![Radio Navigation Signals
output, or to the digitized signal obtained after (optional) downconversion and
filtering of the received signal.
The broadcast navigation signal structure is known to the receiver and will be
characterized in the following paragraphs. A good way to do this is to merge the
frontend filter into the output filter (i.e., into a combined filter) and to describe the
signal after the output filter. This simplification is useful for signal processing pur-
poses and is valid, because both antennas, the propagation path, and the frontend
filter act as linear systems to the signal. Nonlinear effects of the amplifiers or mixer
are not considered.
Without the combined filter (or output filter), the transmitted signal rS is writ-
ten as
( ) ( ) ( )cos(2 )
S S RF
r t a d t c t f t
π
¥
= (1.1)
Here, fRF is the carrier frequency in hertz, c (t) is the infinite bandwidth signal
representation at baseband, and d(t) represents a broadcast navigation message.
The transmitted signal is described in the signal-in-space (SIS) interface control doc-
ument (ICD) [1, 2]. The symbol aS denotes the signal amplitude in arbitrary units.
Other signals might be transmitted on the same carrier frequency by the same
satellite. For example, the Global Positioning System (GPS) C/A signal is broadcast
in phase quadrature with the P(Y) signal on the L1 (1,575.42 MHz) carrier fre-
quency. The access to the different signals is controlled via c (t). Different wave-
forms of c∞(t) can be used to realize code division multiple access (CDMA), time
division multiple access (TDMA), or—by including a carrier into c (t)—frequency
division multiple access (FDMA) schemes.
With the combined filter, the signal is mathematically written as
= - +
( ) ( ) ( )sin(2 ) ( ) ( )cos(2 )
S S Q RF S I RF
r t a d t c t f t a d t c t f t
π π (1.2)
with
( ) ( ) ( )
I Q
c t c t ic t
= + (1.3)
representing the filtered baseband representation of the signal obtained by applying
(1.10). The signal cQ(t) is an artifact generated by the filter. The filter does not affect
d(t), as long as the data rate (e.g., 50 bit/s) is much smaller than the filter bandwidth
(e.g., 2 MHz). If the radio frequency (RF) is much higher than the filter bandwidth,
cQ(t) is small and can be ignored.
Using a complex notation, the signal after the combined filter is given as
( ) Re{ ( ) ( )exp(2 )}
S S RF
r t a d t c t if t
π
= (1.4)
1.2 Signal Propagation
According to Fermat’s principle, the line-of-sight component of the signal rS(t)
travels along the shortest propagation path from the transmitter to the receiver.
Eventually, one or more reflected signals [i.e., copies of rS(t)] are superimposed with
the line-of-sight signal. Each copy has a different amplitude, Doppler, carrier phase](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-22-320.jpg)
![1.3 Signal Conditioning
delay, or code phase delay. Furthermore, the line-of-sight component is delayed by
the atmosphere. Accurate modeling of the propagation effects is one of the key ele-
ments to obtain a precise position and this topic is covered in many satellite navi-
gation text books [3, 4]. We will not go into the modeling details, but will instead
focus on a model for the received signal that can be used as a basis for delay and
Doppler estimation.
Delays affect the carrier and the data-modulating signals c(t), d(t) differently.
Frequency-independent (nondispersive) delays are caused by the geometric distance
and by the electrically neutral part of the atmosphere (troposphere). Frequency-
dependent (dispersive) delays are caused by the ionosphere and by the receiver and
transmitter hardware. Dispersive and nondispersive delays add up and result in the
group delay τG, delaying c(t) and d(t). The carrier is delayed by the phase delay τP.
The delays τG and τP are not equal because the ionospheric delay contributes to
each of them with a different sign. The difference is typically less than 100m and
varies with a frequency not more than 0.05 Hz if no scintillations are present. Also,
hardware group and phase delay are generally different.
The group delay, τG, affects c(t) via
,0
( ) ( ( )) ( )
G D G
c t c t t c t
τ α τ
® - » - (1.5)
and similarly for d(t). The phase delay, τP, affects the carrier via
,0
exp(2 ) exp(2 ( ( ))) exp(2 ( ))
RF RF P RF D P
if t if t t if t
π π τ π α τ
® - » - (1.6)
with
/ ,0 / 0 0
( ) (1 )
G P G P D
t t
τ τ α
= - - (1.7)
The coefficient αD
1
D
v
c
α = - (1.8)
is the Doppler effect, caused by the change in the group/phase delay, expressed as
velocity v in meters per second. The linearization is carried out around the epoch t0.
If the signal duration Tcoh under consideration is short, the Doppler effect on c(t),
d(t) can be ignored, as in
,0 0
( ) ( ( ))
coh D G
c
T c t c t t
vB
α τ τ
Þ - » - (1.9)
and similarly for d(t), where B denotes the signal (or data message) bandwidth in
hertz.
1.3 Signal Conditioning
After signal reception by the antenna, the signal is amplified, filtered, and eventually
downconverted. Amplification changes the amplitude of the signal (from aS to a),
but leaves the signal structure invariant.](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-23-320.jpg)
![1.5 Sampling
pulsed signals are included because they are being used by pseudolites, by LORAN-C,
or RADAR-like ranging systems. In principle, the theory can also be adapted for
sonar ranging systems and, with some limitations, for optical ranging systems. Only
one frequency band is considered (e.g., GPS L1); a generalization from a sampling
of one frequency band to multiple frequency bands is obvious, but one should be
take care that all bands are sampled synchronously.
No assumption on the relation of the signal bandwidth to the sampling rate is
made and, in particular, sub-Nyquist sampling rates can be used. No assumption on
the modulation scheme is made as long as the received signal waveform at baseband
c(t) is known a priori to the receiver. Eventually, filters influence the theory via the
waveform c(t). The more narrow the filter bandwidth, the smoother the waveform
will be.
The filter bandwidth and characteristics define how much noise power is be-
ing received: the wider the bandwidth, the higher the noise power. For simplicity, a
unity noise power is assumed in (1.16) and it is important to keep in mind that only
ratios between power levels have meaningful values, as described in Section 1.8.1.
In that sense, (1.16) defines the power scale.
The Nyquist criterion does not fully apply because the waveform c(t) is known
to the receiver beforehand. Consequently, there is no need to reconstruct the signal
waveform from the received samples [5]. As shown in Section 6.5, a good choice
for the sample rate is exactly equal to the Nyquist rate (e.g., being equal to the noise
bandwidth). Lower sample rates yield less-independent signal samples, thereby gen-
erally decreasing the accuracy of the obtained estimates. The accuracy decrease can
be modeled as an effective signal power loss. This observation is also true when
multiple reflections of the same signal are received.
In the rest of this chapter, we will formulate generic conditions for navigations
signals. Later, we illustrate the conditions with two GNSS signals and two pulsed
terrestrial navigation signals. In addition to those parameters, the number of re-
ceived signals, the amplitude, and the geometric placement of the transmitters affect
the positioning result (see Sections 4.1.3 and 8.4).
1.5 Sampling
In the following, a number of L signal samples are considered, indexed by m. The
index m assumes, in general, the values
{ }
1, ,L
µ Î … (1.13)
The sampling epoch tm in seconds for the sample m is given by
s
t
f
µ
µ
= (1.14)
and fs defines the ADC sample rate in samples per second. Optionally, the sampling
epochs can be offset by a fixed amount of time, which will not explicitly be men-
tioned here.
For each sampling epoch tm, a complex-valued (i.e., I plus Q) signal sample is
generated. We chose a complex signal representation to work with more compact](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-25-320.jpg)
![passed away without any arrest, trial, or tidings, most of the curious had
left, and only a few more persevering remained, past the hour of afternoon
recreation in the public gardens. But just before the captive arrived, a fresh
knot of spectators came in, and stood near one of the side-doors, from
which they could see all.
As Corvinus had prepared his father for what he was to expect, Tertullus,
moved with some compassion, and imagining there could be little difficulty
in overcoming the obstinacy of a poor, ignorant, blind beggar, requested the
spectators to remain perfectly still, that he might try his persuasion on her,
alone, as she would imagine, with him; and he threatened heavy penalties
on any one who should presume to break the silence.
It was as he had calculated. Cæcilia knew not that any one else was
there, as the prefect thus kindly addressed her:
“What is thy name, child?”
“Cæcilia.”
“It is a noble name; hast thou it from thy family?”
“No; I am not noble; except because my parents, though poor, died for
Christ. As I am blind, those who took care of me called me Cæca,[153] and
then, out of kindness, softened it into Cæcilia.”
“But now, give up all this folly of the Christians, who have kept thee
only poor and blind. Honor the decrees of the divine emperors, and offer
sacrifice to the gods; and thou shalt have riches, and fine clothes, and good
fare; and the best physicians shall try to restore thee thy sight.”
“You must have better motives to propose to me than these; for the very
things for which I most thank God and His Divine Son, are those which you
would have me put away.”
“How dost thou mean?”
“I thank God that I am poor and meanly clad, and fare not daintily;
because by all these things I am the more like Jesus Christ, my only
Spouse.”
