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6. Digital Filters Design for Signal and Image Processing

7. This page intentionally left blank

8. Digital Filters Design for
Signal and Image Processing












Edited by
Mohamed Najim

9. First published in France in 2004 by Hermès Science/Lavoisier entitled “Synthèse de filtres
numériques en traitement du signal et des images”
First published in Great Britain and the United States in 2006 by ISTE Ltd
Apart from any fair dealing for the purposes of research or private study, or criticism or
review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may
only be reproduced, stored or transmitted, in any form or by any means, with the prior
permission in writing of the publishers, or in the case of reprographic reproduction in
accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction
outside these terms should be sent to the publishers at the under mentioned address:
ISTE Ltd ISTE USA
6 Fitzroy Square 4308 Patrice Road
London W1T 5DX Newport Beach, CA 92663
UK USA
www.iste.co.uk



© ISTE Ltd, 2006
© LAVOISIER, 2004
The rights of Mohamed Najim to be identified as the author of this work have been asserted
by him in accordance with the Copyright, Designs and Patents Act 1988.

___________________________________________________________________________
Library of Congress Cataloging-in-Publication Data

Synthèse de filtres numériques en traitement du signal et des images.
English
Digital filters design for signal and image processing/edited by Mohamed Najim.
p. cm.
Includes index.
ISBN-13: 978-1-905209-45-3
ISBN-10: 1-905209-45-2
1. Electric filters, Digital. 2. Signal processing--Digital techniques.
3. Image processing--Digital techniques. I. Najim, Mohamed. II. Title.
TK7872.F5S915 2006
621.382'2--dc22
2006021429

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 10: 1-905209-45-2
ISBN 13: 978-1-905209-45-3
Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.

10. Table of Contents
Introduction........................................ x iii

Chapter 1. Introduction to Signals and Systems................. 1
Yannick BERTHOUMIEU, Eric GRIVEL and Mohamed NAJIM
1.1. Introduction.................................... 1
1.2. Signals: categories, representations and characterizations........ 1
1.2.1. Definition of continuous-time and discrete-time signals....... 1
1.2.2. Deterministic and random signals.................... 6
1.2.3. Periodic signals............................... 8
1.2.4. Mean, energy and power.......................... 9
1.2.5. Autocorrelation function.......................... 12
1.3. Systems...................................... 15
1.4. Properties of discrete-time systems...................... 16
1.4.1. Invariant linear systems.......................... 16
1.4.2. Impulse responses and convolution products.............. 16
1.4.3. Causality................................... 17
1.4.4. Interconnections of discrete-time systems............... 18
1.5. Bibliography................................... 19

Chapter 2. Discrete System Analysis........................ 21
Mohamed NAJIM and Eric GRIVEL

2.1. Introduction.................................... 21
2.2. The z-transform................................. 21
2.2.1. Representations and summaries ..................... 21
2.2.2. Properties of the z-transform....................... 28
2.2.2.1. Linearity................................. 28
2.2.2.2. Advanced and delayed operators................... 29
2.2.2.3. Convolution............................... 30

11. vi Digital Filters Design for Signal and Image Processing
2.2.2.4. Changing the z-scale.......................... 31
2.2.2.5. Contrasted signal development.................... 31
2.2.2.6. Derivation of the z-transform..................... 31
2.2.2.7. The sum theorem............................ 32
2.2.2.8. The final-value theorem........................ 32
2.2.2.9. Complex conjugation ......................... 32
2.2.2.10. Parseval’s theorem.......................... 33
2.2.3. Table of standard transform........................ 33
2.3. The inverse z-transform ............................ 34
2.3.1. Introduction................................. 34
2.3.2. Methods of determining inverse z-transforms............. 35
2.3.2.1. Cauchy’s theorem: a case of complex variables.......... 35
2.3.2.2. Development in rational fractions.................. 37
2.3.2.3. Development by algebraic division of polynomials........ 38
2.4. Transfer functions and difference equations ................ 39
2.4.1. The transfer function of a continuous system ............. 39
2.4.2. Transfer functions of discrete systems ................. 41
2.5. Z-transforms of the autocorrelation and intercorrelation functions . . . 44
2.6. Stability...................................... 45
2.6.1. Bounded input, bounded output (BIBO) stability........... 46
2.6.2. Regions of convergence.......................... 46
2.6.2.1. Routh’s criterion............................ 48
2.6.2.2. Jury’s criterion ............................. 49

Chapter 3. Frequential Characterization of Signals and Filters....... 51
Eric GRIVEL and Yannick BERTHOUMIEU

3.1. Introduction.................................... 51
3.2. The Fourier transform of continuous signals................ 51
3.2.1. Summary of the Fourier series decomposition of continuous
signals........................................ 51
3.2.1.1. Decomposition of finite energy signals using an orthonormal
base......................................... 51
3.2.1.2. Fourier series development of periodic signals .......... 52
3.2.2. Fourier transforms and continuous signals............... 57
3.2.2.1. Representations............................. 57
3.2.2.2. Properties ................................ 58
3.2.2.3. The duality theorem.......................... 59
3.2.2.4. The quick method of calculating the Fourier transform..... 59
3.2.2.5. The Wiener-Khintchine theorem................... 63
3.2.2.6. The Fourier transform of a Dirac comb............... 63
3.2.2.7. Another method of calculating the Fourier series development
of a periodic signal................................ 66

12. Table of Contents vii
3.2.2.8. The Fourier series development and the Fourier transform . . . 68
3.2.2.9. Applying the Fourier transform: Shannon’s sampling theorem . 75
3.3. The discrete Fourier transform (DFT).................... 78
3.3.1. Expressing the Fourier transform of a discrete sequence....... 78
3.3.2. Relations between the Laplace and Fourier z-transforms...... 80
3.3.3. The inverse Fourier transform...................... 81
3.3.4. The discrete Fourier transform...................... 82
3.4. The fast Fourier transform (FFT)....................... 86
3.5. The fast Fourier transform for a time/frequency/energy representation
of a non-stationary signal.............................. 90
3.6. Frequential characterization of a continuous-time system ........ 91
3.6.1. First and second order filters....................... 91
3.6.1.1. 1
st
order system............................. 91
3.6.1.2. 2
nd
order system............................. 93
3.7. Frequential characterization of discrete-time system ........... 95
3.7.1. Amplitude and phase frequential diagrams............... 95
3.7.2. Application ................................. 96

Chapter 4. Continuous-Time and Analog Filters................. 99
Daniel BASTARD and Eric GRIVEL
4.1. Introduction.................................... 99
4.2. Different types of filters and filter specifications.............. 99
4.3. Butterworth filters and the maximally flat approximation........ 104
4.3.1. Maximally flat functions (MFM)..................... 104
4.3.2. A specific example of MFM functions: Butterworth polynomial
filters......................................... 106
4.3.2.1. Amplitude-squared expression.................... 106
4.3.2.2. Localization of poles.......................... 107
4.3.2.3. Determining the cut-off frequency at –3 dB and filter orders . . 110
4.3.2.4. Application ............................... 111
4.3.2.5. Realization of a Butterworth filter.................. 112
4.4. Equiripple filters and the Chebyshev approximation........... 113
4.4.1. Characteristics of the Chebyshev approximation........... 113
4.4.2. Type I Chebyshev filters.......................... 114
4.4.2.1. The Chebyshev polynomial...................... 114
4.4.2.2. Type I Chebyshev filters........................ 115
4.4.2.3. Pole determination........................... 116
4.4.2.4. Determining the cut-off frequency at –3 dB and the filter order 118
4.4.2.5. Application ............................... 121
4.4.2.6. Realization of a Chebyshev filter.................. 121
4.4.2.7. Asymptotic behavior.......................... 122
4.4.3. Type II Chebyshev filter.......................... 123

13. viii Digital Filters Design for Signal and Image Processing
4.4.3.1. Determining the filter order and the cut-off frequency...... 123
4.4.3.2. Application ............................... 124
4.5. Elliptic filters: the Cauer approximation................... 125
4.6. Summary of four types of low-pass filter: Butterworth, Chebyshev
type I, Chebyshev type II and Cauer........................ 125
4.7. Linear phase filters (maximally flat delay or MFD): Bessel and
Thomson filters.................................... 126
4.7.1. Reminders on continuous linear phase filters............. 126
4.7.2. Properties of Bessel-Thomson filters.................. 128
4.7.3. Bessel and Bessel-Thomson filters.................... 130
4.8. Papoulis filters (optimum (O
n))........................ 132
4.8.1. General characteristics........................... 132
4.8.2. Determining the poles of the transfer function............. 135
4.9. Bibliography................................... 135

Chapter 5. Finite Impulse Response Filters.................... 137
Yannick BERTHOUMIEU, Eric GRIVEL and Mohamed NAJIM
5.1. Introduction to finite impulse response filters............... 137
5.1.1. Difference equations and FIR filters................... 137
5.1.2. Linear phase FIR filters.......................... 142
5.1.2.1. Representation.............................. 142
5.1.2.2. Different forms of FIR linear phase filters............. 147
5.1.2.3. Position of zeros in FIR filters.................... 150
5.1.3. Summary of the properties of FIR filters................ 152
5.2. Synthesizing FIR filters using frequential specifications......... 152
5.2.1. Windows................................... 152
5.2.2. Synthesizing FIR filters using the windowing method........ 159
5.2.2.1. Low-pass filters............................. 159
5.2.2.2. High-pass filters............................. 164
5.3. Optimal approach of equal ripple in the stop-band and passband.... 165
5.4. Bibliography................................... 172

