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Digital signal processors architecture programming and
applications 2nd Edition B. Venkataramani Digital
Instant Download
Author(s): B. Venkataramani; M. Bhaskar
ISBN(s): 9780070702561, 007070256X
Edition: 2
File Details: PDF, 34.90 MB
Year: 2011
Language: english

DIGITAL SIGNAL
PROCESSORS
Architecture, Programming and Applications
Second Edition

About the Authors
B Venkataramani is presently working as Professor in the Department of Electronics and
Communication Engineering. He has been a faculty of the National Institute of Technology (previously
referred to as Regional Engineering College), Tiruchirappalli, since 1987 . Prior to that, he worked for
BEL, Bangalore, and IIT Kanpur for about three years each. He has executed various development
and research projects in DSP and VLSI and has authored two books and published/presented several
research papers in various international and national journals/conferences respectively. He has guided
ﬁve PhD scholars and mentored several students in presenting their M Tech and MS theses. His research
interests include design of FPGA as well as P-DSP based speech-recognition systems, software-deﬁned
radio and sensor networks.
M Bhaskar is currently working as Associate Professor in the Department of Electronics and
Communication Engineering and teaches the subjects of low power VLSI and DSP architecture,
programming and applications. He has been a faculty of the National Institute of Technology (previously
referred to as Regional Engineering College), Tiruchirappalli, since 1997. He has authored one book and
published/presented several research papers in various international and national journals/conferences
respectively. He has guided several M Tech students in presenting projects on ‘C54X and ‘C6X based
system designs. He has also executed several development projects in DSP. His research interests include
design of FPGA based coprocessors for P-DSP and design of low-power interconnects for high-speed
VLSI.

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DIGITAL SIGNAL
PROCESSORS
Architecture, Programming and Applications
Second Edition
B Venkataramani
Professor
Department of Electronics and Communication Engineering
National Institute of Technology
Tiruchirappalli, Tamil Nadu
M Bhaskar
Associate Professor
Department of Electronics and Communication Engineering
National Institute of Technology
Tiruchirappalli, Tamil Nadu

Tata McGraw-Hill
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Digital Signal Processors, 2e
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CONTENTS
Preface xiii
1. An Overview of Digital Signal Processing and its Applications 1
1.1 Signals and Their Origin 1
1.2 Noise 1
1.3 Filters and Noise 1
1.4 Correlators 2
1.5 Convolution and Inverse Filtering 3
1.6 Fourier Transform and Convolution Theorem 3
1.7 Sampling Theorem and Discrete Time System 4
1.8 Linearity, Shift Invariance, Causality and Stability of Discrete Time Systems 4
1.9 Z Transform 5
1.10 Frequency Response of LTI Discrete Time System 7
1.11 Digital Signal Processing 8
1.12 Advantages of Digital Signal Processing (DSP) 8
1.13 DSP in the Sample and Transform Domain 9
1.14 Fast Fourier Transform (FFT) 10
1.15 Digital Filters 12
1.16 Finite Word Length Effect in Digital Filters 20
1.17 Power Spectrum Estimation 20
1.18 Short Time Fourier Transform 21
1.19 Multirate Signal Processing 22
1.20 Discrete Wavelet Transform 38
1.21 Adaptive Filters 43
1.22 Image Data Compression 44
1.23 Linear Predictive Coder and Speech Compression 50
Review Questions 51
Self Test Questions 55
2. Introduction to Programmable DSPs 57
2.1 Multiplier and Multiplier Accumulator (MAC) 57
2.2 Modiﬁed Bus Structures and Memory Access Schemes in P-DSPs 58
2.3 Multiple Access Memory 59
2.4 Multiported Memory 59

viii Contents
2.5 VLIW Architecture 60
2.6 Pipelining 60
2.7 Special Addressing Modes in P-DSPs 62
2.8 On-Chip Peripherals 64
Review Questions 68
3. Architecture of TMS320C5X 70
3.1 Introduction 70
3.2 Bus Structure 71
3.3 Central Arithmetic Logic Unit (CALU) 72
3.4 Auxiliary Register ALU (ARAU) 73
3.5 Index Register (INDX) 73
3.6 Auxiliary Register Compare Register (ARCR) 73
3.7 Block Move Address Register (BMAR) 74
3.8 Block Repeat Registers (RPTC, BRCR, PASR, PAER) 74
3.9 Parallel Logic Unit (PLU) 74
3.10 Memory-Mapped Registers 74
3.11 Program Controller 74
3.12 Some Flags in the Status Registers 75
3.13 On-Chip Memory 76
Review Questions 79
4. TMS320C5X Assembly Language Instructions 81
4.1 Assembly Language Syntax 81
4.2 Addressing Modes 82
4.3 Load/Store Instructions 88
4.4 Addition/Subtraction Instructions 90
4.5 Move Instructions 91
4.6 Multiplication Instructions 93
4.7 The NORM Instruction 96
4.8 Program Control Instructions 97
4.9 Peripheral Control 100
Review Questions 108
5. Instruction Pipelining in C5X 110
5.1 Pipeline Structure 110
5.2 Pipeline Operation 110
5.3 Normal Pipeline Operation 110

Contents ix
6. Application Programs in C5X 120
6.1 ¢C50-Based DSP Starter Kit (DSK) 120
6.2 Programs for Familiarisation of the Addressing Modes 126
6.3 Program for Familiarisation of Arithmetic Instructions 130
6.4 Programs in C5X for Processing Real Time Signals 135
7.1 Introduction 168
7.2 An Overview of TMS320C3X Devices 168
7.3 Internal Architechture 169
7.4 Central Processing Unit (CPU) 170
7.5 CPU Register File 172
7.6 Memory Organisation 177
7.7 Cache Memory 179
7.8 Peripherals 181
8. Addressing Modes and Language Instructions of ¢C3X 187
8.1 Data Formats 187
8.3 Groups of Addressing Modes 196
8.4 Assembly Language Instructions 198
9.1 TMS320C3X Starter Kit (DSK) 218
9.2 Example Programs for Addressing Modes 222
9.3 Generation and Finding the Sum of Series 230
9.4 Convolution of Two Sequences 233
9.5 Processing Real Time Signals with C3X Kit 236
9.6 Serial Port 240
9.7 Capture and Display of Sine Wave 248
10. An Overview Of TMS320C54X 257
10.2 Architecture of 54X 257
10.3 ¢54X Buses 260
10.4 Internal Memory Organisation 260

x Contents
10.5 Central Processing Unit (CPU) 261
10.6 Arithmetic Logic Unit (ALU) 266
10.7 Barrel Shifter 268
10.8 Multiplier/Adder Unit 268
10.9 Compare, Select and Store Unit (CSSU) 270
10.10 Exponent Encoder 270
10.11 The C54X Pipeline 271
10.13 External Bus Interface 272
10.14 Data-Addressing 273
10.15 Program Address Generation Logic (PAGEN) 273
11.1 Data Addressing in ¢C54X 290
11.2 Arithmetic Instructions 301
11.3 Move Instructions of ¢54X 309
11.4 Load/Store Instructions of ¢54X 310
11.5 Logical Instructions 311
11.6 Control Instructions 313
11.7 Conditional Store Instructions 314
11.8 Repeat Instructions of ¢54X 315
11.9 Instructions for Bit Manipulations 315
11.10 Some Special Control Instructions 316
11.11 I/O Instructions of ¢54X 318
11.12 Parallel Instructions 319
11.13 LMS Instruction 319
12.2 Code Composer Studio 328
12.3 An Overview of the ¢C5402-Based DSK 334
12.4 Introduction to C54X Assembly Language Programming 335
12.5 Applications Programs in C54X 340
13.2 Features of ¢C6X Processors 356
13.3 Internal Architecture 357

Contents xi
13.4 CPU 357
13.5 General-Purpose Register Files 358
13.6 Functional Units and Operation 359
13.7 Data Paths 361
13.8 Control Register File 365
14.1 Functional Units and Its Instructions 370
14.3 Fixed Point Instructions 378
14.4 Conditional Operations 388
14.5 Parallel Operations 388
14.6 Floating Point Instructions 390
14.8 Interrupts 397
15. TMS320C6X Application Programs and Peripherals 402
15.1 Code Composer Studio (CCS) 402
15.2 Application Programs in ¢C64X 407
15.3 Application Programs in ¢C67X 416
15.4 Internal Memory 428
15.5 External Memory 430
16. Architecture of TMS320C55X Processors 438
16.2 Features of ¢C55X Processors 438
16.3 CPU Architecture of ¢C55X 439
16.4 Memory Architecture 446
16.6 Assembly Language Instructions 455
16.8 Interrupts 479
16.9 Peripherals 480

xii Contents
17. Recent Trends in DSP System Design 486
17.1 An Overview of the Application Notes on DSP Systems 486
17.2 An Overview of Open Multimedia Applications Platform (OMAP) 488
17.3 Evolution of FPGA Based DSP System Design 491
17.4 An Introduction To FPGA 491
17.5 Design Flow for an FPGA Based System Design 503
17.6 Cad Tools for FPGA Based System Design 504
17.7 Softcore Processors 506
17.8 FPGA Based DSP System Design 511
17.9 New Algorithms for Implementation of Filters in VLSI 515
17.10 Distributed Arithmetic Algorithm 515
17.11 Case Studies 519
17.12 Comparison of the Performances of the Systems Designed
Using FPGAs and Digital Signal Processors 525
18. FPGAs in Telecommunication Applications 527
18.1 Evolution of the Radio Receiver 527
18.2 DDFS with Phase Accumulator and ROM 530
18.3 Coordinate Rotation Digital Computer (CORDIC)
Algorithm and Its Applications 535
18.4 Case Study of an FPGA Based Digital Receiver 546
Answers to Selected Questions 554
Bibliography 559
Index 563