“Foolish girl!” interrupted the judge, losing patience a little; “hast thou
learnt all these silly delusions already? at least thou canst not thank thy God
that He has made thee sightless.”
“For that, more than all the rest, I thank Him daily and hourly with all
my heart.”](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-29-320.jpg)
![turn would have done, sufficed to inflict an excruciating, or more truly, a
racking pain, through all her frame. Far more grievous was this, from the
preparation and the cause of it being unseen, and from that additional
suffering which darkness inflicts. A quivering of her features and a sudden
paleness alone gave evidence of her torture.
“Ha! ha!” the judge exclaimed, “thou feelest that? Come, let it suffice;
obey, and thou shalt be freed.”
She seemed to take no heed of his words, but gave vent to her feelings in
prayer: “I thank Thee, O Lord Jesus Christ, that Thou hast made me suffer
pain the first time for Thy sake. I have loved Thee in peace; I have loved
Thee in comfort; I have loved Thee in joy,—and now in pain I love Thee
still more. How much sweeter it is to be like Thee, stretched upon Thy
Cross, even than resting upon the hard couch at the poor man’s table!”
“Thou triflest with me,” exclaimed the judge, thoroughly vexed, “and
makest light of my lenity. We will try something stronger. Here, Catulus,
apply a lighted torch to her sides.”[154]
The Martyr Cæcilia.
A thrill of disgust and horror ran through the assembly, which could not
help sympathizing with the poor blind creature. A murmur of suppressed
indignation broke out from all sides of the hall.
Cæcilia, for the first time, learnt that she was in the midst of a crowd. A
crimson glow of modesty rushed into her brow, her face, and neck, just](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-31-320.jpg)
![before white as marble. The angry judge checked the rising gush of feeling;
and all listened in silence, as she spoke again, with warmer earnestness than
before:
“O my dear Lord and Spouse! I have been ever true and faithful to Thee!
Let me suffer pain and torture for Thee; but spare me confusion from
human eyes. Let me come to Thee at once; not covering my face with my
hands in shame when I stand before Thee.”
Another muttering of compassion was heard.
“Catulus!” shouted the baffled judge in fury; “do your duty, sirrah! what
are you about, fumbling all day with that torch?”
The executioner advanced, and stretched forth his hand to her robe, to
withdraw it for the torture; but he drew back, and, turning to the prefect,
exclaimed in softened accents:
“It is too late. She is dead!”
“Dead!” cried out Tertullus; “dead with one turn of the wheel?
impossible!”
Catulus gave the rack a turn backwards, and the body remained
motionless. It was true; she had passed from the rack to the throne, from the
scowl of the judge’s countenance to her Spouse’s welcoming embrace. Had
she breathed out her pure soul, as a sweet perfume, in the incense of her
prayer? or had her heart been unable to get back its blood, from the
intensity of that first virginal blush?[155]
In the stillness of awe and wonder, a clear bold voice cried out, from the
group near the door: “Impious tyrant, dost thou not see, that a poor blind
Christian hath more power over life and death, than thou or thy cruel
masters?”
“What! a third time in twenty-four hours wilt thou dare to cross my
path? This time thou shalt not escape.”
These were Corvinus’s words, garnished with a furious imprecation, as
he rushed from his father’s side round the enclosure before the tribunal,
towards the group. But as he ran blindly on, he struck against an officer of
herculean build, who, no doubt quite accidentally, was advancing from it.
He reeled, and the soldier caught hold of him, saying:
“You are not hurt, I hope, Corvinus?”
“No, no; let me go, Quadratus, let me go.”](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-32-320.jpg)
![again upon his ear. He raised himself up; it was becoming distinct. So sweet
it sounded, so like a chorus of angelic voices, but in another sphere, that he
said to himself: “Who would have thought that Heaven was so near to hell!
Or are they accompanying the fearful Judge to try me?”
And now a faint glimmer of light appeared at the same distance as the
sounds; and the words of the strain were clearly heard:
“In pace, in idipsum, dormiam et requiescam.”[156]
“Those words are not for me. They might do at a martyr’s entombment;
they cannot at a reprobate’s burial.”
The light increased; it was like a dawn glowing into day; it entered the
gallery and passed across it, bearing in it, as in a mirror, a vision too distinct
to be unreal. First, there came virgins robed and holding lamps; then four
who carried between them a form wrapped up in a white linen cloth, with a
crown of thorns upon the head; after them the youthful acolyte Tarcisius
bearing a censer steaming with perfumed smoke; and, after others of the
clergy, the venerable Pontiff himself, attended by Reparatus, and another
deacon. Diogenes and his sons, with sorrowful countenances, and many
others, among whom he could distinguish Sebastian, closed the procession.
As many bore lamps or tapers, the figures seemed to move in an
unchanging atmosphere of mildest light.
And as they passed before him, they chanted the next verse of the psalm:
“Quoniam Tu Domine singulariter in spe constituisti me.”[157]
“That,” he exclaimed, rousing himself up, “that is for me.”
With this thought he had sprung upon his knees; and by an instinct of
grace words which he had before heard came back to him like an echo;
words suited to the moment; words which he felt that he must speak. He
crept forward, faint and feeble, turned along the gallery through which the
funeral procession was passing, and followed it, unobserved, at a distance.
It entered a chamber and lighted it up, so that a picture](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-40-320.jpg)
![“Father, I have sinned before heaven, and against Thee, and I am not
worthy to be called Thy child.”
The Pontiff raised him up kindly, and pressed him to his bosom, saying,
“Welcome back, my son, whoever thou art, to thy Father’s house. But thou
art weak and faint, and needest rest.”
Some refreshment was immediately procured. But Torquatus would not
rest till he had publicly avowed the whole of his guilt, including the day’s
crimes; for it was still the evening of the same day. All rejoiced at the
prodigal’s return, at the lost sheep’s recovery. Agnes looked up to heaven
from her last affectionate glance on the blind virgin’s shroud, and thought
that she could almost see her seated at the feet of her Spouse, smiling, with
her eyes wide open, as she cast down a handful of flowers on the head of
the penitent, the first-fruits of her intercession in heaven.
Jesus cures the Blind
Man, from a picture in
the Cemetery of St.
Domitilla.
Diogenes and his sons took charge of him. An humble lodging was
procured for him, in a Christian cottage near, that he might not be within the
reach of temptation, or of vengeance, and he was enrolled in the class of
penitents, where years of expiation, shortened by the intercession of
confessors—that is, future martyrs—would prepare him for full re-
admission to the privileges he had forfeited.[158]
C H A P T E R X I X .
TWOFOLD REVENGE.](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-42-320.jpg)
![EBASTIAN’S visit to the cemetery had been not merely
to take thither for sepulture the relics of the first martyr,
but also to consult with Marcellinus about his safety. His
life was too valuable to the Church to be sacrificed so
early, and Sebastian knew how eagerly it was sought.
Torquatus now confirmed this, by communicating
Fulvius’s designs, and the motive of his attendance at the
December ordination. The usual papal residence was no
longer safe; and a bold idea had been adopted by the
courageous soldier,—the “Protector of the Christians,” as
his acts tell us he had been authoritatively called. It was to lodge the Pontiff
where no one could suspect him to be, and where no search would be
dreamt of, in the very palace of the Cæsars.[159] Efficiently disguised, the
holy Bishop left the cemetery, and, escorted by Sebastian and Quadratus,
was safely housed in the apartments of Irene, a Christian lady of rank, who
lived in a remote part of the Palatine, in which her husband held a
household office.
Early next morning Sebastian was with Pancratius. “My dear boy,” he
said, “you must leave Rome instantly, and go into Campania. I have horses
ready for you and Quadratus; and there is no time to be lost.”
“And why, Sebastian?” replied the youth, with sorrowful face and tearful
eye. “Have I done something wrong, or are you doubtful of my fortitude?”
“Neither, I assure you. But you have promised to be guided by me in all
things, and I never considered your obedience more necessary than now.”
“Tell me why, good Sebastian, I pray.”
“It must be a secret as yet.”
“What, another secret?”
“Call it the same, to be revealed at the same time. But I can tell you what
I want you to do, and that I think will satisfy you. Corvinus has got orders
to seize on Chromatius and all his community, yet young in the faith, as the
wretched example of Torquatus has shown us; and, what is worse, to put
your old master Cassianus, at Fundi, to a cruel death. I want you to hasten
before his messenger (perhaps he may go himself), and put them on their
guard.”
Pancratius looked up brightly again; he saw that Sebastian trusted him.
“Your wish is enough reason for me,” said he, smiling; “but I would go the](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-43-320.jpg)
![“I cannot possibly spare them at present. They are worth six men, or a
pair of horses, at least, to me. Wait till the heavy work is over, and then they
are at your service.”
“What are their names, that I may take a note of them? And mind, keep
them up in good condition.”
“They are called Largus and Smaragdus; they are young men of
excellent family, but work like plebeians, and will go with you nothing
loth.”
“They shall have their wish,” said Corvinus, with great glee. And so they
had later.
As they went through the works, however, they picked out a number of
captives, for many of whom Rabirius made resistance, but generally in vain.
