scispace - formally typeset

Journal ArticleDOI

A Compressed Sensing Parameter Extraction Platform for Radar Pulse Signal Acquisition

24 Sep 2012-IEEE Journal on Emerging and Selected Topics in Circuits and Systems (IEEE)-Vol. 2, Iss: 3, pp 626-638

TL;DR: A complete (hardware/ software) sub-Nyquist rate (× 13) wideband signal acquisition chain capable of acquiring radar pulse parameters in an instantaneous bandwidth spanning 100 MHz-2.5 GHz with the equivalent of 8 effective number of bits (ENOB) digitizing performance is presented.

AbstractIn this paper we present a complete (hardware/ software) sub-Nyquist rate (× 13) wideband signal acquisition chain capable of acquiring radar pulse parameters in an instantaneous bandwidth spanning 100 MHz-2.5 GHz with the equivalent of 8 effective number of bits (ENOB) digitizing performance. The approach is based on the alternative sensing-paradigm of compressed sensing (CS). The hardware platform features a fully-integrated CS receiver architecture named the random-modulation preintegrator (RMPI) fabricated in Northrop Grumman's 450 nm InP HBT bipolar technology. The software back-end consists of a novel CS parameter recovery algorithm which extracts information about the signal without performing full time-domain signal reconstruction. This approach significantly reduces the computational overhead involved in retrieving desired information which demonstrates an avenue toward employing CS techniques in power-constrained real-time applications. The developed techniques are validated on CS samples physically measured by the fabricated RMPI and measurement results are presented. The parameter estimation algorithms are described in detail and a complete description of the physical hardware is given.

Topics: Effective number of bits (53%), Signal reconstruction (53%), Signal (52%), Compressed sensing (51%), Radar (51%)

Summary (5 min read)

Introduction

  • A principal goal in the design of modern electronic systems is to acquire large amounts of information quickly and with little expenditure of resources.
  • In light of the ever growing demand to capture higher bandwidths, it would seem that a solution at the fundamental system level is needed to address these challenges.
  • The requisite sampling rate is merely proportional to the information level, and thus CS provides a framework for sub-Nyquist rate signal acquisition.
  • The authors complete system is capable of recovering radar pulse parameters within an effective instantaneous bandwidth (EIBW) spanning 100 MHz—2.5 GHz with a digitizing performance of 8 ENOB.

A. Compressed Sensing

  • CS at its heart relies on two concepts: sparsity and incoherence [5].
  • Incoherence captures the idea of dissimilarity between any two representations; two bases are said to be incoherent if any signal having a sparse expansion in one of them must be dense in the other.
  • Fig. 1a shows the block diagram of the IC containing the input buffer driving the common node of the four RD channels, and the timing generator.
  • After the integrator the signal bandwidth is reduced and the circuits are design to meet settling requirements.
  • Finally, an output buffer is designed to the drive the ADC with the correct swing and common-mode voltage.

B. A Brief History and Description of the RMPI

  • Almost simultaneously with the introduction of CS [2], a number of CS-based signal-acquisition architectures were proposed.
  • The rows of each block contain ±1 entries, and the overall matrix will be composed of NTNyq/Tint = 20 blocks (one for each integration window).
  • For the measurements presented in this paper, the authors construct a model of their system’s Φ matrix by feeding in sinusoidal tones and using the output measurements to characterize the system’s impulse response.

A. Architecture and Operation

  • The RMPI presented in this work was realized with the proprietary Northrop Grumman (NG) 450 nm InP HBT bipolar process [26].
  • The timing generator is responsible for generating the pseudo-random bit sequences (PRBS) and the clocking waveforms to coordinate the track-and-hold (T/H) and integration operations.
  • The circuits following the integrator are designed to meet the settling requirements of the reduced bandwidth.
  • In operation, the RMPI circuit takes the analog input signal, buffers it, and distributes the buffered signal to each of the 4 channels.
  • Immediately after the signal is sampled, the capacitor begins discharging and the second capacitor begins integrating the next frame (see Fig. 1b).

B. Analog Signal Path

  • The input buffer is a differential pair with emitter degeneration and 50 Ω termination at each single-ended input.
  • Emitter degeneration is used on the bottom differential pair to improve linearity.
  • At the end of the integration period the signal is sampled and then held for 26 CLKin cycles to allow the external ADC to digitize the signal for post-processing.
  • The diodes act as switches to configure capacitors for integration or reset it based on the level of the control signal (SEL).
  • The input clock buffer was biased with a relatively high power to reduce jitter and it has 4 separate output emitter followers to limit cross-talk.

C. PRBS & Timing Generator

  • A master clock is applied to CLKin, from which all required timing signals are generated.
  • The output pulse from PN6B is re-clocked with the pulse from PN6A to pr duce a sync pulse that is 52 CLKin cycles long once every 3276 cycles.
  • Special attention was paid to the routing of the PRBS, T/H clocks, and select signals to inimize clock/data coupling among th four channels.
  • Shown are (4) quadrature clocks for the T/H and (4) select signals for the interleaved capacitors.

D. Performance Analysis

  • Simulation validation was done by performing transientbased two-tone inter-modulation distortion simulations in the Cadence design environment.
  • Noise simulations were perform d using the peri i steady-state (PSS) mode of spectre.
  • The RMPI sam ling system, including the off-chip ADCs consumed 6.1 W of power.
  • The authors point out that this system was designed as a proof-of-concept and was not optimized for power.
  • 4 Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841].

IV. PULSE-DESCRIPTOR WORD (PDW) EXTRACTION

  • Having described the acquisition system, the authors now present algorithms for detecting radar pulses and estimating their parameters, referred to as pulse-descriptor words (PDW), from randomly modulated pre-integrated (RMPI) samples.
  • The detection process is based on familiar principles employed by detectors that operate on Nyquist samples.
  • The authors algorithms use a combination of template matching, energy thresholding, and consistency estimation to determine the presence of pulses.
  • The remainder of the section elaborates on the procedure and is arranged as follows.
  • After describing how the authors can reliably estimate the carrier frequency of a signal from compressive measurements, they then explain how they use such estimations to form a detection algorithm that jointly uses energy detection and consistency of their frequency measurements.

A. General Parametric Estimation

  • The authors general parameter estimation problem can be stated as follows.
  • Given the measurements y = Φ[x0]+noise, the authors search for the set of parameters corresponding to the subspace which contains a signal which comes closest to explaining the measurements y.
  • Tα is the orthogonal projector onto the column space of Vα.
  • When the noise is correlated, the authors may instead pose the optimization in terms of a weighted least squares problem.
  • In Sec. IV-B and Sec. IV-C below, the authors will discuss the particular cases of frequency estimation for an unknown tone, and time-of-arrival estimation for a square pulse modulated to a known frequency.