PREFACE
Brief Overview
Digital Signal Processors (DSPs) are microprocessors specifically designed to handle Digital Signal
Processing tasks and are deployed in a variety of applications like hard-disk controllers, cellular phones,
speech-recognition systems, image processing, wireless communication systems, and so on. They are
replacing conventional microprocessors in several applications. DSPs from companies such as Analog
Devices, Motorola and Texas Instruments are deployed in all these applications.
This book presents details of DSPs from Texas Instruments (TI) in greater depth as compared to the
DSPs from other vendors. The TI processors are used in major universities, institutes and in the industry.
TI has been donating DSP kits and literature to universities periodically under the University Program
called UNITI. The individual institutions have supplemented this with their own funding and have set
up DSP labs.
Courses on Digital Signal Processing have undergone a gradual change during the last decade. The
focus is shifting gradually from the design of DSP systems and algorithms to efficient implementation
of the systems and algorithms. To facilitate this, the subject of DSP Architecture and Programming is
now included by many leading institutions in the main curriculum. However, students have to generally
rely on the data manuals of various companies for their study since formal textbooks are not readily
available. This book has fulfilled the student’s requirements and has been used extensively in universities
and leading institutions.
Aim of the Revision
The first edition of this text was published eight years back in 2002. Since then, a lot of new DSPs have
evolved due to continuous research and development in this field.
The objective of this revision is to include the recent developments in the field of Digital Signal
Processors including TMSC6X Series and FPGA based system design methodology. We also aim to
bridge the gap in topical coverage in the current edition and improve the pedagogical features to meet
the students’ requirements.
New to this Edition
In the revised edition, the introductory chapter is expanded with more real-world applications. This
includes power spectrum estimation, orthogonal frequency division multiplexing, algorithm for the
computation of 1D and 2D discrete wavelet transforms and JPEG2000.
Some of the digital signal processors such as 55X and 6X were treated in brief in the first edition. Since
a large number of systems are implemented using these processors, a more detailed treatment of these
chapters are given in this edition.

The last chapter in the previous edition had a brief introduction on the FPGA based system. However,
FPGAs are now deployed in many high-speed applications such as network routers and front ends of
software-defined as well as cognitive radio. In view of these, more details of the FPGA based system
design including implementation of system on programmable chips are presented in this revised
edition.
In order to illustrate the use of FPGAs and PDSPs in Digital radio receiver, a separate chapter is devoted
for the presentation of the design details of the various blocks of a radio receiver with digital hardware
and the case study of a software-defined spread spectrum transmitter and receiver is presented. The
pedagogy is also refreshed with inclusion of new review questions, multiple choice questions and new
programs. The chapter on ‘Motorola DSP563XX Processors’ is uploaded on the website.
To summarise, the changes made to this edition are the following:
New Chapters
v TMS320C6X Assembly Language Instructions (Chapter 14)
v Architecture and Application Programs of TMS320C55X (Chapter 16)
v FPGAs in Telecommunication Applications (Chapter 18)
New Topical Inclusion
v Convolution and real time filtering using FFT
v OFDM Using FFT
v Data Paths in TMS320C6X
Organization of the Book
Chapter 1 presents an overview of DSP principles, algorithms and applications. At the beginning of this
chapter, a simple treatment on DSP theory, algorithms and applications is presented for students having
no prior knowledge of DSPs. Introduction to DSP architecture and comparison of this with that of μPs,
DSPs and RISC processors is given in Chapter2. Chapters 3, 4, 5 and 6 present the detailed architecture,
addressing modes, instruction sets, and pipelining and application programs on TMS320C5xDSP. The
corresponding details on TMS320C3X are presented in chapters 7, 8, and 9. Chapter 10 introduces
the TMS30C54X and presents a comparison of the features of 5X with that of 54X. The instruction set
and addressing modes of 54X are discussed in Chapter 11. Application programs on 54X and program
development using Code Composer Studio are presented in Chapter 12. Architecture, assembly-
language instructions, application programs and peripherals of TMS320C6X are given in chapters 13,
14 and 15 respectively. Architecture of TMS320C55X is explained in Chapter 16. Chapter 17 gives a
list of some of the recent DSP application case studies and introduces an alternate DSP system design
approach using programmable logic devices and FPGAs. Examples of architectures of two leading
FPGA families and hardcore as well as softcore processors for these families are explained in this
chapter. Algorithms for efficient implementation of DSP systems in FPGA are also given here. Chapter
18 explains the different applications of FPGAs in telecommunication.
Web Supplements
The web supplements can be accessed at, http://www.mhhe.com/venkataramani/dsp2 and contain the
following material:
xiv Preface

Instructor Resources
v Solution manual
v PowerPoint lecture slides
Student Resources
v Interactive quiz
v Chapter on Overview of Motorola DSP563XXX Processors
Acknowledgements
We are thankful to Texas Instruments for providing the DSP kits on various processors. We thank our
alumni in TI who have been helping us to upgrade our DSP laboratory from time to time. A special note
of thanks goes to the UNITI coordinator and his team for extending his wholehearted encouragement
and necessary approval to use the data books of TI. Motorola was equally cooperative and allowed us
to include the material on their DSPs. Hence, we would like to express our gratitude to the Motorola
team. Xilinx and Altera also deserve a special mention for letting us use their material for the revised
edition.
We are indebted to our former principal, Dr M Arumugam, and former head of the department, Dr N
Kalyanasundaram, who extended all the necessary support and encouragement to offer the laboratory
course based on Digital Signal Processors not only to our undergraduate and postgraduate students but
also to the students from other colleges under the continuing education programme twice a year. The
ﬁrst edition of this book is borne out of the experience we gained by offering this course for ﬁve years to
the regular students and to the students under the continuing education programme. The projects carried
out by them have been helpful in clearing many of the practical implementation issues. We would like
to express our gratitude to all our students who in the process of learning have also taught us. In this
regard, we would especially like to mention the names of Vaidyanathan,Amudha, Balaji,Arun, Praveen,
Karthikeyan, Jeyendran, Radhakrishnan who have contributed to this project in many ways.
Revision of this book has been made possible because of the experience gained by the implementation
and study of various systems on both programmable DSPs and FPGAs by undergraduate, graduate, PhD
students and project staff in addition to our experience through teaching. We would like to thank the
following students and staff who have been responsible for carrying out the above work: Dr V Amudha,
Dr G Seetharaman, Dr S Ramasamy, J Manikandan, S Mohanasundaram,AGeethanath, L Govinda Rao,
G Chaitanya, T Dhirendrakumar, H Reshmi, D S Prasada Reddy and S Ashish. We would like to thank
our Director Dr M Chidambaram for encouraging us to bring out the revised edition of this book.
A number of experts have provided invaluable suggestions and feedback for the revision of this book.
Our heartfelt gratitude goes out to all of them.
Finally, we thank the editorial and production staff of Tata McGraw-Hill for the initiative and interest
shown by them in bringing out this work in a short span of time.
B Venkataramani
M Bhaskar
Feedback
Tata McGraw-Hill invites comments, views and suggestions from readers, all of which can be sent to
tmh.ecefeedback@gmail.com, mentioning the title and author’s names in the subject line.
Preface xv

SIGNALS AND THEIR ORIGIN 1.1
A signal refers to any continuous function f( ) which is a function of one or more variables like time,
space, frequency, etc. Some common examples of signals are the voltage across a resistor, the velocity
of a vehicle, light intensity of an image, temperature, pressure inside a system, etc., as a function of time,
space or any other independent variable. These signals are processed in order to either monitor or control
one or more parameters of a system. Detection of the average, RMS or peak values of a parameter,
separation between adjacent peaks or zero crossings are examples of some processing carried out on the
signal. In some applications, the processing may be done in order to produce another signal which has
better characteristics than the original signal. The processing of the signal is carried out efﬁciently and
with ease if these signals are converted to equivalent electrical voltages or currents using transducers.
Hence, the emphasis in this book will be restricted to processing signals in electrical form.
NOISE 1.2
Processing of the signal is made complex by the presence of other signals called noise. The noise
signals are generated from man made and natural objects. Electrical appliances/machinery, lightening
and thunderstorms are some of the sources of noise. In addition to this, any signal which interferes
with the detection of the desired signal may be called as an intereference or noise. The ﬁrst step in
signal processing is to combat the effect of noise. When the noise and the desired signal have different
characteristics, the signal can be completely separated from the noise before processing the signal
further. Even though this step is an overhead, this may be mandatory. Let us consider the following
example to illustrate this.
FILTERS AND NOISE 1.3
Frequency shift keying is a technique adopted for transmitting binary data from one place to another.
For example, the transmitted sinusoidal signal may be chosen to be 1025 Hz or 1225 Hz depending upon
whether 0 or 1 is to be transmitted.
1
AN OVERVIEW OF DIGITAL
SIGNAL PROCESSING AND ITS
APPLICATIONS

