subspaces and sparsity on the continuum -...
TRANSCRIPT
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Subspaces and sparsity
on the continuum
Mark A. Davenport
School of Electrical and Computer Engineering
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SanthoshKarnik
RakshithSharma
JustinRomberg
ZhihuiZhu
MichaelWakin
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Traditional view of signal processing
[Wikipedia]
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Modern signal processing
measurements sampled
signal
• numerical linear algebra
• optimization
• subspaces
• sparsity
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In many applications, the most natural signal models are
inherently continuous
Translating this to a discrete, finite setting can be subtle
Modeling on the continuum
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Bandlimited functions
Perhaps the most basic model is that is bandlimited
The continuous-time Fourier transform of a function
is given by
We say that is bandlimited (with bandlimit ) if
for
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Sampling bandlimited functions
More specifically, let denote the sampling period and let
denote the sequence of samples we obtain
The sampling theorem shows us that no information is lost
provided
“If we sample a signal at twice its highest
frequency, then we can recover it exactly.”
Whittaker-Nyquist-Kotelnikov-Shannon
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Windows of samples
To simplify our notation, we will assume without loss of
generality that so that
: sampling at the Nyquist rate
: sampling faster than the Nyquist rate
degrees of freedom
degrees of freedom?
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Models for bandlimited signals
If , we expect that has degrees of freedom
How can we represent this mathematically?
From bandlimitedness we have
The discrete Fourier transform (DFT) gives a representation of
the form
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Models for bandlimited signals
If , we expect that has degrees of freedom
How can we represent this mathematically?
The DFT should be sparse
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Models for bandlimited signals
If , we expect that has degrees of freedom
How can we represent this mathematically?
The DFT should be sparse – but it usually isn’t…
spectral
leakage
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A better model
The DFT is simply the wrong basis for compactly representing
this structure
A much better choice: discrete prolate spheroidal sequences
Slepian basis. Defined by the vectors that
satisfy the eigenvalue equation
The first eigenvalues .
The remaining eigenvalues .
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Suppose that we wish to minimize
over all subspaces of dimension
Another perspective: Subspace fitting
Optimal subspace is spanned by
the first Slepian basis elements
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The prolate matrix
From either perspective, it is not hard to show that the
Slepian basis elements are the eigenvectors of the prolate
matrix
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Bottom line
Windowed and sampled bandlimited signals live in
a subspace with an effective dimension of
Other frequency bands handled by simply modulating the
Slepian basis elements to different center frequencies
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Narrowband DOA
Consider the problem of estimating the direction-of-arrival
(DOA) of a narrowband source using a linear array of sensors
sensors
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Narrowband DOA – Far field
Assume we can approximate the source as a plane wave
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Narrowband DOA – Far field
Assume we can approximate the source as a plane wave
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Narrowband DOA – Far field
Assume we can approximate the source as a plane wave
Sinusoid at frequency Sinusoid at frequency
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DOA as spectral estimation
What happens when we have multiple sources?
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Three sources, equal magnitude
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DOA as spectral estimation
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Magnitudes 1.0, 0.05, 0.01
What happens when we have multiple sources?
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Iterative source localization
Essential procedure
1. Find a source
2. Null it out
(remove its effect from the measurements)
3. Repeat until no more sources
Nulling must be performed with care…
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Iterative source localization
angle-2
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Iterative source localization
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Iterative source localization
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“Nulling” procedure
Given several sources, we observe their sum
We do not expect to estimate the angle (or equivalently,
the frequency ) corresponding to source exactly
Suppose that we have an estimate
Then lives in a subspace of dimension , spanned by
the Slepian basis elements modulated to
We can null by projecting onto the orthogonal complement of
this subspace
• choose slightly more than basis elements to null
nearly all of the energy
• can choose to account for non-point sources
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“Rolling Slepian” spectral estimation
We may also wish to use a Slepian-inspired approach to
estimating the source angles as follows:
1. Project onto the “Slepian subspace” corresponding to an
interval of bandwidth
2. Compute the energy in this projection
3. Sweep over all frequencies
By choosing slightly fewer than basis elements, we can
nearly eliminate spectral leakage
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Example
Two 10 MHz bandwidth sources:
One centered at 5.04 GHz … … one 100x lower at 5.0 GHz
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Example
Residual when we “null” using DFT vectors
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Example
“Rolling Slepian” spectral estimate
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Example
Slepian projection cleanly reveals smaller source
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Thomson’s multitaper method
Is the “Rolling Slepian” spectral estimate related to
Thomson’s multitaper method?