At length they came near one of those chambers which flanked the eastern
side of the longer arm of the hall. In one of them they saw a number of
convicts (if we must use the term) resting after their labor. The centre of the
group was an old man, most venerable in appearance, with a long white
beard streaming on his breast, mild in aspect, gentle in word, cheerful in his
feeble action. It was the confessor Saturninus, now in his eightieth year, yet
loaded with two heavy chains. At each side were the more youthful
laborers, Cyriacus and Sisinnius, of whom it is recorded, that, in addition to
their own task-work, one on each side, they bore up his bonds. Indeed, we
are told that their particular delight was, over and above their own assigned
portion of toil, to help their weaker brethren, and perform their work for
them.[160] But their time was not yet come; for both of them, before they
received their crowns, were ordained deacons in the next pontificate.
Several other captives lay on the ground, about the old man’s feet, as he,
seated on a block of marble, was talking to them, with a sweet gravity,
which riveted their attention, and seemed to make them forget their
sufferings. What was he saying to them? Was he requiting Cyriacus for his
extraordinary charity, by telling him that, in commemoration of it, a portion
of the immense pile which they were toiling to raise, would be dedicated to
God, under his invocation, become a title, and close its line of titulars by an
illustrious name?[161] Or was he recounting another more glorious vision,
how this smaller oratory was to be superseded and absorbed by a glorious
temple in honor of the Queen of Angels, which should comprise the entire
of that superb hall, with its vestibule, under the directing skill of the](https://image.slidesharecdn.com/38743-250723221715-21afe087/85/Navigation-Signal-Processing-for-GNSS-Software-Receivers-1st-Edition-Thomas-Pany-52-320.jpg)
Navigation Signal Processing for GNSS Software Receivers 1st Edition Thomas Pany
- 1. Navigation Signal Processing for GNSS Software Receivers 1st Edition Thomas Pany - PDF Download (2025) https://ebookultra.com/download/navigation-signal-processing-for- gnss-software-receivers-1st-edition-thomas-pany/ Visit ebookultra.com today to download the complete set of ebooks or textbooks
- 2. We have selected some products that you may be interested in Click the link to download now or visit ebookultra.com for more options!. Digital satellite navigation and geophysics a practical guide with GNSS signal simulator and receiver laboratory 1st Edition Ivan G Petrovski https://ebookultra.com/download/digital-satellite-navigation-and- geophysics-a-practical-guide-with-gnss-signal-simulator-and-receiver- laboratory-1st-edition-ivan-g-petrovski/ Signal Processing with Free Software Practical Experiments 1st Edition François Auger https://ebookultra.com/download/signal-processing-with-free-software- practical-experiments-1st-edition-francois-auger/ Signal Processing for Digital Communications Artech House Signal Processing Library George J. Miao https://ebookultra.com/download/signal-processing-for-digital- communications-artech-house-signal-processing-library-george-j-miao/ Signal Processing for Wireless Communications 1st Edition Boccuzzi https://ebookultra.com/download/signal-processing-for-wireless- communications-1st-edition-boccuzzi/
- 3. Signal Processing for Neuroscientists 1st Edition Wim Van Drongelen https://ebookultra.com/download/signal-processing-for- neuroscientists-1st-edition-wim-van-drongelen/ Signal Processing for Mobile Communications Handbook 1st Edition Mohamed Ibnkahla https://ebookultra.com/download/signal-processing-for-mobile- communications-handbook-1st-edition-mohamed-ibnkahla/ Digital Signal Processing Mathematical and Computational Methods Software Development and Applications Second Edition Jonathan M Blackledge https://ebookultra.com/download/digital-signal-processing- mathematical-and-computational-methods-software-development-and- applications-second-edition-jonathan-m-blackledge/ Genomic Signal Processing 1st Edition Ilya Shmulevich https://ebookultra.com/download/genomic-signal-processing-1st-edition- ilya-shmulevich/ Signal processing noise 1st Edition Vyacheslav Tuzlukov https://ebookultra.com/download/signal-processing-noise-1st-edition- vyacheslav-tuzlukov/
- 5. Navigation Signal Processing for GNSS Software Receivers 1st Edition Thomas Pany Digital Instant Download Author(s): Thomas Pany ISBN(s): 9781608070275, 1608070271 Edition: 1 File Details: PDF, 8.36 MB Year: 2010 Language: english
- 8. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalog record for this book is available from the British Library. ISBN-13: 978-1-60807-027-5 Cover design by Greg Lamb Accompanying MATLAB and assembler programs are available at www.artechhouse.com. © 2010 ARTECH HOUSE 685 Canton Street Norwood, MA 02062 All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, includ ing photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this informa tion. Use of a term in this book should not be regarded as affecting the validity of any trade mark or service mark. 10 9 8 7 6 5 4 3 2 1
- 9. Contents Preface xiii Acknowledgments xvii Chapter 1 Radio Navigation Signals 1 1.1 Signal Generation 1 1.2 Signal Propagation 2 1.3 Signal Conditioning 3 1.4 Motivation for a Generic Signal Model 4 1.5 Sampling 5 1.6 Deterministic Received Signal Model 6 1.7 Stochastic Noise Model 6 1.8 Short-Period Signal Model 7 1.8.1 Zeroth-Order Moment of Signal Power 8 1.8.2 First-Order Moment of Signal Power 8 1.8.3 Second-Order Moment of Signal Power 9 1.8.4 First-Order Moment of Signal Power Variations 9 1.8.5 Separation of Code and Carrier Correlation 10 1.9 Exemplary Signals 11 1.9.1 A Model for the GPS C/A-Code Signal 11 1.9.2 A Model for the Galileo E1 Open-Service Signal 13 1.9.3 Pulsed GNSS Signals 14 1.9.4 Gaussian Double Pulse 15 References 16 Chapter 2 Software-Defined Radio 17 2.1 Definitions 17 2.2 Communication Radios 19 2.2.1 GNU Radio 19 2.2.2 Joint Tactical Radio System 19 2.3 GNSS Software Receivers 22 2.3.1 Front Ends 22 2.3.2 Illustrative Applications 25 2.3.3 High-End GNSS Software Receivers 28 2.4 Technology Evaluation and Discussion 30 References 30
- 11. Contents vii 4.3.1 Model for One or More Propagation Paths 73 4.3.2 Single Propagation Path 76 4.3.3 Correlation Point 91 4.3.4 Linearization 97 4.3.5 Multiple Propagation Paths 98 4.3.6 Two Propagation Paths: Code-Phase CRLB 100 4.3.7 Two Propagation Paths: Doppler CRLB 104 4.3.8 Two Propagation Paths: Remark on Other Bounds 104 4.4 Data Reduction 106 4.4.1 Sufficient Statistics 106 4.4.2 Multicorrelator Approach 107 4.4.3 First-Derivative Approach 107 4.4.4 Colored Noise 108 4.5 Bayesian Approach 108 4.5.1 Minimum Mean-Squared Error Estimation 109 4.5.2 Kalman–Bucy Filter 110 4.5.3 Other Filters 112 4.5.4 Use of Kalman Filters in GNSS Signal Processing 113 4.6 Squaring Loss Revisited 114 4.7 Numerical Simulation 117 4.7.1 Evaluation of Bounds 118 4.7.2 Cost Function 119 4.7.3 LSQ Solution 120 4.8 Discussion 124 References 125 Chapter 5 Signal Detection 129 5.1 Detection Principles 129 5.1.1 Simple Hypothesis Testing 130 5.1.2 Composite Hypothesis Testing 131 5.2 Detection Domains 133 5.2.1 Pseudorange Domain Detection 133 5.2.2 Position Domain Detection 133 5.3 Preprocessing 133 5.4 Clairvoyant Detector for Uniformly Distributed Phase 134 5.5 Energy Detector 137 5.6 Bayesian Detector 138 5.7 Generalized Likelihood-Ratio Detector 140 5.7.1 Single Coherent Integration 141 5.7.2 Multiple Coherent Integrations 142 5.7.3 Considering Navigation Signal Interference 147 5.7.4 Data and Pilot 149 5.8 System-Detection Performance 154 5.8.1 Idealized Assumptions 155 5.8.2 Mean Acquisition Time 155
- 12. 5.8.3 System Probabilities 156 5.8.4 Independent Bin Approximation 156 5.8.5 Code-Phase and Doppler Losses 157 5.9 Long Integration Times and Differential Detectors 158 5.10 Discussion 159 References 161 Chapter 6 Sample Preprocessing 163 6.1 ADC Quantization 163 6.1.1 Quantization Rule 163 6.1.2 Matched Filter 165 6.1.3 Evaluation of Expected Values 167 6.1.4 Infinite Number of Bits 169 6.1.5 Numerical Evaluation 170 6.2 Noise-Floor Determination 174 6.3 ADC Requirements for Pulse Blanking 174 6.3.1 Front-End Gain and Recovery Time 175 6.3.2 Pulse Blanking 175 6.3.3 ADC Resolution 176 6.4 Handling Colored Noise 178 6.4.1 Spectral Whitening 178 6.4.2 Modified Reference Signals 179 6.4.3 Overcompensation of the Incoming Signal 180 6.4.4 Implementation Issues 180 6.5 Sub-Nyquist Sampling 180 References 182 Chapter 7 Correlators 185 7.1 Correlator and Waveform-Based Tracking 185 7.2 Generic Correlator 187 7.2.1 Expected Value 188 7.2.2 Covariance 189 7.2.3 Variance 191 7.3 Correlator Types with Illustration 191 7.3.1 P-Correlator 192 7.3.2 F-Correlator 193 7.3.3 D-Correlator 194 7.3.4 W-Correlator 194 7.4 Difference Correlators 197 7.4.1 Single-Difference P-Correlators 197 7.4.2 Double-Difference P-Correlators 199 7.5 Noisy Reference Signal for Codeless Tracking 200 7.5.1 Expected Value 202 7.5.2 Covariance 202 viii Contents