2 Digital Signal Processors
On the receiver, the signal frequency is determined in order to decode the transmitted data to be 0 or
1. One of the techniques adopted for determining the frequency (also called frequency detection) is to
count the number of zero crossings of the signal in a given period of time. This can be done efficiently
if the transmitted signal is received without noise. However, one of the common noise which appears at
the receiver input is the power supply ripple. A noise of 60/50 Hz frequency may appear at the input to
the receiver. Alternately, if the receiver site has any high frequency oscillator, it may get leaked through
the power supply lines and will appear at the input to the receiver.
Both these types of noise will make the detector to take wrong decisions. However, its performance
can be improved if these two types of noise are removed from the received signal before it is fed to
detector. This can be achieved by using a low pass filter for removing the high frequency signal and
high pass filter for removing the low frequency power supply ripple. The ideal characteristics of low
pass, high pass and band pass filters are shown in Fig. 1.1. These filters may be constructed using
discrete components and their frequency response do not have the ideal characteristics. The ideal filters
pass the signals in the pass band without any attenuation. The signals in the stop band have infinite
attenuation. The transition from pass band to stop band occurs instantaneously as the frequency of the
signal is swept. The frequency at which this occurs is called as the cut off frequency fc. These filters by
themselves qualify to be called as signal processors as they remove the unwanted frequency components.
Additional processing may be required to carry out a particular task like frequency detection.
CORRELATORS 1.4
However, separation of the signal from the noise cannot always be achieved using simple filters shown
above. The desired signal and the interfering noise may both lie in the same frequency range (bandwidth).
In this case, the signal may be retrieved from noise by exploiting its behaviour in the time domain. The
effect of the interference signal may be minimized by multiplying the received signal with the replica of
all possible transmitted signals with suitable delay and then integrating them over one period (for e.g.,
1 bit duration in the case of FSK) of the transmitted signal. This operation is called correlation.
H( )
f
- 0 Frequency ( )
fc fc f
H f
( )
- 2
f - 1
f 0 f 1 f 2 Frequency ( )
f
H f
( )
(a) Low pass filter response (b) Bandpass filter response
H f
( )
- 1
f 0 f1 Frequency ( )
f
(c) Highpass filter response
Fig. 1.1

An Overview of Digital Signal Processing and its Applications 3
CONVOLUTION AND INVERSE FILTERING 1.5
On its transit to the receiver, the transmitted signal may pass through several systems and each of these
systems may modify it. The output y(t) of a linear time invariant (LTI) causal system can be expressed
as the convolution of the input x(t), with the impulse response h(t) which is given by the expression:
0
( ) ( ) ( ) ( )* ( )
t
y t x h t d x t h t
t t t
= - =
Ú (1.1)
Here ∗ denotes the convolution operation .
Asystem is said to be linear if the superposition principle is true. In other words if y1(t) is the response
to the input x1(t) and y2(t) is the response to the input x2(t), then for any scalar a, b let the response of
the system to the input a x1(t) + b x2(t) be y(t). If the system is linear then
y(t) = a y1(t) + b y2(t) (1.2)
A system is said to be time invariant if a shift in the input causes a corresponding shift in the output.
In other words if y(t) is the response of the system to x(t), then for a time invariant system the response
to the time shifted input x(t – t) is given by y(t – t).
A system is said to be causal if the output at any instant is determined only by its present input and
its past input/outputs but not by its future inputs. Only causal system can be realised in practice and this
requires the impulse response of the system h(t) to be zero for t < 0
For simplicity and to a good accuracy, the systems can be modelled to be LTI and causal in majority
of signal processing applications. However, there are applications such as image processing system
where causality is not true.
In order to nullify the effect of a system on the transmitted signal, the received signal may be passed
through another system whose transfer function (Laplace transform of the impulse response) is the
inverse of the original system. Inverse filters and equalisers use this principle.
FOURIER TRANSFORM AND CONVOLUTION THEOREM 1.6
The signals may be processed either in the time domain or in the transform domain, e.g., computation of
the output y(t) using the convolution integral given by (1.1) is an example of time domain processing.
The output may also be computed using transform domain techniques. This can be explained as follows:
A signal is said to be of finite energy if
2
| ( ) |
f d
t t
•
-•
< •
Ú (1.3)
Any finite energy signal f(t) can be represented using the Fourier transform F(w) given by
( ) ( ) j t
F f t e dt
w
w
•
-
-•
= Ú (1.4)
f(t) can be obtained from F(w) using the inverse Fourier transform given by
( )
1
( )
2
j t
f t F e d
w
w w
p
•
-•
= Ú (1.5)

4 Digital Signal Processors
When the lower limit in (1.4) and (1.5) are 0 and jw is replaced by s, we get the expression for the
Laplace transform of f(t) and inverse Laplace transform of F(s) respectively.
Convolution theorem relates h(t), x(t) and the convolved output y(t) to their Fourier and Laplace
Transforms. Let the Fourier transform and the Laplace Transform of (x(t), h(t), y(t)) be (X(w), H(w),
Y(w)) and (X(s), H(s), Y(s)) respectively.
Then if y(t) = x(t) ∗ h(t) then Y(w) = X(w) H(w) and Y(s) = X(s) H(s).
Similarly if Y(w) = A(w) ∗ B(w) then y(t) = a(t). b(t)
and Y(s) = A(s) ∗ B(s) fi y(t) = a(t). b(t)
As mentioned earlier ∗ denotes the convolution operation. H(s), the Laplace Transform of the impulse
response h(t) of a causal LTI system is called as the transfer function of the system and is given by
( )
( )
( )
Y s
H s
X s
= (1.6)
where X(s), Y(s) are the Laplace transforms of x(t) and y(t) respectively. Laplace Transforms have a no.
of applications like studying the stability of the system, solving the differential equations and finding the
initial value and final value of a system.
SAMPLING THEOREM AND DISCRETE TIME SYSTEM 1.7
Various signal processing operations explained above can be carried out either directly on the continuous
signal or indirectly using the samples of the input signal and impulse responses. Accordingly, they
are called as continuous time signal processing and discrete time signal processing respectively. The
discrete time signal offers no. of advantages. Firstly, it permits a no. of such signals to be transmitted
using the same channel by sending them at disjoint time intervals. This technique is called as the time
division multiplexing. However, in order to transmit the information without any loss of information the
discrete time signal should satisfy the Nyquist’s sampling theorem which states:
Any signal bandlimited to a maximum frequency of fm can be perfectly reconstructed from its
samples if the sampling rate, fs is greater than or equal to 2fm. If fs is equal to 2fm, then it is called the
Nyquist rate.
If the sampling rate is less than 2fm, then any signal component fh which is greater than fs/2 by Df
(i.e., fh = fs/2 + Df) gets mapped to a frequency fs/2 – Df after sampling and appears as a low frequency
signal. This is called as aliasing. To avoid this, either the sampling rate should be chosen to be above
Nyquist rate or the sampler should be preceded by a low pass filter with cut off frequency, fc = fs/2.
LINEARITY, SHIFT INVARIANCE, CAUSALITY AND STABILITY
OF DISCRETE TIME SYSTEMS 1.8
Some of the properties of the continuous time system discussed in Section 1.5 can be extended for the
discrete time system as follows: A discrete time system is said to be LSI if the superposition property
holds and a shift in the input causes a corresponding shift in the output.
In other words if y1(n) is the response of a discrete time system to the input x1(n) and y2(n) is the
response to the input x2(n), then for any scalar a, b let the response of the system to the input a x1(n) +
b x2(n) be y(n). Then if the system is linear
y(n) = a y1(n) + b y2(n) (1.7)

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her own room.
"Won't the childern be surprised an' pleased to see you back, in
the mornin'," she was saying heartily.
Cora, bringing up the rear, remarked with importance, "Mother
sent'm to bed sooner'n usual 'cause to-morrow morning we all got to
get up early. We're going with Miss Claire, in the la'nch, across the
lake, to see a blue herring, she's got there in a cove."
"A blue herring, is it? Well, well!" said Ma abstractedly.
Cora went on. "Mother said when Francie told her, firstoff, you'd
gone away for good, an' wasn't coming back—Mother said, 'No
matter how much I feel my loss, I must try to be cheerful.' Mother
said it was a shock, but you mustn't let the world see your suffering.
The world's got troubles of its own."
Ma's dull eyes brightened. She gazed up searchingly into her
daughter-in-law's face. "And, did you say that indeed, Martha?" she
questioned.
Martha punched a pillow pugilistically. "Very likely," she
answered holding the ticking with her teeth, while she pulled the
clean slip over it. "Yes, I said it."
The old woman slowly, tremulously undressed.
After Cora had gone, and Ma was in bed, Martha lingered a
moment, before turning out the light.
"I'm sorry you had such disappointment," she said. "But
doncher care, Ma. Sometime us two'll go down to New York
together, an' I'll give you the time o' your life."
For a moment Ma made no response. Then her quavering voice
shook out the words, as if they had been stray atoms, falling from a
sieve: "It ain't the disappointment I'm after mindin' so much," she

lamented. "I could thole that, itself—but—(perhaps it's a silly old
woman I am)! but the notion it's got into me head that—that—
maybe the lot o' them—didn't want me!"
Martha extinguished the light with a jerk. "Oh, go to sleep, Ma,
an' quit your foolishness. I'll say to you what I say to the childern. If
you cry about nothin', look out lest the Lord'll be givin' you somethin'
to cry for."
"Then you don't think——?"
"Oh, go to sleep, Ma," repeated Martha, as if the question were
not debatable.
The sun was barely up when the children began to stir.
"Say, Sabina," Cora whispered, "I bet you don't know what's in
Ma's room."
A quick sortie, and Sabina did know. Then Sammy knew, and
Francie knew.
"Come, come!" cried Martha, appearing on the threshold, "get
yourselves dressed, the whole of you. Don't use up all your joy at
the first go-off. Leave some to spread over the rest of the time. Ma's
goin' to stop, you know. Besides—we can't keep Miss Claire waitin'."
"In my da'," observed Ma thoughtfully, "it wouldn't 'a' been
thought well of, for a lady like that to be la'nchin' out, just before
——"
"It's not my picnic," Martha interrupted. "I said all I could to
pervent it in the first place. But her heart's fixed, an' I couldn't say
her no, 'specially when Lord Ronald said he saw no harm, an'd go
along too."
"Well, if he sees no harm—and is goin' along too——" Ma
murmured, as if her consent were to be gained on no other grounds.