B. Carrier Frequency, Amplitude, and Phase

  • The authors consider the task of estimating the frequency of a pure tone from the observed RMPI measurements.
  • The algorithms developed here play a central role in the detection process as well, as the frequency estimation procedure is fundamental to determining the presence of pulses.
  • The authors also describe how to estimate the amplitude and phase once the carrier frequency (CF) is known.
  • The subspaces Since the authors have discretized the signal x(t) through its Nyquist samples, their measurement process is modeled through a matrix Φ, the rows of which are the basis elements φk.
  • Rather than dealing with continuously variable frequency, the authors define a fine grid of frequencies between 0 and fnyq/2.

D. Pulse Detection

  • With their estimation techniques explained the authors next describe a pulse detection algorithm that takes a stream of RMPI samples and classifies each as either having a “pulse present” or “no pulse present.”.
  • The authors task is to determine how many pulses are present and to estimate the parameters of each pulse the authors find.
  • If neighboring blocks have CF estimates that are consistent in value and tones at these frequencies account for a considerable portion of their measurement energies, then the authors are confident that a pulse is indeed present.
  • Once the TOA, TOD, and CF estimates have been refined, the authors calculate their amplitude and phase estimates.
  • Merge any segments that are close together in time and have similar CF estimate, also known as 7.

E. Complexity

  • In fact, since Φ contains repetitions of Φ0 the authors need only take the FFT of the 252 rows of Φ0, which they may then shift through complex modulation.
  • This calculation does not depend on the measurement data, and therefore can be done offline as a precomputation.
  • For each window length L (typically between 5-11 RMPI samples) and each window shift, the authors have to compute.
  • The authors can explicitly invert V Tf Vf using the 2×2 matrix inverse formula, and all other calculations involve a small number of 4L-point inner products.
  • The number of frequencies the authors test is proportional to the number of Nyquist 8 Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841] samples N , and thus the cost of the frequency estimation for each sliding window shift is O(NL).

F. Cancellation and multiple pulses

  • The authors focus on how to remove contributions from certain frequency bands in the RMPI measurements.
  • The operator Ψ can be used to remove the contributions of the interfering band in the measurements y when the authors run their estimation methods.
  • Fig. 9 shows how the use of the nulling operator aids in removing interfering bands.
  • During this second 1These are also known as Slepian sequences.
  • Fig. 10 shows an example of the two-stage detector for overlapping pulse data.

A. Measurement Test Setup Description

  • In order to test the performance of the radar-pulse parameter estimation system-composed of the RMPI sampling hardware and the PDW extraction algorithm §IV, the authors ran a set of over 686 test radar pulses composed of permutations of A0, θ0, CF, TOA and TOD through the RMPI and estimated the varied parameters from the compressed-samples digitized by the RMPI.
  • 9 Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841] Fig. 12 shows a block diagram for the test setup used for the RMPI.
  • An Arbitrary Waveform Generator AWG with an output sampling rate of 10 Gsps was used to output the pulses of interest.
  • The stimulus was input into the RMPI whose outputs were then sampled by external ADCs located on a custom digitizing PCB shown in Fig. 11b: the RMPI IC is mounted on a a low-temperature co-fired ceramic (LTCC) substrate shown in Fig. 11a which is placed in the center of the digitizing board.
  • The digitized samples were then transferred to a PC where the PDW extraction algorithm was used to estimate the signal parameters.

B. Parameter Estimation

  • For each pulse, the authors estimated the CF only from measurements corresponding to times when the signal was active.
  • The authors then estimated the TOA from RMPI samples corresponding to noise followed by the front end of the pulse.
  • The authors repeated the procedure for the TOD, using RMPI samples corresponding to the end of the pulse followed by noise only.
  • Fig. 13 shows the distribution of their estimation errors for CF, TOA, and TOD.
  • Additionally, Table I shows statistics on the errors for each of the three parameters.

C. Pulse Detection

  • The authors tested the pulse detection system by generating 60 test cases containing 12 pulses each (for a total of 720 pulses) with varying amplitudes (ranging over 60 dB), phases, 10 Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841] durations (100 ns—1 µs), carrier frequencies (100 MHz— 2.5 GHz), and overlaps.
  • All pulse rise times were approximately 10 ns.
  • Table III shows the detection rate and standard deviation of the parameter estimate errors as a function of the pulse amplitudes.
  • To test the robustness of the detection and estimation system, the authors repeated their detection experiment and included a constant-frequency interferer at set amplitudes in each experiment.
  • The authors tested 6 interferer strengths, running 60 experiments with 12 pulses per experiment, for a total of 720 pulses per interferer strength.

VI. CONCLUSION

  • The authors have presented a detailed overview of the design of both hardware and software used in a novel radar-pulse receiver in which information is extracted without performing full signal reconstruction.
  • This novel approach obtains desired information with high accuracy while considerably reducing the back-end computational load.
  • The system was validated using parameter estimates obtained from testing with a large and exhaustive set of realistic radar pulses spanning the parameter space.
  • The physically measured results generated from this prototype proof-ofconcept system demonstrates the feasibility of the approach.
  • In addition, the data obtained provides ample motivation for further investigation of the merit of CS-based signal acquisition schemes in general.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