Chapter 6. Infinite Impulse Response Filters................... 173
Eric GRIVEL and Mohamed NAJIM
6.1. Introduction to infinite impulse response filters.............. 173
6.1.1. Examples of IIR filters........................... 174
6.1.2. Zero-loss and all-pass filters....................... 178
6.1.3. Minimum-phase filters........................... 180
6.1.3.1. Problem ................................. 180
6.1.3.2. Stabilizing inverse filters....................... 181
6.2. Synthesizing IIR filters............................. 183
6.2.1. Impulse invariance method for analog to digital filter conversion . 183

14. Table of Contents ix
6.2.2. The invariance method of the indicial response............ 185
6.2.3. Bilinear transformations.......................... 185
6.2.4. Frequency transformations for filter synthesis using low-pass
filters......................................... 188
6.3. Bibliography................................... 189

Chapter 7. Structures of FIR and IIR Filters................... 191
Mohamed NAJIM and Eric GRIVEL
7.1. Introduction.................................... 191
7.2. Structure of FIR filters............................. 192
7.3. Structure of IIR filters.............................. 192
7.3.1. Direct structures............................... 192
7.32. The cascade structure............................ 209
7.3.3. Parallel structures.............................. 211
7.4. Realizing finite precision filters........................ 211
7.4.1. Introduction................................. 211
7.4.2. Examples of FIR filters.......................... 212
7.4.3. IIR filters................................... 213
7.4.3.1. Introduction............................... 213
7.4.3.2. The influence of quantification on filter stability......... 221
7.4.3.3. Introduction to scale factors...................... 224
7.4.3.4. Decomposing the transfer function into first- and
second-order cells ................................ 226
7.5. Bibliography................................... 231

Chapter 8. Two-Dimensional Linear Filtering.................. 233
Philippe BOLON
8.1. Introduction.................................... 233
8.2. Continuous models............................... 233
8.2.1. Representation of 2-D signals ...................... 233
8.2.2. Analog filtering............................... 235
8.3. Discrete models................................. 236
8.3.1. 2-D sampling ................................ 236
8.3.2. The aliasing phenomenon and Shannon’s theorem.......... 240
8.3.2.1. Reconstruction by linear filtering (Shannon’s theorem)..... 240
8.3.2.2. Aliasing effect.............................. 240
8.4. Filtering in the spatial domain......................... 242
8.4.1. 2-D discrete convolution.......................... 242
8.4.2. Separable filters............................... 244
8.4.3. Separable recursive filtering ....................... 246
8.4.4. Processing of side effects......................... 249
8.4.4.1. Prolonging the image by pixels of null intensity.......... 250

15. x Digital Filters Design for Signal and Image Processing
8.4.4.2. Prolonging by duplicating the border pixels............ 251
8.4.4.3. Other approaches............................ 252
8.5. Filtering in the frequency domain....................... 253
8.5.1. 2-D discrete Fourier transform (DFT).................. 253
8.5.2. The circular convolution effect...................... 255
8.6. Bibliography................................... 259

Chapter 9. Two-Dimensional Finite Impulse Response Filter Design .... 261
Yannick BERTHOUMIEU
9.1. Introduction.................................... 261
9.2. Introduction to 2-D FIR filters......................... 262
9.3. Synthesizing with the two-dimensional windowing method....... 263
9.3.1. Principles of method............................ 263
9.3.2. Theoretical 2-D frequency shape..................... 264
9.3.2.1. Rectangular frequency shape..................... 264
9.3.2.2. Circular shape.............................. 266
9.3.3. Digital 2-D filter design by windowing................. 271
9.3.4. Applying filters based on rectangular and circular shapes...... 271
9.3.5. 2-D Gaussian filters............................ 274
9.3.6. 1-D and 2-D representations in a continuous space.......... 274
9.3.6.1. 2-D specifications............................ 276
9.3.7. Approximation for FIR filters...................... 277
9.3.7.1. Truncation of the Gaussian profile.................. 277
9.3.7.2. Rectangular windows and convolution............... 279
9.3.8. An example based on exploiting a modulated Gaussian filter.... 280
9.4. Appendix: spatial window functions and their implementation..... 286
9.5. Bibliography................................... 291

Chapter 10. Filter Stability.............................. 293
Michel BARRET
10.1. Introduction................................... 293
10.2. The Schur-Cohn criterion........................... 298
10.3. Appendix: resultant of two polynomials.................. 314
10.4. Bibliography.................................. 319

Chapter 11. The Two-Dimensional Domain.................... 321
Michel BARRET

11.1. Recursive filters................................ 321
11.1.1. Transfer functions............................. 321
11.1.2. The 2-D z-transform........................... 322
11.1.3. Stability, causality and semi-causality................. 324

16. Table of Contents xi
11.2. Stability criteria................................ 328
11.2.1. Causal filters................................ 329
11.2.2. Semi-causal filters............................. 332
11.3. Algorithms used in stability tests...................... 334
11.3.1. The jury Table............................... 334
11.3.2. Algorithms based on calculating the Bezout resultant ....... 339
11.3.2.1. First algorithm............................. 340
11.3.2.2. Second algorithm........................... 343
11.3.3. Algorithms and rounding-off errors.................. 347
11.4. Linear predictive coding ........................... 351
11.5. Appendix A: demonstration of the Schur-Cohn criterion........ 355
11.6. Appendix B: optimum 2-D stability criteria................ 358
11.7. Bibliography.................................. 362

List of Authors...................................... 365

Index............................................ 367

17. This page intentionally left blank

18. Introduction
Over the last decade, digital signal processing has matured; thus, digital signal
processing techniques have played a key role in the expansion of electronic products
for everyday use, especially in the field of audio, image and video processing.
Nowadays, digital signal is used in MP3 and DVD players, digital cameras, mobile
phones, and also in radar processing, biomedical applications, seismic data
processing, etc.
This book aims to be a text book which presents a thorough introduction to
digital signal processing featuring the design of digital filters. The purpose of the
first part (Chapters 1 to 9) is to initiate the newcomer to digital signal and image
processing whereas the second part (Chapters 10 and 11) covers some advanced
topics on stability for 2-D filter design. These chapters are written at a level that is
suitable for students or for individual study by practicing engineers.
When talking about filtering methods, we refer to techniques to design and
synthesize filters with constant filter coefficients. By way of contrast, when dealing
with adaptive filters, the filter taps change with time to adjust to the underlying
system. These types of filters will not be addressed here, but are presented in various
books such as [HAY 96], [SAY 03], [NAJ 06].
Chapter 1 provides an overview of various classes of signals and systems. It
discusses the time-domain representations and characterizations of the continuous-
time and discrete-time signals.
Chapter 2 details the background for the analysis of discrete-time signals. It
mainly deals with the z-transform, its properties and its use for the analysis of linear
systems, represented by difference equations.

19. xiv Digital Filters Design for Signal and Image Processing
Chapter 3 is dedicated to the analysis of the frequency properties of signals and
systems. The Fourier transform, the discrete Fourier transform (DFT) and the fast
Fourier transform (FFT) are introduced along with their properties. In addition, the
well-known Shannon sampling theorem is recalled.
As we will see, some of the most popular techniques for digital infinite impulse
response (IIR) filter design benefit from results initially developed for analog
signals. In order to make the reader’s task easy, Chapter 4 is devoted to continuous-
time filter design. More particularly, we recall several approximation techniques
developed by mathematicians such as Chebyshev or Legendre, who have thus seen
their names associated with techniques of filter design.
The following chapters form the core of the book. Chapter 5 deals with the
techniques to synthesize finite impulse response (FIR) filters. Unlike IIR filters,
these have no equivalent in the continuous-time domain. The so-called windowing
method, as a FIR filter design method, is first presented. This also enables us to
emphasize the key role played by the windowing in digital signal processing, e.g.,
for frequency analysis. The Remez algorithm is then detailed.
Chapter 6 concerns IIR filters. The most popular techniques for analog to digital
filter conversion, such as the bilinear transform and the impulse invariance method,
are presented. As the frequency response of these filters is represented by rational
functions, we must tackle the problems of stability induced by the existence of poles
of these rational functions.
In Chapter 7, we address the selection of the filter structure and point out its
importance for filter implementation. Some problems due to the finite-precision
implementation are listed and we provide rules to choose an appropriate structure
while implementing filter on fixed point operating devices.
In comparison with many available books dedicated to digital filtering, this title
features both 1-D and 2-D systems, and as such covers both signal and image
processing. Thus, in Chapters 8 and 9, 2-D filtering is investigated.
Moreover, it is not easy to establish the necessary and sufficient conditions to
test the stability of 2-D signals. Therefore, Chapters 10 and 11 are dedicated to the
difficult problem of the stability of 2-D digital system, a topic which is still the
subject of many works such as [ALA 2003] [SER 06]. Even if these two chapters are
not a prerequisite for filter design, they can provide the reader who would like to
study the problems of stability in the multi-dimensional case with valuable
clarifications. This contribution is another element that makes this book stand out.

20. Introduction xv
The field of digital filtering is often perceived by students as a “patchwork” of
formulae and recipes. Indeed, the methods and concepts are based on several
specific optimization techniques and mathematical results which are difficult to
grasp.
For instance, we have to remember that the so-called Parks-McClellan algorithm
proposed in 1972 was first rejected by the reviewers [PAR 72]. This was probably
due to the fact that the size of the submitted paper, i.e., 5 pages, did not enable the
reviewers to understand every step of the approach [McC 05].
In this book we have tried, at every stage, to justify the necessity of these
approaches without recalling all the steps of the derivation of the algorithm. They
are described in many articles published during the 1970s in the IEEE periodicals
i.e., Transactions on Acoustics Speech and Signal Processing, which has since
become Transactions on Signal Processing and Transactions on Circuits and
Systems.