"Certaintly," said Martha.
Everything was in readiness in and about the trim little Moth,
when Claire Ronald appeared on the dock.
"Where's Mr. Frank?" Mrs. Slawson asked.
"He got a message late last night from Boston, about some stuff
for the electric-plant. They've sent it on, and he had to go to
Burbank to examine it, so, in case it wasn't right, it could be shipped
straight back. He said it would save time and cartage, and he wants
the work put through as soon as possible."
"Then, o' course we'll put off our trip!"
"Oh, no!"
"Did he say we could go, an' him not here to go along too?"
"No—but——"
"Then, I guess we'll call it off."
Claire's mouth set, in quite an uncharacteristic way.
"No, indeed! We'll go! We couldn't have a better morning."
"Well, I do' know, but I wisht I had my long-handled feather-
duster here to brush away some o' them flims o' dust off'n the
ceilin'."
"Why, those are darling little clouds!" Miss Claire exclaimed
reproachfully. "When the sun gets high, it will draw them out of sight
entirely, and the sky will be as clear as crystal."
"It's as you think, not as I do," Mrs. Slawson rejoined. "If you're
shooted, I'm shot!"
"In with you, children. Steady now!" commanded Claire.
Martha being already at the wheel, her husband had only to
stow Mrs. Ronald and the girls safely amidships, see Sammy

stationed in the stern in charge of the rudder-ropes, release the boat
from its moorings, and The Moth was ready for flight.
"Take care of yourselves!" he called after them.
"Sure!" Martha shouted back, and they were off.
Now she was fairly in the line of having her own way, Claire was
radiant.
"The idea of finding fault with this day!" she taunted laughingly.
"Why, I couldn't have made it better, myself!"
"Why don't those birds fly up in the sky, mother?" asked
Francie. "What makes 'em fly so low down, right over the water?"
"They are gulls," Mrs. Ronald answered, as if that explained the
mystery.
It was a tremendous surprise to find the blue heron a bird
instead of "a delicatessen."
For a couple of hours after her first introduction to the new
acquaintance Martha kept exclaiming at intervals. "Well, what do you
think o' that!" as a sort of gentle indication of her amazement.
"Say, mother, the way the herring walks, it'd make you think o'
folks goin' up the church-aisle to get married—steppin' as slow, as
slow. Bridesmaids an' things."
Martha winked solemnly across at Claire.
"Nothin' interests Cora so much these days, as the loverin'
business. She's got it on the brain."
"Dear me! But there are no lovers around here, I'm sure," Claire
said, amused.
"Oh, yes, there are. There's you an' Lord Ronald, an' there's Dr.
Ballard an' Miss Katherine—an'——"
"Say, young lady, you talk too much——"

"Well, mother, it's true. I know he likes her a lot, 'cause——"
"That's enough, Cora. You're too tonguey. Go along an' play
with your little brothers an' sisters."
When they were alone Mrs. Ronald turned to Martha. "Is it
really true, Martha? Is Dr. Ballard interested in Miss Crewe?"
Mrs. Slawson laughed. "Like that advertisement says the baby's
interested in the soap: 'He won't be happy till he gets it!'"
"And does she——?"
"Certaintly. You couldn't help it. But the little ol' lady has her
face set against it. You got such pretty, tackful ways with you—
sometime, when you're with the little Madam you might kind o' work
around to help the young folks some, if you'd be so good."
Cora came wandering back. The play of the younger children
did not divert her. She watched the blue heron as it silently,
delicately paced up and down the beach, picking its way among the
submerged stones, suddenly darting its head beneath the surface of
the water, bringing up a bull-head, perhaps, and swallowing it whole.
"Ain't he perfectly killin'?" she murmured. "The way he acts like
he's too dainty to live? And see that yellow flower over there! We
had loads and loads of it last fall, and I used to take it to the teacher
till one of the girls laughed at me 'cause she said the woods's full o'
them, an' besides it gave the teacher hey? fever. That's a joke. It
means, it'd make her ask more questions than she does already. Ann
Upton said that. Ann is awful smart. Once, when her composition
was all marked up with red ink, 'cause the teacher had corrected it
so much, Ann said 'she didn't care. It was the pink of perfection.'"
"That yellow weed is goldenrod," explained Miss Claire. "Do you
remember the names of any of the other wild-flowers I taught you a

year ago, Martha?"
"Well, not so's you'd notice it. Lemme see! P'raps I do. Wasn't
there a sort o' purple flower you called Johnny-pie-plant?"
Mrs. Ronald laughed. "Joepyeweed, yes. You got the idea."
"An' then, there was wild buckwheat, an' Jewel-weed an'—now,
what's the matter with me, for a farmer? Don't I know a thing or two
about the country?"
"You certainly do."
"An' I know the name o' some too," asserted Cora. "Brides-lace,
and Love-in-a-mist, and——"
"Sweet Sibyl of the Sweat-shop, or——"
"Mother, I think you're real mean!" Cora cried, anxious to
prevent further betrayal.
"Say, ladies an' gen'lmen, I hate to break up this pleasant
ent'tainment, but I guess you don't realize how long we been
dreamin' the happy hours away, like Miss Frances Underwood used
to sing, before she married Judge Granville—which they ain't so
happy now, not on your life, poor dear! I think we better get a move
on, or we'll get soaked good and plenty. It's my opinion we're goin'
to have a shower."
Claire did not attempt to argue the point. It was too evident
that something was really going to happen.
"Yes, let's hurry," was all she said. "It's later than I thought."
Martha summoned her straying flock, and they made for the
boat.
The little clouds, no bigger than a man's hand, had turned gray.
Francie's friends, the gulls, were darting excitedly to and fro, as if

without direction, very close to the face of the water. Here and there
the lake showed a white-cap.
Martha stood at the wheel, in the bow, and steered straight for
the opposite shore.
For a while Mrs. Ronald kept up a careless chatter with the
children, then, as if by common consent, there was silence.
A sharp wind had risen out of nowhere, apparently, and begun
to lash the water into frothy fringes that tossed their beads of spray
high over the side of the boat. Suddenly Francie screamed. This time
it was not the spray, but the wave itself that the blast rushed before
it to break full upon The Moth, drenching the child to the skin.
Martha glanced around to see what the trouble was.
"There's some tarpaulin under the seats," she shouted back
over her shoulder, "wrap it about you an'—dry up!"
Again there was silence, while the clouds massed themselves
into granite barricades, shutting out the light, and the gale gathered
force and fury with every second. It was impossible, now, to see the
farther shore. The little Moth seemed blindly fluttering in a dense
mesh of gray mist impossible to penetrate.
"We're going every which way!" moaned Cora.
At the same instant—"The rudder-ropes, Sammy!" shouted
Martha.
The boy slipped from his place, and, by sense of touch alone,
found the cause of the obstruction, and freed the ropes.
The Moth gave a leap forward into the mist.
"I'm afraid!" roared Sabina in no uncertain voice.
"What you afraid of?" came back from the bow. "Don't you
know, if there was any danger I'd get out!"