1
A Compressed Sensing Parameter Extraction
Platform for Radar Pulse Signal Acquisition
Juhwan Yoo, Christopher Turnes, Eric Nakamura, Chi Le, Stephen Becker, Emilio Sovero, Michael Wakin,
Michael Grant, Justin Romberg, Azita Emami-Neyestanak, and Emmanuel Cand
`
es
Abstract—In this paper we present a complete (hard-
ware/software) sub-Nyquist rate (×13) wideband signal acqui-
sition chain capable of acquiring radar pulse parameters in an
instantaneous bandwidth spanning 100 MHz2.5 GHz with the
equivalent of 8 ENOB digitizing performance. The approach
is based on the alternative sensing-paradigm of Compressed-
Sensing (CS). The hardware platform features a fully-integrated
CS receiver architecture named the random-modulation pre-
integrator (RMPI) fabricated in Northrop Grumman’s 450 nm
InP HBT bipolar technology. The software back-end consists
of a novel CS parameter recovery algorithm which extracts
information about the signal without performing full time-
domain signal reconstruction. This approach significantly re-
duces the computational overhead involved in retrieving desired
information which demonstrates an avenue toward employing
CS techniques in power-constrained real-time applications. The
developed techniques are validated on CS samples physically
measured by the fabricated RMPI and measurement results are
presented. The parameter estimation algorithms are described
in detail and a complete description of the physical hardware
is given.
Index Terms—Compressed sensing, Indium-Phosphide, Pa-
rameter Estimation, Random-Modulation Pre-Integration
I. INTRODUCTION
A principal goal in the design of modern electronic systems
is to acquire large amounts of information quickly and with
little expenditure of resources. In the wireless technology
sector, the goal of maximizing information throughput is
illustrated by the strong interest in RF sensing and spectral
applications that require instantaneous bandwidths of many
GHz. Such systems have applications ranging from scientific
instrumentation to electronic intelligence. Although some
J. Yoo and A. Emami-Neyestanak are with the Department of Electrical
Engineering at the California Institute of Technology, Pasadena, CA, e-mail:
juhwan@caltech.edu
C. Turnes and J. Romberg are with the School of Electrical and
Computer Engineering at the Georgia Institute of Technology, Atlanta, GA,
e-mail: cturnes@gatech.edu
M. Wakin is with the Department of Electrical Engineering and
Computer Science, Colorado School of Mines, Golden, CO
S. Becker and M. Grant are with the Department of Applied and Com-
putational Mathematics at the California Institute of Technology, Pasadena,
CA; S. Becker is also with the Laboratoire Jacques-Louis Lions at Paris 6
University, Paris, France
C. Le, E. Nakamura, and E. Sovero are with the Northrop Grum-
man Corporation, Redondo Beach, CA, e-mail:eric.nakamura@ngc.com;
Emilio Sovero is now at Waveconnex Inc., Westlake Village, CA, e-mail:
emilio.sovero@waveconnex.com
E. Cand
`
es is with the Departments of Mathematics and Statistics at
Stanford University, Stanford, CA
solutions already exist, their large size, weight, and power
consumption make more efficient solutions desirable.
At present, realizing high bandwidth systems poses two
primary challenges. The first challenge comes from the
amount of power required to operate back-end ADCs at the
necessary digitization rate. This issue is so significant that
the remaining elements of the signal chain (RF front-end,
DSP core, etc.) are often chosen based upon an ADC that is
selected to be compatible with the available power budget [1].
The second challenge comes from need to store, compress,
and post-process the large volumes of data produced by such
systems. For example, a system that acquires samples at a
rate of 1 Gsps with 10 bits of resolution will fill 1 Gb of
memory in less than 1 s. In light of the ever growing demand
to capture higher bandwidths, it would seem that a solution
at the fundamental system level is needed to address these
challenges.
Some promise for addressing these challenges comes from
the theory of compressed sensing (CS) [2]–[6]. CS has
recently emerged as an alternative paradigm to the Shannon-
Nyquist sampling theorem, which at present is used implicitly
in the design of virtually all signal acquisition systems. In
short, the CS theory states that signals with high overall
bandwidth but comparatively low information level can be
acquired very efficiently using randomized measurement pro-
tocols. The requisite sampling rate is merely proportional to
the information level, and thus CS provides a framework for
sub-Nyquist rate signal acquisition. As we discuss further in
Sec. II, aliasing is avoided because of the random nature of
the measurement protocol.
The emergence of the CS theory is inspiring a fundamen-
tal re-conception of many physical signal acquisition and
processing platforms. The beginning of this renaissance has
already seen the re-design of cameras [7], medical imaging
devices [8], and RF transceivers [9]–[11]. However, the bene-
fits of CS are not without their costs. In particular, the task of
reconstructing Nyquist-rate samples from CS measurements
requires solving an inverse problem that cannot be addressed
with simple linear methods. Rather, a variety of nonlinear
algorithms have been proposed (see, e.g., [12]–[14]). While
the speed of these methods continues to improve, their
computational cost can still be appreciably greater than many
conventional algorithms for directly processing Nyquist-rate
samples. This matter of computation, if not addressed, poten-
tially limits the wide-spread application of CS architectures
“The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government.”
This is in accordance with DoDI 5230.29, January 8, 2009.
Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841]

in power constrained real-time applications.
In this paper we address these issues by presenting a
complete, novel signal acquisition platform (both hardware
and software) that is capable—in certain applications—of
estimating the desired signal parameters directly from CS
measurements [15]. In the spirit of compressive signal pro-
cessing [16], our approach takes the principal motivation of
CS one step further and aims to eliminate the overhead of first
reconstructing the Nyquist-rate signal samples before apply-
ing conventional DSP techniques for parameter extraction.
On the hardware side, we present a fully integrated
wideband CS receiver called the random-modulation pre-
integrator (RMPI) [9,10,12,17]. We fabricate this device with
Northrop Grumman’s 450 nm InP HBT bipolar process. On
the software side, we focus on signal environments consisting
of radar pulses and present a novel algorithm for extracting
radar pulse parameters—carrier frequency (CF), phase θ
0
,
amplitude A
0
, time-of-arrival (TOA), and time-of-departure
(TOD)—directly from CS measurements. (The exact signal
model is described in Sec. IV.)
Our complete system is capable of recovering radar
pulse parameters within an effective instantaneous bandwidth
(EIBW) spanning 100 MHz2.5 GHz with a digitizing per-
formance of 8 ENOB. We validate the system by feeding the
fabricated RMPI with radar pulses and using the physically
digitized CS measurements to recover the parameters of
interest.
An outline of this paper is as follows: Sec. II provides
a brief background on CS and a description of the high-
level operation of the RMPI, Sec. III provides a complete
description of the hardware platform used to encode the CS
samples, Sec. IV provides details of the parameter estimation
algorithms, and Sec. V presents measurement results.
II. THE RMPI
A. Compressed Sensing
CS at its heart relies on two concepts: sparsity and
incoherence [5]. Sparsity captures the idea that many high-
dimensional signals can be represented using a relatively
small set of coefficients when expressed in a properly chosen
basis. Incoherence captures the idea of dissimilarity between
any two representations; two bases are said to be incoherent if
any signal having a sparse expansion in one of them must be
dense in the other. An example of an incoherent pair comes
from the classical time-frequency duality. A sparse signal
time—e.g., a Dirac-delta function—has a dense spectrum.
Similarly, a single tone is sparse in the Fourier domain but
dense in time.
The key observation underlying CS is that when a signal
is sparse in some basis, it can be acquired by taking a
small number of measurements that are incoherent with
its sparse basis [18]. Often this incoherence is achieved
RevisedHardwareDescription
A. ArchitectureandOperation
The RMPI presented in this work was realized with the proprietary Northrop Grumman (NG) 450nm InP HBT bipolar process [25].
The process features a 4-layer metal stack with an fT and fmax > 300GHz. Fig. 1a shows the block diagram of the IC containing the
input buffer driving the common node of the four RD channels, and the timing generator. The timing generator is responsible for
generating the pseudo-random bit sequences (PRBS) and clocking waveforms to coordinate the track-and-hold and integration
operations. All analog and digital signal paths are implemented differentially to improve common-mode noise rejection and increase
linearity of the system. The analog path up to the integrator was designed for a 2.5 GHz bandwidth. After the integrator the signal
bandwidth is reduced and the circuits are design to meet settling requirements. A 5 GHz master clock reference (CLKin) is used to
clock the PRBS generators and sets the 2.5 GHz unambiguous RF bandwidth. The track-and-holds (T/H) operate at 1/52 the master
clock frequency or roughly 96 MHz and a switch capacitor integrator is used so that one capacitor can be reset while the second is
integrating the mixer output. Finally, an output buffer is designed to the drive the ADC with the correct swing and common-mode
voltage. The chip was designed for a full-scale input amplitude of 0.5V p-p differential and 1V p-p differential at the output.
(a) SystemBlockDiagram
(b) RDChannelBlockDiagram
Fig. 1: (a) Simplified block diagram of 4-channel RMPI. The analog-signal path of each RD channel is identical; however the
timing signals they receive in operation are different. (b) Functional diagram of the mixer and integrator circuit.

