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Waveform Design For Active Sensing Systems – A Computational Approach

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Outline

• Introduction

• Waveform design – Correlation Single sequence Sequence set Correlation lower bound

• Waveform design – Correlation & Doppler

• Concluding remarks

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Outline

• Introduction

• Waveform design – Correlation constraint Single sequence Sequence set Correlation lower bound

• Waveform design – Correlation & Doppler

• Concluding remarks

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Active Sensing System

• Radar, Sonar, Medical imaging, Wireless Channel Estimation

The goal is to determine properties of targets or propagation medium by transmitting waveforms and analyzing returned ones

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Christian Hülsmeyer

Reginald Fessenden

Telemobiloscope designed in 1904

First acoustic communication and echo ranging experiment in 1914

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• Better target detection

Why Waveform Design

Pulse compression

Correct detection

plain pulse Two targets

Pulse compression

chirp

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• Interference reduction

Why Waveform Design

CDMA system

Data bits

PN code

Transmit bits

Low correlations of PN codes => low inter-user interference

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Why Waveform Design

• More flexible beampattern

Ultrasound hyperthermia treatment for breast cancer

Focal point of the acoustic power needs to match the tumor region

A ‘bad’ beampattern

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Outline

• Introduction

• Waveform design – Correlation Single sequence Sequence set Correlation lower bound

• Waveform design – Correlation & Doppler

• Concluding remarks

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Waveform Model

• Received waveform

We want to estimate

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Design Criterion

• Matched filter estimateAuto-correlation of {x(n)}

correlation sidelobes

We aim to minimize correlation sidelobes to reduce interference

Unit-modulus constraint

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Existing Waveforms

• Binary Barker code

Auto-correlation of Barker-7

Best binary code in terms of low correlation. But lengths <= 13

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• Binary M sequence, aka., PN (pseudo noise) code

• Polyphase Golomb sequence

Easy to generate. Low correlation sidelobes

Closed-form formula. Low correlation sidelobes.

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Wanted: Lower Correlation Sidelobe

Can we get lower correlation sidelobes?

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• Arbitrary phases in [0,2π]

I

Q

An AWG (arbitrary waveform generator), B&K Precision

Unit-modulus Constraint

We aim to develop computational algorithms, which generate unit-modular sequences with lower correlation sidelobes

More degrees of freedom => better control of correlation sidelobes

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CAN (Cyclic Algorithm New)

• Minimize the ISL (integrated sidelobe level) metric

auxiliary phases

From time to frequency domain

From quartic to quadratic

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CAN

• Phase retrieval in optics Gerchberg & Saxton, 1972

Computationally efficient. Local convergence. Dependent on Initializations.

Dr. W. Owen Saxton

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Example – Merit Factor

• Random-phase sequence, M-sequence, Golomb vs. CAN(G)

Merit Factor

CAN gives the largest Merit Factor, i.e., the smallest correlation sidelobes

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Example – Correlation Level

M-seq & Golomb Random-phase & CAN

CAN gives the lowest correlation sidelobes

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WeCAN (Weighted CAN)

• Extend CAN to WeCAN

e.g., make small

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Example – Channel Estimation

Matched filter estimate

The significant channel taps can occur up to a certain max delay P (P < N)

r(1), …, r(P-1) can be minimized by WeCAN

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• Comparison of Golomb and WeCAN

Example – Channel Estimation

WeCAN provides a lower estimation error than Golomb

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Outline

• Introduction

• Waveform design – Correlation Single sequence Sequence set Correlation lower bound

• Waveform design – Correlation & Doppler

• Concluding remarks

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A Set of Sequences

Auto- & cross-correlation

MIMO Radar

CDMA System

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Multi-CAN & Multi-WeCAN

• Multi-CAN minimizes ISL (auto-correlation sidelobes and all cross-correlations)

From time to frequency domain

• Multi-WeCAN minimizes weighted ISL

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Example – MIMO Radar Imaging

Sequence length N=256, M=4 antennas,

Targets in P=30 range bins

Use a “plain” waveform

Use Multi-WeCAN waveform

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Outline

• Introduction

• Waveform design – Correlation Single sequence Sequence set Correlation lower bound

• Waveform design – Correlation & Doppler

• Concluding remarks

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Correlation Lower Bound

ISL lower bound, 1999

Dr. Dilip Sarwate

Multi-CAN sequence sets approach the lower bound closely

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Outline

• Introduction

• Waveform design – Correlation Single sequence Sequence set Correlation lower bound

• Waveform design – Correlation & Doppler

• Concluding remarks

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Correlation + Doppler

Doppler effect

• Ambiguity function (AF)

AF is a two-dimensional extension of the auto-correlation function

Time delay & Doppler shifts

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Maximum value at (0,0) Symmetry Constant volume

Properties of Ambiguity Function (AF)

where

3D 2D

AF of a chirp signal (T=10 s, B=5 Hz)

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Ambiguity Function (AF)

• Desired AF shape Doppler-tolerant (a high ridge) Doppler-sensitive (thumbtack)

A heartfelt statement…

“The reader may feel some disappointment, not unshared by the writer, that the basic question of what to transmit remains substantially unanswered.”

Dr. Philip Woodward

“Probability and Information Theory, with Applications to Radar”, 1953

But we can still analyze…

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AF of Golomb and CAN(G)

Golomb

CAN(G)

Doppler-tolerant

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AF of Random-phase and CAN(R)

Random-phase

CAN(R)

Doppler-sensitive

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Minimize AF Sidelobes in a Region

• Minimization of discrete-AF sidelobes in a region

All values of are contained in

Minimizing AF sidelobes minimizing correlation sidelobes

Previous CAN-type algorithms can be used

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Example – Minimize AF Sidelobes

• Design a unit-modulus sequence of N=100. K=10, P=3

Low sidelobes in the central rectangular region

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Outline

• Introduction

• Waveform design – Correlation Single sequence Sequence set Correlation lower bound

• Waveform design – Correlation & Doppler• (Waveform design – other constraints)

• Concluding remarks

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Waveform for Spectrum constraints

Avoid reserved frequency bands

Avoid the jamming frequency band

track

jam

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Waveform for Wideband Beampattern

Phased array

Waveform diversity leads to more flexible beampattern

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Outline

• Introduction

• Waveform design – Correlation Single sequence Sequence set Correlation lower bound

• Waveform design – Correlation & Doppler

• Concluding remarks

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Concluding Remarks

• Importance of waveform design for active sensing Range compression, CDMA, channel estimation, beampattern

• New computational algorithms of waveform design Correlation, correlation + Doppler, correlation + spectrum Unit-modulus (arbitrary phases => more degrees of freedom) Better performance than existing waveforms

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Thanks much