- 13. Contents ix 7.5.3 Variance 204 7.5.4 L2 P(Y)-Code Carrier-Phase Discriminator Noise 204 7.6 Incorporating Colored Noise 206 7.6.1 White-Noise Transformation 206 7.6.2 Early–Late Code Discriminator with Infinite Sample Rate 208 7.7 Comparison of Finite and Infinite Sample Rates 212 References 214 Chapter 8 Discriminators 217 8.1 Noncoherent Discriminators 217 8.1.1 Code Discriminator 217 8.1.2 Doppler Discriminator 221 8.1.3 Phase Discriminator 223 8.1.4 Clipping 225 8.2 S-Curve Shaping 225 8.2.1 Code-Discriminator Performance Characteristics 226 8.2.2 Optimum S-Curve 227 8.2.3 Frequency-Domain S-Curve Shaping 228 8.2.4 Discussion 231 8.3 Multipath Estimating Techniques 231 8.3.1 The LSQ Equations 232 8.3.2 Calibration 235 8.3.3 General Procedure 235 8.3.4 Correlator Placement 236 8.3.5 Initial Values 236 8.3.6 Number of Required Iterations 237 8.3.7 Multipath Detection 237 8.3.8 Discussion 238 8.4 From Discriminator Noise to Position Accuracy 238 References 239 Chapter 9 Receiver Core Operations 241 9.1 Test-System Configuration 241 9.2 Signal-Sample Bit Conversion 242 9.2.1 Algorithm 243 9.2.2 Numerical Performance 244 9.2.3 Discussion and Other Algorithms 245 9.3 Resampling 245 9.3.1 Algorithm 245 9.3.2 Numerical Performance 245 9.3.3 NCO Resolution 246 9.3.4 Discussion and Other Algorithms 248 9.4 Correlators 248 9.4.1 SDR Implementation 249
- 14. 9.4.2 Discussion and Other Algorithms 250 9.5 Fast Fourier Transform 251 9.5.1 Algorithm 251 9.5.2 Convolution Theorem 252 9.5.3 Time-Domain Correlation and Data Preparation 253 9.5.4 Spectral Shifting 256 9.5.5 Limited-Size Inverse FFT 257 9.5.6 Circular Correlation with Doppler Preprocessing 260 9.5.7 Handling Secondary Codes 263 9.5.8 Asymptotic Computational Performance 267 9.5.9 Reported FFT Performance Values 267 9.5.10 Discussion and Number of Correlators 269 9.6 Reality Check for Signal Tracking 271 9.7 Power Consumption 272 9.8 Discussion 274 References 275 Chapter 10 GNSS SDR RTK System Concept 277 10.1 Technology Enablers 277 10.1.1 Ultra-Mobile PCs 277 10.1.2 Cost-Effective High-Rate Data Links 278 10.2 System Overview 279 10.2.1 Setup 279 10.2.2 Sample Applications 280 10.2.3 Test Installation and Used Signals 280 10.3 Key Algorithms and Components 281 10.4 High-Sensitivity Acquisition Engine 281 10.4.1 Doppler Search Space 282 10.4.2 Correlation Method 284 10.4.3 Clock Stability 284 10.4.4 Line-of-Sight Dynamics 287 10.4.5 Flow Diagram and FFT Algorithms 287 10.4.6 Acquisition Time 288 10.5 Assisted Tracking 289 10.5.1 Vector-Hold Tracking 290 10.5.2 Double-Difference Correlator 291 10.6 Low-Cost Pseudolites 297 10.6.1 Continuous-Time Signals 299 10.6.2 Pulsed Signals 299 10.7 RTK Engine 304 References 305 Chapter 11 Exemplary Source Code 307 11.1 Intended Use 307 Contents
- 15. Contents xi 11.2 Setup 307 11.2.1 Required Software 307 11.2.2 Preparing the Simulation 308 11.2.3 Signal Selection and Simulation Parameters 308 11.3 Routines 308 11.3.1 True Cramér-Rao Lower Bound 308 11.3.2 Discriminator Noise Analysis 308 11.3.3 FFT Acquisition 308 11.3.4 Simplified Vector Tracking with Multipath Mitigation and Spectral Whitening 309 Appendix A.1 Complex Least-Squares Adjustment 311 A.1.1 Definitions 311 A.1.2 Probability Density Function 312 A.1.3 The Adjustment 312 A.1.4 Real- and Complex-Valued Estimated Parameters 314 A.1.5 A Posteriori Variance of Unit Weight 315 A.1.6 Example 318 A.1.7 Discussion 320 A.2 Representing Digital GNSS Signals 320 A.2.1 Complex-Valued Input Signal 320 A.2.2 Real-Valued Input Signal 321 A.2.3 Comparing Real- and Complex-Valued Signals 322 A.3 Correlation Function Invariance 326 A.4 Useful Formulas 329 A.4.1 Fourier Transform 329 A.4.2 Correlation Function 331 A.4.3 Correlation with an Auxiliary Function 332 A.4.4 Correlation with Doppler 333 A.4.5 Correlation in Continuous Time 334 A.4.6 Probability Density Functions 336 References 338 Abbreviations 339 List of Symbols 343 About the Author 345 Index 347
- 16. xiii Preface The continuous developments of software-defined radio technology resulted in the appearance of the first real-time GPS software radios at the beginning of this cen- tury. For the first time, it was possible to realize a complete GNSS receiver without going into the depths of cumbersome hardware development that requires develop- ment or programming of low-level digital circuitry. The hardware development ef- forts were indeed so high that only a very limited number of companies or research institutes could afford them. Furthermore, the implementation constraints were so severe, especially for the first generation of GPS receivers, that often crude signal- processing approximations were necessary to allow a real implementation. Cur- rently, software-defined radio technology not only allows receiver implementations by a larger research community, but also drastically increases the signal-processing capabilities. It also has the potential to become, in certain navigation areas, a com- mercial success. Software radio technology provides an opportunity to design a new class of GNSS receivers, being more flexible and easier to develop than their FPGA- or ASIC-based counterparts. Therefore, this text reviews navigation signal detection and estimation algorithms and their implementation in a software radio. A focus is put on high-precision applications for GNSS signals and an innovative RTK re- ceiver concept based on difference correlators is proposed. This text makes extensive use of the least-squares principle. The least-squares principle is the typical basis for the calculation of a navigation solution. An adjust- ment or a Kalman filter calculates positions from pseudorange observations in vir- tually any GNSS receiver. Within this text, the least-squares principle is consistently extended to also allow signal samples as observations. In contrast to the pseudorange- observation equation, the signal sample model is highly non-linear, causing a num- ber of difficulties that are discussed. Furthermore, signal sample observations can be complex-valued. In the author’s opinion, the development of a navigation receiver does not nec- essarily require an in-depth theoretical knowledge of signal-estimation and signal- detection theory. The basic algorithms like correlation and tracking can also be understood on an intuitive basis. Indeed many textbooks skip the highly theoretical signal-estimation and signal-detection framework and focus on engineering aspects. The question arises: What can we learn from the theoretical treatment that is pre- sented here? First, the theory allows a performance assessment without building a receiver. By providing benchmarks like the Cramér-Rao lower bound or the clair- voyant detector, the theory serves also as a reference with which to compare a real implementation. This text attempts to generalize the existing theory for arbitrary navigation signal waveforms, going beyond existing GNSS signals. The theoretical
- 17. treatment also gives hints for optimal algorithms; useful examples that are discussed are spectral whitening and the least-squares-based multipath-estimating discrimina- tor. Efficient algorithms are found in the frequency domain for signal acquisition, which itself would justify the effort of going into theoretical details. Furthermore, the theoretical analysis points out that new developments could be expected in the field of direct-position estimation (in a single-step procedure, instead of estimating a position via pseudoranges), which should give advantages in terms of interference robustness and sensitivity. Sensitivity might be further increased by using Bayesian techniques (like a particle filter) that do not rely on a linearized signal model. Unfortunately, the existing navigation signal-processing theory has limits and does not always provide an optimal algorithm for detection or estimation. Ex amples are the nonexistence of a uniform most powerful detector for acquisition and the nonexistence of a minimum variance unbiased code-phase or Doppler estimator for finitely received signal power. In addition, the practical usability of Bayesian techniques within signal processing (apart from the Kalman filter) is not completely assessed. Overall, it seems that a theoretically optimal navigation receiver is out-of- reach today, even if only signal processing is considered. However, software radio technology closes the gap between existing theory and real implementation. Overview Within this text, the navigation signal processing theory is described for generic navi- gation signals to allow a broad range of applications, beyond that of GNSS. Require- ments for navigation signals are introduced in Chapter 1 and are illustrated with one GPS, one Galileo, and two pulsed signals. Software-defined radio technology will be introduced in Chapter 2, together with the architecture and the data flow of a per- manent GNSS reference station in Chapter 3. Chapters 4 and 5 focus on theoretical signal-processing aspects and Chapters 6 through 9 shift the focus to implementation. An innovative high-precision software radio concept is presented in Chapter 10 using double-difference correlators, in addition to double-difference pseudorange and carrier- phase observations to increase carrier-phase tracking stability for real-time kinematic applications. Finally, on the Artech House Web site, www.artechhouse.com, this book has some MATLAB and assembler programs that illustrate the core signal-processing concepts of a navigation receiver. Chapter 11 describes this software. Summary of Presented Signal-Processing Theory In Chapter 1, requirements are formulated that a generic navigation signal has to fulfill to allow for the simultaneous estimation of the code phase, the Doppler and the carrier phase. Based on those requirements, Chapter 4 reviews the estimation theory for navigation signals using a consistent mathematical notation and derives the theory from first principles. The presented mathematical derivations are very detailed and with the finite sample rate approach, a reader should be able to adapt the theory for his or her purposes easily. The finite sample rate description is also chosen to enable a software receiver developer to establish a one-to-one correspon- dence of theory and implementation at every stage of signal processing. xiv Preface