InP RMPI 4 Channel Sampler
CLKin
PRBS1-4
CLK1-4SEL1-4
SYNC
T/H
RD 4
RD 3
RD 2
RD 1
Ibuf
SW Cap
Integrator
Sel
COTS
ADC
COTS
ADC
COTS
ADC
COTS
ADC
Obuf
RFin
Gain
PN/Timing Generator
(a) System Block Diagram
RevisedHardwareDescription
A. ArchitectureandOperation
The RMPI presented in this work was realized with the proprietary Northrop Grumman (NG) 450nm InP HBT bipolar process [25].
The process features a 4-layer metal stack with an fT and fmax > 300GHz. Fig. 1a shows the block diagram of the IC containing the
input buffer driving the common node of the four RD channels, and the timing generator. The timing generator is responsible for
generating the pseudo-random bit sequences (PRBS) and clocking waveforms to coordinate the track-and-hold and integration
operations. All analog and digital signal paths are implemented differentially to improve common-mode noise rejection and increase
linearity of the system. The analog path up to the integrator was designed for a 2.5 GHz bandwidth. After the integrator the signal
bandwidth is reduced and the circuits are design to meet settling requirements. A 5 GHz master clock reference (CLKin) is used to
clock the PRBS generators and sets the 2.5 GHz unambiguous RF bandwidth. The track-and-holds (T/H) operate at 1/52 the master
clock frequency or roughly 96 MHz and a switch capacitor integrator is used so that one capacitor can be reset while the second is
integrating the mixer output. Finally, an output buffer is designed to the drive the ADC with the correct swing and common-mode
voltage. The chip was designed for a full-scale input amplitude of 0.5V p-p differential and 1V p-p differential at the output.

(a) SystemBlockDiagram
(b) RDChannelBlockDiagram
Fig. 1: (a) Simplified block diagram of 4-channel RMPI. The analog-signal path of each RD channel is identical; however the
timing signals they receive in operation are different. (b) Functional diagram of the mixer and integrator circuit.

InP RMPI 4 Channel Sampler
CLKin
PRBS1-4
CLK1-4SEL1-4
SYNC
T/H
RD 4
RD 3
RD 2
RD 1
Ibuf
SW Cap
Integrator
Sel
COTS
ADC
COTS
ADC
COTS
ADC
COTS
ADC
Obuf
RFin
Gain
PN/Timing Generator
(b) RD Channel Block Diagram
Fig. 1: (a) Simplified block diagram of 4-channel RMPI. The analog-signal
path of each RD channel is identical, however the timing signals they receive
in operation are different. (b) Functional diagram of the mixer and integrator
circuits.
by incorporating randomness into the measurement process.
There are many possibilities for implementing incoherent
random measurements; a convenient and admissible choice
for hardware implementation is to correlate the input signal
(in our case, a time-windowed version of the input signal)
with a pseudo-random binary sequence (PRBS) [4]. We
refer the reader to [5] and references therein for additional
information about the mathematical theory of CS.
B. A Brief History and Description of the RMPI
Almost simultaneously with the introduction of CS [2],
a number of CS-based signal-acquisition architectures were
proposed. Some of the more well-known proposals include:
the Random Demodulator (RD) [17,19,20], the Random-
Modulation Pre-Integrator (RMPI) [9,10,21], the Non-
Uniform Sampler (NUS) [4,22], Random Convolution [23],
the Modulated Wideband Converter (MWC) [24], and many
others [25]—for a comprehensive overview see [12]. The
basic function that all of these systems implement is to cor-
2
Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841]

