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Contents
List of Contributors page xi
Introduction xiv
List of Symbols xx
1 Sub-Nyquist Radar: Principles and Prototypes 1
Kumar Vijay Mishra and Yonina C. Eldar
1.1 Introduction 1
1.2 Prior Art and Historical Notes 3
1.3 Temporal Sub-Nyquist Radar 5
1.4 Doppler Sub-Nyquist Radar 15
1.5 Cognitive Sub-Nyquist Radar and Spectral Coexistence 18
1.6 Spatial Sub-Nyquist: Application to MIMO Radar 29
1.7 Sub-Nyquist SAR 39
1.8 Summary 43
References 44
2 Clutter Rejection and Adaptive Filtering in Compressed Sensing Radar 49
Peter B. Tuuk
2.1 Introduction 49
2.2 Problem Formulation 50
2.3 Interference Sources 53
2.4 Signal Processing Treatment of Clutter 55
2.5 Measurement Compression 58
2.6 Estimating Interference Statistics from Compressed Measurements 59
2.7 Mitigating Clutter in Compressed Sensing Estimation 66
2.8 Summary 68
References 69
3 RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging 72
Tianyi Zhang, Jiaying Ren, Jian Li, David J. Greene, Jeremy A. Johnston, and Lam H. Nguyen
3.1 Introduction 72
3.2 RPCA for RFI Mitigation 75
3.3 CLEAN-BIC for RFI Mitigation 82
vii

viii Contents
3.4 Enhanced Algorithms for RFI Mitigation 91
3.5 Performance Evaluations 92
3.6 Conclusions 101
3.7 Acknowledgment 102
References 102
4 Compressed CFAR Techniques 105
Laura Anitori and Arian Maleki
4.1 Introduction 105
4.2 Radar Signal Model 105
4.3 Classical Radar Detection 106
4.4 CS Radar Detection 110
4.5 Complex Approximate Message Passing (CAMP) Algorithm 112
4.6 Target Detection Using CAMP 115
4.7 Adaptive CAMP Algorithm 118
4.8 Simulation Results 120
4.9 Experimental Results 127
4.10 Conclusions 131
References 132
5 Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays 135
Haley H. Kim and Alexander M. Haimovich
5.2 STAP Radar Concepts 137
5.3 STAP Detection Problem 145
5.4 Compressive Sensing CFAR Detection 148
5.5 Numerical Results 157
5.6 Summary 161
References 162
6 Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter 165
Xiaopeng Yang, Yuze Sun, Xuchen Wu, Teng Long, and Tanpan K. Sarkar
6.2 Signal Models 166
6.3 Sparsity Principle Analysis of STAP 168
6.4 Fast and Robust Sparsity-Based STAP Methods 172
6.5 Conclusions 190
References 190
7 Super-Resolution Radar Imaging via Convex Optimization 193
Reinhard Heckel

Contents ix
7.2 Signal Model and Problem Statement 195
7.3 Atomic Norm Minimization and Associated Performance Guarantees 199
7.4 Super-Resolution Radar on a Fine Grid 204
7.5 Proof Outline 207
7.6 MIMO Radar 211
7.7 Discussion and Current and Future Research Directions 219
References 222
8 Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix 225
Yujie Gu, Nathan A. Goodman, and Yimin D. Zhang
8.2 Adaptive Beamforming Criterion 228
8.3 Covariance Matrix Reconstruction-Based Adaptive Beamforming 234
8.4 Simulation Results 240
8.5 Conclusion 252
References 252
9 Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation 257
Augusto Aubry, Vincenzo Carotenuto, Antonio De Maio, and Mark A. Govoni
9.2 System Model and Problem Formulation 259
9.3 2-D Radio Environmental Map Recovery Strategies 263
9.4 Performance Analyses 270
9.5 Conclusions 280
References 280
10 Cooperative Spectrum Sharing between Sparse Sensing-Based
Radar and Communication Systems 284
Bo Li and Athina P. Petropulu
10.2 MIMO Radars Using Sparse Sensing 286
10.3 Coexistence System Model 293
10.4 Cooperative Spectrum Sharing 297
10.5 Numerical Results 309
References 316
11 Compressed Sensing Methods for Radar Imaging in the Presence of Phase Errors
and Moving Objects 321
Ahmed Shaharyar Khwaja, Naime Ozben Onhon, and Mujdat Cetin
11.1 Introduction and Outline of the Chapter 321
11.2 Compressed Sensing and Radar Imaging 322

x Contents
11.3 Synthetic Aperture Radar Autofocus and Compressed Sensing 328
11.4 Synthetic Aperture Radar Moving Target Imaging and Compressed Sensing 333
11.5 Inverse Synthetic Aperture Radar Imaging and Compressed Sensing 341
References 349
Index 355

Contributors
Laura Anitori
Netherlands Organisation for Applied Scientific Research (TNO)
Augusto Aubry
University of Naples Federico II
Vincenzo Carotenuto
Mujdat Cetin
University of Rochester; Sabanci University
Antonio De Maio
Yonina C. Eldar
Weizmann Institute of Science
David J. Greene
University of Florida
Nathan A. Goodman
University of Oklahoma
Mark A. Govoni
US Army Research Laboratory
Yujie Gu
Temple University
Alexander M. Haimovich
New Jersey Institute of Technology
xi

xii List of Contributors
Reinhard Heckel
Rice University
Jeremy A. Johnston
Ahmed Shaharyar Khwaja
Sabanci University
Haley H. Kim
New Jersey Institute of Technology
Bo Li
Qualcomm
Jian Li
Teng Long
Beijing Institute of Technology
Arian Maleki
Columbia University
Kumar Vijay Mishra
Technion Israel Institute of Technology
Lam H. Nguyen
US Army Research Laboratory
Naime Ozben Onhon
Turkish-German University
Athina P. Petropulu
Rutgers, State University of New Jersey
Jiaying Ren
Tapan K. Sarkar
Syracuse University
Yuze Sun
Tsinghua University

List of Contributors xiii
Peter B. Tuuk
Georgia Tech Research Institute
Xuchen Wu
Xiaopeng Yang
Tianyi Zhang
Yimin D. Zhang
Temple University

Introduction
Digital signal processing (DSP) is a revolutionary paradigm shift that enables processing
of physical data in the digital domain, where design and implementation are consider-
ably simplified. The success of DSP has driven the development of sensing and pro-
cessing systems that are more robust, flexible, cheaper, and, consequently, more widely
used than their analog counterparts. As a result of this success, the amount of data gener-
ated by sensing systems has grown considerably. Furthermore, in modern applications,
signals of wider bandwidth are used in order to convey more information and to enable
high resolution in the context of imaging. Unfortunately, in many important and emerg-
ing applications, the resulting sampling rate is so high that far too many samples need to
be transmitted, stored, and processed. In addition, in applications involving very wide-
band inputs it is often very costly, and sometimes even physically impossible, to build
devices capable of acquiring samples at the necessary rate. Thus, despite extraordinary
advances in sampling theory and computational power, the acquisition and processing
of signals in application areas such as radar, wideband communications, imaging, and
medical imaging continue to pose a tremendous challenge.
Recent advances in compressed sensing (CS) and sampling theory provide a frame-
work to acquire a wide class of analog signals at rates below the Nyquist rate, and
to perform processing at this lower rate as well. Together with the theory, various
prototypes have been developed that demonstrate the feasibility of sampling and pro-
cessing signals at sub-Nyquist rates in a robust and cost-effective fashion. More specif-
ically, CS is a framework that enables acquisition and recovery of sparse vectors from
underdetermined linear systems. This research area has seen enormous growth over the
past decade and has been explored in many areas of applied mathematics, computer
science, statistics, and electrical engineering. At its core, CS enables recovery of sparse
high-dimensional vectors from highly incomplete measurements using very efficient
optimization algorithms. More specifically, consider a vector x of length n. The vector
is said to be k-sparse if it has at most k nonzero components. More generally, CS results
apply to signals that are sparse in an appropriate basis or overcomplete representation.
The main idea underlying CS is that the vector x can be recovered from measurements
y = Ax, where y is of length m n as long as A satisfies certain mathematical
properties that render it a suitable CS matrix. The number of measurements m can be
chosen on the order of k log n, which in general is much smaller than the length of
the vector x. A large body of work has been published on a variety of optimization
algorithms that can recover x efficiently and robustly when m ≈ k log n. Loosely
xiv