- 19. com/static/reslib/pany/pany1.html. The source code includes the FFT acquisition methods and multipath-estimating tracking. Furthermore, MATLAB scripts for the true Cramér-Rao lower bound as well as for the thermal-noise analysis of the noncoherent discriminators are included. The scripts run with the four exemplary navigation signals of Chapter 1 and are outlined in Chapter 11. Book’s Usage for Practical Receiver Implementation This book should help in building advanced navigation software receivers. It is not a beginner’s book and the reader should be familiar with the architecture of a GNSS software receiver, which is, for example, excellently described by Borre’s book mentioned in Chapter 2. Borre’s book also comes with a complete MATLABreceiverandourtextmayhelpyoutoextendthisreceiverforhigh-sensitivity applications using efficient FFT techniques or for high-precision applications ap- plying multipath-mitigation schemes or stable double-difference carrier-phase tracking. To build a navigation software receiver, you need there things: navigation sig- nal samples, a software framework that handles the data flow, and efficient core algorithms. Signal samples can be obtained by one of the GNSS front ends described in Chapter 2 or you can use the single-channel signal generator of Chapter 11. Some- times, the front-end manufacturers can provide you with exemplary signal-sample streams, too. Writing a software receiver framework from scratch can be a quite tedious work. The framework handles the enormous amount of signal samples, synchro- nizes the different receiver channels, computes the position, and provides some standard output formats. You can short-cut this by adapting the MATLAB source code mentioned above. Another possibility to avoid this cumbersome work is to use a commercial software receiver having an application programming interface, which you can plug into your own source code. Chapter 11 provides you with a single-channel framework that demonstrates how to convert the sample stream into pseudorange measurements. Finally, the core algorithms can actually be found within this text. They are derived in a way that they can be adapted easily for a specific framework and the assembler code should help to realize them efficiently. xvi Preface
- 20. xvii Acknowledgments This work would not have been possible without the support from numerous co- workers and colleagues. I am especially grateful to the researchers of the University of Federal Armed Forces in Munich, to the researchers at the IFEN GmbH, and to many colleagues from research institutes from all over the world. I am grateful to Professor Günter W. Hein for continually encouraging me to enter this field, for his uninterrupted belief in technology, and for showing me ways of going beyond limits. Professor Bernd Eissfeller established the basis for GNSS receiver technology research at the Institute of Geodesy and Navigation. His con- tribution to this work cannot be overvalued. I would also like to thank Professor Jörn Thielecke for fruitful discussions. With his knowledge on communication and navigation signal processing, he showed me several important links between both fields.
- 22. Radio Navigation Signals output, or to the digitized signal obtained after (optional) downconversion and filtering of the received signal. The broadcast navigation signal structure is known to the receiver and will be characterized in the following paragraphs. A good way to do this is to merge the frontend filter into the output filter (i.e., into a combined filter) and to describe the signal after the output filter. This simplification is useful for signal processing pur- poses and is valid, because both antennas, the propagation path, and the frontend filter act as linear systems to the signal. Nonlinear effects of the amplifiers or mixer are not considered. Without the combined filter (or output filter), the transmitted signal rS is writ- ten as ( ) ( ) ( )cos(2 ) S S RF r t a d t c t f t π ¥ = (1.1) Here, fRF is the carrier frequency in hertz, c (t) is the infinite bandwidth signal representation at baseband, and d(t) represents a broadcast navigation message. The transmitted signal is described in the signal-in-space (SIS) interface control doc- ument (ICD) [1, 2]. The symbol aS denotes the signal amplitude in arbitrary units. Other signals might be transmitted on the same carrier frequency by the same satellite. For example, the Global Positioning System (GPS) C/A signal is broadcast in phase quadrature with the P(Y) signal on the L1 (1,575.42 MHz) carrier fre- quency. The access to the different signals is controlled via c (t). Different wave- forms of c∞(t) can be used to realize code division multiple access (CDMA), time division multiple access (TDMA), or—by including a carrier into c (t)—frequency division multiple access (FDMA) schemes. With the combined filter, the signal is mathematically written as = - + ( ) ( ) ( )sin(2 ) ( ) ( )cos(2 ) S S Q RF S I RF r t a d t c t f t a d t c t f t π π (1.2) with ( ) ( ) ( ) I Q c t c t ic t = + (1.3) representing the filtered baseband representation of the signal obtained by applying (1.10). The signal cQ(t) is an artifact generated by the filter. The filter does not affect d(t), as long as the data rate (e.g., 50 bit/s) is much smaller than the filter bandwidth (e.g., 2 MHz). If the radio frequency (RF) is much higher than the filter bandwidth, cQ(t) is small and can be ignored. Using a complex notation, the signal after the combined filter is given as ( ) Re{ ( ) ( )exp(2 )} S S RF r t a d t c t if t π = (1.4) 1.2 Signal Propagation According to Fermat’s principle, the line-of-sight component of the signal rS(t) travels along the shortest propagation path from the transmitter to the receiver. Eventually, one or more reflected signals [i.e., copies of rS(t)] are superimposed with the line-of-sight signal. Each copy has a different amplitude, Doppler, carrier phase
- 23. 1.3 Signal Conditioning delay, or code phase delay. Furthermore, the line-of-sight component is delayed by the atmosphere. Accurate modeling of the propagation effects is one of the key ele- ments to obtain a precise position and this topic is covered in many satellite navi- gation text books [3, 4]. We will not go into the modeling details, but will instead focus on a model for the received signal that can be used as a basis for delay and Doppler estimation. Delays affect the carrier and the data-modulating signals c(t), d(t) differently. Frequency-independent (nondispersive) delays are caused by the geometric distance and by the electrically neutral part of the atmosphere (troposphere). Frequency- dependent (dispersive) delays are caused by the ionosphere and by the receiver and transmitter hardware. Dispersive and nondispersive delays add up and result in the group delay τG, delaying c(t) and d(t). The carrier is delayed by the phase delay τP. The delays τG and τP are not equal because the ionospheric delay contributes to each of them with a different sign. The difference is typically less than 100m and varies with a frequency not more than 0.05 Hz if no scintillations are present. Also, hardware group and phase delay are generally different. The group delay, τG, affects c(t) via ,0 ( ) ( ( )) ( ) G D G c t c t t c t τ α τ ® - » - (1.5) and similarly for d(t). The phase delay, τP, affects the carrier via ,0 exp(2 ) exp(2 ( ( ))) exp(2 ( )) RF RF P RF D P if t if t t if t π π τ π α τ ® - » - (1.6) with / ,0 / 0 0 ( ) (1 ) G P G P D t t τ τ α = - - (1.7) The coefficient αD 1 D v c α = - (1.8) is the Doppler effect, caused by the change in the group/phase delay, expressed as velocity v in meters per second. The linearization is carried out around the epoch t0. If the signal duration Tcoh under consideration is short, the Doppler effect on c(t), d(t) can be ignored, as in ,0 0 ( ) ( ( )) coh D G c T c t c t t vB α τ τ Þ - » - (1.9) and similarly for d(t), where B denotes the signal (or data message) bandwidth in hertz. 1.3 Signal Conditioning After signal reception by the antenna, the signal is amplified, filtered, and eventually downconverted. Amplification changes the amplitude of the signal (from aS to a), but leaves the signal structure invariant.