Mohamed NAJIM
Bordeaux






[ALA 2003] ALATA O., NAJIM M., RAMANANJARASOA C. and TURCU F., “Extension
of the Schur-Cohn Stability Test for 2-D AR Quarter-Plane Model”, IEEE Trans. on
Information Theory, vol. 49, no. 11, November 2003.
[HAY 96] HAYKIN S., Adaptive Filter Theory, 3
rd
edition, Prentice Hall, 1996.
[McC 05] McCLELLAN J.H. and PARKS W. Th., “A Personal History of the Parks-
McClellan Algorithm” IEEE Signal Processing Magazine , pp 82-86, March 2005.
[NAJ 06] NAJIM M., Modélisation, estimation et filtrage optimale en traitement du signal,
forthcoming, 2006, Hermes, Paris.
[PAR 72] PARKS W. Th. and McCLELLAN J.H., “Chebyshev Approximation for
Nonrecursive Digital Filters with Linear Phase,” IEEE Trans. Circuit Theory, vol. CT-19,
no. 2, pp 189-194, 1972.
[SAY 03] SAYED A., Fundamentals of Adaptive Filtering, Wiley IEEE Press, 2003.
[SER 06] SERBAN I., TURCU F., NAJIM M., “Schur Coefficients in Several Variables”,
Journal of Mathematical Analysis and Applications, vol. 320, issue no. 1, August 2006,
pp 293-302.

21. This page intentionally left blank

22. Chapter 1

Introduction to Signals and Systems
1.1. Introduction
Throughout a range of fields as varied as multimedia, telecommunications,
geophysics, astrophysics, acoustics and biomedicine, signals and systems play a
major role. Their frequential and temporal characteristics are used to extract and
analyze the information they contain. However, what importance do signals and
systems really hold for these disciplines? In this chapter we will look at some of the
answers to this question.

First we will discuss different types of continuous and discrete-time signals,
which can be termed random or deterministic according to their nature. We will also
introduce several mathematical tools to help characterize these signals. In addition,
we will describe the acquisition chain and processing of signals.

Later we will define the concept of a system, emphasizing invariant discrete-time
linear systems.
1.2. Signals: categories, representations and characterizations
1.2.1. Definition of continuous-time and discrete-time signals
The function of a signal is to serve as a medium for information. It is a
representation of the variations of a physical variable.


Chapter written by Yannick BERTHOUMIEU, Eric GRIVEL and Mohamed NAJIM.

23. 2 Digital Filters Design for Signal and Image Processing
A signal can be measured by a sensor, then analyzed to describe a physical
phenomenon. This is the situation of a tension taken to the limits of a resistance in
order to verify the correct functioning of an electronic board, as well as, to cite one
example, speech signals that describe air pressure fluctuations perceived by the
human ear.

Generally, a signal is a function of time. There are two kinds of signals:
continuous and discrete-time.

A continuous-time or analog signal can be measured at certain instants. This
means physical phenomena create, for the most part, continuous-time signals.


Figure 1.1. Example of the sleep spindles of
an electroencephalogram (EEG) signal


The advancement of computer-based techniques at the end of the 20
th
century led
to the development of digital methods for information processing. The capacity to
change analog signals to digital signals has meant a continual improvement in
processing devices in many application fields. The most significant example of this
is in the field of telecommunications, especially in cell phones and digital
televisions. The digital representation of signals has led to an explosion of new
techniques in other fields as varied as speech processing, audiofrequency signal
analysis, biomedical disciplines, seismic measurements, multimedia, radar and
measurement instrumentation, among others.

Time (s)

24. Introduction to Signals and Systems 3

The signal is said to be a discrete-time signal when it can be measured at certain
instants; it corresponds to a sequence of numerical values. Sampled signals are the
result of sampling, uniform or not, of a continuous-time signal. In this work, we are
especially interested in signals taken at regular intervals of time, called sampling
periods, which we write as
1
=
s
s
T
f where f s is called the sampling rate or the
sampling frequency. This is the situation for a temperature taken during an
experiment, or of a speech signal (see Figure 1.2). This discrete signal can be written
either as x(k) or x(kT
s). Generally, we will use the first writing for its simplicity. In
addition, a digital signal is a discrete-time discrete-valued signal. In that case, each
signal sample value belongs to a finite set of possible values.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
x 10
4

Figure 1.2. Example of a digital voiced speech signal
(the sampling frequency f
s is at 16 KHz)



The choice of a sampling frequency depends on the applications being used and
the frequency range of the signal to be sampled. Table 1.1 gives several examples of
sampling frequencies, according to different applications.





Time (s)

25. 4 Digital Filters Design for Signal and Image Processing
Signal f
s T
s
Speech:
Telephone band – telephone-
Broadband – audio-visual conferencing-
8 KHz
or 16 KHz
125 µs
62.5 µs
Audio: Broadband (Stereo) 32 KHz
44.1 KHz
48 KHz
31.25 µs
22.7 µs
20.8 µs
Video 10 MHz 100 ns

Table 1.1. Sampling frequencies according to processed signals


In Figure 1.3, we show an acquisition chain, a processing chain and a signal
restitution chain.

The adaptation amplifier makes the input signal compatible with the
measurement chain.

A pre-filter which is either pass-band or low-pass, is chosen to limit the width of
the input signal spectrum; this avoids the undesirable spectral overlap and hence, the
loss of spectral information (aliasing). We will return to this point when we discuss
the sampling theorem in section 3.2.2.9. This kind of anti-aliasing filter also makes
it possible to reject the out-of-band noise and, when it is a pass-band filter, it helps
suppress the continuous component of the signal.

The Analog-to-Digital Converter (A/D) partly carries out sampling, and then
quantification, at the sampling frequency f
s, that is, it allocates a coding to each
sampling on a certain number of bits.

The digital input signal is then processed in order to give the digital output
signal. The reconversion into an analog signal is made possible by using a D/A
converter and a smoothing filter.

Many parameters influence sampling, notably the quantification step and the
response time of the digital system, both during acquisition and restitution.
However, by improving the precision of the A/D converter and the speed of the
calculators, we can get around these problems. The choice of the sampling
frequency also plays an important role.

26. Introduction to Signals and Systems 5


Figure 1.3. Complete acquisition chain and digital processing of a signal


Different types of digital signal representation are possible, such as functional
representations, tabulated representations, sequential representations, and graphic
representations (as in bar diagrams).

Looking at examples of basic digital signals, we return to the unit sample
sequence represented by the Kronecker symbol δ(k), the unit step signal u(k), and
the unit ramp signal r(k). This gives us:
Unit sample sequence:
()
()01
1for 0kkδ
δ=⎧
⎪
⎨
=≠⎪⎩

Physical
variable
Digital
input
signal
Processing
A/D converter
Low-pass filter
or pass-band
Adaptation
amplifier
Sampling
blocker

Smoothing
filter
Processed
signal
D/A
converter
Digital
output
signal
Digital system
Analog
signal
Sensor

27. 6 Digital Filters Design for Signal and Image Processing
Unit step signal:
()
()1for 0
0for 0
uk k
uk k
⎧ =≥⎪
⎨
=<⎪⎩

Unit ramp signal:
()
() for 0
0for 0.
rk k k
rk k
⎧ =≥⎪
⎨
=<⎪⎩


-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
Scale
amplitude
impulse unity
-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
indices
amplitude
unity
Figure 1.4. Unit sample sequence δ (k) and unit step signal u(k)
1.2.2. Deterministic and random signals
We class signals as being deterministic or random. Random signals can be
defined according to the domain in which they are observed. Sometimes, having
specified all the experimental conditions of obtaining the physical variable, we see
that it fluctuates. Its values are not completely determined, but they can be evaluated
in terms of probability. In this case, we are dealing with a random experiment and
the signal is called random. In the opposite situation, the signal is called
deterministic.

28. Introduction to Signals and Systems 7


Figure 1.5. Several realizations of a 1-D random signal


EXAMPLE 1.1.– let us look at a continuous signal modeled by a sinusoidal function
of the following type.
() ( )sin 2π
xta ft=×
This kind of model is deterministic. However, in other situations, the signal amplitude and the signal frequency can be subject to variations. Moreover, the signal
can be disturbed by an additive noise b(t); then it is written in the following form:
() () () () ()sin 2π
xtat fttbt=× ×+
where a(t), f(t) and b(t) are random variables for each value of t. We say then that
x(t) is a random signal. The properties of the received signal x(t) then depends on the
statistical properties of these random variables.
samples
realization no.5 realization no.4 rea lization no.3 realizati on no.2 realization no.1

29. 8 Digital Filters Design for Signal and Image Processing


Figure 1.6. Several examples of a discrete random 2-D process
1.2.3. Periodic signals
The class of signals termed periodic plays an important role in signal and image
processing. In the case of a continuous-time signal, a signal is called periodic of
period T
0 if T0 is the smallest value verifying the relation:
()() txTtx=+
0 , t∀.
And, for a discrete-time signal, the period of which is N
0, we have:
()() kxNkx=+
0 , k∀.
EXAMPLE 1.2.– examples of periodic signals:
() ( )
0
sin 2π
xtft= ,() ( )
k
kx1−=,() ⎟
⎠
⎞
⎜
⎝
⎛
=
8
cos
πk
kx
.

30. Introduction to Signals and Systems 9

1.2.4. Mean, energy and power
We can characterize a signal by its mean value. This value represents the
continuous component of the signal.

When the signal is deterministic, it equals:
()
1
1
1()
1
µlim
T
T
xtdt
T
→+∞
= ∫
where T 1 designates the integration time. (1.1)
When a continuous-time signal is periodic and of period T
0, the expression of the
mean value comes to:
()
0
0()
1
µ
T
xtdt
T
=∫
(1.2)
PROOF – we can always express the integration time T
1 according to the period of
the signal in the following way:
10
TkT=+ξ where k is an integer and ξ is chosen so that 0 < ξ ≤ T 0.
From there,
() ()
1
10
10() ( )
11
lim lim
Tk
Tk T
xtdt xtdt
Tk Tµ
→+∞ →+∞
== ∫∫
, since ξ becomes
insignificant compared to kT
0.