Introduction xv
speaking, the theory of CS deals with conditions under which the recovery of informa-
tion has vanishing or small errors. The mathematical framework of CS has inspired new
acquisition methods and new signal processing applications in a large variety of areas,
including image processing, analog to digital conversion, communication systems, and
radar processing. In many of these examples the basic ideas underlying CS need to be
extended to include, for example, continuous-time inputs, practical sampling methods,
other forms of structure on the input, computational aspects, noise affects, different
metrics for recovery performance, nonlinear acquisition methods, and more.
Two books devoted to this topic have been published recently, which focus on
many of these aspects, as well as on the underlying mathematical results [1,2]. Their
main emphasis is on the basic underlying theory and its generalizations, optimization
methods, as well as applications primarily to image processing and analog-to-digital
conversion. The latter is also covered in depth in [3].
Radar signal processing represents a fertile field for CS applications. By their very
nature, radars collect data about surveillance volumes (search radars), targets (tracking
radars), terrain and ground targets (imaging radars), or buried objects (radar tomogra-
phy). From radar’s early days in World War II, through the emergence of digital radar in
the 1970s, to today’s advanced systems, the amount of data a radar system has to handle
has increased by orders of magnitude. While early digital radars had to contend with 10s
and 100s of kbps, today’s radars may be faced with data rates in the Gbps range or more,
leading to demanding requirements in cost, hardware, data storage, and processing. The
implications of applying CS to radar are potentially enormous: sampling rates could
be lowered, the number of antenna elements in large arrays might be reduced and the
computers required to handle the data may be downsized.
This book aims to present the latest theoretical and practical advances in radar signal
processing using tools from CS. In particular, this book offers an up-to-date review of
fundamental and practical aspects of sparse reconstruction in radar and remote sensing,
demonstrating the potential benefits achievable with the CS paradigm. We take a wider
scope than previous edited books on CS-based radars: we do not restrict ourselves to
specific disciplines (such as earth observation as in [4]) or applications (such as urban
sensing as in [5]), but discuss a variety of diverse application fields, including clutter
rejection, constant false alarm rate (CFAR) processing, adaptive beamforming, random
arrays for radar, space–time adaptive processing (STAP), multiple input multiple output
(MIMO) systems, radar super-resolution, cognitive radar [6] applications involving sub-
Nyquist sampling and spectrum sensing, radio frequency interference (RFI) suppres-
sion, and synthetic aperture radar (SAR).
The book is aimed at postgraduate students, PhD students, researchers, and engi-
neers working on signal processing and its applications to radar systems, as well as
researchers in other fields seeking an understanding of the potential applications of
CS. To read and fully understand the content it is assumed that the reader has some
background in probability theory and random processes, matrix theory, linear algebra,
and optimization theory, as well as radar systems. The book is organized into eleven
chapters broadly cathegorized into five areas: sub-Nyquist radar (Chapter 1); detection,
clutter/interference mitigation, and CFAR techniques (Chapters 2–6); super-resolution

xvi Introduction
and beamforming (Chapters 7 and 8); radar spectrum sensing/sharing (Chapters 9 and
10); radar imaging (Chapter 11). Each chapter is self-contained and typically covers
three main aspects: fundamental theoretical principles, overview of the current state of
the art, and emerging/future research directions. Some chapters are also complemented
with analyses on real data. Since the chapters are independent, there is flexibility in
selecting material both for university courses and short seminars.
In Chapter 1, the authors review several sub-Nyquist pulse-Doppler radar systems
based on the Xampling framework. Contrary to other CS-based designs, their formu-
lations directly address the reduced-rate analog sampling in space and time, avoid a
prohibitive dictionary size, and are robust in the face of noise and clutter. The chapter
begins by introducing temporal sub-Nyquist processing for estimating the target loca-
tions using less bandwidth than conventional systems. This paves the way to cognitive
radars, which share their transmit spectrum with other communication services, thereby
providing a robust solution for coexistence in spectrally crowded environments. Next,
without impairing Doppler resolution, the authors reduce the dwell time by transmitting
interleaved radar pulses in a scarce manner within a coherent processing interval or
slow time. Then, they consider MIMO array radars and demonstrate spatial sub-Nyquist
processing, which allows the use of few antenna elements without degradation in
angular resolution. Finally, they demonstrate application of sub-Nyquist and cognitive
radars to imaging systems such as SAR. For each setting, the authors present a state-
of-the-art hardware prototype designed to demonstrate the real-time feasibility of
sub-Nyquist radars.
Chapter 2 discusses the problem of clutter mitigation, which has posed challenges to
radar designers and engineers since the early days of radar. Early techniques matured to
current approaches like STAP, which use a coherently processed data cube to estimate
clutter statistics and to perform adaptive filtering. This chapter examines CS techniques
for the mitigation of structured interference, such as clutter. The author first introduces
the relevant sensing model and describes results in uncompressed adaptive filtering.
This paves the way to the development of models for measurement compression of the
coherent data cube and of approaches to estimate and filter clutter from compressed
measurements. The chapter includes recent results showing how clutter second-order
statistics can be reliably estimated from compressed measurements if the clutter has
well-controlled eigenspectrum. Additionally, the covariance of the interference can be
incorporated into the CS estimation process to improve performance.
RFIs pose serious threats to the proper operations of ultra wideband (UWB) radar
systems due to severely degrading their imaging and target detection capabilities. RFI
mitigation is a challenging problem, since dynamic RFI sources utilize diverse mod-
ulation schemes, hence they are difficult to model precisely. Fortunately, RFI sources
possess certain unique properties that can be exploited for their mitigation. In Chapter 3
the authors propose several sparse signal recovery methods for effective RFI mitigation.
They first show that the RFI sources possess a low rank property and are sparse in the
frequency domain, while in contrast the desired UWB radar echoes are sparse in the time
domain. Therefore, robust principal component analysis (RPCA) can be used to simul-
taneously exploit these properties for effective RFI mitigation. RPCA, however, requires

Introduction xvii
a fine tuning of a user parameter, which is dependent on the signal-to-interference ratio
(SIR). This parameter tuning is not straightforward in practice due to the lack of prior
knowledge on the RFI sources and on the desired UWB radar echoes. To avoid the
user parameter tuning problem, the authors consider modeling the RFI sources within
a pulse repetition interval (PRI) as a sum of sinusoids. The CLEAN algorithm can
then be used with the Bayesian information criterion (BIC) to determine the number
of sinusoids and to estimate their parameters. They show that CLEAN-BIC is user-
parameter-free and can be used to remove dominant RFI sources effectively. However,
since the sparse property of the UWB radar echoes are not utilized by CLEAN-BIC, the
resulting SAR images appear noisy, especially for low SIR values. To take advantage
of the merits of both RPCA and CLEAN-BIC algorithms, the authors consider using
CLEAN-BIC to estimate SIR, and the estimated SIR value is then used to determine
the user parameter for the RPCA algorithm. Finally, the algorithms are applied to both
simulated and experimentally measured data for performance evaluation.
Chapter 4 is focused on target detection from a set of compressive radar measure-
ments corrupted by additive white Gaussian noise. The complications in the calculation
of false alarm and detection probabilities that are caused by the nonlinear nature of target
recovery schemes in CS have impeded the application of such systems in practice. In
this chapter, the authors aim to show how recent advances in the asymptotic analysis of
CS recovery algorithms help to overcome this challenge. Fully adaptive and practical
CS target detection schemes are provided together with a detailed analysis of their
performance through extensive simulated and experimental data.
In Chapter 5, the authors present CFAR detectors for STAP random arrays. The
problem is formulated as detection of sparse targets given space–time observations
from thinned random arrays. The observations are corrupted by colored Gaussian noise
of an unknown covariance matrix, but secondary data are available for estimating the
covariance matrix. It is shown that the number of elements required to constrain the
peak sidelobe level scales logarithmically with the array aperture, whereas the number
of elements of a uniform linear array (ULA) scales linearly with the array aperture. New
adaptive detectors are developed that cope with the high sidelobes of random arrays.
Performance and complexity analysis demonstrate high performance at a reasonable
computation cost with significantly fewer elements than a ULA.
In Chapter 6, sparse-based STAP methods are developed by exploiting the intrinsic
sparsity of the clutter spatial-temporal power spectrum and of the space–time adaptive
weight vectors. First, the signal model of received space–time data for an airborne
phased array radar is introduced, and the intrinsic model sparsity for radar STAP is
analyzed. Second, leveraging on the sparsity of clutter spatial-temporal power spectrum,
a robust and fast iterative sparse recovery method is introduced. It can not only alleviate
the effect of noise and dictionary mismatch but can also reduce the computational com-
plexity via recursive inverse matrix calculation. Finally, based on the sparsity of space–
time adaptive weight vectors, a fast STAP method based on projection approximation
subspace tracking (PAST) with a sparse constraint is discussed. It provides a robust
and stable estimation of the clutter subspace when a small set of training samples is
available. Based on both the simulated and actual airborne phased array radar data, it is