- 24. Radio Navigation Signals The frontend filter limits the bandwidth of the received navigation signals and of the received noise. It also rejects out-of-band signals. The frontend filter is typi- cally of lower bandwidth than the output filter and neglecting the output filter is a reasonable approximation. The resulting filter is a band pass filter and is described by its baseband equivalent H via ¥ = ( ) ( ( )) c t H c t (1.10) For signal estimation and detection, it is largely irrelevant at which center fre- quency the filter operates; it can be placed at the RF, at the IF, or at baseband. Nor- mally, filters with discrete components or SAW filters operating at the IF are used, but discrete polyphase filters at baseband have also been utilized. Global navigation satellite system (GNSS) receivers normally do not integrate the filter into a chip solution. Downconversion changes the signal carrier model by ,0 ,0 exp(2 ( )) exp(2 ( ) 2 ) RF D P RF D P LO if t if t if t π α τ π α τ π - ® - - (1.11) where fLO is the local oscillator frequency in hertz. For each transmitter and propagation path, a signal of the form 0 ,0 ( ) ( ( ))exp(2 ( ) 2 ) rec RF D P LO r t ac t t if t if t τ π α τ π = - - - (1.12) arrives at the front end’s ADC(s). The ADC(s) either quantize(s) the real and the imaginary part of the signal or quantizes only one of them (see Appendix A.2). 1.4 Motivation for a Generic Signal Model By specifying core signal elements, it is possible to reduce the amount of information that is necessary to analyze a navigation system. The baseband representation of the navigation signal c(t), the time over which d(t) remains constant, and the carrier frequency represent core elements that determine to a large extent how precisely a receiver can estimate code and carrier delay, the Doppler shift, and, consequently, its position. They are summarized in Table 1.1. The assumptions imposed on the received signals are kept as general as pos- sible to allow the application of the developed theory to a wide range of signals. It focuses mostly on conventional GNSS continuous-time CDMA signals using binary phase shift keying (BPSK), binary offset carrier (BOC), multiplexed BOC (MBOC), alternative BOC (AltBOC), or any other spread-spectrum technique. Furthermore, Table 1.1 Influence of Navigation Signal Elements on Signal Processing Parameters Signal Element Affected Signal Processing Parameter c(t) Code correlation function Rc,c, Doppler correlation function d(t) Coherent integration time Tcoh, Doppler correlation function
- 25. 1.5 Sampling pulsed signals are included because they are being used by pseudolites, by LORAN-C, or RADAR-like ranging systems. In principle, the theory can also be adapted for sonar ranging systems and, with some limitations, for optical ranging systems. Only one frequency band is considered (e.g., GPS L1); a generalization from a sampling of one frequency band to multiple frequency bands is obvious, but one should be take care that all bands are sampled synchronously. No assumption on the relation of the signal bandwidth to the sampling rate is made and, in particular, sub-Nyquist sampling rates can be used. No assumption on the modulation scheme is made as long as the received signal waveform at baseband c(t) is known a priori to the receiver. Eventually, filters influence the theory via the waveform c(t). The more narrow the filter bandwidth, the smoother the waveform will be. The filter bandwidth and characteristics define how much noise power is be- ing received: the wider the bandwidth, the higher the noise power. For simplicity, a unity noise power is assumed in (1.16) and it is important to keep in mind that only ratios between power levels have meaningful values, as described in Section 1.8.1. In that sense, (1.16) defines the power scale. The Nyquist criterion does not fully apply because the waveform c(t) is known to the receiver beforehand. Consequently, there is no need to reconstruct the signal waveform from the received samples [5]. As shown in Section 6.5, a good choice for the sample rate is exactly equal to the Nyquist rate (e.g., being equal to the noise bandwidth). Lower sample rates yield less-independent signal samples, thereby gen- erally decreasing the accuracy of the obtained estimates. The accuracy decrease can be modeled as an effective signal power loss. This observation is also true when multiple reflections of the same signal are received. In the rest of this chapter, we will formulate generic conditions for navigations signals. Later, we illustrate the conditions with two GNSS signals and two pulsed terrestrial navigation signals. In addition to those parameters, the number of re- ceived signals, the amplitude, and the geometric placement of the transmitters affect the positioning result (see Sections 4.1.3 and 8.4). 1.5 Sampling In the following, a number of L signal samples are considered, indexed by m. The index m assumes, in general, the values { } 1, ,L µ Î … (1.13) The sampling epoch tm in seconds for the sample m is given by s t f µ µ = (1.14) and fs defines the ADC sample rate in samples per second. Optionally, the sampling epochs can be offset by a fixed amount of time, which will not explicitly be men- tioned here. For each sampling epoch tm, a complex-valued (i.e., I plus Q) signal sample is generated. We chose a complex signal representation to work with more compact
- 26. Other documents randomly have different content
- 27. Torquatus?” He heard the account of his sudden disappearance, told in as many ways as the Dacian guard’s adventure: but it annoyed him greatly. He had no doubt whatever, in his own mind, that he had been duped by his supposed victim, who had escaped into the unsearchable mazes of the cemetery. If so, this captive would know, and he determined to question her. He stood before her, therefore, put on his most searching and awful look, and said to her sternly, “Look at me, woman, and tell me the truth.” “I must tell you the truth without looking at you, sir,” answered the poor girl, with her cheerfullest smile and softest voice; “do you not see that I am blind?” “Blind!” all exclaimed at once, as they crowded to look at her. But over the features of Fulvius there passed the slightest possible emotion, just as much as the wave that runs, pursued by a playful breeze, over the ripe meadow. A knowledge had flashed into his mind, a clue had fallen into his hand. “It will be ridiculous,” he said, “for twenty soldiers to march through the city, guarding a blind girl. Return to your quarters, and I will see you are well rewarded. You, Corvinus, take my horse, and go before to your father, and tell him all, I will follow in a carriage with the captive.” “No treachery, Fulvius,” he said, vexed and mortified. “Mind you bring her. The day must not pass without a sacrifice.” “Do not fear,” was the reply. Fulvius, indeed, was pondering whether, having lost one spy, he should not try to make another. But the placid gentleness of the poor beggar perplexed him more than the boisterous zeal of the gamester, and her sightless orbs defied him more than the restless roll of the toper’s. Still, the first thought that had struck him he could yet pursue. When alone in a carriage with her, he assumed a soothing tone, and addressed her. He knew she had not overheard the last dialogue. “My poor girl,” he said, “how long have you been blind?” “All my life,” she replied. “What is your history? Whence do you come?” “I have no history. My parents were poor, and brought me to Rome when I was four years old, as they came to pray, in discharge of a vow made for my life in early sickness, to the blessed martyrs Chrysanthus and Daria.
- 28. They left me in charge of a pious lame woman, at the door of the title of Fasciola, while they went to their devotions. It was on that memorable day, when many Christians were buried at their tomb, by earth and stones cast down upon them. My parents had the happiness to be of the number.” “And how have you lived since?” “God became my only Father then, and His Catholic Church my mother. The one feeds the birds of the air, the other nurses the weaklings of the flock. I have never wanted for any thing since.” “But you can walk about the streets freely, and without fear, as well as if you saw.” “How do you know that?” “I have seen you. Do you remember very early one morning in the autumn, leading a poor lame man along the Vicus Patricius?” She blushed and remained silent. Could he have seen her put into the poor old man’s purse her own share of the alms? “You have owned yourself a Christian?” he asked negligently. “Oh, yes! how could I deny it?” “Then that meeting was a Christian meeting?” “Certainly; what else could it be?” He wanted no more; his suspicions were verified. Agnes, about whom Torquatus had been able or willing to tell him nothing, was certainly a Christian. His game was made. She must yield, or he would be avenged. After a pause, looking at her steadfastly, he said, “Do you know whither you are going?” “Before the judge of earth, I suppose, who will send me to my Spouse in heaven.” “And so calmly?” he asked in surprise; for he could see no token from the soul to the countenance, but a smile. “So joyfully rather,” was her brief reply. Having got all that he desired, he consigned his prisoner to Corvinus at the gates of the Æmilian basilica, and left her to her fate. It had been a cold and drizzling day like the preceding evening. The weather, and the incident of the night, had kept down all enthusiasm; and while the prefect had been compelled to sit in-doors, where no great crowd could collect, as hours had
- 29. passed away without any arrest, trial, or tidings, most of the curious had left, and only a few more persevering remained, past the hour of afternoon recreation in the public gardens. But just before the captive arrived, a fresh knot of spectators came in, and stood near one of the side-doors, from which they could see all. As Corvinus had prepared his father for what he was to expect, Tertullus, moved with some compassion, and imagining there could be little difficulty in overcoming the obstinacy of a poor, ignorant, blind beggar, requested the spectators to remain perfectly still, that he might try his persuasion on her, alone, as she would imagine, with him; and he threatened heavy penalties on any one who should presume to break the silence. It was as he had calculated. Cæcilia knew not that any one else was there, as the prefect thus kindly addressed her: “What is thy name, child?” “Cæcilia.” “It is a noble name; hast thou it from thy family?” “No; I am not noble; except because my parents, though poor, died for Christ. As I am blind, those who took care of me called me Cæca,[153] and then, out of kindness, softened it into Cæcilia.” “But now, give up all this folly of the Christians, who have kept thee only poor and blind. Honor the decrees of the divine emperors, and offer sacrifice to the gods; and thou shalt have riches, and fine clothes, and good fare; and the best physicians shall try to restore thee thy sight.” “You must have better motives to propose to me than these; for the very things for which I most thank God and His Divine Son, are those which you would have me put away.” “How dost thou mean?” “I thank God that I am poor and meanly clad, and fare not daintily; because by all these things I am the more like Jesus Christ, my only Spouse.” “Foolish girl!” interrupted the judge, losing patience a little; “hast thou learnt all these silly delusions already? at least thou canst not thank thy God that He has made thee sightless.” “For that, more than all the rest, I thank Him daily and hourly with all my heart.”