To the children, accustomed as they were to accept their
mother's word without question, the statement carried instant
reassurance. Sabina stopped roaring, and Francie only screamed
when each new wave broke over her, threatening to swamp the
boat.
"Hush, Francie!" called Miss Claire at length in a tense, strained
voice. "You'll make your mother nervous."
Martha, hearing, answered back, "She don't make me nervous.
There's nothing to be nervous about. Let her scream, if it makes her
happy."
Francie stopped screaming.
All the while the throbbing of the little engine had been steady,
incessant. But now Martha noticed that, at intervals, it missed a
beat. She waited to see if it would right itself. A minute, and it had
ceased altogether.
"Sammy!"
It only needed that to send the boy crawling, on his hands and
knees, to start it up afresh, if he could—working, as his father had
taught him to work.
The Moth spun around and around, in the trough of the waves.
Martha "knew what she knew," but her hands never left the
wheel for an instant. What if the engine could not be made to go?
What could she say to Mr. Frank if——? No, there was this comfort, if
the worst came to the worst she would be the last to have a chance
to say anything, to any of those waiting on the shore....
She heard the steady heart-beat start afresh.... The boy was
back in his place. Martha, with new courage, strained her vision to

pierce through the curtain of mist and rain, could see nothing, but
clung to her wheel.
At length she realized she was steering toward something that
she, alone of all the little group, could see—a faint adumbration,
showing dark through the pall of enveloping gray.
But now the wind and the water were so high it was impossible
to steer straight for the home-shore—she could only make it by slow
degrees.
The storm had whipped her thick hair out of its customary coils.
It blew about her face and shoulders in long, wet strands, buffeting
her, blinding her. She never lifted a hand to save herself the stinging
strokes.
Little by little the dark line widened, the way was made plain.
Little by little Martha wheedled The Moth shoreward.
"I see somepn'," shouted Francie, at last. "I see our dock!" After
an interval: "I see folks on our dock!" Later still: "I see father, 'n' Mr.
Ronald, 'n' Ma, 'n'—oh! lots o' folks!"
The Moth fluttered forward. The waves beat her back. She
seemed to submit with meekness, but a second later, seeing her
chance, she dodged neatly, and sped on again, so, at last, gaining
the quiet water of the little bay.
Mr. Ronald and Sam Slawson, in silence, made her fast to her
moorings. In silence, Martha gave Claire into her husband's arms. He
wrapped the shaking little figure about, in warm dry coverings, and
carried her home, as he would a child.
The second they were out of sight and hearing, a babel of
voices rose, Ma's shrill, high treble piping loud above the rest:

"When we saw the tempest gatherin', an' youse out in it, on the
deep, an' not a boat could make to get to youse, the fear was in me
heart, I didn't have a limb to move."
A burly form shoved her unceremoniously aside,
Joe Harding approached Martha, implanted a sounding kiss on
her cheek.
"By gum, you're a cracker-jack, Mrs. Slawson, and no mistake!"
he announced.
One by one the little knot of men and women followed suit,
Fred Trenholm, Nancy Lentz, Mr. Peckett—all who, by the wireless
telegraph that, in the country, flashes the news from house to
house, had heard of The Moth's danger, and had come over to help
if they could, and—couldn't.
Martha looked from one to the other in surprise.
"Well, what do you think o' that!" she managed to articulate
through her chattering teeth, and then could say no more.
"Come along home, Martha," urged Sam gently.

CHAPTER XII
At first it seemed as if no one was to be any the worse for the
morning's adventure.
As soon as she had attended to the children, had changed her
own cold, drenched garments for dry, Martha hastened over to the
big house.
Tyrrell, the butler, informed her that Mrs. Ronald was resting
quietly enough now, but they had been uncommonly anxious about
her at the start. The shock had unnerved her. When her husband
carried her in, she was crying like a baby.
"Well, you know where to find me, if, when she wakes, she
seems the least bit ailin'. All you have to do is ring me up, an' I'll be
over in the shake of a lamb's tail."
But when the day passed, and there was no summons, when
supper was over and the children, including Cora and Ma, in bed,
Martha could stand it no longer.
"I just got to go over, an' see for myself how the land lays," she
explained to Sam. "I know it's silly, but I just got to."
"All right. Come along," said Sam.
Martha shook her head. "No, you don't. Somebody's needed
here in case, while I'm between this an' the big house, the
telephone'd ring."
Patient Sam acquiesced at once. "Have it your own way. You've
earned the right to have notions, and be fidgety if you want to. But

no news is good news, an' what you'll make by running over there at
this hour of night, when they said they'd 'phone if anything was
needed, I don't know."
"I'll sleep better if I see for myself," was all the explanation
Martha could give.
It was very dark, outside, once she got beyond the light from
the Lodge windows. In her haste she had forgotten to bring the
lantern with her, but she did not go back for it, because she felt she
knew every inch of the ground, and, moreover, the impulse that
drew her forth was so strong that she could not endure the idea of
delay for a moment. She had discovered a short-cut across the
grounds and meant to use it, though she knew Sam disapproved any
trespassing on his adored lawns, hedges, and shrubberies, and, as a
general rule, she respected his wishes. But now she made straight
for the thicket of bushes walling in her kitchen-garden, meaning to
push through it, at the point of least resistance, strike across the
roadway and so slice off a good quarter of a mile, by bisecting the
lawn sweeping up to the big house. Just within the thicket she stood
as if at attention. For the life of her she could not have said what
brought her to a standstill, but also, for the life of her, she could not
go on until she knew what was on the other side of that wall of
bushes.
Listening, she could hear nothing but the common-place night-
sounds, now grown familiar to her ears. The stirrings of leaves,
when the wind sighed through them, the surreptitious whirr of wings
aloft, up over the tree-tops, the lowly meanderings of insects among
the grass, the soft pad-pad of tiny, furry feet scampering to safety.
But there was still another sound, an unusual, unfamiliar sound. It

came to Martha in a flash what it was. A fox, caught in one of Sam's
traps.
"Oh, you poor devil, you!" she heard herself exclaim.
The words were echoed by a human groan, so close at hand,
she fairly started.
"Who are you?" Her question rang out sharply.
"None of your damned business!" came back in instant answer.
"But since you're here, curse you! come, and get me out of this ——
—— trap."
A light flashed, by which Martha made out a man's figure
crouching on the ground the other side of the hedge. His face was
completely hidden, not alone by the drooping brim of his soft hat,
but by a sort of black mask he wore. Without a moment's hesitation
she forced her way through the hedge. Now she could see more
plainly, she made out that the man was on his hands and knees.
One hand was free—the other, caught in the fox-trap, was bleeding
cruelly. On the ground, within easy reach lay a pistol, a bundle of
fagots, and a bull's-eye electric torch. The man's uninjured left hand
was clutching the torch.
"Doncher stir a muscle, Mr. Buller," Martha said imperatively, "till
I make out how this thing works. I don't want to hurt you more than
I got to, unspringin' the trap."
Buller swore violently as he bade her, "Go ahead then, and be
quick about it!"
A moment, and the mangled hand was free. Instantly, its owner
listed over on the grass in a dead faint, in total darkness.
Martha felt about in the darkness for the torch, set it glowing
and, by aid of its light, found a flask in Buller's pocket, some of the

contents of which she forced between his lips. When he was fully
conscious, she bade him pick up his belongings, and come along
home with her, where she could look after his hand, and, if
necessary, telephone for the doctor.
Clutching at her shoulder, he staggered to his feet.
"Don't forget your gun," warned Martha drily.
"Damn the gun!" returned Buller.
Somehow they reached the Lodge. Sam, hearing footsteps,
came to the door with an anxious face.
"Martha," he whispered, before he had made out she was not
alone, "hurry back to the big house. Mr. Ronald's just called you up
this minute. His wife wants you, and—I'm going for the doctor."
Martha pushed Buller forward into the entry.
"Look after'm, Sam. He was on his way to give us a call. With
his pistol an' a bunch o' kindlin's to fire the house. He heard me
comin', an' lay low for a minute, an' got caught in the trap you set
for—the other fox. But take care of'm," she said, and vanished into
the night.
Neither Sam nor Buller spoke for a moment. Then Sam opened
the sitting-room door.
"Come in," he invited the other. "Let's take a look at your hand."
The tortured Buller thrust it forward where the lamplight could
fall upon it. Sam shook his head.
"That's beyond me," he explained. "But I tell you what, I'm
going for Dr. Driggs, anyhow. You get in the car and come along
with me. Only, I better take that black dingus off your face, hadn't
I?"

Buller made a clumsy effort to detach it himself, but his left
hand alone could not manage it. Sam did it for him.
"Now, as soon as I get the car," he explained, "we can start."
While he was gone Buller paced the floor like a caged animal,
writhing with pain, crying, cursing. Sam was gone but a few
minutes. It seemed an eternity to the poor, waiting wretch. Then
away they sped through the cool, calming darkness of the night.
In the extremity of his anguish, nothing really signified to Buller,
yet again and again he found himself wondering if Slawson would
"split" on him. As a matter of fact, Sam never opened his lips,
beyond delivering his message to the doctor from Mr. Ronald, then
turning Buller over to him for immediate attention.
The old physician scowled through his spectacles when he saw
the wound.
"How did you manage this job?" he asked in his blunt,
uncompromising way.
Buller winced. "Trap. Foxes after my hens. I set a trap to catch
them."
"And got caught in it yourself! Huh! That's sometimes the way.
Here, swallow this down. It'll dull the pain some. Now is the time
you may wish you weren't a drinking man, Buller. I'll do the best I
can for you, but you've given yourself a nasty hurt, and your blood's
not in a state to help the healing along much. However, we'll see
what we'll see. I'll give you these extra drops to take home with you.
Use them if the pain comes back. Don't meddle with my bandage,
d'you hear. Leave it alone. And, let me see you in the morning. Now,
Mr. Slawson—— Ready!"
Again that swift, almost silent speeding through the night.