xviii Introduction
verified that the developed methods can provide satisfactory performance with a small
training sample support in a practical complex nonhomogeneous environment.
Chapter 7 considers the use of CS techniques for the resolution of multiple targets.
Estimating the relative angles, delays, and Doppler shifts from the received signals
allows for the determination of the locations and velocities of objects. However, due to
practical constraints, the probing signals have finite bandwidth B, the received signals
are observed over a finite time interval of length T only, and in addition, a radar typically
has only one or a few transmit and receive antennas. Those constraints fundamentally
limit the resolution up to which objects can be localized: the delay and Doppler reso-
lution is proportional to 1/B and 1/T, and a radar with NT transmit and NR receive
antennas can only achieve an angular resolution proportional to 1/(NT NR). The author
shows that the continuous angle-delay-Doppler triplets and the corresponding attenua-
tion factors can be resolved at much finer resolution, using ideas from CS. Specifically,
provided the angle-delay-Doppler triplets are separated either by factors proportional
to 1/(NT NR − 1) in angle, 1/B in delay, or 1/T in Doppler direction, they can be
recovered at significantly smaller scale or higher resolution.
Traditional adaptive beamformers are very sensitive to model mismatch, especially
when the training samples for adaptive beamformer design are contaminated by the
desired signal. In Chapter 8, the authors propose a strategy to reconstruct a signal-
free interference-plus-noise covariance matrix for adaptive beamformer design. Using
the sparsity of sources, the interference covariance matrix can be reconstructed as a
weighted sum of the tensor outer products of the interference steering vectors, and the
corresponding parameters are estimated from a sparsity-constrained covariance matrix
fitting problem. In contrast to classical CS and sparse reconstruction problems, the for-
mulated sparsity-constrained covariance matrix fitting problem can be effectively solved
by using the a priori information on array structure rather than using convex relaxation.
Simulation results demonstrate that the proposed adaptive beamformer almost always
provides near-optimal performance.
Chapter 9 deals with two-dimensional (2-D) spectrum sensing in the context of a
cognitive radar to gather real-time space–frequency electromagnetic awareness. Assum-
ing a sensor equipped with multiple receive antennas, a formal discrete-time sensing
signal model is developed, and two signal processing techniques capable of recovering
the space–frequency occupancy map via block sparsity exploitation are presented. The
former relies on the iterative adaptive algorithm (IAA) and incorporates a BIC-based
stage to foster block-sparsity in the recovery process. The latter resorts to the regularized
maximum likelihood (RML) estimation paradigm, which automatically promotes block-
sparsity in the 2-D profile evaluation. Some illustrative examples (both on simulated and
real data) are provided to compare the different strategies and highlight the effectiveness
of the developed approaches.
In Chapter 10, a cooperative spectrum-sharing scheme for a MIMO communication
system and a sparse sensing-based MIMO radar is presented. Both the radar and the
communication systems use transmit precoding. The radar transmit precoder, the radar
subsampling scheme, and the communication transmit covariance matrix are jointly
designed in order to maximize the radar SIR, while meeting certain communication

Introduction xix
rate and power constraints. The joint design is implemented at a control center, which
is a node with which both systems share physical layer information, and which also
performs data fusion for the radar. Efficient algorithms for solving the correspond-
ing optimization problem are presented. The cooperative design significantly improves
spectrum sharing performance, and the sparse sensing provides opportunities to control
interference.
Chapter 11 discusses applications of CS to radar imaging problems with reference
to SAR and inverse synthetic aperture radar (ISAR) sensors. The authors first provide
the relevant mathematical expressions for CS and SAR necessary to formulate the prob-
lem of CS SAR imaging. Thereafter, they consider the case where unknown motion
errors are present during the SAR acquisition process. Autofocusing, i.e., the blind
compensation of the aforementioned errors, is discussed, and general CS solutions are
presented. The chapter ends with a survey of CS methods for ISAR imaging of targets
with unknown motion.
References
[1] Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications. Cambridge
University Press, 2012.
[2] S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing. Birkhäuser
Basel, 2013, vol. 1, no. 3.
[3] Y. C. Eldar, Sampling Theory: Beyond Bandlimited Systems. Cambridge University Press,
2015.
[4] C.-H. Chen, Compressive Sensing of Earth Observations. CRC Press, 2017.
[5] M. Amin, Compressive Sensing for Urban Radar. CRC Press, 2014.
[6] A. Farina, A. De Maio, and S. Haykin, The Impact of Cognition on Radar Technology. Scitech
Publishing, Radar, Sonar Navigation, 2017.

Symbols
A unified notation is used throughout the book.
z column vector (lower case)
Z matrix (upper case)
zi ith element of z
Zi,l (i,l)-th entry of Z
A sensing matrix
sparsity matrix
= A product
y observed measurement vector
x original signal vector
k sparsity
n ambient dimension
m number of measurements
· p p-norm
(·)T transpose operator
(·)∗ conjugate operator
(·)H conjugate transpose operator
(·)† pseudo inverse of the matrix argument
tr (·) trace of the square matrix argument
Rank (·) rank of the square matrix argument
λmax(·) maximum eigenvalue of the square matrix argument
λmin(·) minimum eigenvalue of the square matrix argument
diag(x) N-dimensional diagonal matrix whose ith diagonal element
is xi, i = 1,. . .,N, with x ∈ CN
Range (A) range span of the column vectors of the matrix A
I identity matrix (its size is determined from the context)
0 matrix with zero entries (its size is determined from the context)
RN set of N-dimensional vectors of real numbers
CN set of N-dimensional vectors of complex numbers
HN set of N × N Hermitian matrices
for any A ∈ HN , A 0 means that A is a positive semidefinite matrix
for any A ∈ HN , A 0 means that A is a positive definite matrix
xx

List of Symbols xxi
T standard notation for sets (uppercase letter)
|T | cardinality of a set T
x̂ result of 1 minimization/recovery algorithm
supp(x) support of vector x
I standard notation for subset of indices
xT length-|T | sub-vector containing the elements of x
corresponding to the indices in T
AT m × |T | sub-matrix containing the columns of
the m × n matrix A indexed by T
j imaginary unit
Re(x) real part of the complex number x
Im(x) imaginary part of the complex number x
|x| modulus of the complex number x
arg(x) argument of the complex number x
E [·] statistical expectation
Hadamard product
⊗ Kronecker product
ẏ, ∂y
∂x , dy
dx first derivative of y with respect to variable x
ÿ, ∂2y
∂x2 , d2y
dx2 second derivative of y with respect to variable x
P[·] probability measure
x(t) continuous time signal
h(t) pulse shape
xi measurements of x(t)
δk = δk(A) restricted isometry constant.
Statement of restricted isometry property (RIP): a matrix A satisfies the RIP of order
K if
(1 − δk)x2 ≤ Ax2 ≤ (1 + δk)x2
for all x with x0 ≤ K.

