ𝐿×1 𝐜 ∈ {0,1}2𝐿×1

𝑦 𝑜 = 𝑕[𝑜] ෍

𝑙=0 𝐿−1

𝑡 𝑙 𝑓𝑘2𝜌 𝑜

𝑋 𝑔𝑙 = 𝑕[𝑜] ෍ 𝑙=0 𝐿−1

𝑡 𝑙 𝑓𝑘2𝜌 𝑜

𝑋 𝑙Δ𝑔 = 𝑕[𝑜] ෍ 𝑙=0 𝐿−1

𝑡 𝑙 𝑓

𝑘2𝜌𝑜𝑙 𝐿

= 𝑕[𝑜] 𝐿 IDFT{𝑡 𝑙 } The baseband OFDM signal:

In-phase quadrature

𝐆𝐿

𝐼𝐭 ∈ ℂ𝐿×1

𝐲 = 𝐔𝑎𝑄𝐆𝐿

𝐼𝐭 ∈ ℂ𝐿+𝑀×1

Sea Tria ial l OFDM Sig ignal

+

𝑀-tap CIR ℎ 𝑜, 𝜐 𝑤[𝑜] 𝑦[𝑜] 𝑧[𝑜] Recording from sea trial in the Mediterranean, December 2016

Research Goal

Develop a carrier frequency offset estimator for underwater acoustic OFDM modems. The solution is required to be computationally efficient and practical for the underwater acoustic channel.

9/

Mult lticarrier UAC Effects

zero padding frequency time Transmitted block

Mult lticarrier UAC Effects

OFDM blocks zero padding frequency time Transmitted block Doppler scaling Received block

Mult lticarrier UAC Effects

OFDM blocks zero padding frequency time Transmitted block Doppler scaling Received block

𝜗Δ𝑔

Received Sig ignal Model

Li et al. ‘08

Sync. Doppler rescaling CFO est. Channel equalizer Demod. packet block by block

𝑧 𝑜 = 𝑓

𝑘2𝜌𝜗0𝑜 𝐿

ℎ 𝑜 ∗ 𝑦 𝑜 + 𝑤[𝑜]

• r in matrix form 𝐳 = 𝚫

𝐿 𝜗0 ด

𝐈𝐲

𝐴

+ 𝐰

Radio Frequency Approaches

Training blocks with periodic characteristics (Classen & Meyer ‘94) Block to block pilot signal cross- correlation (Schmidl & Cox ‘97)

Estimate CFO Apply Estimation Estimate CFO

In UAC – CFO varies between adjacent blocks

UAC Approaches

Null Carriers Minimum variance (Li et al. ‘08) Pilot aided Maximum power (Li et al. ‘06)

Requires exhaustive grid search

extract pilots\ nulls frequency shift max

𝜗

𝐷(𝜗𝑗) Hypothesized 𝜗𝑗Δ𝑔 single block Ƹ 𝜗

Pil ilot Based Estim imation

𝑦 𝑜 = 1 𝐿 ෍

𝑙=0 𝐿−1

𝑡 𝑙 𝑓

𝑘2𝜌𝑜𝑙 𝐿

G carriers

Pil ilot Based Estim imation

𝑦 𝑜 = 1 𝐿 ෍

𝑙=0 𝐿−1

𝑡 𝑙 𝑓

𝑘2𝜌𝑜𝑙 𝐿

= 1 𝐿 ෍

𝑙∈𝑇𝐸

𝑡 𝑙 𝑓

𝑘2𝜌𝑜𝑙 𝐿

G carriers

data signal ~𝒪 0,

𝐿−𝑅 𝑅

Pil ilot Based Estim imation

1 𝐿 ෍

𝑟=0 𝑅−1

𝑡 𝑟𝐻 𝑓

𝑘2𝜌𝑜𝑟 𝑅

𝑦 𝑜 = 1 𝐿 ෍

𝑙=0 𝐿−1

𝑡 𝑙 𝑓

𝑘2𝜌𝑜𝑙 𝐿

= 1 𝐿 ෍

𝑙∈𝑇𝐸

𝑡 𝑙 𝑓

𝑘2𝜌𝑜𝑙 𝐿

+

G carriers

data signal ~𝒪 0,

𝐿−𝑅 𝑅

pilot signal 𝑅-periodic

Idea: Use correlation between periods of the pilot signal Problem: Low “SNR” – Pilot to Data Ratio (PDR) Solution: Design pilot signal with “Good” auto-correlation