Since Buller's torture had ceased, the motion seemed for him
part of a blissful dream, by which he was being gradually lulled to
deeper and deeper peace. At first he started in to babble fatuously,
but Dr. Driggs brusquely bade him, "Shut up! This is no time for
merrymaking!" and he dropped back into himself, subdued but not
suppressed.
At the big house Sam stopped his car.
"I'll take Buller home, and come back for you," he explained to
Dr. Driggs.
"Better dump him out on the road," was the harsh, whispered
rejoinder. "I know him from the ground up. He lied to me about his
hand. He was up to deviltry of some kind, other than trapping foxes,
depend upon it! Between you and me, that's a fierce hand he's got. I
don't envy him his dance with it."
In the meantime, Martha had found Claire Ronald feverish and
excited. It did not take her long to decide she would not leave the
big house that night. When Sam returned to take him home, Dr.
Driggs was not ready to go. Neither was Martha.
"But you'd better turn in, Slawson," advised Mr. Ronald. "No use
in everybody's getting worn out. If I should need you, I'll call you
up."
Early next morning the young kitchen-maid from the big house
appeared at the Lodge door for certain necessaries Martha wanted
and could not be spared long enough to come, herself, and fetch.
"Eh, now! You don't say so! Things must be pretty bad over
there!" observed Ma.
The girl nodded dumbly. She adored Mrs. Ronald.

"If I was you, beggin' pardon for the liberty," Martha addressed
Mr. Frank, "I'd get a-holt of those doctors an' nurses from the city
you have engaged. They was comin' up in two weeks, anyhow. You
never can tell. This might be a false alarm, but then again it
mightn't. Either way, we don't want to take no risks."
"I'll telegraph," said Francis Ronald dully.
"What's the matter with the telefoam? Ain't you got a long-
distance connection here?"
While Central was clearing the wire, Katherine Crewe was
ushered in. She hesitated on the library threshold, then came
forward rapidly, her face more lovely than Martha had ever seen it,
in its softened expression of human sympathy.
"I'm so sorry—I've just heard—I came to see if I could do
something—be of any help," she stammered shyly.
Frank Ronald had risen and was about to reply, when Dr. Driggs
pushed through the doorway, interrupting gruffly.
"I'm not quite satisfied with the way things are going. Nothing
to be uneasy about, you know, but, under the circumstances, I'd like
another man to talk the case over with."
"I've just called up the New York specialist. He and the nurses
——"
"Lord! I don't mean that! It'll take them a full day to get here.
We can't wait that long. I want some one now."
"Now?" Frank Ronald echoed, without any appearance of
understanding what the word meant.
"Now," repeated Dr. Driggs. "I'd like to call in——"
Tinkled the telephone-bell with irritating insistence.

Frank Ronald's cold hand gripped the thing as if he would choke
it.
"Hello! Is this New York? Is this Dr. Webster? 'Morning, Dr.
Webster! This is F. B. Ronald speaking. Yes—I've called you up,
because my wife—— Can you hear me now? Is this better?—My wife
—I'm worried about my wife. I've called in Dr. Driggs of this village.
He wants more advice.... Yes, by all means come on at once, and
bring the nurses. But Driggs says he can't wait. Must have some one
immediately.... Eh? ... Who, do you say? ... Boston? Yes, I get that
... Ballard of Boston? ... There's a young fellow here from Boston
named Ballard, but he ... I don't believe he's the man. Wait a
minute.... Please repeat that! ... You say he's the best skill in New
England? National repute? ... I'm afraid.... Hello! Dr. Webster ...
Driggs, here, says 'tis the man you mean. He says he was just trying
to tell me, when ... yes ... I'm sure we can get him. Yes, we are in
luck! ... Very well ... Burbank Junction ... midnight.... Good-by!"
Francis Ronald's words and manner were painfully precise.
Thought Martha, "I've seen parties none too steady on their
pins, just that kind o' mincin' about their steps. As if they'd dare you
say they couldn't walk a chalk-line. Poor fella. He's so crazed with
worry he can't see straight, but he's goin' to prove anybody thinks
so, is another!"
When Katherine reached home she found Madam Crewe
awaiting her.
"Well, and how are things going? You had your tramp for
nothing, eh? Young Sammy's account of Mrs. Ronald's danger was
hocus-pocus, of course!"

"No. Dr. Driggs is very anxious. He wants a consultation. While I
was there Mr. Ronald called up Dr. Webster—Elihu Webster, from
home. He's coming up with two nurses——"
"And Mrs. Ronald is going to wait for him? That's obliging of her,
I'm sure!"
"Dr. Driggs had asked Mr. Ronald to let him have Dr. Ballard. He
had asked, before they got Dr. Webster on the wire. Then, the first
name Dr. Webster suggested was Dr. Ballard's. He called him 'the
best skill in New England.' Said he was of 'national repute.'"
"You mean Driggs did. Well, what then? Driggs is getting old. He
sometimes muddles. He's probably got this young sprig here
confused with the great one."
"No, grandmother. Dr. Webster said it. Dr. Driggs only repeated
what Dr. Webster said."
During the pause following Katherine's statement, Madam
Crewe sat quite still, apparently absorbed in contemplation of her
two, tiny hands, lying folded and motionless in her lap. When, at
length, she looked up, a curious ghost of a smile curled the corners
of her mouth.
"Really I am uncommonly gratified. You see, I can't help
thinking, how barely I missed the honor of being this young man's
grandmother. I'd have liked to have a grandchild of whom I could be
proud."
Katherine winced. "I'm sorry I've disappointed you," she said
bitterly.
"Don't mention it. It's not the first disappointment I've had in
my life. It probably won't be the last. Moreover, now that you know,
undoubtedly you'll think better of your decision to give him up. You'll

marry him, after all, in spite of the loss of me and my money. So I'll
have my eminent grandson, whether I want him or not."
"Grandmother!"
"Well, won't I? It seems to me, you have quite a keen eye for
the main chance. At least, that's how I've made it out, judging from
your behavior. At first, you were all for marrying him, when you
thought you could do it on the sly, without sacrificing your interests
with me. Then, on the impulse of the moment, for Norris's benefit,
maybe, you played tragedy-queen and forswore your fortune for the
sake of the man you love. All of which would have been very pretty
and romantic—if you had stuck to it. But, when you had had time to
calculate—presto! it's your lover you repudiate, to hang on to the
money. Now you're fairly certain he's got all you'll need—doctors
fleece one abominably, nowadays! Come and feel your pulse, and
give you a soothing-syrup, and send in a bill for ten dollars, and
that's no placebo, I tell you! Oh, there's no doubt you'll be rich, if
you marry a doctor—— Where was I?"
"You were running down doctors, grandmother, and I don't see
how you can, when you know what those you've had have done for
you. I——"
"There, there! I don't need you to inform me, young miss. What
I was saying is, nobody would doubt, for a minute, you'll take him
now. I don't."
"Grandmother," the girl began, with the same kind of
exaggerated punctilio Martha had observed in Mr. Ronald.
"Grandmother, I want to be very respectful to you. I don't want to
say one word that will excite you, or make you ill. But I think you

take unfair advantage of me. You taunt me, and jeer at me because
you know I can't hit back, without being an unutterable coward."
Madam Crewe made a clicking sound with her tongue.
"On the whole, I think I'd like it better if you did hit back,
providing you hit back in the right way. No temper, you understand.
No rage, no rumpus and that sort of vulgarity. But real dexterous
thrusting and parrying. Now, for example, you missed an opportunity
a few moments ago. When I said I'd have liked to have a grandchild
I could be proud of, you might have retorted, 'I'm sorry I disappoint
you, grandmother, but, perhaps, if you had been Dr. Ballard's
grandmother, his distinction might not have been so great.' That
would have been a silencer, because,—it would have been true. I'm
afraid you're not very clever, my dear."
"If that sort of thing—slashing people with one's tongue, is
clever, I'm glad I'm stupid."
"There! That's not so bad! Try again!" applauded the old
woman.
Katherine turned away, with a gesture of discouragement.
"It never occurred to me before," Madam Crewe meditated, "but
what you really need is a sense of humor. You're quite without
humor. You've brains enough, but you have about as much dash and
sparkle as one of your husband-that-is-to-be's mustard-plasters.
Only the mustard-plaster has the advantage of you in sharpness."
The girl wheeled about abruptly. "He is not my husband that-is-
to-be. I have told you that before."
"But the circumstances have changed. Now you know he is
distinguished—probably well-to-do——"

