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DELAY-DOPPLER
COMMUNICATIONS
Principles and
Applications
Yi Hong
Department of Electrical and Computer Systems Engineering
Monash University
Melbourne, VIC, Australia
Tharaj Thaj
Monash University
Emanuele Viterbo
Monash University

We dedicate this work to our parents
Guomei, Guifa, Mariamma, Thajudeen, Mariella, and Davide

List of figures
Figure 1.1 Evolution of the wireless systems. 3
Figure 1.2 An example of a high-mobility environment. 4
Figure 1.3 How the transmit basis functions ΨTX used to multiplex information
symbols are transformed to the receiver basis functions ΨRX by
static multipath and high-mobility channels for (a) CP-OFDM,
(b) PS-OFDM, and (c) OTFS. TF: time-frequency, DD: delay-Doppler. 10
Figure 2.1 Paths with different propagation delays. 15
Figure 2.2 Paths with different Doppler shifts due to the different angles of arrival. 16
Figure 2.3 An example of high-mobility wireless channel scenarios showing
how the delay-Doppler channel response changes when the
geometry of the scene changes. 19
Figure 2.4 The TDL models for static and high-mobility channels with integer
delay taps L = {0,1,2}. 23
Figure 2.5 The continuous delay-Doppler vs time-frequency channel
representation of a high-mobility multipath channel (linear
time-varying). 24
Figure 2.6 The continuous delay-Doppler vs. time-frequency channel
representation of a static multipath channel (linear time-invariant). 25
Figure 2.7 Different domain representations of a time-variant multipath
channel impulse response g(τ,t), also denoted as the delay-time
channel response. 25
Figure 3.1 The discrete time-frequency grid (Λ). 31
Figure 3.2 The magnitude of the cross-ambiguity function of square pulses. 33
Figure 3.3 The magnitude of the cross-ambiguity function of square pulses for
different values of f . 34
Figure 3.4 The magnitude of the cross-ambiguity function of square pulses for
different t/T . 34
Figure 3.5 The zeros of the cross-ambiguity function in the time-frequency plane. 35
Figure 3.6 OFDM transmitter. 36
Figure 3.7 OFDM receiver. 37
Figure 3.8 OFDM power spectrum. 40
Figure 4.1 OTFS system diagram in its original form. 51
Figure 4.2 The discrete time-frequency grid (Λ) and delay-Doppler grid (Γ ). 51
Figure 4.3 OTFS system diagram using the discrete Zak transform. 54
Figure 4.4 Received 2D signal (c) for ideal pulses as the 2D circular convolution
of the input (a) and the channel (b). 56
Figure 4.5 Time-frequency domain representation of transmitted signals with
PS-OFDM. 57
Figure 4.6 OTFS transmitter based on the IDZT using rectangular pulse
shaping waveform (M = 8,N = 6). 58
Figure 4.7 OTFS receiver based on the DZT using rectangular pulse shaping
waveform (M = 8,N = 6). 60
Figure 4.8 Time domain channel matrix G with three delay paths represented
by the three green (light gray in print version) subdiagonals
partitioned into M × M submatrices. 64
xiii

xiv List of figures
Figure 4.9 Time-frequency channel matrix Ȟ partitioned into M × M submatrices. 66
Figure 4.10 The row-column interleaving operation to generate the time domain
samples from the delay-time samples using the permutation matrix
P for an OTFS frame of size 12 with M = 3 and N = 4. 68
Figure 4.11 Delay-time domain channel matrix H̃ = PT · G · P with three delay
paths partitioned into N × N submatrices. 69
Figure 4.12 Delay-Doppler domain channel matrix
H = (IM ⊗ FN ) · H̃ · (IM ⊗ F†
N ) with three delay paths partitioned
into N × N submatrices. 70
Figure 4.13 The time domain frame for different variants of OTFS (CPs are
prepended and ZPs are appended). Option (a): a single CP/ZP is
added to the frame; option (b): a CP/ZP is added to each block;
option (c): a CP/ZP is included within each block. 71
Figure 4.14 RZP-OTFS delay-time domain channel matrix H̃ = PT · G · P with
three delay paths partitioned into N × N submatrices. 74
Figure 4.15 RZP-OTFS delay-Doppler domain channel matrix H with three
delay paths partitioned into N × N submatrices. 75
Figure 4.16 Continuous (ν(κ)) vs. discrete (νm,l(k)) Doppler response for
integer Doppler K = {2} (top) and fractional Doppler K = {2.5}
(bottom). 77
Figure 4.17 RCP-OTFS time domain channel matrix G with three delay paths
Figure 4.18 CP-OTFS time domain channel matrix G with three delay paths
Figure 4.19 ZP-OTFS time domain channel matrix G with three delay paths
Figure 4.20 Factor graph representation of the delay-Doppler domain I/O relation. 86
Figure 5.1 The Fourier transform viewed as a change of coordinate system. 96
Figure 5.2 Periodicity and quasiperiodicity of the Zak transform. 100
Figure 5.3 1
√
T
|Φτ0,ν0 (t)| for normalized delay (
τ0
T = 0) and normalized
Doppler shift (
ν0
f = 0.3) (modulated by the red (light gray in print
version) dashed line) and (
ν0
f = 0.6) (modulated by the blue (dark
gray in print version) dashed line). 101
Figure 5.4 |
√
T
M ΦB,T
τ0,ν0
(t)| for normalized delay (
τ0
T = 0) and normalized
Doppler shift (
ν0
f = 0.3) for M = 16. 105
Figure 5.5 | 1
MN ZB,T
T (τ,ν)| vs. normalized delay ( τ
T ) and normalized Doppler
shift ( ν
f ) for τ0 = 0.5T and ν0 = 0.4 f . 108
Figure 5.6 Contour plot of |Z̄B,T
T [s(t)](τ,ν)| vs. normalized delay ( τ
T ) and
normalized Doppler shift ( ν
f ) for M = 10, N = 10 for one
information symbol placed at τ0 = 3/M f and ν0 = 2/NT . 110
Figure 5.7 Contour plot of |ZT [r(t)](τ,ν)| vs. normalized delay ( τ
T ) and
f ) for M = 10, N = 10, τ0 = 3/M f , and
ν0 = 2/NT and channel parameters (τ1,ν1) = (0/M f,0/NT ) and
(τ2,ν2) = (3/M f,3/NT ). 111
Figure 5.8 Contour plot of |ZT [r(t)](τ,ν)| vs. normalized delay ( τ
T ) and
f ) for M = 20, N = 20, τ0 = 3/M f , and
ν0 = 2/NT and channel parameters (τ1,ν1) = (0/M f,0/NT ) and
(τ2,ν2) = (3/M f,3/NT ). 111