relate of the input signal x(t) with an incoherent, randomly
generated set of “basis” elements over a fixed time window.
The RMPI is one of the most direct physical implemen-
tations of the CS concept; it is composed of a parallel set
of RDs driven by a common input. (See Fig. 1a, which is
described more fully in Sec. III.) Each RD is driven by
a distinct PRBS p(t); it uses this PRBS to modulate the
incoming signal x(t), integrates the result over a time interval
of length T
int
, and then digitizes the output at a rate f
ADC
=
1/T
int
f
nyq
. In our RMPI, T
int
= 52 · T
nyq
with f
nyq
=
1/T
nyq
= 5 GHz. Thus f
ADC
= 1/(52 · T
nyq
) = 96.154 MHz;
the aggregate back-end sampling rate f
s
= 384.616 Msps,
which corresponds to undersampling the Nyquist rate by a
factor of 13. Aliasing is avoided in this measurement scheme
because (i) the modulation with the PRBS will spread the
spectrum of any tone (including high-frequency ones) across
the entire band so that one can effectively subsample, and
(ii) the input signal is again assumed to obey some model
(aside from merely being bandlimited).
Letting x denote a time-windowed vector of Nyquist-rate
samples of the input signal x(t), we can implicitly model
the RMPI measurement process as multiplication of x by a
matrix Φ having 13× fewer rows than columns. Each row of
this matrix corresponds to a portion of the PRBS sequence
used in a specific integration window from a specific channel.
As an example, if we consider a sample vector x of length
N = 1040, the matrix Φ will be block-diagonal, with each
block having 4 rows (representing the parallel operation of
the 4 channels) and T
int
/T
Nyq
= 52 columns (representing
an integration window of 52 Nyquist bins). The rows of
each block contain ±1 entries, and the overall matrix will
be composed of NT
Nyq
/T
int
= 20 blocks (one for each
integration window). Denoting the vector of measurements as
y, the RMPI mode of acquisition can be modeled as y = Φx
where Φ R
80×1040
.
We point out that high-fidelity recovery/extraction of in-
formation from CS measurements requires precise knowledge
of the system transfer function Φ. Thus, practical deviations
from the block-diagonal ±1 model described above must be
taken into account. For the measurements presented in this
paper, we construct a model of our system’s Φ matrix by
feeding in sinusoidal tones and using the output measure-
ments to characterize the system’s impulse response.
III. HARDWARE IMPLEMENTATION DESCRIPTION
A. Architecture and Operation
The RMPI presented in this work was realized with the
proprietary Northrop Grumman (NG) 450 nm InP HBT
bipolar process [26]. The process features a 4-layer metal
stack with an f
T
and f
max
> 300 GHz. Fig. 1a shows the
block diagram of the IC containing the input buffer driving
the common node of the four RD channels and the timing
generator. The timing generator is responsible for generating
the pseudo-random bit sequences (PRBS) and the clocking
waveforms to coordinate the track-and-hold (T/H) and inte-
gration operations. All analog and digital signal paths are
implemented differentially to improve common-mode noise
rejection and increase linearity of the system. The analog path
up to the integrator was designed for a 2.5 GHz bandwidth.
The ensuing integration reduces the bandwidth containing
significant energy content. The circuits following the inte-
grator are designed to meet the settling requirements of the
reduced bandwidth. A 5 GHz master clock reference (CLKin)
is used to toggle the PRBS generators and is chosen to be
the Nyquist-rate of the input bandwidth [12,17]. The T/H
operate at 1/52 the master clock frequency (= 96.154 MHz).
A switched-capacitor interleaving integrator [27] is used so
that one capacitor can be reset while the second integrates the
mixer output. Finally, an output buffer is designed to drive the
ADC with the correct swing and common-mode voltage. The
chip was designed for a full-scale input amplitude of 0.5 V
pp
differential and 1 V
pp
differential output. In operation, the
RMPI circuit takes the analog input signal, buffers it, and
distributes the buffered signal to each of the 4 channels. In
each channel, the signal is multiplied by one of 4 orthogonal
PRBS—each of which is a 3276 bit long Gold code [28]. The
resulting product is integrated by one of two sets of inter-
leaved capacitors for exactly one frame (52 CLKin cycles).
At the end of the integration period the signal is sampled and
then held for 26 CLKin cycles to allow the external ADC
to digitize the signal for post-processing. Immediately after
the signal is sampled, the capacitor begins discharging and
the second capacitor begins integrating the next frame (see
Fig. 1b). The interleaved integration capacitors are used to
avoid missing frames due to the reset operation. Additionally,
the sampling instants for each channel are staggered to create
more diversity in the windowed integrations obtained.
B. Analog Signal Path
The input buffer is a differential pair with emitter degen-
eration and 50 termination at each single-ended input. It
has a gain of 3 dB, a 2.5 GHz bandwidth, 70 dB SFDR,
and a full-scale differential input amplitude of 0.5 V
pp
. The
random modulation is performed by a standard differential
Gilbert mixer with the PRBS generator driving the top pair
and the analog input driving the bottom differential pair.
Emitter degeneration is used on the bottom differential pair
to improve linearity. To reduce noise, the mixer was designed
to have about 20 dB gain to offset the attenuation from
the integrator. The output of the mixer is integrated using
interleaved switched capacitors as shown in Fig. 1b and Fig. 2
and has a 12.5 MHz pole frequency. Input and output select
switches are closed to route the mixer output current to the
integration capacitor as well as to read out the capacitors
with the T/H circuit. When the reset switch is on, the
integration capacitor voltage is reset to zero. At the end of
each integration cycle (one frame = 10.4 ns), the output of
3
Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841]