- 30. “How so? dost thou think it a blessing never to have seen the face of a human being, or the sun, or the earth? What strange fancies are these?” “They are not so, most noble sir. For in the midst of what you call darkness, I see a spot of what I must call light, it contrasts so strongly with all around. It is to me what the sun is to you, which I know to be local from the varying direction of its rays. And this object looks upon me as with a countenance of intensest beauty, and smiles upon me ever. And I know it to be that of Him whom I love with undivided affection. I would not for the world have its splendor dimmed by a brighter sun, nor its wondrous loveliness confounded with the diversities of others’ features, nor my gaze on it drawn aside by earthly visions. I love Him too much not to wish to see Him always alone.” “Come, come! let me have no more of this silly prattle. Obey the emperors at once, or I must try what a little pain will do. That will soon tame thee.” “Pain?” she echoed innocently. “Yes, pain. Hast thou never felt it? hast thou never been hurt by any one in thy life?” “Oh, no! Christians never hurt one another.” The rack was standing, as usual, before him; and he made a sign to Catulus to place her upon it. The executioner pushed her back on it by her arms; and as she made no resistance, she was easily laid extended on its wooden couch. The loops of the ever-ready ropes were in a moment passed round her ankles, and arms drawn over the head. The poor sightless girl saw not who did all this; she knew not but it might be the same person who had been conversing with her. If there had been silence hitherto, men now held their very breath; while Cæcilia’s lips moved in earnest prayer. “Once more, before proceeding further, I call on thee to sacrifice to the gods, and escape cruel torments,” said the judge, with a sterner voice. “Neither torments nor death,” firmly replied the victim tied to the altar, “shall separate me from the love of Christ. I can offer up no sacrifice but to the one living God: and its ready oblation is myself.” The prefect made a signal to the executioner, and he gave one rapid whirl to the two wheels of the rack, round the windlasses of which the ropes were wound; and the limbs of the maiden were stretched with a sudden jerk, which, though not enough to wrench them from their sockets, as a further
- 31. turn would have done, sufficed to inflict an excruciating, or more truly, a racking pain, through all her frame. Far more grievous was this, from the preparation and the cause of it being unseen, and from that additional suffering which darkness inflicts. A quivering of her features and a sudden paleness alone gave evidence of her torture. “Ha! ha!” the judge exclaimed, “thou feelest that? Come, let it suffice; obey, and thou shalt be freed.” She seemed to take no heed of his words, but gave vent to her feelings in prayer: “I thank Thee, O Lord Jesus Christ, that Thou hast made me suffer pain the first time for Thy sake. I have loved Thee in peace; I have loved Thee in comfort; I have loved Thee in joy,—and now in pain I love Thee still more. How much sweeter it is to be like Thee, stretched upon Thy Cross, even than resting upon the hard couch at the poor man’s table!” “Thou triflest with me,” exclaimed the judge, thoroughly vexed, “and makest light of my lenity. We will try something stronger. Here, Catulus, apply a lighted torch to her sides.”[154] The Martyr Cæcilia. A thrill of disgust and horror ran through the assembly, which could not help sympathizing with the poor blind creature. A murmur of suppressed indignation broke out from all sides of the hall. Cæcilia, for the first time, learnt that she was in the midst of a crowd. A crimson glow of modesty rushed into her brow, her face, and neck, just
- 32. before white as marble. The angry judge checked the rising gush of feeling; and all listened in silence, as she spoke again, with warmer earnestness than before: “O my dear Lord and Spouse! I have been ever true and faithful to Thee! Let me suffer pain and torture for Thee; but spare me confusion from human eyes. Let me come to Thee at once; not covering my face with my hands in shame when I stand before Thee.” Another muttering of compassion was heard. “Catulus!” shouted the baffled judge in fury; “do your duty, sirrah! what are you about, fumbling all day with that torch?” The executioner advanced, and stretched forth his hand to her robe, to withdraw it for the torture; but he drew back, and, turning to the prefect, exclaimed in softened accents: “It is too late. She is dead!” “Dead!” cried out Tertullus; “dead with one turn of the wheel? impossible!” Catulus gave the rack a turn backwards, and the body remained motionless. It was true; she had passed from the rack to the throne, from the scowl of the judge’s countenance to her Spouse’s welcoming embrace. Had she breathed out her pure soul, as a sweet perfume, in the incense of her prayer? or had her heart been unable to get back its blood, from the intensity of that first virginal blush?[155] In the stillness of awe and wonder, a clear bold voice cried out, from the group near the door: “Impious tyrant, dost thou not see, that a poor blind Christian hath more power over life and death, than thou or thy cruel masters?” “What! a third time in twenty-four hours wilt thou dare to cross my path? This time thou shalt not escape.” These were Corvinus’s words, garnished with a furious imprecation, as he rushed from his father’s side round the enclosure before the tribunal, towards the group. But as he ran blindly on, he struck against an officer of herculean build, who, no doubt quite accidentally, was advancing from it. He reeled, and the soldier caught hold of him, saying: “You are not hurt, I hope, Corvinus?” “No, no; let me go, Quadratus, let me go.”
- 33. “Where are you running to in such a hurry? can I help you?” asked his captor, still holding him fast. “Let me loose, I say, or he will be gone.” “Who will be gone?” “Pancratius,” answered Corvinus, “who just now insulted my father.” “Pancratius!” said Quadratus, looking round, and seeing that he had got clear off; “I do not see him.” And he let him go; but it was too late. The youth was safe at Diogenes’s, in the Suburra. While this scene was going on, the prefect, mortified, ordered Catulus to see the body thrown into the Tiber. But another officer, muffled in his cloak, stepped aside and beckoned to Catulus, who understood the sign, and stretched out his hand to receive a purse held out to him. “Out of the Porta Capena, at Lucina’s villa, an hour after sunset,” said Sebastian. “It shall be delivered there safe,” said the executioner. “Of what do you think did that poor girl die?” asked a spectator from his companion, as they went out. “Of fright, I fancy,” he replied. “Of Christian modesty,” interposed a stranger who passed them. The Woman of Samaria, from a picture in the Cemetery of St. Domitilla. C H A P T E R X V I I I . RETRIBUTION.
- 34. HE prefect of the city went to give his report on the untoward events of the day, and do what was possible to screen his worthless son. He found the emperor in the worst of moods. Had Corvinus come in his way early in the day, nobody could have answered for his head. And now the result of the inroad into the cemetery had revived his anger, when Tertullus entered into the audience-chamber. Sebastian contrived to be on guard. “Where is your booby of a son?” was the first salutation which the prefect received. “Humbly waiting your divinity’s pleasure outside, and anxious to propitiate your godlike anger, for the tricks which fortune has played upon his zeal.” “Fortune!” exclaimed the tyrant; “fortune indeed! His own stupidity and cowardice: a pretty beginning, forsooth; but he shall smart for it. Bring him in.” The wretch, whining and trembling, was introduced; and cast himself at the emperor’s feet, from which he was spurned, and sent rolling, like a lashed hound, into the midst of the hall. This set the imperial divinity a- laughing, and helped to mollify its wrath. “Come, sirrah! stand up,” he said, “and let me hear an account of yourself. How did the edict disappear?” Corvinus told a rambling tale, which occasionally amused the emperor; for he was rather taken with the trick. This was a good symptom. “Well,” he said at last, “I will be merciful to you. Lictors, bind your fasces.” They drew their axes forth, and felt their edges. Corvinus again threw himself down, and exclaimed: “Spare my life; I have important information to furnish, if I live.” “Who wants your worthless life?” responded the gentle Maximian. “Lictors, put aside your axes; the rods are good enough for him.” In a moment his hands were seized and bound, his tunic was stripped off his shoulders, and a shower of blows fell upon them, delivered with well- regulated skill, till he roared and writhed, to the great enjoyment of his imperial master. Smarting and humbled, he had to stand again before him.