List of figures xv
Figure 5.9 The rectangular transmit wtx[q] (black), delayed transmit wtx[q − li]
(blue (dark gray in print version)), receiver wrx[q] (red (mid gray in
print version)), and effective window functions wtx[q] (green (light
gray in print version)) for RCP-OTFS (left) and wtx[q − li] (green
(light gray in print version)) for RZP-OTFS (right) for M = 5, N = 3,
LCP = LZP = 4, and li = 2. 118
Figure 6.1 Messages exchanged in the factor graph. 131
Figure 6.2 MRC delay-Doppler domain operation for M = 7, M = 5, and the set
of discrete delay indices L = {0,1,2}. [Copyright permission obtained.] 135
Figure 6.3 OTFS iterative rake turbo decoder operation. 144
Figure 6.4 The BER performance of OTFS using 4-QAM and 16-QAM signaling
and different detectors for a frame size of N = M = 64 (MP has 10
iterations for 4-QAM and 15 iterations for 16-QAM; MRC has 5
iterations for 4-QAM and 10 for 16-QAM). 146
Figure 6.5 The BER performance of OTFS using MP and MRC detectors in the
EVA channel model with different frame sizes (MP has 10 iterations
for 4-QAM and 15 iterations for 16-QAM; MRC has 5 iterations for
4-QAM and 10 for 16-QAM). 147
Figure 6.6 The BER performance of OTFS with 4-QAM signaling and the MRC
detector (5 iterations) for EPA, EVA, and ETU channel models with
max Doppler spread corresponding to a maximum speed of
120 km/h for N = 64 and M = 64,512. 148
detector for EVA at different UE speeds for N = M = 64. 149
detector for N = 64, M = 64 and different numbers of Doppler paths
|Kl| per delay bin l using the synthetic wireless channel model
described in Section 2.5.2. 150
Figure 7.1 (a) Transmitted pilot, guard, and data symbols. (b) Received symbols. 156
Figure 7.2 Pilot and data placement in ZP-OTFS (N = 8,M = 9) systems for
time domain channel estimation. (a) ZP-OTFS transmitter. (b) Time
domain operation. 162
Figure 7.3 Reconstruction of the real part of the l-th delay tap channel from the
estimated channel ĝs[l,mp + l + nM] using linear and spline
interpolation for N = 8, M = 64 and UE speed = 500 km/h at
SNRd = 20 dB and β = 0 dB. 164
Figure 7.4 BER performance of 4-QAM OTFS using an iterative time domain
detector for the EVA channel with speeds of 500 km/h and 1000 km/h. 166
Figure 7.5 OTFS receiver impairment effects on the pilot (magnitude) in the
indoor wireless channel (received SNR = 25 dB) (paths associated
with the same Doppler shift [in the same row] are shaded with the
same color). 168
Figure 7.6 OTFS SDR modem setup. 170
Figure 7.7 Bit and frame error rates vs. transmitter gain for 4-QAM and
16-QAM OTFS modulation. 172
Figure 7.8 Bit error rates vs. transmitter gain for 4-QAM OTFS and OFDM
modulation. 172
Figure 8.1 MIMO-OTFS system. 181
Figure 8.2 MIMO-OTFS embedded pilot channel estimation schemes for (a)
moderate and (b) high Doppler spread channel estimation. 192
Figure 8.3 Multiuser OTFS embedded pilot channel estimation schemes for
uplink communication with four users. 193

xvi List of figures
Figure 8.4 Multiuser OTFS embedded pilot channel estimation scheme for
downlink communication with four users. 194
Figure 8.5 OTFS vs. OFDM BER performance using 4-QAM for block-wise
LMMSE detection for a frame size of N = M = 32 for different
number of antennas. 196
Figure 8.6 OTFS uncoded BER performance using 4-QAM for block-wise
LMMSE vs. MP vs. MRC detection for a frame size of N = M = 32
for different number of antennas. 196
Figure 8.7 OTFS BER performance with MRC detector for a frame size of
N = M = 32 for different modulation sizes and number of antennas. 197
Figure 8.8 2 × 2 MIMO-OTFS BER performance using 4-QAM for block-wise
LMMSE vs. MRC detector for a frame size of N = M = 32 for
different UE speeds with channel estimation in Fig. 8.2(a) at an SNR
of 15 dB. 197
Figure 8.9 2 × 2 vs. 4 × 4 MIMO-OTFS BER performance using 4-QAM for
block-wise LMMSE vs. MRC (maximum 30 iterations) detection for
a frame size of N = M = 32 for different UE speeds with channel
estimation in Fig. 8.2(a) at an SNR of 15 dB. 198
Figure B.1 Example of a doubly circulant block matrix B formed by X with
M = 3 and N = 2. 222

Biography
Yi Hong
Dr. Yi Hong is an Associate Professor at the Department of Electrical
and Computer Systems Engineering, Monash University, Australia. She
obtained her PhD in Electrical Engineering and Telecommunications from
the University of New South Wales (UNSW), Sydney, and received the
NICTA-ACoRN Earlier Career Researcher Award at the 2007 Australian
Communication Theory Workshop, Adelaide. Yi Hong served as a mem-
ber of the Australian Research Council College of Experts in 2018–2020.
She was an Associate Editor for IEEE Wireless Communication Letters and
Transactions on Emerging Telecommunications Technologies (ETT). She was
the Tutorial Chair of the 2021 IEEE International Symposium on Infor-
mation Theory, held in Melbourne, and the General Co-Chair of the 2021
IEEE International Conference on Communications Workshop on Orthog-
onal Time Frequency Space Modulation (OTFS) for 6G and Future High-
mobility Communications, held in Montreal. She was the General Co-
Chair of the 2014 IEEE Information Theory Workshop, held in Hobart;
the Technical Program Committee Chair of the 2011 Australian Communi-
cations Theory Workshop, held in Melbourne; and the 2009 Publicity Chair
at the IEEE Information Theory Workshop, held in Sicily. Her research inter-
ests include communication theory, coding, and information theory with
applications to telecommunication engineering.
Tharaj Thaj
Mr. Tharaj Thaj received his B.Tech. degree in Electronics and Commu-
nication Engineering from the National Institute of Technology, Calicut,
India, in 2012 and his M.Tech. degree in Telecommunication Systems En-
gineering from the Indian Institute of Technology, Kharagpur, India, in
2015. He is currently pursuing a PhD with the Department of Electrical
and Computer Systems Engineering, Monash University, Australia. From
2012 to 2013, he worked at Verizon Data Services India, as a Software En-
gineer, focusing on network layer routing algorithms and protocols. From
2015 to 2017, he worked as a Senior Engineer in the Communication, Nav-
igation, and Surveillance (CNS) Department of Honeywell Technology
Solutions Lab, Bengaluru. His current research interests include physical
xvii

xviii Biography
layer design and implementation of wireless communication systems for
next-generation wireless networks.
Emanuele Viterbo
Dr. Emanuele Viterbo is a Professor in the Department of Electrical
and Computer Systems Engineering at Monash University, Australia. He
served as Head of Department and Associate Dean Graduate Research in
the Faculty of Engineering at Monash University.
Prof. Viterbo obtained his degree and a PhD in Electrical Engineer-
ing, both from the Politecnico di Torino, Turin, Italy. From 1990 to 1992,
he worked at the European Patent Office, The Hague, The Netherlands,
as a patent examiner in the field of dynamic recording and error-control
coding. Between 1995 and 1997, he held a post-doctoral position at Politec-
nico di Torino. In 1997–1998, he was a post-doctoral research fellow in the
Information Sciences Research Center of ATT Research, Florham Park,
NJ, USA. He later joined the Dipartimento di Elettronica at Politecnico di
Torino. From 2006 to August 2010, he was a Full Professor in DEIS at the
University of Calabria, Italy. In September 2010, he joined the ECSE De-
partment at Monash University as a Professor, where he is continuing his
research.
Prof. Viterbo is a Fellow of the IEEE, an ISI Highly Cited Researcher, and
a Member of the Board of Governors of the IEEE Information Theory Soci-
ety (2011–2013 and 2014–2016). He served as an Associate Editor for IEEE
Transactions on Information Theory, European Transactions on Telecommunica-
tions, and the Journal of Communications and Networks. His main research
interests are in lattice codes for Gaussian and fading channels, algebraic
coding theory, algebraic space-time coding, digital terrestrial television
broadcasting, and digital magnetic recording.