By using the periodicity property of the continuous signal x(t), we deduce that
() ()
00
00 () ()
11
µ
kTT
xtdt xtdt
kT T
==∑∫∫ .
When the signal is random, the statistical mean is defined for a fixed value of t,
as follows:
() () ( )µ , ,tEXt xpxtdx
+∞
−∞
==⎡⎤
⎣⎦ ∫
(1.3)
where E[.] indicates the mathematical expectation and p(x, t) represents the
probability density of the random signal at the instant t. We can obtain the mean
value if we know p(x, t); in other situations, we can only obtain an estimated value.

31. 10 Digital Filters Design for Signal and Image Processing
For the class of signals called ergodic in the sense of the mean, we assimilate the
statistical mean to the temporal mean, which brings us back to the expression we
have seen previously:
()
1
1
1()
1
µlim
T
T
xtdt
T
→+∞
= ∫
.
Often, we are interested in the energy ε of the processed signal. For a
continuous-time signal x(t), we have:
()
2
ε xtdt
+∞
−∞
=∫
. (1.4)
In the case of a discrete-time signal, the energy is defined as the sum of the
magnitude-squared values of the signal x(k):
()
2
ε
k
xk=∑ (1.5)
For a continuous-time signal x(t), its mean power P is expressed as follows:
()dttx
T
P
T
T
∫
+∞→
=
)(
2
1
lim
. (1.6)
For a discrete-time signal x(k), its mean power is represented as:
()∑
=
+∞→
=
N
k
N
kx
N
P
1
2
1
lim
(1.7)
In signal processing, we often introduce the concept of signal-to-noise ratio
(SNR) to characterize the noise that can affect signals. This variable, expressed in
decibels (dB), corresponds to the ratio of powers between the signal and the noise. It
is represented as:
10
SNR 10log
signal
noise
P
P⎡⎤
= ⎢⎥
⎣⎦ (1.8)
where
signal
Pand
noise
Pindicate, respectively, the powers of the sequences of the
signal and the noise.

EXAMPLE 1.3.– let us consider the example of a periodic signal with a period of
300 Hz signal that is perturbed by a zero-mean Gaussian additive noise with a
signal-to-noise ratio varying from 20 to 0 dB at each 10 dB step. Figures 1.7 and 1.8
show these different situations.

32. Introduction to Signals and Systems 11

0 0.01 0.02 0.03 0.04 0.05 0.06
-5
0
5
time, in seconds
signal without additive noise
0 0.01 0.02 0.03 0.04 0.05 0.06
0
5
time, in seconds
signal with additive noise SNR=20dB

Figure 1.7. Temporal representation of the original signal and of the signal with
additive noise, with a signal-to-noise ratio equal to 20 dB


0 0.01 0.02 0.03 0.04 0.05 0.06
-5
0
5
time, in seconds
SNR=10dB
0 0.01 0.02 0.03 0.04 0.05 0.06
-5
0
5
time, in seconds
SNR=0 dB
signal with additive noise
signal with additive noise


Figure 1.8. Temporal representation of signals with additive noise,
with signal-to-noise ratios equal to 10 dB and 0 dB

33. 12 Digital Filters Design for Signal and Image Processing
1.2.5. Autocorrelation function
Let us take the example of a deterministic continuous signal x(t) of finite energy.
We can carry out a signal analysis from its autocorrelation function, which is
represented as:
() ( )
*
(τ)
xx
Rxtxtdt τ
+∞
−∞
=−∫
(1.9)
The autocorrelation function allows us to measure the degree of resemblance
existing between x(t) and
()
τ−tx. Some of these properties can then be shown from
the results of the scalar products.
From the relations shown in equations (1.4) and (1.9), we see that R
xx(0)
corresponds to the energy of the signal. We can easily demonstrate the following
properties:
)()(
*
ττ−=
xxxxRR
τ∀∈ \ (1.10)
)0()(
xxxx
RR≤τ τ∀∈ \ (1.11)
When the signal is periodic and of the period T
0, the autocorrelation function is
periodic and of the period T
0. It can be obtained as follows:
() ( )
0
*
0
01
(τ) τT
xx
Rxtxtd t
T
=− ∫
(1.12)
We should remember that the autocorrelation function is a specific instance of
the intercorrelation function of two deterministic signals x(t) and y (t), represented as:
() ( )
*
(τ) τ
xy
Rxtytdt
+∞
−∞
=−∫
(1.13)
Now, let us look at a discrete-time random process {x(k)}. We can describe this
process from its autocorrelation function, at the instants k
1 and k 2, written R xx (k1, k2)
and expressed as
⎥
⎦
⎤
⎢
⎣
⎡
= )()(),(
2
*
121kxkxEkkR
xx 12
(, ) ,kk∀∈× ]] (1.14)
where
)(
2
*
kxdenotes the conjugate of )(
2kxin the case of complex processes.

34. Introduction to Signals and Systems 13

The covariance (or autocovariance) function C xx taken at instants k 1 and k 2 of the
process is shown by:
[] [ ]
⎥
⎦
⎤
⎢
⎣
⎡
−−= ))()(())()((),(
*
22
*
1121
kxEkxkxEkxΕkkC
xx
, (1.15)
where
[])(
1kxE indicates the statistical mean of )(
1kx.

We should keep in mind that, for zero-mean random processes, the
autocovariance and autocorrelation functions are equal.
),(),(
2121kkRkkC
xxxx=
12
(, )kk∀ . (1.16)
The correlation coefficient is as follows:
12
12
11 2 2
(, )
ρ(, )
(,) (, )
xx
xx
xx xx
Ckk
kk
CkkCkk
=

12
(, ) kk∀∈× ]]. (1.17)
It verifies:
12
ρ(, ) 1
xx
kk≤
12
(, ) kk∀∈× ]]. (1.18)
When the correlation coefficient
12
ρ(, )
xx
kk takes a high and positive value, the
values of the random processes at instants k
1 and k 2 have similar behaviors. This
means that the elevated values of x(k
1) correspond to the elevated values of x(k 2).
The same holds true for the lowered values k
1; the process takes the lowered values
of k
2. The more
12
ρ(, )
xx
kk tends toward zero, the lower the correlation. When
12
ρ(, )
xx
kk equals zero for all distinct values of k 1 and k 2, the values of the process
are termed decorrelated. If
12
ρ(, )
xx
kk becomes negative, x(k 1) and x(k 2) have
opposite signs.
In a more general situation, if we look at two random processes x(k) and y(k),
their intercorrelation function is written as:
⎥
⎦
⎤
⎢
⎣
⎡
= )()(),(
2
*
121kykxEkkR
xy (1.19)
As for the intercovariance function, it is shown by:
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−=
*
22
*
1121
)()()()(),(kyEkykxEkxEkkC
xy (1.20)

35. 14 Digital Filters Design for Signal and Image Processing
[] ()( )
*
212121
)()(,),(kyEkxEkkRkkC
xyxy −= (1.21)
The two random process are not correlated if
0),(
21=kkC
xy
),(
21kk∀ (1.22)
A process is called stationary to the 2
nd
order, even in a broad sense, if its
statistical mean
[]µ()Exk= is a constant and if its autocorrelation function only
depends on the gap between k
1 and k 2; that is, if:
)(),(
2121kkRkkR
xxxx−= . (1.23)
From this, in stationary processes, the autocorrelation process verifies two
conditions.

The first condition relates to symmetry. Given that:
⎥
⎦
⎤
⎢
⎣
⎡
+= )()()(
*
kxmkxEmR
xx (1.24)
we can easily show that:
)()(
*
mRmR
xxxx
=− m∀ ∈ }. (1.25)
For the second condition, we introduce the random vector x
consisting of M+1
samples of the process {x(k)}:
()
()
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
Mx
x
x #
0
. (1.26)
The autocorrelation matrix R
M is represented by
{}
H
Exx where
H
x indicates
the hermetian vector of.
H
x This is a Toeplitz matrix that is expressed in the
following form:
() ( ) ( ) ( )
() ( ) ( )
() ()
() ( ) () ()
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
−−
+−
−+−−
=
011
11
101
110
xxxxxxxx
xxxx
xxxxxx
xxxxxxxx
M
RRMRMR
RMR
MRRR
MRMRRR
R
"
%%
#%%%#
%
"
(1.27)

36. Introduction to Signals and Systems 15

NOTE.– vectoral and matricial approaches can often be employed in signal
processing. As well, using autocorrelation matrices and, more generally,
intercorrelation matrices, can be effective. This type of matrix plays a role in the
development of optimal filters, notably those of Wiener and Kalman. It is important
to implement decomposition techniques in signal and noise subspaces used for
spectral analysis, speech enhancement, determining the number of users in a
telecommunication cell, to mention a few usages.
1.3. Systems
A system carries out an operation chain, which consists of processing applied to
one or several input signals. It also provides one or several output signals. A system
is therefore characterized by several types of variables, described below:
– inputs: depending on the situation, we differentiate between the commands
(which are inputs that the user can change or manipulate) and the driving processes
or excitations which usually are not accessible;
– outputs;
– state variables that provide information on the “state” of the system. By the
term “state” we mean the minimal number of parameters, stored usually in a vector,
that can characterize the development of the system, where the inputs are supposed
to be known;
– mathematical equations that link input and output variables.

In much the same way as we classify signals, we speak of digital systems
(respectively analog) if the inputs and outputs are digital (respectively analog).