1 Sub-Nyquist Radar: Principles
and Prototypes
Kumar Vijay Mishra∗ and Yonina C. Eldar∗∗
1.1 Introduction
Radar remote sensing has advanced tremendously since 1950 and is now applied to
diverse areas such as military surveillance, meteorology, geology, collision avoidance,
and imaging [1]. In monostatic pulse-Doppler radar systems, an antenna transmits a
periodic train of known narrowband pulses within a defined coherent processing interval
(CPI). When the radiated wave from the radar interacts with moving targets, the ampli-
tude, frequency, and polarization states of the scattered wave change. By monitoring this
change, it is possible to infer the targets’ size, location, and radial Doppler velocity. The
reflected signal received by the radar antenna is a linear combination of echoes from
multiple targets; each of these is an attenuated, time-delayed, and frequency-modulated
version of the transmit signal. The delay in the received signal is linearly proportional
to the target’s range or its distance from the radar. The frequency modulation encodes
the Doppler velocity of the target. The complex amplitude or target’s reflectivity is a
function of the target’s size, geometry, propagation, and scattering mechanism. Radar
signal processing is aimed at detecting the targets and estimating their parameters from
the output of this linear, time-varying system.
Traditional radar signal processing employs matched filtering (MF) or pulse com-
pression [2] in the digital domain, wherein the sampled received signal is correlated
with a replica of the transmit signal in the delay-Doppler plane. The MF maximizes
the signal-to-noise ratio (SNR) in the presence of additive white Gaussian noise. In
some specialized systems, this stage is replaced by a mismatched filter with a different
optimization metric such as minimization of peak-to-sidelobe ratio of the output. Here,
the received signal is correlated with a signal that is close but not identical to the
transmit signal [3–5]. While all of these techniques reliably estimate target parameters,
their resolution is inversely proportional to the support of the ambiguity function of the
transmit pulse, thereby restricting ability to super-resolve targets that are closely spaced.
The digital MF operation requires the signal to be sampled at or above the Nyquist
sampling rate, which guarantees perfect reconstruction of a bandlimited analog signal
[6]. Many modern radar systems use wide bandwidths, typically ranging from hundreds
∗ K.V.M. acknowledges partial support via the Andrew and Erna Finci Viterbi Postdoctoral Fellowship and
the Lady Davis Postdoctoral Fellowship.
∗∗ This work is supported by the European Union’s Horizon 2020 research and innovation program under
grant agreement no. 646804-ERC-COG-BNYQ.
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now, I can no longer blame you for having observed so much
mystery: Oh! no—on the contrary, I have rewarded all your kindness
towards me, with the blackest ingratitude.”
“We will pardon and forgive each other,” said Mr. Hatfield, solemnly:
“you shall pardon and forgive me for the stigma that attaches itself
to your birth—you shall likewise pardon me your mother’s wrongs,
even as she herself has long, long since pardoned me: and I, on my
part, will think no more of all that you have lately done—save to
extricate you from the cruel embarrassments in which by your hasty
conduct, your imprudence, and your misconceptions, you have
become involved. In a word, I will be to you as a kind friend and
adviser;—and if henceforth I may not hope for your affection—at
least I may reckon upon your gratitude.”
“Yes—both, both!” cried Charles Hatfield, again embracing his father
tenderly. “Oh! how wicked—how criminal I have been! A veil has
fallen from my eyes—my soul has lost its dogged obstinacy—and I
now perceive how ungrateful I have been to my dear mother and
yourself. But if it be not too late to repair the past,” he continued,
retreating a few paces, and addressing his parent with a tone and
manner of solemn earnestness,—“if it be not too late to regain my
mother’s love and yours also,—oh! then the remainder of my life
shall be wholly and solely devoted to that one object! Yes—I will
reinstate myself in your esteem—I will prove by years of affection
and obedience how bitter is my remorse and how sincere is my
repentance for the follies and indiscretions of a few weeks! But in

the meantime, father—in the meantime, how am I to act towards
the vile—the guilty woman, whom I lately loved so madly?”
“Where is she at present?” demanded Mr. Hatfield, profoundly
touched by the contrition and altered feelings now manifested by his
son.
“I left her asleep in a chamber belonging to this suite,” was the
reply. “Oh! I dare not meet her again—for I fear that I should spring
upon her like a tiger, and sacrifice her to my resentment! For all my
affection has now turned to a bitter—burning hatred,—a hatred
against herself and her more vile mother; and I am astounded when
I reflect how completely I have been deluded by them. It appears to
me a dream—a vision! I can scarcely bring myself to conceive that I
could possibly have been so insensate—so mad—so blind—so
besotted! Oh! I could dash my head against the wall, to punish
myself for this atrocious folly!”
And the young man struck his clenched fists forcibly against his
forehead.
“Compose yourself—in the name of God! compose yourself,” said his
parent, rushing in upon him and restraining him from the
commission of farther violence. “Give not way to despair, my dear
son—meet your misfortune with courage——”
“Oh! it is easy thus to recommend patience and endurance,”
exclaimed Charles, bitterly: “but think how cruelly I have been
deceived! I was fascinated as by the eyes of a serpent;—the magic
of her charms, the melody of her voice, the sophistry of her tongue,
and the excitement of her caresses, threw spells of an irresistible
nature upon me: I was enchanted—held captive in silken chains—
dazzled by the almost superhuman beauty of that prodigy of deceit
and wantonness! I was not allowed time for reflection—suspicion
had no leisure to rise up in my bosom, much less to fix its habitation
there;—for I was whirled along, as in a delirious dream, from the

first instant that I met that woman until the instant when your
revelations of this morning dispelled the entire illusion. The
artfulness of that designing creature sustained a constant elysian
excitement in my soul: a perpetual succession of insidious wiles, of
apparent proofs of deep tenderness, and of caresses that would
enthral the heart of a saint,—such—such was the magic course in
which I was hurried madly along. Endowed with a wondrous
presence of mind, she had a ready answer for every question that I
put to her—even to the explanation of her singular name;—and, with
a guile as profound as it was ravishing—with an artfulness as deep
as it was calculated to enchant and captivate—she invested the
history of her early days with a mystery which only increased my
admiration, and made her appear more interesting in my eyes.”
“You cannot wonder, then, that you were so completely deceived,
my poor boy,” said Mr. Hatfield, who had listened with great, though
mournful interest to the eloquent delineation of causes and effects
which the impassioned language of the young man had so
graphically shaped. “But as for the designing creature’s name, I
heard its origin from the officers whom I met at Dover. She is called
Perdita, or ‘The Lost One,’ because she was born in Newgate—and
her mother, in the moment of repentance for her own crimes, gave
her that appellation as a memorial and a warning——”
“Heavens!” ejaculated Charles; “and I believed the specious—the
plausible explanation which the artful girl gave me relative to her
name! Oh! she is made up of deceit: the world has never known her
equal in that respect. I have read of Circe, with her spells—and of
the Syrens, with their perilous allurements;—I have read also of
those Mermaids—with the heads and busts of beauteous women,
and with the tails of monsters—and whose melting looks and
ravishing songs enticed sailors to their coasts, only to fall victims to
these unnatural devourers of human flesh:—but all these wonders of
heathen mythology are surpassed by this modern Circe—this Syren
of the nineteenth century—this Mermaid who preys, not on mortal
flesh, but upon immortal souls!”

There was a terrible earnestness in the tone and manner of Charles,
as he gave utterance to these words:—and his father perceived that
the heart of the young man was painfully lacerated by the conviction
of Perdita’s tremendous duplicity.
“Yes,” resumed Charles,—and Mr. Hatfield allowed him to speak on,
knowing that feelings so powerfully excited as his had been and still
were, must have a proper vent, in order that the soul might regain
something approaching to the equilibrium of calmness:—“yes,”
exclaimed the young man, passionately,—“she, whom I believed to
be the mirror in which all excellent qualities were reflected, is the
embodiment of every possible vice—every earthly iniquity. Oh! what
a splendid personification of Sin would she make for the painter or
the architect! But it must be a bold pencil or a powerful pen that
could do justice to her,—aye, and a man deeply read in the
mysteries of human life, to pourtray her character with accuracy!
And that character I can read now;—and I know her to be a creature
who has studied sensuality, with all the ardour of a glowing
temperament—with all the vivid sensibility that could enhance the
joys of amorous enchantment! Oh! mine was an idolatry such as a
rapt enthusiasm pays, in its blind belief, to the Spirit of Evil,
conceiving it to be the source of every virtue! Fatal mistake—
deplorable error: shall I ever surmount the terrible consequences?”
“Yes—by taking courage, following my counsel, and placing me in
full possession of all the minutest details of this distressing and
perplexing case,” said Mr. Hatfield, assuming the part of a comforter,
now that the indignation of his son had in some degree expended
itself in those passionate outpourings which we have endeavoured to
describe.
“Oh! fear not, my beloved father—my only friend,” cried Charles,
warmly,—“fear not that I shall now conceal aught from you! I have
obeyed the impulses of my own wrongheadedness—and I am
suffering terribly in consequence: I have followed the dictates of my
own wilfulness—and I have gone lamentably astray! The result is