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Contents
List of figures xiii
Biography xvii
Preface xix
1. Introduction
1.1 High-mobility wireless channels 2
1.2 Waveforms for high-mobility wireless channels 3
1.3 Bibliographical notes 11
References 11
2. High-mobility wireless channels
2.1 Input–output model of the wireless channel 14
2.1.1. Geometric model 14
2.1.2. Delay-Doppler representation 17
2.2 Continuous-time baseband channel model 20
2.3 Discrete-time baseband channel model 22
2.4 Relation among different channel representations 23
2.5 Channel models for numerical simulations 26
2.5.1. Standard wireless mobile multipath propagation scenarios 26
2.5.2. Synthetic propagation scenario 27
References 27
3. OFDM review and its limitations
3.1 Introduction 30
3.2 OFDM system model 30
3.2.1. Generalized multicarrier modulation 31
3.2.2. OFDM transmitter 35
3.3 OFDM frequency domain input–output relation 38
3.4 Advantages and disadvantages of OFDM 40
3.4.1. High PAPR 40
3.4.2. High OOB 42
3.4.3. Sensitivity to CFO 42
3.5 OFDM in high-mobility multipath channels 43
References 44
vii

viii Contents
4. Delay-Doppler modulation
4.1 System model 49
4.1.1. Parameter choice for OTFS systems 50
4.1.2. OTFS modulation 50
4.1.3. High-mobility channel distortion 52
4.1.4. OTFS demodulation 53
4.2 OTFS input–output relation with ideal waveforms 54
4.2.1. Time-frequency domain analysis 55
4.2.2. Delay-Doppler domain analysis 55
4.3 Matrix formulation for OTFS 57
4.3.1. OTFS modulation 57
4.3.2. OTFS modulation via the IDZT 58
4.3.3. OTFS demodulation 59
4.3.4. OTFS demodulation via the DZT 60
4.4 OTFS input–output relations in vectorized form 61
4.4.1. Time domain input–output relation 63
4.4.2. Time-frequency input–output relation 64
4.4.3. Delay-time input–output relation 67
4.4.4. Delay-Doppler input–output relation 69
4.5 Variants of OTFS 70
4.5.1. Reduced ZP OTFS 71
4.5.2. Reduced CP-OTFS 77
4.5.3. CP-OTFS 79
4.5.4. ZP-OTFS 83
4.6 Summary of channel representations and input–output relations for OTFS
variants 85
4.6.1. Channel representations for OTFS variants 85
4.6.2. Delay-Doppler input–output relations for OTFS variants 87
4.6.3. Comparison of OTFS variants 89
References 90
5. Zak transform analysis for delay-Doppler communications
5.1 A brief review of the different Fourier transforms 94
5.2 The Zak transform 96
5.2.1. Properties of the Zak transform 97
5.2.2. The inverse Zak transform 99
5.3 The delay-Doppler basis functions 100
5.4 Zak transform in delay-Doppler communications 102
5.4.1. Single path delay-Doppler channel 102
5.4.2. Multipath and general delay-Doppler channel 102
5.4.3. Band- and time-limited delay-Doppler basis functions 104
5.4.4. Communications using band- and time-limited signals 109

Contents ix
5.5 The discrete Zak transform 112
5.5.1. The inverse discrete Zak transform 113
5.5.2. Properties of the DZT 113
5.6 DZT in delay-Doppler communications 114
5.6.1. Receiver sampling 114
5.6.2. Time-windowing at RX and TX 115
5.6.3. RCP-OTFS with rectangular Tx and Rx window 117
5.6.4. RZP-OTFS with rectangular Tx and Rx window 119
References 121
6. Detection methods
6.1 Overview of OTFS input–output relation 124
6.2 Single-tap frequency domain equalizer 125
6.2.1. Single-tap equalizer for RCP-OTFS 126
6.2.2. Block-wise single-tap equalizer for CP-OTFS 127
6.2.3. Complexity 128
6.3 Linear minimum mean-square error detection 129
6.3.1. Delay-Doppler domain LMMSE detection 129
6.3.2. Time domain LMMSE detection 129
6.4 Message passing detection 130
6.4.1. Message passing detection algorithm 130
6.5 Maximum-ratio combining detection 134
6.5.1. Delay-Doppler domain MRC detection 135
6.5.3. Reduced complexity delay-time domain implementation 138
6.5.5. Low complexity initial estimate 141
6.5.6. MRC detection for other OTFS variants 143
6.6 Iterative rake turbo decoder 144
6.7 Illustrative results and discussion 146
References 150
7. Channel estimation methods
7.1 Introduction 154
7.2 Embedded pilot delay-Doppler channel estimation 155
7.2.1. The integer Doppler case 155
7.2.2. The fractional Doppler case 158
7.2.3. Effect of channel estimation on spectral efficiency 160
7.3 Embedded pilot-aided delay-time domain channel estimation 161
7.3.1. Pilot placement 162

x Contents
7.3.2. Delay-time channel estimation 163
7.3.3. Channel estimation complexity 166
7.3.4. Extension to other OTFS variants 167
7.4 Real-time OTFS software-defined radio implementation 167
7.4.1. Effect of DC offset on channel estimation 169
7.4.2. Effect of carrier frequency offset on channel estimation 169
7.4.3. Experiment setup, results, and discussion 170
References 173
8. MIMO and multiuser OTFS
8.1 Introduction 178
8.2 System model for MIMO-OTFS 178
8.2.1. Transmitter and receiver 178
8.2.2. Channel 179
8.2.3. Input–output relation for MIMO-OTFS 180
8.3 Detection methods 184
8.3.1. Linear minimum mean-square error detector 184
8.3.2. Message passing detector 185
8.3.3. Maximum-ratio combining detector 185
8.4 MIMO-OTFS channel estimation 189
8.5 Multiuser OTFS channel estimation 192
8.6 Numerical results and discussion 195
References 199
9. Conclusions and future directions
9.1 OTFS key advantages 202
9.2 Pros and cons of OTFS variants 204
9.3 Other research directions 204
9.3.1. Channel estimation and PAPR reduction 205
9.3.2. Channels with fast time-varying delay-Doppler paths 206
9.3.3. Multiuser communications 206
9.3.4. Massive MIMO-OTFS 207
9.3.5. OTFS for RadCom 208
9.3.6. Orthogonal time sequency multiplexing and precoding design 208
9.3.7. Machine learning for OTFS 209
References 210
A. Notation and acronyms
B. Some useful matrix properties
B.1 The DFT matrix 219

Contents xi
B.2 Permutation matrices 219
B.3 Circulant matrices 220
B.4 Linear and circular convolutions 220
B.5 2D transforms, doubly circulant block matrices, and 2D circular convolution 221
C. Some MATLAB® code and examples
C.1 Transmitter 223
C.2 Channel 225
C.3 Receiver 227
C.4 Generate G matrix and received signal for OTFS variants 229
Index 233
Please visit the book’s companion site for additional materials (MATLAB
package code): https://www.elsevier.com/books-and-journals/book-
companion/9780323850285

Preface
Orthogonal frequency division multiplexing (OFDM) has been the
waveform of choice for most wireless communications systems in the
past 25 years. “What comes next?” – The book will address this question
by illustrating the current limitations of OFDM when dealing with high-
mobility environments and presenting the fundamentals of a new recently
proposed waveform known as “orthogonal time-frequency space” (OTFS).
The OTFS waveform is based on the idea that mobile wireless channels
can be effectively modeled in the delay-Doppler domain. The information is
encoded in such domain to combat the Doppler shifts in multipath propa-
gation channels that are typically found in high-mobility environments.
This book has been developed from a number of tutorial presentations
delivered by the authors at major conferences on wireless communications
between 2018 and 2021. It expands and integrates several key research
papers by the authors intending to provide a taxonomy for the different
flavors of delay-Doppler communications.
We have chosen to present the fundamentals of delay-Doppler commu-
nications following two approaches: (i) The first one is oriented to readers
who are familiar with multicarrier techniques and is based on the idea that
the delay-Doppler domain and the time-frequency domain are related by a pre-
coding operation, based on the two-dimensional symplectic Fourier transform.
(ii) The second one is for readers interested in the underlying mathemat-
ical tools that provide a direct relation between the (two-dimensional)
delay-Doppler domain and the (one-dimensional) time domain signals,
effectively bypassing the time-frequency domain interpretation. This is
based on the theory of the Zak transform and its specific properties.
We believe that both approaches should be followed and mastered by
the reader interested in further developing delay-Doppler communica-
tions systems in high-mobility environments. As a first approach to the
subject, we recommend Chapters 1–4 and 6 to cover the basic delay-
Doppler modulation and demodulation. For readers interested in the Zak
transform approach and the more specialized topics related to channel
estimation, MIMO, and multiuser systems, Chapters 5, 7, and 8 will be
of relevance. Finally, Chapter 9 touches upon some new research direc-
tions in high-mobility communications and outlines a more general two-
dimensional scheme beyond the delay-Doppler.
We would like to express thanks and appreciation to our colleagues Dr.
Raviteja Patchava (Qualcomm), Ezio Biglieri (Universitat Pompeu Fabra,
Barcelona), Saif Mohammad Khan (IIT Delhi), and Ananthanarayanan
Chokalingham (IISc. Bangalore), who have pioneered with us the inter-
xix

