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
The goal is to determine properties of targets or propagation
medium by transmitting waveforms and analyzing returned ones
• Radar, Sonar, Medical imaging, Wireless Channel Estimation
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Christian Hülsmeyer
Telemobiloscope designed in 1904
Reginald Fessenden
First acoustic
communication and
echo ranging
experiment in 1914
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Why Waveform Design
• Better target detection
plain
pulse
Two targets
Pulse
compression
chirp
Pulse
compression
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Correct detection
Why Waveform Design
• Interference reduction
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
A ‘bad’ beampattern
Ultrasound hyperthermia treatment for breast cancer
Focal point of the acoustic power needs to match the tumor region
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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 estimate
Auto-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
Easy to generate. Low correlation sidelobes
• Polyphase
 Golomb sequence
Closed-form formula. Low correlation sidelobes.
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Wanted: Lower Correlation Sidelobe
Can we get lower correlation sidelobes?
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Unit-modulus Constraint
• Arbitrary phases in [0,2π]
Q
An AWG (arbitrary waveform generator), B&K Precision
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More degrees of freedom => better control of correlation sidelobes
We aim to develop computational algorithms, which generate
unit-modular sequences with lower correlation sidelobes
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CAN (Cyclic Algorithm New)
• Minimize the ISL (integrated sidelobe level) metric
From time to frequency domain
From quartic to quadratic
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auxiliary phases
CAN
• Phase retrieval in optics
 Gerchberg & Saxton, 1972
Dr. W. Owen Saxton
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Computationally efficient. Local convergence.
Dependent on Initializations.
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
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small
Example – Channel Estimation
The significant channel taps can occur up to a certain max delay P (P < N)
Matched filter estimate
r(1), …, r(P-1) can be minimized by WeCAN
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Example – Channel Estimation
• Comparison of Golomb and WeCAN
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
CDMA System
MIMO Radar
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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
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Use Multi-WeCAN
waveform
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)
Time delay & Doppler shifts
AF is a two-dimensional extension of the auto-correlation function
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Properties of Ambiguity Function (AF)
 Maximum value at (0,0)
 Symmetry
 Constant volume
where
AF of a chirp signal (T=10 s, B=5 Hz)
3D
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2D
Ambiguity Function (AF)
• Desired AF shape
 Doppler-tolerant (a high ridge)
 Doppler-sensitive (thumbtack)
“Probability and Information Theory,
with Applications to Radar”, 1953
A heartfelt statement…
Dr. Philip Woodward
“The reader may feel some disappointment, not unshared by
the writer, that the basic question of what to transmit remains
substantially unanswered.”
But we can still analyze…
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AF of Golomb and CAN(G)
Golomb
Doppler-tolerant
CAN(G)
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AF of Random-phase and CAN(R)
Random-phase
Doppler-sensitive
CAN(R)
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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
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Previous CAN-type algorithms can be used
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
track
Avoid the jamming
frequency band
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
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