that I have no more confidence in myself: from the pinnacle of that
proud independence which I sought to assume, I am dashed down
into a state of childish helplessness. If you abandon me—I should
not have courage even to attempt to extricate myself from this maze
of embarrassments in which I am so cruelly involved: I should resign
myself to my fate—I should sink into despair!”
“Cheer up, my beloved son—and think not for a moment of these
dreadful alternatives,” said Mr. Hatfield: “but answer me a few
questions, and I shall then know better how to act. Did you not find
certain papers in a secret recess in the Earl’s library——”
“Yes—and those papers are safe,” replied Charles: “at least—Perdita
has them secure in her writing-desk, and we will make her surrender
them presently.”
“As her husband—alas! that I should have to speak of you as such,—
you may break open that desk and take them by force,” said Mr.
Hatfield! “Does the young woman know their contents?”
“Unfortunately she does,” was the mournful answer.
“And her mother——”
“Is equally well acquainted with them,” said Charles. “Even to save
you a pang,—and heaven knows I would now do much to spare you
any additional uneasiness,—I will not deceive nor mislead you in a
single detail.”
“No—this is not a time nor a case for trifling, Charles,” observed Mr.
Hatfield. “Then both these women know who I am?” he added, in a
low and hoarse voice.
“Oh! my God!” cried Charles, giving vent to his deep vexation and
obeying the impulse of his self-accusing spirit: “to what humiliations
have I not exposed you, my dearest father? Can you—will you ever
forgive me for all this?”

“Have we not had much to pardon—much to explain, on either side,
already?” asked Mr. Hatfield, his voice now regaining its mildness—a
mildness that was, however, mournfully subdued. “Well, then, my
dear boy, give not way to these self-reproaches; for if I be anxious
to obtain a certain knowledge of the full extent of these evils, it is
only with the view of falling into no error and committing no
oversight in extricating both yourself and me from the
embarrassments that surround us. To return, then, to the immediate
subject of our discourse—those women know all?”
“All—every thing,” replied Charles. “In that blind infatuation——”
“Compose yourself, my dear boy,” said Mr. Hatfield, in a voice slightly
indicative of paternal authority. “Respecting the promissory note you
gave the money-lender Percival——”
“Oh! now I shrink indeed from telling you the truth,” interrupted
Charles, his countenance glowing with shame and confusion; “and
yet—faithful to my promise—I will not mislead you. The note of hand
to which you allude was signed—Viscount Marston!”
“If I recollect aright,” said Mr. Hatfield, “the account of the murder,
as reported in the newspapers, states distinctly that no papers nor
documents of any kind were found in the victim’s house—the tin-
box, in which such things were probably kept, having been emptied
of its contents. The assassin or assassins, then, whoever they may
be, possessed themselves of all the poor man’s papers—and your
note doubtless amongst the rest. In this case, we shall probably
never hear of it again. But—knowing the two women as you do—can
you believe that they were the murderesses?”
“No—I cannot think it!” exclaimed Charles. “What motive could they
have had? Certainly not to recover my promissory note, since they
believed me to be the heir to immense wealth;—and as they no
doubt fancied that their connexion with me would place ample
resources at their command, they were not likely to peril their lives

by killing the man for the sake of the money which he might have
had in the house. Besides, when I saw them on the following
morning, there was no confusion—nothing on their part to denote
that they had so recently committed a horrible crime; and, depraved
—wicked—unscrupulous as they evidently are, I cannot bring myself
to imagine that they could meet me with calm and unruffled
countenances, only a few hours after having accomplished a
midnight murder.”
“Let us hope that they are indeed innocent,” said Mr. Hatfield
solemnly. “And now I will explain to you the manner in which I
propose to deal with this Perdita.”
The interest and attention of Charles redoubled, if possible, as his
father uttered these words.
“Thank heaven,” continued Mr. Hatfield, “I possess wealth; and by
means of gold, every thing can be accomplished with such
mercenary adventuresses as these. Perdita shall receive a handsome
sum of ready money, and a suitable income allowed her so long as
she shall consent to dwell upon the continent, take any other name
than that which you have unfortunately given her, and never more
molest you.”
But scarcely had Mr. Hatfield uttered these words,—and before his
son had time to offer a single comment upon the proposed plan to
be adopted,—the door opened, and Perdita entered the room.

CHAPTER CLII.
THE FATHER, THE SON, AND THE SON’S WIFE.
The magnificent creature whom Mr. Hatfield now beheld for the first
time, had perhaps never shone to greater advantage than on the
present occasion.
She was absolutely dazzling—radiant—supernally grand, in all the
glory of her queen-like beauty.
A French cambric wrapper, worked, and trimmed with costly lace,
enveloped her form—fitting loosely, yet defining all the rich contours
of her voluptuous shape;—and, though—having risen hurriedly
almost immediately after awakening—she had no stays on, the
natural firmness of her bust maintained its rounded proportions
without any artificial support.
We have before said that her early initiation in a career of
wantonness and the licentious course which she had pursued in
Australia, had marred nothing of the first freshness of youth in
respect to her;—and thus, though her wrapper was so far open at
the bosom as to show that the glowing orbs of snowy whiteness
were unsustained by the usual article of apparel, their contours were
of virgin roundness.
Her dark brown hair had been hastily gathered up in two massive
bands, silken and glossy, and serving as a frame to set off the height

and width of the fine forehead, which rose above brows arching
majestically, and almost meeting between the temples.
Her cheeks were slightly flushed with a carnation hue;—her large
grey eyes shone brilliantly, and appeared to give a halo of light to
her whole countenance;—her moist red lips, parted with a smile of
happiness and satisfaction, revealed the teeth so perfectly regular
and of such pearly whiteness;—and her neck arched proudly and
with swan-like grace.
One arm hung negligently, but slightly rounded, by her side: the
other, thrown across her form just above the waist, kept the folds of
the wrapper together;—and from beneath the skirt of that elegant,
tasteful garment, of almost gauzy lightness and transparency,
peeped forth the beautifully-modelled ankles in their flesh-coloured
silk stockings, and the charming feet in their embroidered slippers of
pale blue satin.
Though, as we have before stated, she was not above the middle
height, yet there was something truly regal and commanding in her
deportment—something more than graceful and less than imperious
in her carriage, and, altogether, she appeared a being to whom it
would not be idolatrous to kneel.
On the contrary,—prejudiced and naturally inveterate as he was
against her, Mr. Hatfield could well comprehend, even at the first
glance which he threw upon her, how a young man of enthusiastic
disposition and keen sensibility might love that enchanting creature
with a devotion amounting to a worship.
The apartment was large and beautifully furnished,—the uncarpetted
floor of oak was polished almost to mirror-like brightness,—vast
looking-glasses, set in splendid frames, were suspended to the walls,
—a massive or-molu time-piece and handsome porcelain vases filled
with flowers freshly gathered that morning, stood on the mantel,—
and through the casements, which reached from the ceiling to the

floor, and which were only partially shaded by muslin curtains,
flowed the gorgeous lustre of the cloudless sun, so that the room
seemed filled with a transparent and impalpable haze of gold-dust.
Thus the whole aspect of that large and lofty apartment was
magnificent and rich, bright and joyous;—and, had the minds of the
father and son at the instant been in a different mood, they would
have felt thrilled with admiration and delight at the presence of the
magnificent creature who now entered an atmosphere so congenially
glorious and sunny.
It seemed as if the beauteous being herself were surrounded with a
golden halo,—as if the perfume of the freshly gathered flowers were
the delicious fragrance of her breath,—as if the delicate feet and
ankles bore her glancingly along a polished surface which she
scarcely appeared to touch; while the immense mirrors multiplied
the voluptuous form, as though other and kindred houris were
moving about in attendance on their queen.
The effulgence of the warm sun played on her shining hair, as if a
glory sate on that exquisitely shaped head,—gave additional
brightness to the eyes that flashed with the natural fire of joy,—and
rendered the fine and faultless countenance radiant and dazzling in
its surpassing beauty.
Were that a room in a palatial dwelling,—were it an empress making
her appearance,—and were the two men courtiers awaiting her
presence, the effect could not have been more grand—more striking,
—and the courtiers would have fallen on their knees in mute
adoration of a being that seemed almost divine!
But, alas! circumstances marred all those fine effects which the
transcendant charms of a lovely woman might have produced;—for
the soul of this woman corresponded not with her captivating
exterior,—it was dark and hideous—inspiring horrible thoughts, and