When we consider continuous physical systems, if we have two inputs and two
outputs, the system is then a quadrupole. We wish to impose a given variation law on
the output according to the input. If the relation between input and output is given in
the form of a differential linear equation with constant coefficients, we then have a
linear system that is time-invariant and continuous. Depending on the situation, we use
physical laws to develop equations; in electronics, for example, we employ
Kirchhoff’s laws and Thévenin’s and Norton’s theorems or others to establish our
equations.

Later in this text, we will discuss discrete-time systems in more detail. These are
systems that transform a discrete-time input signal x(k) into a discrete-time output
signal y(k) in the following manner:
()kx⇒() ()[]kxTky=. (1.28)

37. 16 Digital Filters Design for Signal and Image Processing
By way of example, we see that () ()kxky=, () ( )1−=kxkyand () ( )1+=kxky
respectively express the identity, the elementary delay and the elementary lead.
1.4. Properties of discrete-time systems
1.4.1. Invariant linear systems
The important features of a system are linearity, temporal shift invariance (or
invariance in time) and stability.

A system represented by the operator T is termed linear if
21
,xx∀
21
,aa∀ so
we get:
[][ ][ ] )()()()(
22112211kxTakxTakxakxaT+=+. (1.29)
A system is called time-invariant if the response to a delayed input of l samples
is the delayed output of l samples; that is:
()kx⇒() ()[]kxTky=, then ()[] ()lkylkxT−=− (1.30)
and this holds, whatever the input signal x (k) and the temporal shift l.

As well, a continuous linear system time-invariant system is always called a
stationary (or homogenous) linear filter.
1.4.2. Impulse responses and convolution products
If the input of a system is the impulse unity δ(k), the output is called the impulse
response of the system h (k), or:
() ()δhk T k=⎡⎤
⎣⎦. (1.31)

Figure 1.9. Impulse response

A usual property of the impulse δ(k) helps us describe any discrete-time signal as
the weighted sum of delayed pulses:
Linear filter δ(k) h(k)

38. Introduction to Signals and Systems 17

() ()( )∑
+∞
−∞=
−=
l
lklxkxδ (1.32)
The output of an invariant continuous linear system can therefore be expressed in
the following form:
() ()[] () ( )
() ( )
[] () ( ).∑∑
∑
∞+
−∞=
∞+
−∞=
+∞
−∞=
−=−=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−==
ll
l
lkhlxlkTlx
lklxTkxTkyδ
δ
(1.33)
The output y(k) thus corresponds to the convolution product between the input
x(k) and the impulse response h(k):
() () () () () ()( )∑
+∞
−∞=
−===
n
nkhnxkxkhkhkxky**. (1.34)
We see that the convolution relation has its own legitimacy; that is, it is not
obtained by a discretization of the convolution relation obtained in continuous
systems. Using the example of a continuous system, we need only two hypotheses to
establish this relation: those of invariance and linearity.
1.4.3. Causality
The impulse response filter h (k) is causal when the output y(k) remains null as
long as the input x(k) is null. This corresponds to the philosophical principle of
causality, which states that all precedent causes have consequences. An invariant
linear system is causal only if its output for every k instant (that is, y (k)), depends
solely on the present and past (x(k), x(k-1),… and so on).

Given the relation in equation (1.34), its impulse response satisfies the following
condition:
()0for 0hk k=< (1.35)
An impulse response filter h (k) is termed anti-causal when the impulse response
filter h(-k) is causal; that is, it becomes causal after inversion in the sense of time.
The output of rank k then depends only on the inputs that are superior, or equal to k.

39. 18 Digital Filters Design for Signal and Image Processing
1.4.4. Interconnections of discrete-time systems
Discrete-time systems can be interconnected either in cascade (series) or in
parallel to obtain new systems. These are represented, respectively, in Figures 1.10
and 1.11.

Figure 1.10. Interconnection in series

For interconnection in series, the impulse response of the resulting system h(k) is
represented by
() () () khkhkh
21*= . Thus, subject to the associativity of the law *,
we have:
() () ()
() () ()()
() () () () () () () ()
.*****
**
*
2112
12
2
kxkhkxkhkhkxkhkh
kxkhkh
kskhky
===
=
=


Figure 1.11. Interconnection in parallel

For a interconnection in parallel, the impulse response of the system h(k) is
written as
() () () khkhkh
21+= .

So we have:
() () ()
() () () ()
() ()
[] () () () .**
**
21
21
21
kxkhkxkhkh
kxkhkxkh
ksksky
=+=
+=
+=

y(k) x(k)
y(k) x(k) s(k)
h1(k) h2(k)
h1(k)
h2(k)
s1(k)
s
2(k)
+

40. Introduction to Signals and Systems 19

1.5. Bibliography
[JAC 86] JACKSON L. B., Digital Filters and Signal Processing, Kluwer Academic
Publishing, Boston, ISBN 0-89838-174-6. 1986.
[KAL 97] KALOUPTSIDIS N., Signal Processing Systems, Theory and Design, Wiley
Interscience, 1997, ISBN 0-471-11220-8.
[ORF 96] ORFANIDIS S. J., Introduction to Signal Processing, Prentice Hall, ISBN 0-13-
209172-0, 1996.
[PRO 92] PROAKIS J and MANOLAKIS D., Digital Signal Processing, Principles,
Algorithms and Applications, 2
nd
ed., MacMillan, 1992, ISBN 0-02-396815-X.
[SHE 99] SHENOI B. A., Magnitude and Delay Approximation of 1-D and 2-D Digital
Filters, Springer, 1999, ISBN 3-540-64161-0.
[THE 92] THERRIEN C., Discrete Random Signals and Statistical Signal Processing,
Prentice Hall, ISBN 0-13-852112-3, 1992.
[TRE 76] TREITTER S. A., Introduction to Discrete-Time Signal Processing, John Wiley &
Sons (Sd), 1976, ISBN 0-471-88760-9.
[VAN 89] VAN DEN ENDEN A. W. M. and VERHOECKX N. A. M., Discrete-Time Signal
Processing: An Introduction, pp. 173-177, Prentice Hall, 1989, ISBN 0-13-216755-7.

41. This page intentionally left blank

42. Chapter 2

Discrete System Analysis
2.1. Introduction
The study of discrete-time signals is based on the z-transform, which we will
discuss in this chapter. Its properties make it very useful for studying linear, time-
invariant systems.

This chapter is organized as follows. First, we will study discrete, invariant linear
systems based on the z-transform, which plays a role similar to that of the Laplace
transform in continuous systems. We will present the representation of this transform,
as well as its main properties; then we will discuss the inverse-z-transform. From a
given z-transform, we will present different methods of determining the corresponding
discrete-time signal. Lastly, the concepts of transfer functions and difference equations
will be covered. We also provide a table of z-transforms.
2.2. The z-transform
2.2.1. Representations and summaries
With analog systems, the Laplace transform X
s(s) related to a continuous
function x(t), is a function of a complex variable s and is represented by:
() ()
∫
+∞
∞−
−
= dtetxsX
st
s
. (2.1)


Chapter written by Mohamed NAJIM and Eric GRIVEL.

43. 22 Digital Filters Design for Signal and Image Processing
This variable exists when the real part of the complex variable s satisfies the
relation:
()RersR<< , (2.2)
with
and ,rR=−∞ =+∞ r and R potentially characterizing the existence of limits
of X
s(s) .

The Laplace transform helps resolve the linear differential equations to constant
coefficients by transforming them into algebraic products.

Similarly, we introduce the z-transform when studying discrete-time signals.

Let {x(k)} be a real sequence. The bilaterial or two-sided z-transform X
z(z) of the
sequence {x(k)} is represented as follows:
∑
+∞
−∞=
−
=
k
k
z
zkxzX)()(, (2.3)
where z is a complex variable. The relation (2.3) is sometimes called the direct z-
transform since this makes it possible to transform the time-domain signal { x(k)}
into a representation in the complex-plane.

The z-transform only exists for the values of z that enable the series to converge;
that is, for the value of z so that X
z(z) has a finite value. The set of all values of z
satisfying these properties is then called the region of convergence (ROC).

DEMONSTRATION 2.1.– we know that the absolute convergence of a series brings
about the basic convergence of the series. By applying the Cauchy criterion to the
series
∑
+∞
=0
)(
k
kx, the series ∑
+∞
=0
)(
k
kx absolutely converges if:
1)(lim
/1
<
+∞→
k
k
kx .
The series diverges if 1)(lim
/1
>
+∞→
k
k
kx . If 1)(lim
/1
=
+∞→
k
k
kx , we cannot be
certain of the convergence.

44. Discrete System Analysis 23

From this, let us express X
z(z) as follows:
∑∑∑
+∞
−
−
−∞=
−
+∞
−∞=
−
+==
0
1
)()()()(
k
k
k
k
k
z
zkxzkxzkxzX.
The series
∑
−
−∞=
−
1
)(
k
k
zkx converges absolutely if:
1)(lim
/1
<−
+∞→
k
k
k
zkx ,
or if:
k
k
kx
z
/1
)(lim
1
−
<
+∞→
.
As well, the series
∑
+∞
−
0
)(
k
zkx converges absolutely if:
1)(lim
/1
<
−
+∞→
k
k
k
zkx ,
or if:
zkx
k
k
<
+∞→
/1
)(lim
If we write
max
/1
)(lim
1
λ=
−
+∞→
k
k
kx
and
min
/1
)(limλ=
+∞→
k
k
kx , the z-transform
X
z(z) converges if:
min max
0 z≤λ < <λ .
The quantities
min
λ and
max
λ now characterize the region of convergence
(ROC) of the series X
z(z). The series ∑
+∞
−∞=
−
k
k
zkx)( diverges towards the strict
exterior of the ROC.