xx Preface
est in delay-Doppler communications. This book has taken shape by the
many interactions and discussions with them. We thank Dr. Viduranga Wi-
jekoon and Dr. Birenjith Sasidharan for their careful proofreading of our
manuscript. A special thanks goes to the inventor of OTFS, Ronny Hadani
(Cohere Technologies and University of Texas, Austin) for some very in-
spiring discussions with the authors. Since the first public disclosure of
OTFS in 2017, he has helped us to appreciate the more abstract interpreta-
tion of OTFS.
Yi Hong, Tharaj Thaj, and Emanuele Viterbo
Monash University, Melbourne, VIC, Australia
September 2021

C H A P T E R
1
Introduction
O U T L I N E
1.1 High-mobility wireless channels 2
References 11
The concept of cellular wireless networks emerged in the 1980s with
the first generation analog radio communications systems (1G). These
were an evolution of trunked radio systems that enabled personal wire-
less phones to become accessible to the general public. The radio spectrum
was accessed using frequency division multiplexing (FDMA) and analog
frequency modulation of the voice signals was used.
The second generation system (2G) Global System for Mobile Communica-
tions (GSM) was introduced in the early 1990s to embrace the benefits of
digital communication technology, namely the possibility to compress and
encrypt the voice signal and better control the quality of the communica-
tion in the presence of radio disturbances. This resulted in an increase in
the number of users per frequency band, and some basic digital data com-
munications became more accessible. A narrowband digital modulation
for compressed digitized voice was adopted, and time division multiple
access (TDMA) was used in conjunction with different frequency bands
for channelization. The notion of time-frequency resources was beginning
to emerge.
Prompted by the explosion of internet browsing, the third generation
(3G) system was standardized at the turn of the century under the In-
ternational Telecommunications Union IMT-2000 standard comprising two
variations: the Universal Mobile Telecommunications System (UMTS) and
Delay-Doppler Communications
https://doi.org/10.1016/B978-0-32-385028-5.00009-8 Copyright © 2022 Elsevier Inc. All rights reserved.
1

2 1. Introduction
the code division multiple access 2000 (CDMA2000) system. Wideband
signaling using the spread spectrum technique of CDMA was adopted to
enable higher digital data rates and radically moved away from narrow-
band signaling in 2G. The CDMA scheme used Walsh–Hadamard spread-
ing sequences as orthogonal basis functions, which effectively spread the
information of different users in both time and frequency. The multiple
access scheme for the uplink was realized by allowing these functions to
overlap in both time and frequency and separating them at the receiver
thanks to the different orthogonal codes (signatures). The channel impair-
ments distort the signals and cause interference between users (multiple
access interference) that needs to be compensated at the receiver. Severe
multiple access interference hinders the quasi-orthogonality of the signa-
ture waveforms at the receiver and calls for more complex processing to
recover the individual user information.
The above drawback led to the introduction of orthogonal frequency
division multiple access (OFDMA) in 4G systems which was preserved
in the 5G systems. Orthogonal frequency division multiplexing (OFDM)
is a wideband signaling technique, where multiple information symbols
overlapping in time are made orthogonal in the frequency domain by
allocating them to suitably spaced subcarriers. It can be used for multi-
ple access by allocating orthogonal time-frequency resources to the users.
With OFDMA, orthogonality is maintained at the receiver even in the pres-
ence of some multipath channel impairments. Among the advantages of
this solution, we have a relatively low processing cost for detection and
channel estimation. Fig. 1.1 summarizes the evolution of the wireless sys-
tems.
As the mobility of terminals in the system increases, the immediate ad-
vantages of multicarrier technology tend to dissipate due to the rapidly
time-varying nature of the channel affecting the orthogonality of the re-
ceived signals and resulting in severe intersymbol interference. In order
to continue using multicarrier techniques in such high-mobility environ-
ments, more complex equalization and more overhead for channel estima-
tion are required.
In this book, we will illustrate how delay-Doppler communications can
offer a solution to the limitations of multicarrier techniques over channels
with high mobility, where Doppler shifts cannot be easily compensated
for.
1.1 High-mobility wireless channels
In a static wireless channel, nothing in the environment is moving, and
both transmitters and receivers are static. The electromagnetic signal,
propagating from a transmit antenna, reaches the receiving antenna via

FIGURE 1.1 Evolution of the wireless systems.
multiple paths due to the presence of reflecting objects (scatterers). The
receiver needs to extract the transmitted information from the superpo-
sition of signals from each path. Due to the different path lengths, such
signals do not add up coherently, which may result in fading of the overall
received signal.
As the use cases for wireless communication networks continuously
evolve, a high mobility scenario is gradually becoming more prominent.
For example, high speed trains, self-driving cars, and flying taxis have the
potential to travel at speeds of several hundreds of kilometers per hour
with passengers requiring high data rates.
We define a high-mobility wireless channel as a channel where transmit-
ters, receivers, and many scatterers are moving at different speeds in dif-
ferent directions. For each path, mobility causes different Doppler shifts
of the carrier frequency fc used in the transmitted signal. A Doppler shift
by fd [Hz] is equivalent to modulating the transmitted signal with ej2πfd t .
The challenge of communicating over a channel like the one illustrated in
Fig. 1.2 is that the transmitted signal traveling over different paths is af-
fected by multiple Doppler shifts and delays. Due to the Doppler shifts,
the receiver will see a superposition of nonlinearly distorted versions of
the transmit signal.
1.2 Waveforms for high-mobility wireless channels
At an abstract level, in a point-to-point communication system, a stream
of information symbols {an} from an alphabet A is multiplexed in time

Exploring the Variety of Random
Documents with Different Content

CHAPTER VI
“ABOUT THAT HALF-DOLLAR.”
At every sound of wheels Jack started; and more than once he
imagined he heard a wagon stop at the gate. Still no deacon; would
he never return? Jack watched the clock, and thought he had never
seen the pointers move so slowly.
Three or four times he went to the door to listen; and at last he
walked down to the gate. It was bright, still moonlight, only the
crickets and katydids were singing, and now and then an owl hooted
in the woods or a raccoon cried.
“There’s a buggy coming!” exclaimed Jack. He could hear it in the
distance; he could see it dimly coming down the moonlit road. “It’s
Mr. Chatford!” He knew the deacon’s peculiar “Ca dep!” (get up) to
the horse.
“That you, Jack?” said the deacon, driving in.
“Yes; thought I’d come down and shut the gate after you,” replied
Jack.
Mr. Chatford stopped at the house, and Jack ran to help him take
out some bundles. Then the deacon drove on to the barn, and Jack
hurried after him. Still not a word about the half-dollar.
“You can go into the house; I’ll take care of Dolly,” said Jack.
“I’ll help; ’t won’t take but a minute,” said Mr. Chatford. “I’ve got
bad news for you.”