suggesting ideas of a nature so sinister, sombre, and gloomy, as to
throw into the shade all the glory of the outward loveliness.
But, unsuspicious of the storm which was about to explode against
her, Perdita entered that room;—and the influence of a night of love
and voluptuousness and of elysian dreams lingered upon her
countenance in the smile that it wore.
She had slept for nearly an hour after Charles Hatfield had risen so
noiselessly from her side in the nuptial couch;—and when she at

length awoke, she imagined that her young husband had been
unwilling to disturb her when he himself arose. Nevertheless, she
determined to seek him ere she passed through the routine of the
toilette;—and hastily fastening up her hair, and assuming a slight
apparel, she had proceeded to the sitting-room where she supposed
him to be.
And there indeed he was: but not alone!
Still, when Perdita, on first entering the apartment, beheld another
person with him whom she sought, she had no suspicion of the real
truth, but imagined it must be some friend who had found out her
husband’s residence in Paris and had perhaps called to congratulate
him on his bridal.
Thus was it that her countenance wore that delicious expression of
pleasure and satisfaction, as she advanced towards Charles and that
other;—and it was not until she was within a few paces of them,
that she observed the foreboding looks which they cast upon her—
even the aversion and the hate with which they both regarded her!
Then she stopped suddenly short, her countenance undergoing an
immediate change—the smile disappearing, and giving place to an
expression of proud defiance and haughty contempt; though she
was still unconscious of the nature of the storm that she saw
lowering so ominously.
“Charles, who is this person?” she demanded, indicating Mr. Hatfield
with a movement of the head, accompanied by a slight inflection of
the whole form—a gesture which would have become a queen.
“My father,” answered the young man quietly;—and he turned away
towards the mantel-piece.
For an instant Perdita seemed shocked by this announcement;—but
in the next moment, as the thought swept across her brain that it
was impossible for Mr. Hatfield to know aught seriously detrimental

to her character, she crossed the room in a majestic manner, and,
laying her long tapered fingers gently upon her husband’s arm, said,
“Is it possible that the remonstrances of your father should have
induced you to repent of this alliance,—you, who have sworn to love
and cherish me in spite of parents and all the world beside?”
“When a man discovers that he has taken a reptile to his bosom,”
said Charles, the words hissing through his almost set teeth, “he
flings it away from him. He ought to crush it beneath his heel!”
The last sentence was added after a moment’s pause, and ere
Perdita, who was astounded at the tone, and manner, and words of
her husband, had regained the power of utterance so as to enable
her lips to shape a comment or a reply.
“Is it to me that this insulting allusion applies?” she demanded at
length—her countenance becoming ashy pale, and her lips quivering
with the rage which she still sought to subdue.
“It is to you that I addressed myself,” exclaimed Charles, now
turning round and confronting the woman whom he had lately loved
with such madness, and whom he now loathed with such savage
aversion. “Vile—polluted—wanton thing,” he cried, unabashed—
undismayed by the lightning glances that flashed from her wildly
dilating orbs: “the mask is torn from your face as the film from my
eyes—and I am no longer your dupe, though, alas! I am perhaps still
your victim! I know all—all—every thing,—the depravity of your past
life—the hypocrisy of your present course:—all—all is now revealed
to me. Your evil fame has followed you from beyond the seas;—it
overtook you on the Marine Parade at Dover;—and it now attaches
itself for ever to your steps, in the capital of France. Oh! my God—
how cruelly, how miserably have I been deceived!”
And the young man darted a glance of savage hatred upon the
woman who, pale and motionless as a marble statue, seemed
petrified by the crushing truths that fell upon her ears.

Meantime Mr. Hatfield stood aloof, with folded arms—listening to the
words that his son addressed to Perdita, and marking their effect.
“That you were born in Newgate—of a woman condemned to death
for felony, and then reprieved,—this was no fault of yours,”
continued Charles, in a slow and measured tone—for he sought as
much as possible to prevent a violent outburst of the rage that
boiled within him:—“that the mystic name of Perdita, or ‘The Lost
One,’ should have proved prophetic of your after life, you also could
not help;—and that, amongst the felonry of New South Wales, you
should have become polluted—contaminated—and indeed lost, was
perhaps a fate for which you are rather to be pitied than blamed.
But here all sympathy ceases for you! Wherefore, on your arrival in
England, did you seek me out to become your victim?—wherefore
did your wretched mother dog my footsteps—accost me—ensnare
me into a discourse to which she imparted a mysterious interest—
and then lead me into your presence? Why did you open the battery
of all your meretricious charms upon me?—why cast your spells
around me—wean my affections from an estimable young lady who
is white as snow compared with the blackness of your soul—and
lead me on until the crowning act of ruin was accomplished
yesterday in the Chapel of the British Embassy?”
“I have heard you with patience—and if you possess the generosity
of a man and an Englishman, you will give me an equal share of
your attention,” said Perdita, who, during her husband’s address, had
recovered all her wonted presence of mind—though her heart was
wounded in its very core. “It is true that I was born in Newgate—
that I deceived you respecting the origin of my Christian name—and
that I escaped not the contamination of a far-off clime into which my
sad destinies threw me. But when my mother, for reasons which I
think she made satisfactorily apparent to you, sought an interview
with you,—and when that circumstance introduced us to each other,
did you not proffer me your friendship of your own accord?—did you
not next assure me that this sentiment had changed to the feeling of
love?—did you not implore me, almost on your knees, to become

your wife at the altar—I, who in the first instance had proposed and
agreed to become your mistress only? And then you dare to speak of
our marriage as the crowning act of you ruin,—that marriage on
which you yourself so imploringly—so earnestly—so solemnly
insisted?”
“Oh! yes—because I deemed you pure and virtuous!” exclaimed
Charles, almost gnashing his teeth as the words of Perdita reminded
him of all the arts which she had practised to ensnare him—all the
sophistry she had used to make herself appear in his eyes every
thing that she was not.
“Was it to be supposed,” she asked, impatiently and haughtily,—that
shameless Perdita—“was it to be supposed that I would reveal to
you the incidents of my past life? And yet, even if I had, I do firmly
and sincerely believe that you would still have made me your wife!”
“No—never, never!” cried Charles, his voice and manner expressing
loathing, abhorrence, and indignation. “But let us not bandy words
thus. I have intelligence which—lost and depraved as you are, and
vilely as you have treated me—I nevertheless grieve to have to
convey to you,—for I cannot, even in my anger and hate, forget that
you are a woman.”
“And that intelligence?” demanded Perdita, suffering not her
countenance nor her manner to betray the deep curiosity and the
suspense which her husband’s words had suddenly excited within
her bosom.
“The intelligence regards your mother, and explains her mysterious
disappearance at Dover,” continued Charles, who, as well as his
father, now intensely watched the young woman’s countenance.
“Speak on!” she said, not a muscle of her face betraying any
emotion:—and still she stood motionless and statue-like.

“Your mother was arrested on suspicion of being concerned in the
murder of Mr. Percival, the money-lender whom you represented to
me as the discounter of my promissory note;”—and, as Charles
uttered these words in slow and measured tones, he maintained his
eyes fixed upon the pale but unchanging features of his wife.
“Then my mother has been accused of that whereof she is innocent,”
said Perdita, in a voice so firm and resolute, yet devoid of passion,
that her hearers felt convinced she was practising no artifice now. “It
is true that Percival discounted your note: I myself received the
money—and you can doubtless give your father a satisfactory
explanation relative to the expenditure of the portion that is gone. If
Percival have indeed met his death by violent means, it was not by
the hands of two weak women that he fell.”
“Thank heaven! this crime at least cannot, then, be attributed to
you,” said Charles. “There must be enough upon your conscience
without that!”
“And have you nothing wherewith to reproach yourself?” demanded
Perdita, still maintaining that majesty of demeanour which, with her
now marble-like features, her motionless attitude, and her fine form
enveloped in drapery that fell in classic plaits and graceful folds
around her, gave her the air of a statue of Diana the Huntress or of
Juno Queen of Heaven. “Have you inflicted no injury upon me?” she
asked. “Yes—yes: and I will convince you that your conduct has
been far from blameless in that respect. You loved me—loved me
almost from the first instant that you beheld me. Yours was not a
tranquil—serene—and sickly sensation: it was a fury—a wild passion
—a delirium—a species of hurricane of the strongest, most fervent
emotions. I was all—every thing to you: parents—family—friends,—
Oh! you cared for none of these in comparison with me. The holiest
ties you would have broken—the most sacred bonds you would have
snapped—the most solemn obligations you would have violated,
sooner than have resigned your hope of possessing me! All this is
true—and you know it. Your love amounted to a madness—a frenzy,