the integrator is sampled by the T/H and held for 5.2 ns. This
ensures that the external ADC has enough time to digitize
the held voltage.
The T/H was implemented using the switched emitter
follower topology with gain 1. To minimize the hold-
mode feed-through, small feed-forward capacitors were in-
serted [29]. The switched emitter follower was chosen in
favor over the more conventional diode bridge switch for
its smaller footprint and comparatively low parasitic capaci-
tance. The amplifier after the T/H has a gain of 2. In addition
to emitter degeneration, diode connected transistors are used
in the output load to cancel the input differential pair V
be
modulation and improve linearity. The output driver was
designed to be DC-coupled to the external ADC and have
70 dB SFDR and a 1 V
pp
swing. In order to save power,
200 on-chip termination resistors were used on each side
to exploit the relatively long settling time.
In operation, the RMPI circuit takes the analog input signal, buffers it, and distributes the buffered signal to each of the 4 channels. In
each channel, the signal is multiplied by one of 4 orthogonal PRBS–each of which is a 3276 bit long Gold code. The resulting product
is integrated by one of two sets of interleaved capacitors for exactly one frame (52 CLKin cycles). At the end of the integration period
the signal is sampled and then held for 26 CLKin cycles to allow the external ADC to digitize the signal for post-processing.
Immediately after the signal is sampled, the capacitor begins discharging and the second capacitor begins integrating the next frame
(see Fig. 1b.) The interleaved integration capacitors are used to avoid missing frames due to the reset operation. Additionally, the
sampling instants for each channel are staggered to create more diversity in the windowed integrations obtained.
B. AnalogSignalPath
The input buffer is a differential pair with emitter degeneration and 50 ohms termination at each single ended input. It has a gain of
3dB, a 2.5GHz bandwidth, 70dB SFDR and can handle a 0.5V p-p differential input. The random modulation is performed by a
standard Gilbert mixer with the PRBS on the top pair and the analog input on the bottom differential pair. Emitter degeneration is
used on the bottom differential pair to improve linearity. To reduce noise, the mixer was designed to have about 20dB gain to offset
the attenuation from the integrator. The output of the mixer is integrated using interleaved switched capacitors as shown in Fig. 1b
and Fig.2 and has a 12.5 MHz pole frequency. Input and output select switches are closed to route the mixer current to the integration
capacitor and to connect it to the track-and-hold circuit. When the reset switch is on, the integration capacitor voltage is reset to zero.
At the end of each integration cycle (one frame or 10.4 ns), the output of the integrator is sampled by the T/H and held at for 5.2 ns.
This ensures that the external ADC has enough time to acquire the data accurately.
The T/H was implemented using the switched emitter follower topology with a gain of approximately 1. To minimize the hold-mode
feed-through, small feed-forward capacitors were inserted [26]. The switch emitter follower was chosen in favor over the more
conventional diode bridge switch for its smaller footprint and comparatively low parasitic capacitance. The amplifier after the T/H
has a gain of 2. In addition to emitter degeneration, diode connected transistors are used in the output load to cancel the input
differential pair Vbe modulation and improve the linearity. The output driver was designed to be DC-coupled to the external ADC
and have 70dB SFDR and a 1V-p-p swing. 200 Ohm on-chip terminations for each side was used to save power by taking advantage
of the relatively long settling time.
Fig. 2: Simplified schematic of the interleaved switch capacitor integrator. The diodes act as switches to configure capacitors for
integration or reset it based on the level of the control signal (SEL). When SEL is high, integrator A is resetting and integrator B is
integrating. When SEL+ is low, integrator B is resetting and integrator A is integrating.
C. PRBSandTimingGenerator
A master clock is applied to CLKin, from which all required timing signals are generated. The input clock buffer was biased with a
relatively high power to reduce jitter and it has 4 separate output emitter followers to limit cross-talk. Each emitter follower provides
a low jitter signal to re-clock the PRBS input before it is mixed with the RF input in each channel.
The PRBS signals are generated with two 6-bit PRBS shift-registers. One PRBS generator (PN6A in Fig. 3) is programmed to cycle
every 52 CLKin cycles while the second (PN6B) is allowed to cycle through all 63 states. The 2 PRBS generator outputs are
combined to generate 4 orthogonal 3276 bit long Gold code sequences. PN6A is also used to generate the T/H clocks (divide by 52)
and select signal (divide by 104) for the switched capacitor integrator. Both PN generators also output a sync pulse used to
synchronize the system. The output pulse from PN6B is re-clocked with the pulse form PN6A to produce a sync pulse that is 52
CLKin cycles long once every 3276 cycles. The synchronization pulse is essential to provide precise knowledge of the chipping
sequence used in each integration window which is necessary for signal-recovery and parameter estimation.
CLKin and the RF input are located on opposite sides of the chip to minimize coupling. Special attention was paid to the routing of
the PRBS, T/H clocks, and select signals to minimize clock/data coupling among the four channels. A simplified block diagram of the
PRBS/timing generator is shown in Fig.3. The timing generator block was designed to operate upto 5GHz and it consumes roughly
2.8W.
DArst+
DArst-
DAin+
DAin-
DAout+
DAout-
DBrst+
DBrst-
DBout+
DBout-
DBin+
DBin-
VCM
Fig. 2: Simplified schematic of interleaved switched capacitor integrator.
The diodes act as switches to configure capacitors for integration or reset it
based on the level of the control signal (SEL). When SEL is high, integrator
A is resetting and integrator B is integrating. When SEL+ is low, integrator
B is resetting and integrator A is integrating.
C. PRBS & Timing Generator
A master clock is applied to CLKin, from which all
required timing signals are generated. The input clock buffer
was biased with a relatively high power to reduce jitter. In
addition, it has 4 separate output emitter followers to mitigate
the deleterious effects of cross-talk on the clock jitter. Each
emitter follower provides a low jitter signal to re-clock the
PRBS input before it is mixed with the RF input in each
channel.
The PRBS signals are generated with two 6 bit PRBS gen-
erating linear-feedback shift-registers (LFSR(s)) [30]. One
PRBS generator (PN6A in Fig. 3) is programmed to cycle
every 52 CLKin cycles while the second (PN6B in Fig. 3)
is allowed to cycle through all 63 states. The 2 PRBS
generator outputs are combined to generate 4 orthogonal
52 × 63 = 3276 bit long Gold code sequences. PN6A is
also used to generate the T/H clocks (divide by 52) and
select signal (divide by 104) for the switched capacitor
integrator. Both PN generators also output a sync pulse used
to synchronize the system. The output pulse from PN6B is
re-clocked with the pulse from PN6A to produce a sync
pulse that is 52 CLKin cycles long once every 3276 cycles.
The synchronization pulse is essential to provide precise
knowledge of the chipping sequence used in each integration
window. This relative alignment information is crucial for
signal-recovery and parameter estimation.
CLKin and the RF input are located on opposite sides of
the chip to minimize coupling. Special attention was paid
to the routing of the PRBS, T/H clocks, and select signals
to minimize clock/data coupling among the four channels.
A simplified block diagram of the PRBS/timing generator is
shown in Fig. 3. The timing generator block was designed to
operate at speeds in excess of 5 GHz and consumes 2.8 W
when operated at the designed rate.
Fig. 3: Simplified block diagram of PRBS/timing generator. Shown are (4) quadrature clocks for the T/H and (4) select signals for
the interleaved capacitors.
D. PerformanceAnalysis
The only change is instead of Fig. 6 for the die photo, it’s Fig. 4
Added References
[26] P. Vorenkamp and J. Verdaasdonk, “Fully Bipolar, 120-Msample/s 10-b Track-and-Hold Circuit,” IEE Journal of Solid-State
Circuits, Vol. 27, No. 7, July 1992.
Fig. 3: Simplified block diagram of PRBS/timing generator. Shown are (4)
quadrature clocks for the S/H and (4) control signals for the interleaved
capacitors. Combinational logic is used to prevent the illegal start up
condition of the PN generators and to generate divide by 52 and divide
by 104 used as control signals in the S/H and in the interleaved integrators
respectively.
D. Performance Analysis
Simulation validation was done by performing transient-
based two-tone inter-modulation distortion simulations in the
Cadence design environment. Noise simulations were per-
formed using the periodic steady-state (PSS) mode of spectre.
The RMPI sampling system, including the off-chip ADCs
consumed 6.1 W of power. We point out that this system
was designed as a proof-of-concept and was not optimized
for power. Thus, caution should be used when comparing
the CS system in this work to conventional counterparts.
For example, the use of an InP process in this work leads
to power penalties compared to the CMOS RMPI (which
consumes 0.5 W) reported in [9,10], which is also similarly
unoptimized, due to the availability of static logic. A die
photo of the fabricated chip is shown in Fig. 4.
4
Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841]