Best auto correla latio ion: id identic ical pil ilots

Amar, Avrashi, Stojanovic ‘16

ǁ 𝑡 𝑜 = 1 𝐿 ෍

𝑟=0 𝑅−1

𝑣𝑓

𝑘2𝜌𝑜𝑟 𝑅

+ 1 𝐿 ෍

𝑙∈𝑇𝐸

𝑡 𝑙 𝑓

𝑘2𝜌𝑜𝑙 𝐿

Q samples

Exploiting In Inter-Segment Correlations

𝐳𝑕

𝐼𝐳𝑕′ = 𝑓−𝑘2𝜌 𝐻 𝜗0 𝛽(𝜗0) 𝑕′−𝑕

𝐴𝑕

𝐼𝐴𝑕′

𝑕 𝑕′

𝑧 𝑜 = 𝑓

𝑘2𝜌𝜗0𝑜 𝐿

ℎ 𝑜 ∗ 𝑦 𝑜

𝑨[𝑜]

𝐒

Eig igen Valu lue Decomposition

The cost function in matrix formulation We look for Ƹ 𝜗 that minimizes (maximizes) 𝑚 = 𝛃(𝜗)𝐼𝐒𝛃(𝜗) Under two constraints:

• 𝛃

= 1

• arg(𝛃) ∝ 𝜗

1 𝐻 1 𝛽−1 ⋯ 𝛽−(𝐻−1) 1 𝛽1 ⋮ 𝛽𝐻−1

𝐒

EVD estim imator

min

𝛃 𝛃𝐼𝐒𝛃

• s. t. 𝛃𝐼𝛃 = 1

➔ෝ 𝛃 = 𝐖min(𝐒)

• Min. Var.

LS 𝐳 Ƹ 𝜗 ෝ 𝛃

arg[ෝ 𝛃] ≈ − 2𝜌 𝐻 𝐡𝜗

Decompose 𝐒→ find the eigenvector of the smallest EV → extract ො 𝝑

➔ Ƹ 𝜗 = − 𝐻 2𝜌 σ𝑕=0

𝐻−1 𝑕 arg ෝ

𝛃

𝑕

σ𝑕=0

𝐻−1 𝑕2

Research objectives

• The EVD-based estimator has two drawbacks:
• High PAPR
• Requires constant CFO during the block
• Our goal: Propose a CFO estimator for

UAC with the following characteristics:

• Low complexity
• Negligible PAPR
• Adjustable for time-varying channels

Signal Space Estimation

Two sid ides of f the same coin in

array processing interpretation Steering 𝜗 in the noise\signal space to achieve lowest\highest SNR

𝐴𝑕

𝐼𝐴𝑕′

Correlated Uncorrelated

Exploiting In Inter-Segment Correlations

𝐳𝑕

𝐼𝐳𝑕′ = 𝑓−𝑘2𝜌 𝐻 𝜗0 𝛽(𝜗0) 𝑕′−𝑕

𝐴𝑕

𝐼𝐴𝑕′

𝑕 𝑕′

𝑧 𝑜 = 𝑓

𝑘2𝜌𝜗0𝑜 𝐿

ℎ 𝑜 ∗ 𝑦 𝑜

𝑨[𝑜]

𝐒

EVD in in Sig ignal Space

max

𝛃

𝛃𝐼𝐒s𝛃

• s. t. 𝛃𝐼𝛃 = 1

➔ ෝ 𝛃s = 𝐖max(𝐒s)

• Max. Var.

LS 𝐳s Ƹ 𝜗s ෝ 𝛃s Decompose 𝐒s → find the eigenvector of the largest EV → extract ො 𝝑

arg[ෝ 𝛃] ≈ − 2𝜌 𝐻 𝐡𝜗 ➔ Ƹ 𝜗 = − 𝐻 2𝜌 σ𝑕=0

𝐻−1 𝑕 arg ෝ

𝛃

𝑕

σ𝑕=0

𝐻−1 𝑕2

Combined LS estim imate

Max.