capable of the most unheard-of sacrifices, and as likely to hurry you
into the most desperate extremes. For had I provoked your jealousy,
you would have murdered me: had I fled and abandoned you, you
would have pined to death—or committed suicide. In fine, yours was
no common love—no ordinary affection. Poets never dreamt and
novelists never depicted a love so boundless—so absorbing—so
immense as yours. And what could result from such a love as this!
The consequence was inevitable;—and that consequence was that I,
who had never loved before, received into my soul a transfusion of
the spirit that animated you. You were so happy in your love, that
my imagination doubtless longed to revel in the same paradise which
you had created for yourself;—and I was taught by you to love as
profoundly and as well. In a word, you ensnared my heart—you
obtained a hold upon my affections; and, as there is a living God
above us! I swear that when you led me to the altar, you loved me
not better than I loved you. And this love which I experienced for
you, would have made me a good wife—a sincere friend—a
conscientious adviser. I should have entered upon a new existence;
and my soul would have become purified. True it is that I gave to
the marriage-bed a body that was polluted and unchaste: but I gave
also a heart that was wholly and solely thine;—and from the instant
that our hands were united by the minister of God, it would have
proved as impossible for me to have played the wanton with another
as that the infant child should harbour thoughts of villainy and
murder. Now you have learnt the antecedents of my life—and your
love is suddenly changed into hatred. But did you not take me for
better or worse?—did you not wed me, because you loved me!—did
you not espouse me for myself alone! Oh! you should pity me for the
past—and cherish me at present and for the future: and your
conscience tells you thus much even now!”
Charles Hatfield, who had listened with deep and solemn interest,—
for his soul was absolutely enchained by this strange display of
natural eloquence,—now shook his head impatiently.

“No! Then mark how fatal your love will have proved to me,”
exclaimed Perdita. “You cast me off—you put me away from you;—
and yet you cannot give me back the heart which you have
ensnared. Wherefore—wherefore did you bring to bear upon me the
influence of your ardent love, unless you were prepared to make
every sacrifice unto the end? I am young—I am beautiful—and I
might gain a high and a proud position by means of marriage: but,
no—I am chained to you—and you are intent upon discarding me!
Now reflect well on the probable consequences of this proceeding on
your part,” continued Perdita, her melodious voice gathering energy,
and a tinge of rose-bud hue appearing on her cheeks and gradually
deepening into a flush,—while her eyes shone with a lustre that
gave an almost unearthly radiance to her entire countenance:
“reflect well, I say,” she repeated, “on the probable consequences of
the resolution which you have taken. As your wife, and dwelling with
you as such, I should have clung to you—loved you with unceasing
devotion—exerted all my powers to retain your esteem. Nay, more—
in time I should have won your good opinion by my actions—as I
had already secured it by my words. Amongst the entire community
of women, there would have been none more exemplary than I;—
and thus your love would have proved a saving influence—valuable
to society at large, and blessed by the Almighty Ruler whom you
worship. But how changed are these prospects! You are prepared to
discard me—to thrust me away from your presence—to push me out
into the great world, where I must battle for myself. There I shall
find my circumstances terribly—-fearfully altered from what they
were before your lips whispered the delicious but fatal tale of love in
mine ears. For if I retain your name, I thereby proclaim myself a
divorced wife: if I pass myself off as an unmarried young lady, I shall
not dare to accept proposals for an alliance, be it never so
advantageous—because the fear of a prosecution for bigamy would
hang over my head. Will you, then, forgive me for the past, and
receive me as an affectionate wife and reformed woman to your
arms?—or will you send me forth, an outcast—with ruined hopes,
blighted prospects, and a damaged character?”

Gradually, as she approached the end of this speech, Perdita had
suffered her voice to lose its energy and its firmness, and grow
tender, pathetic, and mournful—until at the close of her appeal, it
became tremulously plaintive and profoundly touching,—while her
form simultaneously relaxed from its statue-like rigidity—the head
slightly inclining, the body bending in the least degree forward, and
the hands joining as the last words fell from her lips.
For an instant Charles was about to yield to the appeal commenced
with a dignity so well assumed, and terminated with a tenderness so
well affected; but, at the critical moment, Mr. Hatfield, who had
hitherto remained a mute spectator of this extraordinary scene,
stepped forward, exclaiming, “No—no; a compromise of such a
nature is impossible! Charles, the sophistry is indeed most specious
—but the peril is likewise tremendous!”
“Yes—yes,” cried the young man, instantly recovering his presence of
mind: “I told you, father, that she was a Circe—a Syren,—and now
you have ample proofs of the assertion.”
While he was yet speaking, the appearance of Perdita underwent a
rapid and signal change. She suddenly seemed to throw off the air
of a suppliant, as if she were discarding a mean garment that was
unbecoming and abhorrent: her cheeks acquired a deeper flush, her
eyes a more dazzling brilliancy;—the blue veins in her forehead grew
more clearly traceable—her nostrils dilated—her lips wreathed into
an expression of sovereign disdain—and her entire form appeared to
expand into more majestic proportions.
A moment before she had seemed a voluptuous beauty, in the
melting softness of an appeal for pardon at love’s shrine: now she
stood in the presence of the father and son,—proud—haughty—and
magnificent as Juno,—and armed with authority to wield the
lightning-shafts and the thunderbolts of Jove.

“Let us think of peace no more,” she exclaimed: “but war—terrible
war,—war to the knife! Cast me off—thrust me from you—denounce
me as the wanton Perdita—proclaim me to be born of a felon, and to
have first seen the light in Newgate,—do all this if you will: I shall
not the less remain your wife, Charles—and, as your wife, I am
ennobled,—I bear the proud title of Viscountess Marston!”
“Miserable woman,” cried Mr. Hatfield: “you deceive yourself—even
as Charles has been by himself deceived! For know that he is
illegitimate——”
“’Tis false! you would delude—you would mislead me!” exclaimed
Perdita, who, in spite of the tone of confidence in which she uttered
these ejaculations, was painfully affected by the revelation that had
elicited them.
“It is true—too true!” cried Charles, with a bitterness that carried
conviction to the mind of Perdita.
“Then if I cannot proclaim myself to be Viscountess Marston,” she
said, concealing with a desperate and painful effort the shock which
she had just experienced,—“I can still have my revenge against you
both;—for if my mother were a felon, Charles, your father was the
same—if I were born in Newgate, the author of your being has
passed through the hands of the public executioner!”
“Fiend—wretch!” ejaculated the young man, springing forward as if
about to dash her on the floor and trample her under foot.
But the hand of his father suddenly grasped him as in an iron vice,
and held him back; and all the while Perdita had maintained her
ground—shrinking not a step, retreating not a pace.
“Coward!” she exclaimed, in a tone of ineffable contempt, as she
kept her eyes—her large, shining grey eyes—fixed with disdain upon
him whom she had lately loved so fervently and so well.