"It only makes another barrier. Can't you see? Can't you
understand?"
"Perhaps I might, if you'd have the goodness to explain. But you
must remember, I'm an old woman. It's a great many years since I
had heroics."
"Perhaps you never had them," Katherine retorted. "Perhaps
you never were young—never cared for any one with all your heart.
Perhaps you never had a heart."
"Perhaps," agreed Madam Crewe. "In which case, don't appeal
to it. Appeal to my imagination. That, at least, I can vouch for."
"I took your word for it, that Dr. Ballard was a young struggling
doctor, poor—with, at best, no more than a problematic future—
that's what you said—a problematic future."
"Well?"
"When I began to suspect he cared for me, I was glad he hadn't
a lot of advantages, to emphasize my want of them. It didn't seem
to me, then, so impossible, that as poor as I should be, and as dull
as you've always said I am, I might marry him some day, if he loved
me. I never cared a rush about that nonsense connected with his
grandfather. I wouldn't have cared, if it had been true. So when you
threw mud at my grandfather and father, I didn't suppose he'd care
—or believe it—either. And, he didn't and—doesn't. So far, we stood
about equal. I could give him as true a love as he could give me. But
——"
"Oho! So that's your idea. I see your point now. You've got the
kind of love that weighs and balances, have you? You won't take
more than you can give! Why, young miss, let me tell you, you may
think that's high-flown and noble—it's no such thing! If you want to

know what it is, it's your great-grandfather's arrogance turned inside
out, that's all! If you refuse to marry the man you love, because you
have nothing to offer him, you're as bad as I was when I refused
because my lover had nothing to offer me. There's a pride of poverty
that's as detestable as the pride of riches. You talk about love! You
don't know what the word means. If you did, you'd see that the real
thing is beyond such mean dickering. In love fair exchange is low
snobbery."
The girl stared silently into her grandmother's face. Two bright
spots were glowing in the withered cheeks, the old woman's eyes
shot forth the fire of youth.
For the second time Katherine felt that the drawbridge was
down. Impulsively, she took a step forward, grasping one of the little
old hands, folding it tight in both her own.
"Grandmother, I want to tell you something—I see what you
mean and—I know it's true. But—but—there's something else——"
Madam Crewe did not withdraw her hand. It almost seemed to
Katherine as if its clasp tightened on hers.
"What else?"
"When he—when Dr. Ballard first spoke to me about his
grandfather, he said, 'But after all, the only thing that really counts is
character.' He said: 'One can afford to whistle at family-trees if one's
own record is clean!' He said: 'After all, what's most important, is to
be straight goods one's self. If I'd lied, or was a coward or had taken
what belonged to some one else, or had any other dirty rag of
memory trailing after me, I'd hesitate to ask any one to share my life
with me, but——'"
"Well?"

"Grandmother—I've the kind of dirty rag of memory, he spoke
about. I'm a coward—I've lied—I've taken what belonged to some
one else."
CHAPTER XIII
Madam Crewe said nothing.
She gazed into Katherine's face blankly for a moment, then
gradually withdrew her eyes to fix them on a bit of sky visible
through the bowed shutters of the open window.
When the silence became unendurable, "Won't you speak to
me, grandmother?" the girl pleaded. "Won't you let me feel you
understand?"
There was a long pause before any answer came.
"Understand? No, I don't understand. How could one
understand one's own flesh and blood being, doing—what you
describe? That story would be perpetually new—perpetually
incomprehensible. But perhaps you're vaporing. Using big words for
insignificant things. A child's trick. Tell me the truth, and be quick
about it."
There was something so formidable in the tiny old woman
sitting there, coldly withdrawn into herself again, controlling any
show of natural emotion with a fairly uncanny skill, that Katherine
quailed before her.
In as few words as possible, she sketched the story of the
recovered pocket.

Madam Crewe heard her through, in silence. In silence, received
the object that had, at one time, been such a determining factor in
her life. Katherine could not see that she betrayed, by so much as
the quiver of an eyelash, the natural interest one might be conceived
as feeling in so significant a link with the past.
"Be good enough to leave me," the old woman said at last. "And
don't open this subject again, unless I bid you. If I need any one I'll
ring for Eunice. Don't you come—for the present. Oh, before you go,
see that you keep a close mouth about this thing, not alone to me,
but to every one. Understand?"
Katherine nodded dumbly. She felt like a child dismissed in
disgrace, or a prisoner returned to his cell. She did not know how
long she remained in her room, but when Eunice came to announce
luncheon, she sent her away, merely explaining that she was not
hungry. And would Eunice kindly answer if Madam Crewe should
ring?
Within her, a hundred impulses of revolt urged to some act of
self-deliverance. She fought them down with appeals to her own
better nature, her grandmother's need of her. It was to escape from
herself, as much as from her environment, that, at last, in
desperation, she caught up her hat and left the house.
She had been gone several hours, and it was twilight, when a
low tap sounded on Madam Crewe's door.
Without waiting for permission to come in, Dr. Ballard did so.
The old woman started up, as if his presence roused her from sleep,
but he could see she had been fully awake.
"You look as if you had been through the wars," she observed
dryly, examining his face with her searching eyes.

He dropped heavily into the chair she indicated.
"I have," he answered.
"You've saved two souls alive? Mother and child?"
He nodded. "But the war's not over. The fight's still on. I've
done all I can. The rest lies with——"
The old woman took him up sharply. "Don't try to talk. Touch
that bell."
Then, when Eunice, responding, stood on the threshold: "Bring
me the leathern case you'll find standing beside the clothes-press in
my dressing-room. Yes ... that's the one. Bring it here to me! Now,
go downstairs and fetch a plateful of hard biscuits. Hurry! ... Stop! ...
Before you go, hand me that glass from my table."
When the girl was gone, Madam Crewe unlocked the case
before her, took from it a flask, and with surprisingly steady hands,
poured a share of its contents into the glass Eunice had placed on
the wide arm of her chair.
"Wine?" asked Dr. Ballard doubtfully, hesitating to drink.
"No, not wine. Drink it down. Now, the biscuits. Don't talk."
She pretended to busy herself with the leathern case upon her
knees—replacing the flask, turning the key in the lock, rather
elaborately fingering the smooth surface, as if all her attention was
concentrated on some imaginary fleck or flaw she had just
discovered.
When, watching covertly, she saw the haggard lines slowly fade
from her companion's face, the blood gradually mount to his cheeks,
she drew an audible breath.
"That's great stuff!" Daniel Ballard observed appreciatively.
"What do you call it?"

Madam Crewe raised her eyebrows. "I don't call it. It has no
name, so far as I know. It's an old stimulant my father picked up
somewhere in the far East. He treasured it like gold."
"It's certainly done the trick. I was all in, and now I feel quite
fit. Mrs. Slawson and I have been on the job since morning. She's a
wonder, that woman! No end of nerve and pluck. I could make a
corking good nurse of her! She's back there now, watching. Firm as
Gibraltar. I couldn't stand it any longer. I had to get away for a
moment, to catch a breath of fresh air, and a glimpse of——"
"Me?" Madam Crewe caught him up.
He corrected her gravely. "No, the evening star."
"Katherine came home from the Ronalds' this morning much
disturbed."
"Over the case?"
"Yes—that, and—the fact of your being what she hadn't
supposed."
Dr. Ballard looked his question.
"She feels overawed, now she's aware what a great man are
you. A bit sheepish, too, I fancy, because, if I remember right, she
has twitted you, more than once, on being worn out waiting for
patients."
"Well, what of it? Suppose she has? I can stand chaffing, I
hope. And besides, she was right. I am worn out waiting for patients
—waiting for patients to 'do the rest' after I've, so to speak, 'pressed
the button.'"
"It's hard to believe you're the Daniel Ballard of Boston there's
so much fuss about. Are you sure you're the man Elihu Webster

meant? The man he called a celebrated specialist—the best skill in
New England—and so forth and so forth?"
"I'm the only M.D. of my name in Boston," the young man said
simply. "But I don't call myself a specialist, much less and-so-forth
and-so-forth!"
"What do you call yourself, then?"
"A physician."
"I wish I had married your grandfather," Madam Crewe
announced.
Daniel Ballard bent his head, acknowledging what was more
than mere compliment, by a silence sincerer than words.
"I must go. Where's Katherine?" he asked, after a moment.
"I don't know. Not at home, I fancy. Will you do me a favor?"
"If I can."
"Don't try to see her for a while. Leave her alone."
He had risen to go, but her words checked him.
"I can't give you any such promise," he said. "It seems a
strange request for you to make."
"You don't trust me?"
"No. Not in this."
"You may."
He hesitated. "Perhaps. Still—I give no promise. I'll think it over.
When I have more time, you'll explain?"
"Perhaps," she echoed.
The next minute she was alone.
However she accomplished it, Madam Crewe had her way.
Katherine did not see Dr. Ballard again before he left for Boston. He

left a brief note explaining that Mr. Ronald refused to release him,
even after Dr. Webster arrived with his brace of nurses.
Katherine read the letter with a bitter smile. Technically, she had
nothing to complain of. She had definitely said she would never
marry him. He had taken her at her word—and yet, his easy
acquiescence hurt her cruelly. It did not explain anything, that Mr.
Ronald himself confessed his dependence on Dr. Ballard.
The saving of his wife and baby (a miracle, Dr. Webster called
it) made Frank Ronald feel that, whoever came or went, "Ballard"
and Martha Slawson could not be spared from Claire's bedside, until
the danger was over, recovery absolutely certain.
It was all perfectly plausible, and yet—
Then came an urgent recall to Boston, which "the best skill in
New England" felt obliged to respond to in person.
"If you didn't have a family, Mrs. Slawson," he said to Martha,
the last evening, as they sat in Claire's sitting-room, gratifying Frank
Ronald's whim that they remain within call,—"If you didn't have a
family I'd urge you to take up nursing. You have an excellent knack
for it. I could make a capital nurse of you."
Martha nodded appreciatively. "Thank you, sir. But there's so
many things I'm, as you might say, billed to be made over into first,
I guess I'll have to cut out the trained nurse. Besides, I might fall
down on a case I was a stranger to. It's dead easy do for anybody
you love, but to go an' pick'm up off'n the roadside——! Well, that's
a differnt proposition. The dirt an' the smell o' some o' them! You
wouldn't believe it!"
"Do you love that scamp Buller?"
"Not on your life! That is,—not so you'd notice it."