“Have you?” said Jack, with sudden faintness of heart. “What?”
“For you and Lion,” added the deacon. “Duffer’s got another dog.
He made his brags of him to-night. Said he could whip any dog in
seven counties.”
“He’d better not let him tackle Lion!” said Jack.
“I told him I hoped he wouldn’t kill sheep, as his other dog did.
Take her out of the shafts; we’ll run the buggy in by hand.”
The broad door of the horse-barn stood open. Jack led the mare
up into the bright square of moonshine which lay on the dusty floor.
There the harness was quickly taken off. Not a word yet concerning
the half-dollar, which Jack was ashamed to appear anxious about,
and which he began to think Mr. Chatford, with characteristic absent-
mindedness, had forgotten.
“By the way, I’ve good news for you!” suddenly exclaimed the
deacon.
Jack’s heart bounded. “Have you?”
“I saw Annie over at the Basin. She wants to go home to her folks
to-morrow. Would you like to drive her over? She spoke of it.”
“And stay till Monday?” said Jack, to whom this would indeed have
been good news at another time.
“Yes; start early, and get back Monday morning in time for her to
begin school. Then she won’t go home again till her summer term is
out.”
“Maybe—I’d better—wait and go then.” Jack felt the importance of
early securing his treasure, and, having set apart Sunday afternoon
for that task (“a deed of necessity,” he called it to his conscience), he
saw no way but to postpone the long-anticipated happiness of a ride
and visit with his dear friend. Yet what if the treasure were no
treasure?
“As you please,” said the deacon, a little surprised at Jack’s choice.
“Moses will be glad enough to go. See that she has plenty of hay in

the rack, and don’t tie the halter so short as you do sometimes. Now
give me a push here,”—taking up the buggy-shafts.
“Oh!” said Jack, as if he had just thought of something,—“I was
going to ask you—about that half-dollar?”
“I didn’t think on ’t,” said Mr. Chatford, standing and holding the
shafts while Jack went behind,—“not till I’d got started for home.
Then I put my hand in my pocket for something, and found your
half-dollar. Help me in with the buggy, and then I’ll tell you.”
The deacon drew in the shafts, Jack pushed behind, and the
buggy went rattling and bounding up into its place.
“Did you go back?” asked Jack, out of breath,—not altogether
from the effort he had just made.
The deacon deliberately walked out of the barn, and carefully shut
and fastened the door; then, while on the way to the house, he
explained.
“I had paid for my purchases out of my pocket-book, or I should
have found that half-dollar before. However, as I had promised you,
I whipped about and drove back to the goldsmith’s. He was just
shutting up shop. I told him what I wanted. He went behind his
counter, lit a lamp, looked at your half-dollar, cut into it, and then
flung it into his drawer.”
“Kept it!” gasped out Jack.
“Yes; ’t was as good a half-dollar as ever came from the mint, he
said. He gave me another in its place.”
Jack could not utter a word in reply to this announcement, which,
notwithstanding his utmost hopes, astonished and overjoyed him
beyond measure. As soon as he had recovered a little of his breath
and self-possession, he grasped the deacon’s arm, and was on the
point of exclaiming, “O Mr. Chatford! I have found a trunk full of just
such half-dollars as that!”—for he felt that he must tell his joy to
some one, and to whom else should he go? But already the deacon’s

other hand was on the latch of the kitchen door, which he opened;
and there sat the family round the table within.
“What is it, my boy?” said Mr. Chatford, as Jack shrank back and
remained silent. “Oh! you want your half-dollar. Of course!” putting
his hand into his pocket.
“I don’t care anything about that,” said Jack. He took it,
nevertheless,—a bright, clean half-dollar in place of the scratched
and tarnished coin he had given Mr. Chatford that afternoon.
Mr. Chatford stood holding the door open.
“Ain’t you coming in?”
“No, sir,—not just yet.”
Jack felt that he must be alone with his great, joyful, throbbing
thoughts for a little while; and he wandered away in the moonlit
night.

CHAPTER VII
HOW JACK WENT FOR HIS TREASURE.
In the forenoon of the following day Annie Felton dismissed her
little school half an hour earlier than she was accustomed to do, and
went to her Aunt Chatford’s house, to dine with her relatives and
prepare for the long afternoon’s ride. She was greatly surprised
when told that Jack was not to accompany her.
“Did Uncle Chatford speak to him about it?” she inquired of her
aunt.
“Yes, but for some reason he didn’t seem inclined to go. That just
suited Moses; he was glad enough of the chance.”
“Jack has found a half-dollar, and it has just about turned his
head,” remarked Mrs. Pipkin.
“A half-dollar?” repeated Annie, wondering if such a trifle could
indeed have so affected her young friend. No, she could not believe
it. Then why had he willingly let slip an opportunity which she had
thought he would be eager to seize?
Soon the men and boys came in to dinner,—Moses in high spirits,
and with his Sunday clothes on; Jack jealous and unhappy.
“Why didn’t I leave that till another Sunday? or get it one of these
moonlight nights?” he said to smile, and moved her lips with some
sweet, inaudible meaning as she passed him; but Moses, good fellow
though he was, cast upon him a look of contempt, and flourished his
whip, driving proudly away beside his beautiful cousin.

“IT’S A GREAT SECRET.”
Jack, much as he thought of his hidden treasure, now for the first
time in his life felt the utter worthlessness of money compared with
the good-will and companionship of those we love,—a truth which it
takes some of us all our lives to discover.
The sight of Annie Felton always stirred the nobler part of his
nature; and now, going back to the house, he began to blame

himself for having taken hitherto a purely selfish view of his
treasure.
“All I’ve thought of has been just the good it was going to do me!”
And he said to himself that he didn’t deserve the good fortune that
had befallen him. Now to bestow it all upon her he felt would be his
greatest happiness.
“And give some to you, precious little Kate!” was his second
thought, as the gay little creature came running with Lion to meet
him. In like manner his benevolence overflowed to all,—even to
sharp-tongued Mrs. Pipkin,—after Annie Felton had stirred the
fountain.
Twenty-four hours seemed long to wait. But the time for securing
his treasure at last came round. He walked to church in the morning
with Phin and Mr. Pipkin, but, without saying a word to anybody of
his intentions, he at noon came home alone across the fields. He
found, as he expected, Mrs. Chatford keeping house.
“Why, Jack!” said she, “why didn’t you stay to Sunday school and
the afternoon services?”
“Don’t you want to go this afternoon?” replied Jack, evasively.
“There will be some of the neighbors riding by, who will carry you.
I’ll take care of the house.”
“You are very kind to think of me,” she said. “But I don’t think of
going. You’d better eat your luncheon, and go right back.”
Jack longed to tell her everything on the spot, but feared she
might disapprove of his going to bring home the treasure on the
Sabbath. “After all’s over, then she’ll say I did right,” thought he. So
he remarked, carelessly, “There’s a new minister to-day; I don’t like
him very well. I guess I’ll go over and see Aunt Patsy a little while
this afternoon.”
“If you do, I’ll send a loaf of bread and one of the pies we baked
yesterday,” said Mrs. Chatford.

This was what Jack expected; and it gave him an excuse for
carrying a basket. He took off his Sunday clothes, putting on an
every-day suit in their place, lunched, and soon after started with
Lion. He made a brief visit to the poor woman, and then set out for
home by way of the woods.
On the edge of Aunt Patsy’s wood-lot he paused and looked
carefully all about him. Not a human being was in sight. A Sabbath
stillness reigned over all the sunlit fields and shadowy woods. There
were Squire Peternot’s cattle feeding quietly in the pasture. A hawk
was sailing silently high overhead. As he turned and walked on, two
or three squirrels, gray and black, ran along the ground,
disappearing around the trunks of trees to reappear in the rustling
tops, and it was all he could do to keep Lion still.
“Look here, old fellow!” said he, “remember, you are not to bark
to-day!”
From Aunt Patsy’s wood-lot he entered the squire’s, stepping over
a dilapidated fence of poles and brush. The snapping of the decayed
branches broke the silence; then, as he listened, he heard, far off,
the bells for the afternoon service begin to ring. It was a strange
sound, in that wildwood solitude, so shadowy and cool, and full of
the fresh odors of moss and fern.
The bells were still ringing, and their faint, slow, solemn toll filled
Jack’s heart with an indefinable feeling of guilt as he reached the log
where his treasure was, and reflected upon the very worldly
business that brought him there.
He did not reflect long,—he was too eager for the exciting work
before him. Having walked on to the farther edge of the woods, to
see that nobody was approaching from that direction, he returned,
and began to pull out the sticks which he had stuffed into the end of
the log.
“Everything’s just as I left it, so far,” thought he. “Wonder if my
money-chest will dodge a fellow, like old Daddy Cobb’s!”