“Charles—Charles,” said Mr. Hatfield, in an imploring voice, as he
held his son firmly by both arms,—“merit not by your actions that
infamous woman’s reproaches. I was prepared for what she dared to
address to me——”
“Oh! my dear father, this is terrible!” murmured the young man, who
felt a faintness coming over him, as the words which Perdita had
spoken concerning his parent still rang in his ears, and as he
observed the deadly pallor which had spread over that parent’s
countenance.
“Compose yourself, Charles,” said Mr. Hatfield, conducting him to a
seat: then, turning round and accosting Perdita, he exclaimed,
“Madam, let us treat this most unpleasant affair as a purely
business-matter: in short, let us effect an arrangement which may
be proper and suitable for both parties—the basis being the
immediate separation of yourself and my son.”
“Yes—I have no longer any objection to offer to that proposal,” said
Perdita; “for after his attempt to strike me, I despise even more than
I hate him.”
“And just now,” exclaimed the young man, starting from his seat,
“you declared that I possessed your heart. Oh! I am rejoiced that
you have admitted your hatred towards me—because I have thereby
received another proof of your boundless duplicity.”
Perdita smiled scornfully—but deigned no reply.
“Leave the affair in my hands, Charles,” said Mr. Hatfield, in an
authoritative tone: then, observing with satisfaction that his son
returned to his seat, the father addressed himself once more to
Perdita, who remained standing near the mantel. “Madam,” he
continued, “you have already heard that the bright hopes in which
your husband had indulged, and the golden visions which he had
conjured up, are all destroyed by the revelation which I have this
morning made to him,—the revelation of the one fatal secret—his

illegitimacy! Instead, then, of being Viscount Marston at present and
Earl of Ellingham in perspective, he is still plain and simple Charles
Hatfield—and so he is likely to remain. By consequence, you,
madam, are Mrs. Hatfield—and not Viscountess Marston now, nor
with any chance of becoming Countess of Ellingham. If you require
proofs of what I am now telling you, I can exhibit them at once;—
for, knowing beforehand the nature of the delusions in which my son
had cradled his fancy, and the necessity of destroying them, I set
out on this journey provided with several papers of importance. For
instance,” continued Mr. Hatfield, taking forth his pocket-book; “here
is the certificate of my marriage with Lady Georgiana Hatfield—and
you may at once perceive by the date how impossible it is that our
son could have been born in wedlock.”
While thus speaking, Mr. Hatfield had sunk his voice to the lowest
audible whisper—so that Perdita alone heard him: for the revelation
he was making was of a most painful nature, although rendered
imperatively necessary under the circumstances.
Perdita glanced rapidly over the certificate, and bit her lip with a
vexation she could no longer conceal;—for that document effectually
set at rest the question of her husband’s legitimacy or illegitimacy;
and she indeed found that instead of gaining a noble title by
marriage, she had formed an alliance with an obscure young man
who was dependant on his parents for even a morsel of bread.
“It now remains for you to decide whether you choose to proclaim
yourself, wherever you go, to be the wife of Mr. Charles Hatfield;—or
whether you will think fit to resume your maiden name—or any
other that may suit your purposes—and maintain a strict silence
henceforth relative to this most unfortunate alliance.”
Thus spoke Mr. Hatfield;—and Perdita appeared to be plunged in
deep thought for a few minutes.

“And what are the conditions you annex to those alternatives?” she
asked at length, fixing her eyes, which now shone with a subdued
and sombre lustre, in a penetrating manner upon Mr. Hatfield’s
countenance—as if she would there read the reply to her question
even before his lips could frame it.
“If you proclaim yourself my son’s wife,” said he, meeting her look
firmly and speaking resolutely, “I shall spare no expense in bringing
the whole transaction before the proper tribunals in England, with
the ultimate view of enabling him to obtain a divorce; and in this
case I should not allow you one single farthing—no, not even to
save you from starvation.”
“And have you not reflected,” asked Perdita, in a tone and with a
gesture indicative of superb disdain,—“have you not reflected that a
judicial investigation must inevitably lay bare all the tremendous
secrets connected with yourself and family?—for you cannot
suppose, that if you commence the part of a persecutor against me,
I shall evince any forbearance towards you! No—it would be, as I
said just now, a terrible warfare—a warfare to the very death,—and
in which human ingenuity would rack itself to discover and set in
motion all possible means of a fearful vengeance.”
“I have weighed all this,” said Mr. Hatfield, calmly; “and I have
resolved to dare exposure of every kind—nay, to sacrifice myself, if
necessary—in order to save my son.”
“And now for the conditions annexed to the second alternative?” said
Perdita, maintaining a remarkable coolness and self-possession,
although in the secret recesses of her soul she harboured the
conviction that the triumph was as yet on the other side, and that
she must end by accepting the best terms she could obtain.
“If you will sign a paper, undertaking never to represent yourself as
my son’s wife,” said Mr. Hatfield,—“never to molest him in any way—
never to return to England, but to fix your abode in some continental

state,—and lastly, that you will retain inviolably secret not only the
fact of this most inauspicious marriage, but likewise all matters
connected with myself and family,—if you affix your name to such a
document,” continued Mr. Hatfield, “I will immediately pay you the
sum of one thousand pounds, and I will allow you five hundred
pounds a year so long as the convention shall be duly kept on your
part.”
“And should you happen to die before me?” said Perdita, her manner
now being of that cold, passionless nature which rendered it
impossible for Mr. Hatfield to conjecture what sort of an impression
his alternatives and their conditions had made upon her mind: “for
you must remember,” she added, “that such an event is to be
reckoned upon in the common course of nature.”
“Granted,” was the prompt reply. “My will shall contain a clause
enjoining and empowering my executors to continue the payment of
your income, from a fund especially sunk for the purpose, so long as
your conduct shall be in accordance with the conditions stipulated.”
“And am I to understand that if I leave your son unmolested, I shall
remain unmolested also!” demanded Perdita.
“I scarcely comprehend you,” said Mr. Hatfield, evidently perplexed.
“I mean,” replied Perdita, in a slow and measured tone, so that her
words could not be misapprehended nor their sense mistaken,—“I
mean that if I go forth into the world again as Miss Fitzhardinge, or
Miss Fitzgerald, or any other name I may choose to take,—and if,
receiving a suitable offer of marriage, I contract such an alliance,—I
mean, then, to ask whether I may calculate upon acting thus with
impunity at your hands?”
“My God! what interest can I have to molest you in any way?” cried
Mr. Hatfield. “Would to heaven that you could both of you sign a
paper effectually emancipating you from any claim on each other in
respect to this accursed—this miserable marriage.”

“You are now speaking with unnecessary excitement, sir, after
having reproved your son for the same fault—and also after having
yourself proposed to discuss this matter in a purely business-like
manner,” said Perdita, her lip curling slightly with an expression of
scornful triumph.
“True, madam,” observed Mr. Hatfield, who, throughout this dialogue
—since his son had remained seated apart—had treated Perdita with
a perfect though frigid courtesy: “I was in error to give way to any
intemperance of tone or manner—and I ask your pardon. You have
now heard all that I have to propose——”
“And I accept the conditions,” she said. “Indeed, I shall be happy for
this scene to terminate as speedily as possible.”
“A few minutes’ more will suffice, madam,” observed Mr. Hatfield. “If
you will have the kindness to provide me with writing-materials, I
shall not be compelled to intrude on you much longer.”
Perdita bowed slightly: and quitted the room,—not in haste—but
with stately demeanour and measured tread, as if she were merely a
consenting party to a business-transaction, and not a vanquished
one on whom conditions had been imposed.
The moment the door closed behind her, Mr. Hatfield said to his son,
“That woman is indeed a prodigy of beauty, and a very demon at
heart. What an angelic creature would she have been were she as
pure and virtuous as she is lovely!”
“Ah! my dear father,” returned Charles, who appeared to be
completely spirit-broken and overwhelmed by the terrible
occurrences and revelations of this memorable morning,—“you can
now comprehend, perhaps,—at least to some extent,—the nature of
that infatuation which I experienced in respect to this singular being.
The world has never seen her equal for beauty and for wickedness.”

“The sooner you are removed from the sphere of her fatal influence,
the better,” observed Mr. Hatfield. “When she re-appears, do you quit
the room, and hasten as much as possible your preparations to
depart with me.”
“Fear not, my dear father,” responded Charles, “that I shall, of my
own accord, interpose any delay. But the papers—she will surrender
them——”
“As a matter of course. You may have observed,” added the parent,
“that, in spite of her haughty coldness, she was subdued and
vanquished.”
At this instant the door opened, and Perdita returned, bearing her
writing-desk in her hands.
Her countenance, though flushed, and thus presenting a striking
contrast to its colourless appearance some time before, gave no
indication of the nature of her feelings: impossible was it to judge of
the emotions that might occupy her bosom, by that which is wont to
be denominated the mirror of the soul.
Her step was still measured and stately, while her attitude was
graceful; and, as she advanced towards the table—passing through
the golden flood of lustre that filled the room—the waving of her
white drapes; gave an additional charm to the undulating nature of
her motion.
From beneath her richly fringed lids, while affecting to keep her eyes
half bent downward as if on the rose-wood desk which she carried,
she darted a rapid glance at Mr. Hatfield—and then her look dwelt
the least thing more lingeringly on her husband, who had risen from
his seat and was leaning on the mantel.
By a natural effect of curiosity,—perhaps also in obedience to a last
remaining particle of that immense love which he had so lately borne
her,—Charles Hatfield likewise glanced towards her from beneath his

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