We should remember that the region of convergence may be empty, as is
sometimes the case where
()
2
() 1xk k=+ .

45. 24 Digital Filters Design for Signal and Image Processing
We can also represent, especially for causal sequences, the monolateral z-
transform,
)(zX
z, from the sequence {x(k)} with:
∑
+∞
=
−
=
0
)()(
k
k
z
zkxzX with
z≤
min
λ .
DEMONSTRATION 2.2.– to establish the absolute convergence of the series, we
can use another approach from the one previously shown with the bilateral transformation. It is based on d’Alembert’s law. We use this law to understand the
relation between two consecutive samples of the analyzed discrete-time signal.

We know that if the sequence
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
+
)(
)1(
kx
kx
converges towards a limit L that is
strictly inferior to 1, the absolute convergence of
∑
+∞
=
0
)(
k
kx is assured. If we apply
this test to the z-transform, we get:
1
)(
)1(
lim
)(
)1(
lim
1
1
<
+
=
+
−
+∞→
−
−−
+∞→
z
kx
kx
zkx
zkx
k
k
k
k
,
which gives us:
min
)(
)1(
lim
λ=
+
>
+∞→ kx
kx
zk

The ROC corresponds to all points in the complex-plane outside the central disk
of radius λ
min.

With discrete-time causal signals, such as:
()0=kxfor 0<k,
the one-sided (or unilateral) and the bilateral z-transforms are reduced to the same
expression:
∑∑
+∞
=
−
+∞
−∞=
−
==
0
)()()(
k
k
k
k
z
zkxzkxzX with
z≤
min
λ

46. Discrete System Analysis 25

Now let us look at two examples of z-transforms.

EXAMPLE 2.1.– the unit step signal u(k) can be represented as:
()0=ku for 0<k and ()1=ku for 0≥k.
Its z-transform is written
∑
+∞
=
−
=
0
)(
k
k
z
zzU. The convergence is assured for
1>z, and we get the closed form expression of the z-transform
11
1
)(
1
−
=
−
=
−
z
z
z
zU
z
with
1>z.

EXAMPLE 2.2.– here we assume that the signal x(k) is represented by:
()
k
kxα=with 1<α
We then get:
∑∑∑
−∞=
−−
+∞
=
−
+∞
−∞=
−
+==
0
1
)(
k
kk
k
kk
k
kk
z
zzzzXααα .
The absolute convergence of the series
∑
+∞
=
−
1
k
kk
zα and ∑
−∞=
−−
0k
kk
zα is assured for
α
α
1
<<z
. We then have:
111
1
1
1
1
)(
−−−
−
−
+
−
=
zz
z
zX
z
αα
α
and
α
α
1
<<z
.
When the signal is causal, we will obtain
()
k
kxα= for 0≥k and ()0=kx. Its
z-transform then equals: 1
1
1
)(
−
−
=
z
zX
z
α
with
z<α .

47. 26 Digital Filters Design for Signal and Image Processing

-10 -8 -6 -4 -2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
indices
amplitude
sequence






Figure 2.1. Representation of x(k)= α
│k│
and of the ROC of its z-transform X
z (z)

α α
1

48. Discrete System Analysis 27

-2 0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
indices
amplitude
sequence






Figure 2.2. Representation of the causal signal x(k)=α
│k│
u(k)
and of the ROC of its z-transform X
z (z)

α
α

49. 28 Digital Filters Design for Signal and Image Processing
2.2.2. Properties of the z-transform
2.2.2.1. Linearity
The z-transform is linear. Actually, with the two sequences
(){}kx
1 and
(){}kx
2.
21
,aa∀, we have:
⎥⎦
⎤
⎢⎣
⎡
Ζ+
⎥⎦
⎤
⎢⎣
⎡
Ζ=
⎥⎦
⎤
⎢⎣
⎡
+Ζ )()()()(
22112211
kxakxakxakxa (2.4)
where Z[.] represents the operator “z-transform”. This result is valid, provided the
intersection of the ROC is not empty.

DEMONSTRATION 2.3.–
[] []
[] []
)()(
)()(
)()()()(
2211
2211
22112211
kxakxa
zkxazkxa
zkxakxakxakxa
k
k
k
k
k
k
Ζ+Ζ=
+=
+=+Ζ
∑∑
∑
∞+
−∞=
−
∞+
−∞=
−
+∞
−∞=
−

The ROC of a sum of transforms then corresponds to the intersection of the
ROCs.

EXAMPLE 2.3.– the linearity property can be exploited in the calculation of the z-
transform of the discrete hyperbolic sinus x(k)=sh(k) u(k):
()[]
() ( )()
() ( )
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−=
−−=Ζ
∑∑
∑
∞+
=
−
∞+
=
−
+∞
=
−
00
0
expexp
2
1
expexp
2
1
k
k
k
k
k
k
zkzk
zkkksh

The ROC is represented by
1)1exp(
1
<
−
z and 1)1exp(
1
<−
−
z , so )1exp(>z.
()[]
()
()
21
1
11
121
12
)1exp(11
1
)1exp(11
1
2
1
−−
−
−−
+−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−−
−
−
=Ζ
zzch
zsh
zz
ksh

for |z| > exp(1).

50. Discrete System Analysis 29

2.2.2.2. Advanced and delayed operators
Let X
z (z) be the z-transform of the discrete-time signal {x(k)}. The z-transform
of
(){}mkx− is:
()[] ()[] ()zXzkxzmkx
z
mm −−=Ζ=−Ζ (2.5)
Delaying the signal by m steps thus brings about a multiplication by z
-m
in the z
domain. The operator z
-1
is called the basic delay operator, then simply the delay
operator. With filters, we often see the following representation:


Figure 2.3. Delayed unitary operator
Usually, the ROC is not modified, except potentially at origin and at infinity.

DEMONSTRATION 2.4.– by definition
()[] ()∑
+∞
−∞=
−
−=−Ζ
k
k
zmkxmkx. By
changing the variables
mkn−=, we get:
()[] ()
()
() () []kxzznxzznxmkx
m
n
nm
n
mn
Ζ===−Ζ
−
+∞
−∞=
−−
+∞
−∞=
+−
∑∑

Advancing the m signal leads to a multiplication by z
m
of the transform in the
domain of z. The operator z is called the advanced unitary operator or, more simply,
the advance operator. The following representation shows this.

Figure 2.4. Advanced unitary operator


EXAMPLE 2.4.– now we look at the z-transform of discrete-time exponential
signals
()
k
ekx
α−
= for k ≥ 0 and x(k) = 0 for k < 0 and y(k) = x(k-m) where m is a
natural integer.
z
()kx

)(zX
z
()1+kx

)(zzX
z
1−
z
()kx

)(zX
z
()1−kx

)(
1
zXz
z
−

51. 30 Digital Filters Design for Signal and Image Processing
()[]
1
1
1
−−
−
−
=Ζ=
ze
ezX
k
z
α
α
for
α
ez>
and
() ()
1
1
−−
−
−
−
==
ze
z
zXzzY
m
z
m
z
α
.
2.2.2.3. Convolution
We know that the convolution between two discrete causal sequences {x
1(k)}
and {x
2(k)} verifies the following relation:
() () () ( ) () ( ) ∑∑
=
+∞
=
−=−=
k
nn
nkxkxnkxkxkxkx
0
21
0
2121
* (2.6)
The z-transform of the convolution product of the two sequences is then the
simple product of the z-transforms of the two sequences:
() ()[] ()[] ()[]kxkxkxkx
2121* ΖΖ=Ζ (2.7)
The ROC of the convolution product is the intersection of the ROC of the z-
transforms of {x
1(k)} and {x 2 (k)}.

We see that this result is very often used in studying invariant linear systems,
since the response of a system corresponds, as we saw in equation (1.34), to the
convolution product of its impulse response by the input signal.

DEMONSTRATION 2.5.– since
()
11
0
()
k
k
Zxk xkz
+∞
−
=
=⎡⎤
⎣⎦∑
and ()
22
0
()
k
k
Zxk xkz
+∞
−
=
=⎡⎤
⎣⎦∑
,
the product
() ()zXzX
21 can be written as:
()[] ()[] () () () () () ()[]
()[] () ()[] kxkxZzkxx
zmkxmx
zmkxmx
zxxxxxxkxZkxZ
k
k
k
k
k
m
k
k
m
21
0
21
00
21
0
21
1
21212121
**
)()(
)()(
011000
==
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−=
+−++
++=
−
∞+
=
−
∞+
==
−
=
−
∑
∑∑
∑
""

on the condition that the intersection of the ROC of the two series must not be
empty.