The opening clear, he put on an old brown frock which he had
brought in the basket, laid his hat and coat on the ground, told Lion
to watch them, and entered the log headforemost. The treasure,
too, was where he had left it. His body stopped the cavity so that he
could see nothing in its depths, but his groping hand felt the little
trunk and the coin that had fallen out of its broken end.
“I’ll take this loose money out of the way first,” thought he; “then
maybe I can move the trunk.”
He had nothing but his pockets to put the coin into, and those his
frock covered. “I’ll find something better,” thought he. Backing out of
the log, he pulled off his shoes, and re-entered with one of them in
his hand. This he filled with all the half-dollars he could find about
the end of the trunk, which he then tried to move.
“It’s stuck in a heap of rotten stuff here,” he muttered, “and I shall
break it more if I pull hard on it.” So he resolved to empty it where it
was.
He was half-way out of the log, bringing after him his shoe
freighted with coin, when he was startled by a sudden bark from
Lion. Leaving his shoe, he tumbled himself out upon the ground in
fearful haste, to find a stray calf in the bushes the innocent cause of
alarm. For keeping guard too faithfully poor Lion got a box on the
ear.
After waiting awhile, to see if anything more dangerous than the
calf was nigh, Jack brought out his shoe, poured its rattling contents
into the basket, which he covered with his coat, and then went back
into the log. This time he took both shoes in with him, which he
filled, and emptied one after the other into the basket. Another
journey, another, and still another, and he began to think there was
more coin than he could carry home.
“I can get it away from here, though, so nobody can tell on whose
land I found it,”—which he seemed to think a very important point to
gain. “I’ll leave the little trunk where it is,—only take out the money.”

He had gone into the log for the last time, and got the last of the
money, filling both shoes quite full, and was bringing them out with
him,—he had actually got them out, leaving one at the entrance to
the opening, and holding the other in his hands,—when Lion,
notwithstanding his previous punishment, uttered a very low,
suppressed growl.
Jack looked up from under his tumbled hair, and there, not three
yards distant, with his horn-headed cane, regarding with grim
amazement the boy and his shoes full of coin, stood Squire Peternot!

CHAPTER VIII
JACK AND THE SQUIRE.
Fearing a raid upon his melon-patch, which bad boys in the
neighborhood were beginning to molest, the squire had stayed at
home to watch it that Sunday afternoon. He had seen Jack with his
dog and basket cross the fields, go to Aunt Patsy’s house, and
afterwards enter the woods; and, feeling the interest of a stern
moral censor in the conduct of all Sabbath-breaking boys, he had
followed him to the hollow log. Lion’s indiscreet barking had at first
served to guide him to the spot; and afterwards his equally
unfortunate silence, in consequence of the punishment he had
suffered for that offence, favored the old man’s stealthy approach.
To have the faintest idea of the emotions that agitated the squire
at sight of Jack and the shoes full of coin,—the wrath, the surprise,
the avarice,—one must have seen him as he stood there, or have
heard Jack (as I have heard him many times) describe the grim and
frowning figure that met his eyes.
“What’s this, what’s this, eh?” cried Peternot, taking a stride
forwards. “Money! on my land!” and the gray eyes glittered. “Ha! ha!
This, then, is the meaning of all that talk about treasure-trove the
other day!”

“BOY! ARE YOU A ROBBER?”
Jack felt so stunned for the moment that he did not attempt to
speak, or even to rise. He sat on the ground, guarding his shoes,
keeping one hand on the rim of the basket and looking up steadily at
the squire with eyes full of mingled fear and defiance.
“So, so! What have you got in your basket?” And the stiff-jointed
old man stooped to remove the coat which Jack had taken the

precaution to spread over it each time when he entered the log.
“Here! you just leave that alone!” exclaimed Jack, while Lion gave
a fierce growl. The squire dropped the garment instantly, but he had
pulled it far enough from the basket to expose its surprising
contents.
“Boy!” said he, in still greater amazement, “are you a robber?”
“Like enough I am,” muttered Jack, quite willing that he should
take that view of the case.
“Boy!” repeated Peternot, with awful severity, “you’ve stolen this
money, and it’s my duty to have you arrested. I am a justice of the
peace.” Jack changed countenance at that.
“I’ve stolen it about as much as I stole Mr. Chatford’s horse and
buggy once, which you were so sure of, when they were all the
while standing under the shed at the Basin, just where Mr. Chatford
left them.”
“Then how did you come by so much money?”
“If you must know, I found it in this log,” said Jack, with a sudden
determination to tell the plain truth, and stand or fall by it.
“How do I know but what you stole it and hid it here, so you could
pretend you’d found it?”
Jack was glad now that he had not removed the trunk.
“If you can’t see by the look of this silver that it’s been hid away
here longer than I’ve been in the town,” he replied, “you can just go
into the log and find the trunk, that you’ll say has been there about
as many years as I am old, that’s all!”
“Is there any more money in there?”
Jack was willing the squire should think there might be, nor was
he sure there were not a few pieces in the rubbish about the trunk;
so he said, “It belongs to me, if there is.”
“Belongs to you? You little scapegrace! By what right?”

“It belongs to me,—that is,” added Jack, “if the real owner doesn’t
turn up,—because I found it.”
“Found it, on my land! You haven’t got it off from my land yet, and
I forbid your taking it off. What’s left in the log you haven’t even had
in your possession. I want nothing but what’s my own by a plain
interpretation of law; but the law’s with me in this. If you had once
fairly got the coin away without my knowledge, there might have
been some question about it; but that you’ve been caught
trespassing, and that you’ve no right to take anything from my
premises in my presence and against my express orders, is common
sense as well as common law.”
Fire and tears rushed into poor Jack’s eyes.
“And do you mean to say you’ll take all this money away from
me?”
“Sartin, I do, since it don’t belong to you, not a dollar on ’t. I’ll
make ye a reasonable reward, however, if you give it up without
making me any unnecessary trouble.”
“What do you call a reasonable reward? Half?”
“Half! of all that money!” exclaimed the squire, in huge
astonishment. “Preposterous! I’ll give ye more than liberal pay for
your trouble. I’ll give ye five dollars.”
Thereupon grief and fury and fierce contempt burst from the soul
of Jack. All the softening influences which had been at work upon
him for the past few months were forgotten in a moment; he was
the vicious, desperate, profane little canal-driver once more. Looking
up through tears of rage at the startled squire, he shouted, “Go to
thunder, you hoary old villain!” and followed up this charge with a
volley of blasphemy and abuse, which lasted for at least a minute.
By that time the squire had recovered his self-possession; so, in a
measure, had Jack; and the hurricane of passion that had swept
everything before it was followed by a lull of sullen hate and despair.