.
Fig. 4: RMPI IC die photo. Die size is 4.0 mm ×4.4 mm.
IV. PULSE-DESCRIPTOR WORD (PDW) EXTRACTION
Having described the acquisition system, we now present
algorithms for detecting radar pulses and estimating their pa-
rameters, referred to as pulse-descriptor words (PDW), from
randomly modulated pre-integrated (RMPI) samples. The
detection process is based on familiar principles employed
by detectors that operate on Nyquist samples. Our algorithms
use a combination of template matching, energy thresholding,
and consistency estimation to determine the presence of
pulses. By using all three of these methods, we gain robust
detection at the cost of a number of tunable parameters that
must be set to appropriate levels depending on the application
and sensing equipment. The general procedure consists of
three steps: first, we estimate the carrier frequency and energy
of a potential pulse segment at various time shifts; second,
based on consistency in frequency estimates and large enough
pulse energies, we apply criteria to determine if a pulse is
present; finally, for detected pulses we use our parameter
estimation methods to refine our carrier frequency, amplitude,
phase, time-of-arrival, and time-of-departure estimates.
The remainder of the section elaborates on the procedure
and is arranged as follows. First, we describe our methods
of parametric estimation, focusing in particular on carrier
frequency estimation. After describing how we can reliably
estimate the carrier frequency of a signal from compressive
measurements, we then explain how we use such estima-
tions to form a detection algorithm that jointly uses energy
detection and consistency of our frequency measurements.
We then describe how we perform parametric estimation on
compressive samples while simultaneously removing a band
in which a known interfering signal is present. Finally, we
combine the detection algorithm we have formulated with
the cancellation technique to present an algorithm capable of
detecting multiple overlapping pulses.
A. General Parametric Estimation
Our general parameter estimation problem can be stated
as follows. We consider signals x
0
(t) drawn from one of a
collection of (low-dimensional) subspaces {S
α
} indexed by
a parameter set α = (α
1
, α
2
, . . . , α
K
). Given the measure-
ments y = Φ[x
0
]+noise, we search for the set of parameters
corresponding to the subspace which contains a signal which
comes closest to explaining the measurements y. We solve
ˆα = arg min
α
min
x∈S
α
ky Φ[x]k
2
2
. (IV.1)
The inner optimization finds the signal in S
α
that is most
consistent with the measurements for a fixed α; the outer
optimization compares these best fits for different α.
The inner optimization program, which is the classical
“closest point in a subspace” problem, has a well-known
closed form solution as it is easily formulated as a least-
squares problem.
Let u
α,1
(t), u
α,2
(t), . . . , u
α,d
(t) be a basis for the space
S
α
, meaning that
x
0
(t) S
α
x
0
(t) = a
1
u
α,1
(t) + a
2
u
α,2
(t) + · · · + a
p
u
α,d
(t),
for some unique a
1
, a
2
, . . . , a
d
R. If we define V
α
to be the
M ×d matrix containing the inner products between each pair
of RMPI test functions φ
m
(t) and basis functions u
α,i
(t),
V
α
=
hφ
1
, u
α,1
i hφ
1
, u
α,2
i · · · hφ
1
, u
α,d
i
hφ
2
, u
α,1
i hφ
2
, u
α,2
i · · · hφ
2
, u
α,d
i
.
.
.
.
.
. · · ·
.
.
.
hφ
M
, u
α,1
i hφ
1
, u
α,2
i · · · hφ
M
, u
α,d
i
(IV.2)
then we can re-write (IV.1) as
ˆα = arg min
α
ky V
α
(V
T
α
V
α
)
1
V
T
α
yk
2
2
= arg min
α
k(I P
α
)yk
2
2
, (IV.3)
where P
α
= V
α
(V
T
α
V
α
)
1
V
T
α
is the orthogonal projector
onto the column space of V
α
. It is worth mentioning that
when the measurement noise consists of independent and
identically distributed Gaussian random variables, the result
ˆα in (IV.3) is the maximum likelihood estimate (MLE). When
the noise is correlated, we may instead pose the optimization
in terms of a weighted least squares problem.
In Sec. IV-B and Sec. IV-C below, we will discuss the
particular cases of frequency estimation for an unknown tone,
and time-of-arrival estimation for a square pulse modulated
to a known frequency. In both of these cases, we are trying
to estimate one parameter and the underlying subspaces S
α
have dimension d = 2. Moreover, the functional k(IP
α
)yk
2
2
5
Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841]

Figures (14)
Citations
More filters

Journal ArticleDOI
TL;DR: A wide bandwidth, compressed sensing based nonuniform sampling (NUS) system with a custom sample-and-hold chip designed to take advantage of a low average sampling rate is presented.
Abstract: We present a wide bandwidth, compressed sensing based nonuniform sampling (NUS) system with a custom sample-and-hold chip designed to take advantage of a low average sampling rate. By sampling signals nonuniformly, the average sample rate can be more than a magnitude lower than the Nyquist rate, provided that these signals have a relatively low information content as measured by the sparsity of their spectrum. The hardware design combines a wideband Indium-Phosphide heterojunction bipolar transistor sample-and-hold with a commercial off-the-shelf analog-to-digital converter to digitize an 800 MHz to 2 GHz band (having 100 MHz of noncontiguous spectral content) at an average sample rate of 236 Ms/s. Signal reconstruction is performed via a nonlinear compressed sensing algorithm, and the challenges of developing an efficient implementation are discussed. The NUS system is a general purpose digital receiver. As an example of its real signal capabilities, measured bit-error-rate data for a GSM channel is presented, and comparisons to a conventional wideband 4.4 Gs/s ADC are made.

127 citations


Cites background or methods from "A Compressed Sensing Parameter Extr..."

  • ...Our implementation is such a nonuniform sampling (NUS) approach, which we describe here and treat in more detail in Section II....

    [...]

  • ...Numerical simulations are required regardless since CS theories rely on possibly conservative constants and also on signal-dependent parameters, such as the sparsity of the signal....

    [...]


Journal ArticleDOI
Abstract: Traditional radar sensing typically employs matched filtering between the received signal and the shape of the transmitted pulse. Matched filtering (MF) is conventionally carried out digitally, after sampling the received analog signals. Here, principles from classic sampling theory are generally employed, requiring that the received signals be sampled at twice their baseband bandwidth. The resulting sampling rates necessary for correlation-based radar systems become quite high, as growing demands for target distinction capability and spatial resolution stretch the bandwidth of the transmitted pulse. The large amounts of sampled data also necessitate vast memory capacity. In addition, real-time data processing typically results in high power consumption. Recently, new approaches for radar sensing and estimation were introduced, based on the finite rate of innovation (FRI) and Xampling frameworks. Exploiting the parametric nature of radar signals, these techniques allow significant reduction in sampling rate, implying potential power savings, while maintaining the system's estimation capabilities at sufficiently high signal-to-noise ratios (SNRs). Here we present for the first time a design and implementation of an Xampling-based hardware prototype that allows sampling of radar signals at rates much lower than Nyquist. We demonstrate by real-time analog experiments that our system is able to maintain reasonable recovery capabilities, while sampling radar signals that require sampling at a rate of about 30 MHz at a total rate of 1 MHz.

98 citations


Proceedings ArticleDOI
08 Jul 2014
TL;DR: BigBand is presented, a technology that can capture GHz of spectrum in realtime without sampling the signal at GS/s - i.e., without high speed ADCs - and an extension of BigBand can perform GHz spectrum sensing even in scenarios where the spectrum is not sparse.
Abstract: We present BigBand, a technology that can capture GHz of spectrum in realtime without sampling the signal at GS/s -i.e., without high speed ADCs. Further, it is simple and can be implemented on commodity low-power radios. Our approach builds on recent advances in the area of sparse Fourier transforms, which show that it is possible to reconstruct a sparse signal without sampling it at the Nyquist rate. To demonstrate our design, we implement it using 3 software radios, each sampling the spectrum at 50 MS/s, producing a device that captures 0.9 GHz — i.e., 6× larger digital bandwidth than the three software radios combined. Finally, an extension of BigBand can perform GHz spectrum sensing even in scenarios where the spectrum is not sparse.

84 citations


Cites methods from "A Compressed Sensing Parameter Extr..."

  • ...Once x̂ is computed, it can recover the time signal x and decode the wireless symbols....

    [...]