52. Discrete System Analysis 31

2.2.2.4. Changing the z-scale
Let us assume that X
z(z) is the z-transform of the discrete-time signal {x(k)}.
With a given constant a, real or complex, the z-transform of the
()
{ }kxa
k
is:
() ⎟
⎠
⎞
⎜
⎝
⎛=
⎥
⎦
⎤
⎢
⎣
⎡
Ζ
−
zaXkxa
z
k 1 with
maxminλλaza≤≤* (2.8)
DEMONSTRATION 2.6.–
()[] () () () () zaXzakxzkxakxa
z
kk
k
kkk
11−
+∞
−∞=
+∞
−∞=
−
−−
∑∑
===Ζ
The ROC is then:
maxminλλaza≤≤
2.2.2.5. Contrasted signal development
Let X
z (z) be the z-transform of the discrete-time signal (){}kx with
maxmin
λλ<<z . We then represent the sequence as (){} (){} kxky−=. The z-
transform of
(){}ky then equals:
()
()
1−
=zXzY
zz . (2.9)
DEMONSTRATION 2.7.–
() ( ) () ()
1−
+∞
−∞=
+∞
−∞=
−
∑∑
==−= zXzkxzkxzY
z
kk
kk
z
The region of convergence is then written as:
minmax
11
λλ
<<z
2.2.2.6. Derivation of the z-transform
By deriving the z-transform in relation to z
-1
and then multiplying it by z
-1
, we
return to the following characteristic result:
()
()
()() () ()() kkxzkkxzkkxz
zd
zdX
z
kk
k
k
z
Ζ===∑∑
+∞
−∞=
+∞
−∞=
−
−
−−
−
−
1
11
1
1
(2.10)

53. 32 Digital Filters Design for Signal and Image Processing
EXAMPLE 2.5.– now we will at the z-transform of the following discrete-time
causal signal:
() () () () () 4123154335−+−=−+−=kkkkkkkx
δδδδ
We can easily demonstrate that the z-transform of δ(k) for all values of z equals
1. By using advanced and delayed linearity properties, we find that:
()()[]
43
354335
−−
+=−+−Ζzzkkδδ for all values of z.
From this
()
( )
43
1
43
1
1215
35
−−
−
−−
−
+=
+
= zz
dz
zzd
zzX
z

2.2.2.7. The sum theorem
If 1 is inside the ROC, we easily find that:
()zXkx
z
k
z∑
+∞
−∞=
→
=
1
lim)( (2.11)
2.2.2.8. The final-value theorem
Here we look at two sequences
(){}kx and (){}ky such as () ( ) () kxkxky−+=1,
by supposing the absolute convergence of the series
∑
+∞
−∞=k
ky)(.
From this we get the sum theorem
()zYky
z
k
z∑
+∞
−∞=
→
=
1
lim)(. Now, we know that
() ( ) ()zXzzY
zz1−= , and, by construction, )(lim)(lim)(kxkxky
k
k
k −∞→
+∞
−∞=
+∞→
−=∑
.
From there, if
0)(lim=
−∞→
kx
k
, we have ()() zXzkx
z
zk
1lim)(lim
1
−=
→+∞→
.
2.2.2.9. Complex conjugation
Here we consider the two sequences
(){}kx and (){}ky such as () ()
*
kxky=
() () ()( ) ()[]
*
*
*
*
* zXzkxzkxzY
z
kk
kk
z
=
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
===∑∑
∞+
−∞=
∞+
−∞=
−−
(2.12)

54. Discrete System Analysis 33

2.2.2.10. Parseval’s theorem
()
∫ ∑
+∞
=
−−
=
C k
zz
kxdzzzXzX
j
0
211
)()(
2
1
π
(2.13)
provided that X
z(z) converges on an open ring containing the circle unity. The energy
does not depend on the representation mode, whether it is temporal or in the z-domain.
2.2.3. Table of standard transform
(){}kx )(zX
z
()δk 1
()kmδ−
m
z
−

()
()
()
0for 0
1for 0
xk k
uk
xk k
⎧ =<⎪
⎨
=≥⎪⎩
1−z
z

() ()kkukx=
()
2
1−z
z

() ()kukkx
2
=
()
3
2
1−
+
z
zz

() ()kukkx
3
=
( )
()
4
2
1
14
−
++
z
zzz

() ()kukkx
4
=
( )
()
5
23
1
11111
−
+++
z
zzzz

()α
k
xk= with α1<.
1
() 1
α1 α
Xz
zzα
=+ +
−−
() ()α
k
xkkuk=
()
2
α
α
z
z−

() ()
2
α
k
xkk uk=
()
()
3
αα
α
zz
z
+
−

() ()
3
α
k
xkk uk=
()
()
22
4
α 4αα
α
zz z
z
++
−

55. Another Random Document on
Scribd Without Any Related Topics

59. The Project Gutenberg eBook of Wanderings in
Ireland

60. This ebook is for the use of anyone anywhere in the United
States and most other parts of the world at no cost and with
almost no restrictions whatsoever. You may copy it, give it away
or re-use it under the terms of the Project Gutenberg License
included with this ebook or online at www.gutenberg.org. If you
are not located in the United States, you will have to check the
laws of the country where you are located before using this
eBook.
Title: Wanderings in Ireland
Author: Michael Myers Shoemaker
Release date: October 29, 2013 [eBook #44066]
Most recently updated: October 23, 2024
Language: English
Credits: Produced by Matthias Grammel, Ann Jury and the
Online
Distributed Proofreading Team at http://www.pgdp.net
(This
file was produced from images generously made
available
by The Internet Archive)
*** START OF THE PROJECT GUTENBERG EBOOK WANDERINGS IN
IRELAND ***

61. By M. M. SHOEMAKER
ISLANDS OF THE SOUTHERN SEAS
 With 80 Illustrations. Second Edition. Large 8vo. Gilt top  $2.25
QUAINT CORNERS OF ANCIENT EMPIRES
 With 47 Illustrations. Large 8vo. Gilt top  $2.25
THE GREAT SIBERIAN RAILWAY FROM PETERSBURG TO
PEKING
 With 30 Illustrations and a Map. Large 8vo net,$2.00
THE HEART OF THE ORIENT
 With 52 Illustrations. Large 8vo net,$2.50
WINGED WHEELS IN FRANCE
 With about 60 Illustrations. Large 8vo net,$2.50
WANDERINGS IN IRELAND
 With 72 Illustrations. Large 8vo net, 
PALACES AND PRISONS OF MARY QUEEN OF SCOTS
 
With about 60 Illustrations. Large 8vo
Large Paper Edition. 4
o
net,
net,
$5.00
$12.00
G. P. PUTNAM'S SONS
New York London

62. "The Harp of Erin"
From the original painting by T. Buchanan Read in possession
of the author

63. WANDERINGS
IN
IRELAND
BY
MICHAEL MYERS SHOEMAKER
Author of "Islands of the Southern Seas,"
"Winged Wheels in France," etc.
Illustrated
G. P. PUTNAM'S SONS
NEW YORK AND LONDON
The Knickerbocker Press
1908
Copyêight , 1908
BY

64. MICHAEL MYERS SHOEMAKER
The Knickerbocker Press, New York
TO MY AUNT
ANNA L. SHOEMAKER
THESE NOTES ARE AFFECTIONATELY
DEDICATED

65. PREFACE
Are you minded for a jaunt through the island of Erin where tears
and smiles are near related and sobs and laughter go hand in hand?
We will walk, and will take it in donkey-cart and jaunting-car—by
train and in motor-cars—and if you suit yourself you will suit me.
Leaving Dublin we will circle northward, with a visit to Tanderagee
Castle and the tomb of St. Patrick—God bless him,—then on past the
Causeway and down to Derry, and so into the County of Mayo, where
in the midst of a fair you will encounter the wildest "Konfusion" and
will be introduced to the gentleman who pays the rent.
In the silence and solitudes of the island of Achill you will see tears
and hear sobs as you listen to the keening for the dead. Near the
island of Clare, Queen Grace O'Malley will almost order you away, as
she did her husband, and your motor with all its wings out will roll
through the grand scenery of the western coast—now down by the
ocean and then far up amidst the sombre mountains—Kylemore
Castle and quaint Galway, Leap Castle—ghost-haunted—and moated
Ffranckfort, Holy Cross and the Rock of Cashel—will pass in stately
array and be succeeded by a glimpse of army life at Buttevant, and a
dinner at Doneraile Court, where you will hear of the only woman
Free Mason. Killarney will follow with its music and legends, and Cork
and Fermoy, and so on and into the County of Wexford, where you
will rush through the lanes and byways and will scare many old
ladies—driving as many donkeys—almost into Kingdom Come. You
will be welcomed at Bannow House and entertained in that quaintest
of all earthly dwellings, "Tintern Abbey," which was a ruin when the
family moved into it more than three centuries ago. You will visit the
buried city of Bannow and pass on to where Moore watched the
"Meeting of the Waters." You will visit in stately mansions, and go
with a wild rush to the races at the Curragh. At Jigginstown House

66. you will be reminded of the cowardice of a king, and as you bid
farewell to Ireland you will lay a wreath on the grave of Daniel
O'Connell,—all this and much more if you are so minded.
M. M. S.   
   Union Club, New Yoêk, January 1, 1908.

67. CONTENTS
PAGE
CHAPTER I
Welcome to Ireland. Quaint People of Dublin.
Packing the Motors. Departure. Tara Hill. Its
History and Legends. Ruins at Trim. Tombs
of the Druids. Battle-field of the Boyne.
1
CHAPTER II
Through Newry to Tanderagee Castle. Life in
the Castle. Excursions to Armagh. Its History.
The English in Armagh.
15
CHAPTER III
Through Newcastle to Downpatrick. Grave of
St. Patrick. His Life and Work. The Old Grave
Digger. Belfast and Ballygalley Bay.
O'Halloran, the Outlaw.
25
CHAPTER IV
Ballycastle to the Causeway. Prosperity of
Northern Ireland. Bundoran. Gay Life in
County Mayo. Mantua House. Troubles in
Roscommon. Wit of the People. Irish Girls.
Emigration to America. Episode of the Horse.
People of the Hills. Chats by the Wayside.
Mallaranny.
34
CHAPTER V