“That’s the kind of boy you are, is it? after all your living among
Christian people!” said the old man, with a sort of grim satisfaction.
“It’s the kind of boy I was, and it’s the kind of boy such Christians
as you are will make me again, if I let you!” said Jack, kindling once
more. “I didn’t mean to swear, but I forgot myself. I haven’t before,
since the first Sunday after I came off from the canal. That’s
because I have been living among Christians,—people who try to
encourage a fellow and help him, by bringing out the good that’s in
him, instead of grinding him down, and keeping him down, by telling
him how bad he’s always been and always will be,—like the kind of
Christian you are!”
“Talk to me about being a Christian, you profane Sabbath-
breaker!” said Peternot, choking with indignation.
“A Sabbath-breaker, am I? And what are you? I own up to what
brought me here to-day, but what brought you here? What keeps
you here? Why ain’t you at church? Guess you consider your worldly
interests worth looking after a little, if ’tis Sunday,—don’t you?”
“Come, come, boy! that kind of talk won’t help matters.”
“Then le’s stop it,” said Jack. “But if you come here on Sunday and
try to get my money away from me, and accuse me of Sabbath-
breaking because I mean to keep it, I shall have just a word to say
back, you better believe!” And, still sitting on the ground, Jack held
his shoes between his legs, and guarded one side of the basket,
while Lion guarded the other.
“What do you want of so much money,—a boy like you?” said the
squire, adopting a more conciliatory tone.
“What do you want of it,—a man like you? without a child in the
world, since you drove your only son away from home by your hard
treatment, and he died a drunkard and a gambler!”
The old man fairly staggered backward at this cruel blow, and
uttered a suppressed groan.

“It was mean in me to say that,” added Jack, relenting; “I didn’t
mean to; but you drove me to it. What do you want of more money
than you’ve got already?—that’s what I meant to ask. You’re a rich
man now. You’ve ten times as much as you need; what do you want
of more? To carry into the next world with ye? one would think so,—
an old man like you!”
“Boy!” said the trembling Peternot, “you don’t know what you’re
talking about!”
“Yes, I do; I’m talking just what a good many other folks talk, only
not to your face. They say, ‘There’s old Squire Peternot, seventy
years old, with one foot almost in the grave,—rich enough in all
conscience,—don’t use even the interest on what money he has, but
lays it up, lays it up,—lives meanly as the poorest farmer in town,—
never gives a dollar, except when he can’t help it, and then you’d
think it hurt him like pulling his teeth,—and yet there he is, trying to
get Aunt Patsy’s little house and lot away from her,—making tight
bargains, screwing his workmen’s wages down to the lowest notch’;
that’s what I’ve heard, every word of it, and you know that every
word of it is true!”
“I have my own ideas about property,” said the squire; “and no
man—no prudent man—likes to squander what’s his own.”
“And so you, with all your wealth, come and grab this money,
which is all I have in the world, and offer me five dollars to give it up
to you! You are a prudent man! I say squander!”
“I’ll give you twenty dollars of it,—and that’s liberal, I’m sure,” said
Peternot, a good deal shaken by what Jack had said, but unable,
from long habit, to take his hand from any worldly goods that it
chanced to cover.
“Twenty dollars!” laughed Jack, with scornful defiance. “I don’t
make bargains on Sunday.”
This cool sarcasm caused the worthy Peternot to wince as at the
taste of some bitter medicine. “I don’t bargain on the Lord’s day,

neither. But I see the necessity of coming to some sort of terms with
you.”
“Very well; then you just walk off and leave me and my dog to
take care of this money; those are the only terms you can come to
with me.”
“But what do you propose to do, if I don’t walk off?”
“Stay here,—Lion and I,—and hang on to our treasure-trove. Your
nephew, who knows so much about law, advised me to keep
possession,—to fight for it,—and I will.”
“And do you think I’m going to give up to you, you renegade?”
cried the squire. He moved to lay his hand on the basket; but there
was something in Lion’s growl he didn’t like. “I’ll beat that beast’s
brains out, if he offers to touch me!” he exclaimed, grasping his cane
menacingly.
“I advise you not to try that little thing,” said Jack. “If you should
miss your stroke, where would you be the next minute?”
The squire thought of that. His tone changed slightly.
“I don’t leave this spot till I git possession of that money!”
“All right, Squire. Sit down,—you’d better. You’ll have some time to
stop, I guess. Have a peach?” And the audacious little wretch took
one out of his coat-pocket. “We shall need refreshments before we
get through!” As Peternot indignantly declined the proffered fruit,
Jack quietly broke it open, and ate, with a relish, the rich yellow
pulp. The old man accepted the invitation to sit down, however, and
reposed his stiff old limbs on the end of the hollow log, not clearly
foreseeing how this little adventure was to end.

CHAPTER IX
THE SQUIRE’S PERPLEXITY AND JACK’S STRATAGEM.
A little calm reflection opened the squire’s mind to a ray of light
which would certainly have dawned upon it before, had not his wits
been clouded by passion. “Boy!” he suddenly exclaimed, “I believe
every dollar of that money is bogus.”
“Then what’s the use of making a row over it?” was the boy’s cool
retort.
“It’s the business of a magistrate to look after counterfeiters and
counterfeit money,” said Peternot. But at the same time he thought,
“He has satisfied himself that it ain’t counterfeit; his whole conduct
shows it.” And the avaricious old man still laid siege to the basket.
Half an hour passed, during which time very little was said. Jack
took out his knife and began to whittle a stick; perhaps he was not
unwilling to show the squire that he was armed. He also put on his
coat, and then his shoes, after emptying their contents into the
basket.
Peternot grew more and more impatient, as he saw the afternoon
gliding away. Another half-hour, and the situation still remained
unchanged. “I may set here till night,” thought he, “and all night,
and all day to-morrow, fur’s I know,—but what’s the use? He’ll stick
as long as I do. He’s tough; he can stand anything; ye can’t starve a
canal-driver. Sakes!” he exclaimed, half aloud, suddenly putting his
hand into his pocket, remembering that the key of his kitchen door
was there.

On leaving home he had carefully made fast all the doors and
windows of his house,—his wife and nephew having gone to
meeting that afternoon; and now, should they return before he did,
they would find themselves locked out!
Still the old man’s cupidity would not suffer him to raise the siege.
He was taken by a fit of coughing; and, fearing to catch cold by
sitting on the damp log, he got up and walked about,—frowning and
striking his cane upon the ground in huge dissatisfaction and
disgust. “You’re the most obstinate, unreasonable boy I ever see!”
he exclaimed angrily.
“Am I?” laughed Jack. “You haven’t begun to see how obstinate I
am. Wonder what you’ll think to-morrow at this time? or the next
day?” And what, he might have added, would the wife and nephew
think?
“Hush!” whispered the old man. “What boys are those?”
There was a crackling of sticks in a not very distant part of the
woods, occasioned by a gang of four or five boys climbing Peternot’s
brush fence. Jack jumped upon the log and looked.
“It’s the Huswick tribe,” said he. “There’s Dock, there’s Hank,
there’s Cub,—there they all are, going over your fence like a flock of
sheep!”
“The Huswicks, Cub and Dock,—Hank with ’em!” ejaculated the
squire, in great excitement. “They’re the wust set of boys in town!”
“Yes, and they’re putting straight towards your house,” observed
Jack.
“They’re after my melons!” said Peternot, brandishing his cane.
“The rogues! I’ll larn ’em!” With a limping stride he started in
pursuit, but turned back immediately. “Promise me you’ll stay here!”
Jack couldn’t help laughing at the old man’s simplicity. “Do you
think I’m such a fool as to make that promise? Or even if I should,

would you trust me to keep it? Come!” cried Jack, “you must have a
better opinion of me than you pretend.”
“I know you have some good traits—the rogues will destroy all my
melons—if I could borrow your dog—leave your basket and go with
me—we’ll settle our diffikilty when we come back,” said the agitated
squire.
“I’ll take care of my basket; you can look after your melons,”
retorted Jack.
“I’d as lives have a passel o’ pigs in my melon-patch!” cried
Peternot, striding to and fro. “Boy! I’m sure this money is bogus!—I
wish I had called to ’em ’fore they got out o’ hearin’!”
“Why didn’t ye?” asked Jack.
“That might ’a’ led ’em to come here, and we don’t want anybody
by the name o’ Huswick to have a hand in this business. But my
melons!—Boy, be reasonable!”
“Be reasonable yourself, Squire Peternot! You’re sure this money is
bogus; then why don’t you leave it and go for your melons?”
“I ain’t sure,” replied the squire. “But you’re sure it’s good money;
I see that, and you’re no fool.”
“Thank ye, sir,” said Jack, politely. And, seeing that the old man’s
cupidity made him ready to believe almost anything, he added, “Now
look here! If I’ll give you what money there is in the basket, will you
be satisfied?”
Peternot started. “Satisfied? Sartin—I can’t tell—explain!”
“Will you take this, and leave me what there is still in the log?
That’s what I mean,” said Jack, with an air of candor.
Peternot, astonished by this strange proposition, but afraid of
being cheated out of a few dollars, asked, “How much is there in the
log?” at the same time stooping with difficulty and peeping into the
cavity.
“That’s my risk. Come, is it a bargain?”