They are equivalent!
windowing
DFT energy
Average over windows
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Thomson’s multitaper method
Is the “Rolling Slepian” spectral estimate related to
Thomson’s multitaper method?
They are equivalent!
Slepian basis element
projection energy
Sum across all basis elements
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Summary so far
The Slepian basis provides a natural tool for working with
(finite windows) of sampled bandlimited signals
• Subspace modeling
projecting a vector in onto span of the first
Slepian basis elements to enforce/exploit bandlimited
model
• Applications in DOA estimation
• Applications in spectral estimation (Thomson’s method)
How can we do this at a speed comparable with the FFT?
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Motivating example
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Motivating example
• Radar array whose goal is to track every object of size 10
cm or larger in low-Earth orbit
• First radar site expected to go online this year in Marshall
Islands, another planned for Australia
• Each radar site has a digital phased array consisting of
~100,000 (S-band) receivers
• TREMENDOUS data volume, need for scalable algorithms
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Towards fast Slepian computations
Recall that the Slepian basis can be computed via an
eigendecomposition of the so-called prolate matrix
Let be the matrix whose columns correspond to the
lowest frequency length- DFT vectors
One can show that , where
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Number of eigenvalues in transition region
The rank of gives as a nonasymptotic bound on the number
of eigenvalues of in the “transition region”
Specifically,
Improves on previous asymptotic bounds by Slepian and
nonasymptotic bounds by Zhu and Wakin
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Fast Slepian Projection
Let be the matrix formed by the first columns of
Theorem
For any , , and such that
, there exist matrices and such
that
where
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Fast Slepian Projection
We can apply the approximation to a vector in
operations
Similar fast algorithms can be developed for Thomson’s
method as well as solving related problems
Toeplitz
low rank
small
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Extensions
• Higher dimensions
– array geometry and source environment can be two- or three-
dimensional
• DOA with unknown frequencies
– given a sequence of samples in time, we can consider a joint
search over both angle-of-arrival and frequency
• Compressive acquisition
– subsample array elements in time
– subsample using spatial coded aperture techniques
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Simple compression: Subsampling
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Compressive beamforming
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Full sample beam sweep Compressive beam sweep
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Iterative (compressive) source localization
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Iterative (compressive) source localization
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Iterative (compressive) source localization
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Large scale simulation
• 64 x 64 antenna array
• 100 sources, each 10 MHz, located in [2 GHz, 12 GHz]
• 0 dB SNR, keep only 12% of samples
These results simply would not be possible without
(fast) Slepian projections
Ground truth and predicted source locations at 0dB noise level
5
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Ground truth
Predicted
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Exploiting bandwidth in active sensing
Recall the linear array setup:
In general, for a target profile at a constant range, our
linear array observes (a warped) version of the Fourier
transform of over the range
Sinusoid at frequency Sinusoid at frequency
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If we discretize we can write our observations as
where the columns of are uniformly spaced complex
exponentials over the range
What happens if we repeat this for many different
frequencies?
A linear model
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Does bandwidth buy us anything?
If , we seemingly do not gain any new
information beyond what is contained in
Effectively, for all
But what if we subsample?
One can show that can be highly dimensionality-reducing
without compromising our ability to estimate
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• 40x40 sensor array, sensors placed 3.75cm apart
• Traditional imaging using excitation wavelength of 7.5cm
would require ~1100 beams
• By exploiting bandwidth (lower frequencies) we can
dramatically reduce the number of “beams”
Simulated results
1100 beams 80 “generic” beams
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Summary
• The Slepian basis is a natural choice in many applications
– any time you are working with finite windows of samples of
bandlimited/narrowband/multiband functions
• We now have fast (approximate) algorithms for working
with the Slepian basis, with complexity that scales
comparably to the FFT
• Can play an important role in large-scale problems,
especially in the context of compressive acquisition
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References
D and Wakin. Compressive sensing of analog signals using discrete prolate
spheroidal sequences. Applied and Computational Harmonic Analysis (2012)
D, Needell, and Wakin. Signal space CoSaMP for sparse recovery with redundant
dictionaries. IEEE Trans. on Information Theory (2013)
Zhu, Karnik, Wakin, D, Romberg, ROAST: Rapid Orthogonal Approximate Slepian
Transform. IEEE Trans. on Signal Processing (2018)
Karnik, Zhu, Wakin, Romberg, D. The Fast Slepian Transform. Applied and
Computational Harmonic Analysis (2019)
Karnik, Romberg, D. Fast multitaper spectral estimation. SAMPTA (2019)
Srinivasa, D, Romberg, Trading beams for bandwidth: Imaging with randomized
beamforming. http://arxiv.org/abs/1902.00075 (2019)