68. The Island of Achill. Picturesque Scenery.
Poverty of the People. "Keening" for the
Dead. "The Gintleman who pays the Rint."
Superstitious Legends.
53
CHAPTER VI
Monastery of Burrishoole. Queen Grace
O'Malley and her Castle of Carrig-a-Hooly.
Her Appearance at Elizabeth's Court.
Dismissal of her Husband. Wild Scenery of
the West Coast. The Ancient Tongue.
Recess. Kylemore Castle. Crazy Biddy.
77
CHAPTER VII
The Ancient City of Galway. Quaint People.
Curious Houses. Vile Hotel. Parsonstown.
Wingfield House. Leap Castle, and its
Ghosts. Ffranckfort Castle. Clonmacnoise.
Holy Cross Abbey.
94
CHAPTER VIII
The Rock of Cashel. Its Cathedral, Palace, and
Round Tower—Its History and Legends.
Kilmalloch, its Ruins and History. The
Desmonds. Horse Fair at Buttevant.
119
CHAPTER IX
Buttevant Barracks. Army Life. Mess-room Talk.
Condition of the Barracks. Balleybeg Abbey.
Old Church. Native Wedding. Kilcoman
Castle, Spenser's Home. Doneraile Court.
Mrs. Aldworth, the only Woman Freemason.
Irish Wit. Regimental Plate. Departure from
the Barracks.
132

69. CHAPTER X
Route to Killarney. Country Estates. Singular
Customs. Picturesque Squalor. Peace of the
Lakes. Innisfallen. The Legend of "Abbot
Augustine." His Grave. "Dennis," the
"Buttons," and his Family Affairs. Motors in
the Gap of Dunloe.
161
CHAPTER XI
Kenmare and Herbert Demesnes. Old Woman
at the Gates. Route to Glengariff. Bantry Bay.
Boggeragh Mountains. Duishane Castle. The
Carrig-a-pooka and its Legend. Macroom
Castle and William Penn. Cork. Imperial
Hotel. "Ticklesome" Car Boy. The Races and
my Brown Hat. Route to Fermoy. Breakdown.
Clonmel and its "Royal Irish." Ride to
Waterford.
170
CHAPTER XII
Ancient Waterford. History. Reginald's Tower.
Franciscan Friary. Dunbrody Abbey. New
Ross. Bannow House. Its "Grey Lady."
Legend of the Wood Pigeon. Ancient Garden.
Buried City of Bannow. Dancing on the
Tombs. Donkeys and Old Women. Tintern
Abbey and its Occupants. Quaint Rooms and
Quainter Stories. Its History and Legends.
The Dead man on the Dinner Table. The
Secret of the Walls. The Illuminated
Parchment. The Sealed Library. Ruined
Chapel. King Charles's Clothes. Is History
False or True?
181

70. CHAPTER XIII
Return to Ireland. Illness. Conditions on the
Great Liners. The Quay at Cork "of a
Saturday Evening." En route once more. The
Old Lady and the Donkey. Barracks at
Fermoy. Killshening House, Abandoned Seat
of the Roche Family. Fethard. Quaint
Customs. The Man in the Coffin.
"Curraghmore House" and its Great Kennels.
Its Legends, Ghosts, and History. Lady
Waterford. Oliver Cromwell at the Castle.
The Marquis in the Dungeon.
209
CHAPTER XIV
Departure from Fethard. A Dead Horse and a
Lawsuit. Approach to Dublin. Estate of
Kilruddery. The Swan as a Fighter.
Glendalough, its Ruins and History. Tom
Moore and his Tree in Ovoca. Advantages of
Motor Travel. Superstition of the Magpie. A
Boy, a Cart, and a Black Sheep. The Goose
and the Motor.
225
CHAPTER XV
The Lunatic. Insanity and its Causes in Ireland.
The Usual Old Lady and Donkey. Sunshine
and Shadow. Clonmines and its Seven
Churches. The Crosses around the Holy Tree.
Baginbun and the Landing of the English.
The Bull of Pope Adrian. Letter of Pope
Alexander. Protest of the Irish Princes.
Legends. Death of Henry II.
243
CHAPTER XVI

71. Wild Times in Ireland. Landlord and Tenant.
Evictions. Boycott at Bannow House. The
Parson and the Legacy. The Priest and the
Whipping. Burial in Cement. Departure from
Bannow House. Kilkenny and her Cats. The
Mountains of Wicklow. Powerscourt and a
Week-End. Run to Dublin and an Encounter
by the Way. The Irish Constabulary. Motor
Runs in the Mountains. Lord H——.
260
CHAPTER XVII
Dublin. Derby Day and the Rush to the
Curragh. An Irish Crowd. The Kildare Street
Club and Club Life. Jigginstown House and
its History. The Cowardice of a King. The Old
Woman on the Tram Car. Parnell. The Grave
of Daniel O'Connell.
276

72. ILLUSTRATIONS
PAGE
The Haêp of Eêin Frontispiece
 
From the original painting by T.
Buchanan Read, in the
possession of the author
 
Statue of St. Patêick on the
Hill of Taêa 4
Castle of King John at Têim 8
Monument on the Battle-field
of the Boyne 12
Tandeêagee Castle, Iêish Seat
of the Duke of Manchesteê 16
Chapel, Tandeêagee Castle 20
Dêawing-êoom, Tandeêagee
Castle 24
Teêêace , Tandeêagee Castle 28
Tomb of St. Patêick at
Downpatêick 32
A Cabin in the Noêth 36
A Woman of the Noêth 40
Mantua House, Roscommon 44
Ballina, a Typical Iêish Town 48
A Glimpse of Achill 52
Slieîemoêe Mountain, and
Dugoêt, Achill 56
Fisheêfolk of Achill 60
A Lonely Road in Connemaêa 64
Kylemoêe Castle, Connemaêa 68

73. Cêazy Biddy 72
The Lynch House, Galway 76
Abbey of St. Dominick , Loêêha,
Ancient Buêial-place of the
Caêêolls 80
Leap Castle, Couêt Side 84
Leap Castle, Paêk Side 88
Moat of Ffêanckfoêt Castle 92
Ffêanckfoêt Castle 96
Clonmacnoise 100
Abbey of the Holy Cêoss 104
Rock of Cashel 108
Coêmac's Chapel, Cashel 112
Cêoss of Cashel, and Thêone of
the Kings
of Munsteê 116
Ancient Gateway, Kilmalloch 120
Dominican Abbey, Kilmalloch 124
Butteîant Baêêacks 128
Dinneê, Butteîant Baêêacks 132
Butteîant, County Coêk 136
Kilcoman Castle, Spenseê 's
Home 140
Doneêaile Couêt, County Coêk 144
Room in Doneêaile Couêt wheêe
Mês. Aldwoêth
Hid 148
The Hon. Mês. Aldwoêth, the
only Woman
Fêeemason 152
The Lake, Doneêaile Paêk 156
Mallow Castle, County Coêk 160
Iêish Cottage, County Keêêy 164

74. Chapel of St. Finian the Lepeê,
Innisfallen 168
Têee oîeê the Abbot's Gêaîe,
Innisfallen 172
Uppeê Lake, Killaêney 176
"Dinnis," Hotel Victoêia 180
The Route to Glengaêiff 184
Caêêig-a-pooka Castle 188
Macêoom Castle 192
Reginald 's Toweê, Wateêfoêd 196
Fêanciscan Fêiaêy, Wateêfoêd 200
Dunbêody Abbey, County
Wexfoêd 204
Bannow House, County
Wexfoêd 208
Teêêace , Bannow House,
County Wexfoêd 212
Coêneê of the Rose Gaêden,
Bannow House,
County Wexfoêd 216
Bannow Chuêch, County
Wexfoêd 220
Tombs in Bannow Chuêch 224
Tinteên Abbey, County
Wexfoêd 228
Kilkenny Castle 232
Deseêted Killshening House,
Feêmoy 236
Cuêêaghmoêe House, Maêquis
of Wateêfoêd 240
Hallway, Cuêêaghmoêe House 244
Dining-êoom, Cuêêaghmoêe
House 248

75. Kilêuddeêy House, Eaêl of
Meath 252
Glendalough 256
Tom Mooêe's Têee, Vale of
Oîoca 260
One of the Seîen Chuêches ,
Clonmines 264
Funeêal Cêosses by the
Wayside, County
Wexfoêd 268
Poweêscouêt House 272
Gêeat Salon, Poweêscouêt
House 276
Ruins of Jigginstown House,
Eaêl of Stêaffoêd 280
Paênell 's Gêaîe, Glasneîin
Cemeteêy,
Dublin 284
Daniel O'Connell 's Monument ,
Glasneîin
Cemeteêy, Dublin 288

76. WANDERINGS IN IRELAND

77. CHAPTER I
Welcome to Ireland—Quaint People of Dublin—Packing the Motor and
Departure—Tara Hill; its History and Legends—Ruins at Trim—Tombs of the
Druids—Battle-field of the Boyne.
"Glory be to God, but yer honour is welcome to Ireland."
An old traveller understands that it is the unexpected which makes
the joy of his days. I had come to Europe with the intention of
spending some conventional weeks in London, followed by an auto
tour with the family through the fair land of France. Fate brings me,
upon my first day in town, to Prince's Restaurant, when out of the
chaos of faces before me rises one whose owner, a son of Erin whom
I had last seen under the cherry blossoms of Japan, advances upon
me. Then the conventional promptly drops off and away, and it is but
a short while before a motor tour is arranged in the Emerald Isle, a
month to be passed amidst its beauties and miseries, its mirth and
its sadness, for all go in one grand company in the land of St.
Patrick.
With Boyse of Bannow I shall follow the fancy of the moment, which
to my thinking is the only true mode of travel.
"Du Cros" has agreed to furnish a perfectly new Panhard for and
upon the same terms which I received in France last year, viz., thirty
pounds sterling per week, and everything found except the board
and lodging of the chauffeur. These very necessary details arranged
we are impatient to be off and leave London on a hot day in June.
The smells, dirt, and dust of her wooden streets, driven in clouds
over all the grand old city, follow us far out into the green meadows
of England until we ask whether the hawthorn blossoms have ever
held any fragrance, and have we not been mistaken as to roses. But
London is not all of England, and we are finally well beyond her
influence and wondering why we remained within her limits with the

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