“I thought you didn’t make bargains on the Sabbath day!”
“Well, I don’t,” laughed Jack, “unless some good man sets me the
example. I’m only a boy,—it’s easy to corrupt me.”
“Corrupt you! you sassy, profane—”
“Sabbath-breaker,” suggested Jack, as Peternot hesitated for a
word bad enough. “What do you say to my offer?”
“I say, if there’s money in the log, it belongs to me, the same as
this belongs to me.” And the squire, impressed by the importance of
having some accurate knowledge on that point, vigorously thrust in
his cane.
“Your stick can’t give ye much information,” said Jack. “You’ll have
to go in yourself.”
“I’m going in myself!” exclaimed the squire, sharply. “Move out of
my way here.”
Jack readily made room for him, tickled to the heart’s core at the
thought of the stiff-jointed old man’s going into the log.
“Grin, will ye?” said Peternot. “I s’pose you think the minute I’m in
there you’ll start to run with your basket. But you can’t run fur with
that weight to carry; I shall ketch ye!”
He leaned his cane by the log, laid his hat beside it, and put his
head and one arm into the cavity. Then he put in his shoulders and
both arms. “I can hear ye, if ye stir to move!” he cried from the
hollow depths, which muffled his voice; and in his body went,
leaving only the long Peternot legs sticking out.
Jack was convulsed with laughter. But all at once the idea
occurred to him that practical advantage might be taken of the
squire’s ludicrous situation. Up he jumped, and seizing the largest of
the sticks with which he had previously stopped the mouth of the
log, began to thrust them in after the squire.
“Here! oh! oh! murder!” cried the voice, now more muffled than
ever, while the old man struggled violently to get out. “Oh! oh!”

“Good by!” screamed Jack, holding him, and thrusting in more
sticks. “You may have what’s in the log, and I’ll take the basket.”
PETERNOT IN THE HOLLOW LOG.
“Help! ho! I’m killed!” said the voice, growing fainter and fainter.

“And buried!” Jack yelled back, laughing with wild excitement. “But
you kick well, for all that!” And in went more rubbish about the old
man’s heels. “How do ye like your bargain? You’ll have plenty of time
to count your dollars before I send Pipkin over to help you out.”
And, having got the old man wedged so tightly into the log that he
could not even kick, Jack, inspired with extraordinary strength for
the occasion, caught up his basket of coin and started to run,
followed by Lion.

CHAPTER X
“THE HUSWICK TRIBE.”
Running quickly behind walls and fences, the Huswick boys made a
rapid raid upon Peternot’s melon-patch, and left it loaded with spoils.
“Say, Dock!” said Hank (nickname for Henry), skulking behind
some bushes, “le’s put for Chatford’s orchard, and scatter rines by
the way, so if we’re tracked the old man’ll think ’t was the deacon’s
boys hooked his melons.”
“Go ahead!” said Dock (nickname for Jehoshaphat), carrying two
fine ripe melons on his left arm while he dug into one of them with a
jack-knife in his right hand. “Stoop, and keep clus to the fence!”
“No danger, old man’s gone to meetin’,” said Cub, whose real
name was Richard,—his odd shape (he was ludicrously short and fat)
having probably suggested the nickname.
“Me an’ Cub can go without stoopin’,” giggled Hod, the youngest
(christened Horace). “See Hank! he looks like a well-sweep!”
And indeed the second of the boys, who was as wonderfully tall
and lank as Cub was short and thick, bore no slight resemblance to
that ornament of country door-yards.
“Hanged if one o’ mine ain’t a green one!” exclaimed Tug (short
for Dwight), dashing to the ground a large watermelon, the sight of
which in ruins would have made old Peternot’s heart ache.
“Guess we made a clean sweep of all the ripe ones,” said Cub.
“No, you don’t!” as Tug offered to relieve him of one of his three. “I

never had my fill o’ melons yit, though I’ve”—cramming his mouth
while he continued to talk—“been in the squire’s patch much as once
afore now.”
“You never had your fill of anything, I believe, Cub!” said Hod,
with his usual giggle. “Remember when we went there in the night
last year?”
“Night’s no time to go for melons,” said Cub. “Ye can’t tell a ripe
one ’thout cuttin’ into ’t.”
“Yes, ye can,” said Tug; “smell on ’t. That’s the best way to tell a
mushmelon.”
“Cub’s terrible petic’lar about slashin’ the ol’ man’s whoppers, all to
once,” said Horace.
“Of course, for if we cut a green one we sha’n’t find it ripe next
time we go,” Cub explained. “Jest look! we’re makin’ a string o’ rines
all the way from Peternot’s to the deacon’s orchard!”
“There now, boys,” said Hank, “throw what rines ye got down here
by the brook, an’ stop eatin’ till we git to the woods.”
Their course had been westward, until they reached the orchard.
They now took the line of stone-wall which divided the squire’s land
from the deacon’s, and which led northward to the corner of
Peternot’s wood-lot,—Hank following Dock, Cub following Hank, Tug
after Cub, and Hod bringing up the rear. In this order they entered
the woods, and were hastening to find a secluded spot where they
could sit and enjoy their melons, when suddenly Dock stopped.
“Thought I heard somebody,” he said to Hank, coming up.
“So did I. Lay low, boys! Git behind this log!”
Down went boys and melons in a heap, each of the brothers, as
he arrived, tumbling himself and his load with the rest. There they
lay, only Hank’s long, crane-like neck being stretched up over the log
to reconnoitre; but presently even he thought it time to duck, and
threw himself flat upon the ground with the rest.

“Keep dark!” he whispered; “it’s that Jack Hazard, that lives to the
deacon’s! him an’ his big dog!”
Jack indeed it was, who had been too intently occupied in
fastening Peternot into the log to notice the approach of the Huswick
boys. He had thought of them, to be sure, but had supposed they
would return through the woods as they went.
He was now running as fast as he could with his basket of
treasure, directing his course towards the orchard, but keeping a
little to the right in order to reach a low length of fence, over which
he intended to climb, and then betake himself to the smoother
ground of the pasture. A log lay in his way. Lion, growling, drew
back from it—too late. Jack, in his headlong haste, sprang upon it,
and leaped down on the other side, alighting on a frightful heap of
legs and heads and watermelons. He jumped on Hank, tripped
against Cub, and, falling, spilt his basket of rattling coin all over Tug
and Dock and Hod. Thereupon the heap rose up as one man,
astonishing poor Jack much as if he had stumbled upon a band of
Indians lying in ambush.
“What in thunder!—Jerushy mighty!—half-dollars!” ejaculated Cub
and Dock and Tug; while Hank stretched himself up to his full
height, and Hod fell vindictively upon Jack.
“Le’ me go!” screamed Jack, taking his knee out of a muskmelon,
and shaking off his assailant.
“That’s my melon,” said Hod, diving at him again furiously, “an’
you’ve smashed it!”
He was butting and striking with blind rage, when Lion bounced
upon him, and actually had him by the collar of his coat, dragging
and shaking him with terrible growls, when Tug and Cub and Dock—
one catching Hod by the heels, one Jack, and the other Lion—
disentangled the combatants.
“Where j’e git all this money?” demanded Cub.

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