"Yet you stood by him like a soldier, when Driggs and I took his
hand off, last night. How's that?"
Martha pondered a moment. "Well, you see, sir, to tell the truth,
I feel kind o' responsible for Buller. 'Twas me made'm mad in the
first place, an' then, when he wanted to get back at me, 'twas our
trap give'm the nip. Poor fella! You couldn't help be sorry for'm, he'll
miss that strong right hand o' his so, which it used to be a reg'lar
pretidigiagitator with the licka—— 'Now you see it an' now you don't
effec'.'"
Dr. Ballard laughed. "His left hand's in training already. Between
the whiskey and the ether, last night, I was almost anesthetized
myself. But joking aside, I'm going to leave Buller in your care. I'll
show you about the bandaging, so when Driggs gets through with
the patient, you can take him up. I wouldn't like to trust to Mrs.
Buller. She's a slipshod creature, sure to neglect. Dr. Driggs tells me,
Buller dreads him like the mischief, so he won't go there any longer
than he has to. May I trust you to keep your eye on him, follow him
up, and let me know if there's any hitch in the healing?"
"Certaintly you may," said Martha.
"Another thing," Dr. Ballard paused. "I'd be glad to feel you are
keeping an eye on—a—Crewesmere."
Mrs. Slawson nodded. "Certaintly, again. But you don't think—
that is, you ain't in doubt about the ol' lady, are you? I'd hate to
think she might have somethin' I ain't used to. I kinda got
accustomed to strokes now, so's if she'd have any more, I'd know
just how to take a-holt, but if she set about gettin' up somethin'
new, it'd sorta rattle me, maybe. You never can tell."
"No, that's it! You never can tell. I can't tell."

"It ain't as if she didn't have a sympton to show you," pleaded
Martha, "so's you'd be workin' in the dark. When ladies is that way,
the doctors says to'mselves: 'Her color's good, an' her pulse is
strong, which proves she's far from a well woman. While I'm waitin'
for somethin' to happen, I'll remove her appendicitis.' Folks has such
funny furbelows inside'm nowadays, I don't wonder the doctors is
puzzled. What's the use o' adenoids now, an' appendicitises, I should
like to know, if it's only to go to the trouble an' expense of havin' 'm
cut out?"
"Quite so," acquiesced Dr. Ballard gravely. "No, I'm not anxious
about Madam Crewe's appendix. I'm anxious about her—
granddaughter."
"Oh!" said Martha. "It's her you want to remove."
Dr. Ballard flushed. "Yes, Mrs. Slawson. That is—I wish to marry
Miss Crewe. You already know of Madam's opposition. I don't mind
that—any more. But something has happened—I don't know what—
to change Miss Crewe, herself. I would never ask her to desert her
grandmother. In fact, I would not respect her if she did desert her,
leave her alone in her infirmity and old age. But I don't want her
mind to be embittered. She is not happy. I wish you'd look after her
—lend her a helping hand, once in a while. Lend her a helping
heart."
"I'll do my best," promised Martha solemnly.
"I've grown attached to this place. I'd like to hear about—
everybody, once in a while. I'd like, so to speak, to keep my finger
on the pulse of the public."
Martha looked up perplexed. "The pulse o' the public? I don't
know as I exackly get what you mean. But, if you want to feel the

pulse o' the public, why—you're the doctor! Anyhow, I'll let you
know how things is goin', if you'll excuse the liberty, and won't mind
my spellin', which Sam says it's fierce."
"I'll deeply appreciate any line you may take the trouble to write
me," Dr. Ballard assured her, with hearty sincerity.
It was September before Mrs. Slawson was actually settled at home
again. The nurses, over at the big house, were altogether capable
and trustworthy, but even after all need of her had passed, Mr.
Ronald liked to feel Martha was within call. He fancied his wife felt
more content when she was by, and, certainly, the baby slept better
on her ample bosom than anywhere else.
It was a tiny creature, very delicate and fragile, a mere scrap of
humanity that Martha could hold in the hollow of her hand.
In the privacy of their own sitting-room, the two trained nurses
confided to Mrs. Slawson: "It's too bad the parents' hearts are so set
on the child. They'll never raise it, never!"
"Now, what do you think o' that!" Martha said mournfully, and
the two uniformed ones never knew that, in her heart, she despised
them, "and their mizrable Bildadin' talk, which nobody could stand
up against it, anyhow, much less a innocent little lamb that hasn't
the stren'th to call'm liars to their faces."
"O' course we'll raise her," she assured Mr. Ronald confidently.
"There's no doubt about it. Yes, I know she ain't very hefty, an' she
ain't very robustic. But what do you expec'? You ain't give her a fair
show yet. You can't take a baby, a few weeks old, 'specially if it had
the tough time gettin' in on the game at all, that this one had, an'
expec' her to be as big an' husky as my Sabina. It wouldn't be

sensible. Besides, look at her mother! Miss Claire's no giantess, nor
ever was, but she's as sound as a nut, an' so'll the baby be, when
she gets her gait on, an' knows it's up to her to keep in step with the
percession. Don't you let nobody discourage you. Believin's half the
battle. You can take it from me, that baby's goin' to live, an' thrive,
like the little thorabred she is. She wouldn't give us all this trouble
for nothin'."
Her invincible confidence was like a tonic to Francis Ronald. It
reinforced his own more flickering faith, so he could meet Claire's
hungrily questioning eyes with reassurance.
And, as the weeks went by, Martha's prediction seemed less and
less preposterous.
"Didn't I tell you?" she exulted. "That baby's a winner! She's
goin' to be standard weight, all right, all right, an' measure up to
requirements too, give her time. But between you an' me, all this
new-fangled business with scales, an' tape-measures, an' suchlike, is
enough to discourage the best-intentioned infant. There's more
notions, nowadays, than you can shake a stick at—an' I'd like to
shake a stick at most of'm, believe me!"
At the time, she was thinking rather more of Miss Crewe, than
of the nurses, whose "queer fandangoes" she never could become
reconciled to.
She was frankly anxious about Katherine.
"If I could do with her, like I do with Buller, I wouldn't say a
word," she ruminated. "I just keep a kinda gener'l line on him, an'
when the time comes, I get a-holt of his collar-band, an' march'm up
to the captain's office, as brave as a lion. He's got so the minute I
tip'm the wink, he comes for his washin' an' ironin'—I should say,

bandidgin', as meek as a lamb to the slaughterhouse. But you can
take it from me, there's no gettin' a line on Miss Katherine. She's
devotin' all her time an' attention to puttin' off flesh an' color. The
trouble is, she's got nothin' to do, an' she does it so thora, she ain't
got time for anything else. Dear me! I wisht I could sort o' set her
an' Buller at each other. It might help'm both to forget their losses.
He certaintly is a queer dick, an' no mistake!"
"In spite o' his sportin' a G.A.R. one, you can take it from me,
Buller ain't got all his buttons!" she told Miss Katherine. "Do you
know what he says? He says everybody's gone back on'm because
he's in trouble. He says, nobody'll look at'm now he's mangled. They
was his friends before, when he had all the limbs was comin' to'm,
but—now he's shy a hand—they're too proud to notice'm. He says
the world's a hard place for cripples."
A faint smile flitted across Katherine's face
"What a perverted point of view," she said, for the sake of
saying something.
"Do you know what I think?" Mrs. Slawson continued. "I think
now is the zoological moment to catch Buller, an' see what kind o'
animal he is—if he's got the makin' of a man in'm. If he could be got
to give up the drink, I do believe he might amount to somethin' yet.
You can't know what a fella reely is, when he's always steepin' in
licka. It's like pickles. You wouldn't know if they're dill, or sweet or
what they are, till you take'm out o' soak an' test'm."
"I should think you might influence him," suggested Miss Crewe
impersonally. "You're so strong and wholesome and steady."
"Land, no! Buller wouldn't listen to me," said Martha. "How
would I be reformin' anybody, when so many is reformin' me?"

"Mrs. Peckett, then?"
"Mrs. Peckett's way o' doin' things makes some folks nervous.
It's like as if she said: 'I'm goin' to raise the tone o' this town, if I
have to raise it by the scruff of its neck!' She's a good woman, Mrs.
Peckett is, more power to her! Yes, she's as good as old gold, and—
just as dull."
Katherine was amused. "Does Mr. Buller require people to be so
very brilliant, then?"
"Land, no! He don't. But his case does. There's a differnce. The
fella that gets the whip-hand of'm is the fella he's goin' to respec'.
No others need apply. If there was anybody in this town could kinda
give'm the fright of his life on the licka question, it'd be dead easy
tame him to'm afterwords."
Miss Crewe's face lost its apathetic expression. A light of interest
shone in her eyes.
"I wonder if an idea that has just occurred to me would be of
any use? Last winter I attended a course of lectures at Columbia
College, and one of the lectures was illustrated by lantern-slides,
showing the effect of alcohol on the body and mind of habitual
drunkards. They were enough to give one the horrors! If Buller could
see those pictures——!"
Mrs. Slawson brought her hands down upon her knees with a
sounding slap. "There, didn't I know you'd strike on just the right
idea, quicker'n, sure'n anybody else? An' you've done it!"
"But it would cost a lot of money to get that lecturer here. We
might not be able to get him at all, even if we could raise the money
to pay——"

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Digital signal processors architecture programming and applications 2nd Edition B. Venkataramani

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