• Var. DR

LS 𝐳s Ƹ 𝜗c ෝ 𝛃s

• Min. Var.

DR 𝐳n ෝ 𝛃n Ƹ 𝜗c = 𝛾 Ƹ 𝜗n + 1 − 𝛾 Ƹ 𝜗s , 0 ≤ 𝛾 ≤ 1

Generali lized EVD

GEVD 𝐳s Ƹ 𝜗g ෝ 𝛃g(𝜗) 𝐳n LS

max

𝛃

𝛃𝐼𝐒s𝛃 𝛃𝐼𝐒n𝛃

• s. t. 𝛃𝐼𝛃 = 1

➔ ෝ 𝛃g = 𝐖max 𝐒n

−1𝐒s

Decompose 𝐒n

−1𝐒s → find the eigenvector of the largest EV → extract ො

𝝑

Computational Complexity

Method Complexity Grid Search 𝒫(𝐿 𝐿) Noise Space EVD 𝒫(𝐻2(𝑅 − 𝑀)) = 𝒫 𝐿𝐻 Signal Space EVD 𝒫(𝐻2𝑀) Combined LS 𝒫(𝐻2 max{𝑅 − 𝑀, 𝑀}) Generalized EVD 𝒫(𝐻2 max{𝑅 − 𝑀, 𝑀})

RMSE vs SNR

K=2048 G=8 CFO= 0.2Δ𝑔 L=100 L=50 Long Delay Spread Short Delay Spread

Effect of f Dela lay Spread

L Q

Pool Tria ial

Pilot Design Optimization

PDR and PAPR tradeoff

High PDR Low PDR Low PAPR High PAPR

Random Pilots Identical Pilots

PAPR = max |𝑦[𝑜]|2 1 𝐿 σ𝑜=1

𝐿

|𝑦[𝑜]|2

data level data level data level pilot level pilot level pilot level

Proposed pil ilot design formulation

Pilot tones: 𝜚 𝑙 = 𝑡 𝑟𝐻 = 𝑓𝑘𝜚𝑙 Pilot signal (one period): 𝜔[𝑜] =

1 𝑅 IDFT{𝜚[𝑙]}

Phase retrieval problem: min

𝛚,𝛠

𝐗𝑀𝑞𝛚 − 𝐆𝑅

𝐼𝛠 2 𝑥𝑀𝑞 𝑜 = ൞ 1 𝑀𝑞 , 𝑜 < 𝑀𝑞 0, 𝑜 ≥ 𝑀𝑞

Unit amplitude Known envelope Find the phases 𝜚𝑙

Pil ilot design – generalized GSA algorithm

Initialize Initialize 𝛠0 = rand 𝑅, 1 , 𝐾 = ∞ while while 𝐾 > 𝜃 do do

𝛚𝑗 = 𝑓𝑘∢ IFFT{𝛠𝑗−1} time domain pilot signal 𝛠𝑗 = 𝑓𝑘∢ FFT{𝐗𝛚𝑗} pilot tones 𝜁 = |IFFT 𝛠𝑗 | − 𝐱 2 envelope error norm 𝑞 =

max |IFFT 𝛠𝑗 |2

1 𝑅 σ𝑜=1 𝑅

|IFFT 𝛠𝑗 |2

PAPR 𝐾 = 𝛽𝜁 + 𝛾𝑞

end while end while return return 𝛠𝑗

Sim imula lation results

Time-Varying CFO Estimation

How can we capture TV-CFO?

Phases accumulated within 1 sub-segment duration Phases accumulated within 5 sub-segment duration

𝐒 𝜗, 𝑕 becomes 𝐒 𝜗, 𝑕, 𝑜 : 𝐒𝑕,𝑕′ = 𝐳𝑕

𝐼𝐳𝑕′ = 𝐴𝑕 𝐼𝚫 𝑕 𝐼(𝜗, 𝑜)𝚫 𝑕′(𝜗, 𝑜)𝐴𝑕′

𝚫

𝑕 𝐼(𝜗, 𝑜)𝚫 𝑕′(𝜗, 𝑜) is a diagonal matrix

For constant CFO the diagonal is 𝑒𝑕,𝑕′ = 𝛽𝑕

∗𝛽𝑕′

Polynomial Model

• Time variations are decomposed to its Taylor series:

𝜗 𝑜 = ෍

𝑚=0 ∞

𝜗𝑚𝑜𝑚 , 0 ≤ 𝑜 ≤ 𝐿 − 1

• The diagonal of 𝚫

𝑕 𝐼(𝜗, 𝑜)𝚫 𝑕′(𝜗, 𝑜) becomes

𝑒𝑕,𝑕′ = 𝛽𝑕

∗ exp 𝑘2𝜌

𝐿 ෍

𝑚=1 ∞

𝜗𝑚 𝑠

𝑚 𝑜, 𝑕′ − 𝑠 𝑚 𝑜, 𝑕

𝛽𝑕′ α𝑕 = exp 2𝜌 𝐻 ෍

𝑚=1 𝐻−1

𝑕𝑚𝑅𝑚−1𝜗𝑚−1 𝑠𝑚 𝑜,𝑕 = ෍

𝑙=1 𝑚−1

𝑚 𝑙 𝑜𝑚−𝑙 𝑕𝑅 𝑙

Approximated solu lution

GEVD 𝐳s Ƹ 𝜗[𝑕] ෝ 𝛃(𝜗) 𝐳n LS

max

𝛃

𝛃𝐼𝐒s𝛃 𝛃𝐼𝐒n𝛃

• s. t. 𝛃𝐼𝛃 = 1

➔ෝ 𝛃 = 𝐖max 𝐒n

−1𝐒s

arg[ෝ 𝛃] ≈ − 2𝜌 𝐻 𝐁𝐑𝛝 ➔ Ƹ 𝜗 = − 𝐻 2𝜌 𝐑−1 𝐁𝑈𝐁 −1𝐁𝑈arg[ෝ 𝛃]

TV neglected TV included

Pie iecewise-Constant Model

• Time variations are represented as piecewise-constant:

𝜗 𝑜 = ෍

𝑕=0 𝐻−1

𝜗𝑕𝑣𝑕 𝑜 , 0 ≤ 𝑜 ≤ 𝐿 − 1

• The diagonal of 𝚫

𝑕 𝐼(𝜗, 𝑜)𝚫 𝑕′(𝜗, 𝑜) becomes

𝑒𝑕,𝑕′ = 𝛽𝑕

∗ exp 𝑘2𝜌

𝐿 ෍

𝑚=1 ∞

𝜗𝑕′ − 𝜗𝑕 𝑜 𝛽𝑕′ , α𝑕 = 𝑓

2𝜌 𝐻 𝜗𝑕𝑕

The time varying component is bounded by exp

𝑘2𝜌 𝐿 |𝜗𝑕′ − 𝜗𝑕|𝑅

Approximated solu lution

GEVD 𝐳s Ƹ 𝜗[𝑕] ෝ 𝛃(𝜗) 𝐳n LS

max

𝛃

𝛃𝐼𝐒s𝛃 𝛃𝐼𝐒n𝛃

• s. t. 𝛃𝐼𝛃 = 1

➔ෝ 𝛃 = 𝐖max 𝐒n

−1𝐒s

arg[ෝ 𝛃] ≈ − 2𝜌 𝐻 𝐇𝛝 ➔ Ƹ 𝜗 = − 𝐻 2𝜌 𝐇−1arg[ෝ 𝛃]

TV neglected TV included

Sim imula lation results

Polynomial model: 𝜗 𝑜 = σ𝑚=0

4 𝑐𝑚 𝐿𝑚 𝑜𝑚

, 𝑐𝑚~𝑉 −0.25,0.25 Sinusoidal model: 𝜗 𝑜 = Δ𝑔 𝐵0 + 𝐵sin 2𝜌𝑜

𝑔sin 𝐿

, 𝐵0, 𝐵~𝑉 −0.25,0.25 , 𝑔

sin~𝑉 0.25,2

Conclusions and Future Research

• A complete Tx-Rx scheme was suggested:
• Reduced complexity closed form CFO estimation
• Pilot design resolves the PAPR problem and makes the solution practical
• Time-varying model allows deployment in harsh environments
• Future Research
• Time Varying CIR
• Combined pilots-data PAPR reduction
• Proving the solution in sea trials
• Model order estimation and channel sensing for the TV estimator