Posted Content
TL;DR: This work presents for the first time a design and implementation of an Xampling-based hardware prototype that allows sampling of radar signals at rates much lower than Nyquist, and demonstrates by real-time analog experiments that the system is able to maintain reasonable recovery capabilities, while sampling radar signals that require sampling at a rate of about 30 MHz at a total rate of 1 MHz.
Abstract: Traditional radar sensing typically involves matched filtering between the received signal and the shape of the transmitted pulse. Under the confinement of classic sampling theorem this requires that the received signals must first be sampled at twice the baseband bandwidth, in order to avoid aliasing. The growing demands for target distinction capability and spatial resolution imply significant growth in the bandwidth of the transmitted pulse. Thus, correlation based radar systems require high sampling rates, and with the large amounts of data sampled also necessitate vast memory capacity. In addition, real-time processing of the data typically results in high power consumption. Recently, new approaches for radar sensing and detection were introduced, based on the Finite Rate of Innovation and Xampling frameworks. These techniques allow significant reduction in sampling rate, implying potential power savings, while maintaining the system's detection capabilities at high enough SNR. Here we present for the first time a design and implementation of a Xampling-based hardware prototype that allows sampling of radar signals at rates much lower than Nyquist. We demostrate by real-time analog experiments that our system is able to maintain reasonable detection capabilities, while sampling radar signals that require sampling at a rate of about 30MHz at a total rate of 1Mhz.

83 citations


Additional excerpts

  • ...X AMPLING OF RADAR SIGNALS...

    [...]


Journal ArticleDOI
Abstract: In structural health monitoring (SHM) systems for civil structures, massive amounts of data are often generated that need data compression techniques to reduce the cost of signal transfer and storage, meanwhile offering a simple sensing system. Compressive sensing (CS) is a novel data acquisition method whereby the compression is done in a sensor simultaneously with the sampling. If the original sensed signal is sufficiently sparse in terms of some orthogonal basis (e.g., a sufficient number of wavelet coefficients are zero or negligibly small), the decompression can be done essentially perfectly up to some critical compression ratio; otherwise there is a trade-off between the reconstruction error and how much compression occurs. In this article, a Bayesian compressive sensing (BCS) method is investigated that uses sparse Bayesian learning to reconstruct signals from a compressive sensor. By explicitly quantifying the uncertainty in the reconstructed signal from compressed data, the BCS technique exhibits an obvious benefit over existing regularized norm-minimization CS methods that provide a single signal estimate. However, current BCS algorithms suffer from a robustness problem: sometimes the reconstruction errors are very large when the number of measurements K are a lot less than the number of signal degrees of freedom N that are needed to capture the signal accurately in a directly sampled form. In this article, we present improvements to the BCS reconstruction method to enhance its robustness so that even higher compression ratios N/K can be used and we examine the trade-off between efficiently compressing data and accurately decompressing it. Synthetic data and actual acceleration data collected from a bridge SHM system are used as examples. Compared with the state-of-the-art BCS reconstruction algorithms, the improved BCS algorithm demonstrates superior performance. With the same acceptable error rate based on a specified threshold of reconstruction error, the proposed BCS algorithm works with relatively large compression ratios and it can achieve perfect loss-less compression performance with quite high compression ratios. Furthermore, the error bars for the signal reconstruction are also quantified effectively.

73 citations


References
More filters

Book
D.L. Donoho1
01 Jan 2004
TL;DR: It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients, and a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing.
Abstract: Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0

18,593 citations


Journal ArticleDOI
Abstract: This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=/spl sigma//sub /spl tau//spl isin/T/f(/spl tau/)/spl delta/(t-/spl tau/) obeying |T|/spl les/C/sub M//spl middot/(log N)/sup -1/ /spl middot/ |/spl Omega/| for some constant C/sub M/>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the /spl lscr//sub 1/ minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C/sub M/ which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|/spl middot/logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N/sup -M/) would in general require a number of frequency samples at least proportional to |T|/spl middot/logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.

13,375 citations


Journal ArticleDOI
TL;DR: The theory of compressive sampling, also known as compressed sensing or CS, is surveyed, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition.
Abstract: Conventional approaches to sampling signals or images follow Shannon's theorem: the sampling rate must be at least twice the maximum frequency present in the signal (Nyquist rate). In the field of data conversion, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation - the signal is uniformly sampled at or above the Nyquist rate. This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use.

8,847 citations


Journal ArticleDOI
TL;DR: Practical incoherent undersampling schemes are developed and analyzed by means of their aliasing interference and demonstrate improved spatial resolution and accelerated acquisition for multislice fast spin‐echo brain imaging and 3D contrast enhanced angiography.
Abstract: The sparsity which is implicit in MR images is exploited to significantly undersample k -space. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain–for example, in terms of spatial finite-differences or their wavelet coefficients. According to the recently developed mathematical theory of compressedsensing, images with a sparse representation can be recovered from randomly undersampled k -space data, provided an appropriate nonlinear recovery scheme is used. Intuitively, artifacts due to random undersampling add as noise-like interference. In the sparse transform domain the significant coefficients stand out above the interference. A nonlinear thresholding scheme can recover the sparse coefficients, effectively recovering the image itself. In this article, practical incoherent undersampling schemes are developed and analyzed by means of their aliasing interference. Incoherence is introduced by pseudo-random variable-density undersampling of phase-encodes. The reconstruction is performed by minimizing the 1 norm of a transformed image, subject to data

5,802 citations


Journal ArticleDOI
TL;DR: It is shown that ‘1 minimization recovers x 0 exactly when the number of measurements exceeds m Const ·µ 2 (U) ·S · logn, where S is the numberof nonzero components in x 0, and µ is the largest entry in U properly normalized: µ(U) = p n · maxk,j |Uk,j|.
Abstract: We consider the problem of reconstructing a sparse signal x 0 2 R n from a limited number of linear measurements. Given m randomly selected samples of Ux 0 , where U is an orthonormal matrix, we show that ‘1 minimization recovers x 0 exactly when the number of measurements exceeds m Const ·µ 2 (U) ·S · logn, where S is the number of nonzero components in x 0 , and µ is the largest entry in U properly normalized: µ(U) = p n · maxk,j |Uk,j|. The smaller µ, the fewer samples needed. The result holds for “most” sparse signals x 0 supported on a fixed (but arbitrary) set T. Given T, if the sign of x 0 for each nonzero entry on T and the observed values of Ux 0 are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about this many samples.

2,036 citations


"A Compressed Sensing Parameter Extr..." refers background in this paper

  • ...Index Terms—Compressed sensing, Indium-Phosphide, Parameter Estimation, Random-Modulation Pre-Integration I. INTRODUCTION A principal goal in the design of modern electronic systems is to acquire large amounts of information quickly and with little expenditure of resources....

    [...]

  • ...In short, the CS theory states that signals with high overall bandwidth but comparatively low information level can be acquired very efficiently using randomized measurement protocols....

    [...]


Frequently Asked Questions (1)
Q1. What have the authors contributed in "A compressed sensing parameter extraction platform for radar pulse signal acquisition" ?

In this paper the authors present a complete ( hardware/software ) sub-Nyquist rate ( ×13 ) wideband signal acquisition chain capable of acquiring radar pulse parameters in an instantaneous bandwidth spanning 100 MHz–2.