- 35. “Now, sir,” said the latter, “what is the wonderful information you have to give?” “That I know who perpetrated the outrage of last night, on your imperial edict.” “Who was it?” “A youth named Pancratius, whose knife I found under where the edict had been cut away.” “And why have you not seized him and brought him to justice?” “Twice this day he has been almost within my grasp, for I have heard his voice; but he has escaped me.” “Then let him not escape a third time, or you may have to take his place. But how do you know him, or his knife?” “He was my school-fellow at the school of Cassianus, who turned out to be a Christian.” “A Christian presume to teach my subjects, to make them enemies of their country, disloyal to their sovereigns, and contemners of the gods! I suppose it was he who taught that young viper Pancratius to pull down our imperial edict. Do you know where he is?” “Yes, sire; Torquatus, who has abandoned the Christian superstition, has told me.” “And pray who is this Torquatus?” “He is one who has been staying some time with Chromatius and a party of Christians in the country.” “Why, this is worse and worse. Is the ex-prefect then, too, become a Christian?” “Yes, and lives with many others of that sect in Campania.” “What perfidy! what treachery! I shall not know whom to trust next. Prefect, send some one immediately to arrest all these men, and the school- master, and Torquatus.” “He is no longer a Christian,” interposed the judge. “Well, what do I care?” replied the emperor peevishly; “arrest as many as you can, and spare no one, and make them smart well; do you understand me? Now begone, all; it is time for my supper.”
- 36. Corvinus went home; and, in spite of medicinal applications, was feverish, sore, and spiteful all night; and next morning begged his father to let him go on the expedition into Campania, that so he might retrieve his honor, gratify his revenge, and escape the disgrace and sarcasm that was sure to be heaped on him by Roman society. When Fulvius had deposited his prisoner at the tribunal, he hastened home to recount his adventures, as usual, to Eurotas. The old man listened with imperturbable sternness to the barren recital, and at last said, coldly: “Very little profit from all this, Fulvius.” “No immediate profit, indeed; but a good prospect in view, at least.” “How so?” “Why, the Lady Agnes is in my power. I have made sure, at last, that she is a Christian. I can now necessarily either win her or destroy her. In either case her property is mine.” “Take the second alternative,” said the old man, with a keen glow in his eye, but no change of face; “it is the shorter, and less troublesome, way.” “But my honor is engaged; I cannot allow myself to be spurned in the manner I told you.” “You have been spurned, however; and that calls for vengeance. You have no time to lose, remember, in foolery. Your funds are nearly exhausted, and nothing is coming in. You must strike a blow.” “Surely, Eurotas, you would prefer my trying to get this wealth by honorable,” (Eurotas smiled at the idea coming into either of their minds) “rather than by foul, means.” “Get it, get it any way, provided it be the surest and the speediest. You know our compact. Either the family is restored to wealth and splendor, or it ends in and with you. It shall never linger on in disgrace, that is, in poverty.” “I know, I know, without your every day reminding me of the bitter condition,” said Fulvius, wringing his hands, and writhing in all his body. “Give me time enough, and all will be well.” “I give you time, till all is hopeless. Things do not look bright at present. But, Fulvius, it is time that I tell you who I am.” “Why, were you not my father’s faithful dependant, to whose care he intrusted me?”
- 37. “I was your father’s elder brother, Fulvius, and am the head of the family. I have had but one thought, but one aim in life, the restoring of our house to that greatness and splendor, from which my father’s negligence and prodigality had brought it down. Thinking that your father, my brother, had greater ability than myself for this work, I resigned my rights and gains to him upon certain terms; one of which was your guardianship, and the exclusive forming of your mind. You know how I have trained you, to care nothing about the means, so that our great ends be carried.” Fulvius, who had been riveted with amazement and deep attention on the speaker, shrunk into himself with shame, at this baring of both their hearts. The dark old man fixed his eyes more intently than ever, and went on: “You remember the black and complicated crime by which we concentrated in your hands the divided remnant of family wealth.” Fulvius covered his face with his hands and shuddered, then said entreatingly, “Oh, spare me that, Eurotas; for heaven’s sake spare me!” “Well, then,” resumed the other, unmoved as ever, “I will be brief. Remember, nephew, that he who does not recoil from a brilliant future, to be gained by guilt, must not shrink from a past that prepared it by crime. For the future will one day be the past. Let our compact, therefore, be straightforward and honest, for there is an honesty even in sin. Nature has given you abundance of selfishness and cunning, and she has given me boldness and remorselessness in directing and applying them. Our lot is cast by the same throw,—we become rich, or die, together.” Fulvius, in his heart, cursed the day that he came to Rome, or bound himself to his stern master, whose mysterious tie was so much stronger than he had known before. But he felt himself spell-bound to him, and powerless as the kid in the lion’s paws. He retired to his couch with a heavier heart than ever; for a dark, impending fate never failed to weigh upon his soul every returning night. The reader will perhaps be curious to know what has become of the third member of our worthy trio, the apostate Torquatus. When, confused and bewildered, he ran to look for the tomb which was to guide him, it so happened, that, just within the gallery which he entered, was a neglected staircase, cut in the sandstone, down to a lower story of the cemetery. The steps had been worn round and smooth, and the descent was precipitous. Torquatus, carrying his light before him, and running heedlessly, fell
- 38. headlong down the opening, and remained stunned and insensible at the bottom, till long after his companions had retired. He then revived, and for some time was so confused that he knew not where he was. He arose and groped about, till, consciousness completely returning, he remembered that he was in a catacomb, but could not make out how he was alone and in the dark. It then struck him that he had a supply of tapers about him, and means of lighting them. He employed these, and was cheered by finding himself again in light. But he had wandered from the staircase, of which, indeed, he recollected nothing, and went on, and on, entangling himself more inextricably in the subterranean labyrinth. He felt sure that, before he had exhausted his strength or his tapers, he should come to some outlet. But by degrees he began to feel serious alarm. One after the other his lights were burnt out, and his vigor began to fail, for he had been fasting from early morning; and he found himself coming back to the same spot, after he had wandered about apparently for hours. At first he had looked negligently around him, and had carelessly read the inscriptions on the tombs. But as he grew fainter, and his hope of relief weaker, these solemn monuments of death began to speak to his soul, in a language that it could not refuse to hear, nor pretend to misunderstand. “Deposited in peace,” was the inmate of one; “resting in Christ” was another; and even the thousand nameless ones around them reposed in silent calm, each with the seal of the Church’s motherly care stamped upon his place of rest. And within, the embalmed remains awaited the sound of angelic trumpet-notes, to awaken them to a happy resurrection. And he, in a few more hours, would be dead like them; he was lighting his last taper, and had sunk down upon a heap of mould; but would he be laid in peace, by pious hands, as they? On the cold ground, alone, he should die, unpitied, unmourned, unknown. There he should rot, and drop to pieces; and if, in after years, his bones, cast out from Christian sepulture, should be found, tradition might conjecture that they were the accursed remains of an apostate lost in the cemetery. And even they might be cast out, as he was, from the communion of that hallowed ground. It was coming on fast; he could feel it; his head reeled, his heart fluttered. The taper was getting too short for his fingers, and he placed it on a stone beside him. It might burn three minutes longer; but a drop filtering through the ceiling, fell upon it, and extinguished it. So covetous did he feel of those three minutes more of light, so jealous was he of that little taper-
- 39. end, as his last link with earth’s joys, so anxious was he to have one more look at things without, lest he should be forced to look at those within, that he drew forth his flint and steel, and labored for a quarter of an hour to get a light from tinder, damped by the cold perspiration on his body. And when he had lighted his remnant of candle, instead of profiting by its flame to look around him, he fixed his eyes upon it with an idiotic stare, watching it burn down, as though it were the charm which bound his life, and this must expire with it. And soon the last spark gleamed smouldering like a glow- worm, on the red earth, and died. Was he dead too? he thought. Why not? Darkness, complete and perpetual, had come upon him. He was cut off for ever from consort with the living, his mouth would no more taste food, his ears never again hear a sound, his eyes behold no light, or thing, again. He was associated with the dead, only his grave was much larger than theirs; but, for all that, it was as dark and lonely, and closed for ever. What else is death? No, it could not be death as yet. Death had to be followed by something else. But even this was coming. The worm was beginning to gnaw his conscience, and it grew apace to a viper’s length, and twisted itself round his heart. He tried to think of pleasant things, and they came before him; the quiet hours in the villa with Chromatius and Polycarp, their kind words, and last embrace. But from the beautiful vision darted a withering flash; he had betrayed them; he had told of them; to whom? To Fulvius and Corvinus. The fatal chord was touched, like the tingling nerve of a tooth, that darts its agony straight to the centre of the brain. The drunken debauch, the dishonest play, the base hypocrisy, the vile treachery, the insincere apostasy, the remorseful sacrileges, of the last days, and the murderous attempt of that morning, now came dancing, like demons hand in hand, in the dark before him, shouting, laughing, jibing, weeping, moaning, gnashing their teeth; and sparks of fire flying before his eyes, from his enfeebled brain, seemed to dart from glaring torches in their hands. He sunk down and covered his eyes. “I may be dead, after all,” he said to himself; “for the infernal pit can have nothing worse than this.” His heart was too weak for rage; it sunk within him in the impotence of despair. His strength was ebbing fast, when he fancied he heard a distant sound. He put away the thought; but the wave of a remote harmony beat
- 40. again upon his ear. He raised himself up; it was becoming distinct. So sweet it sounded, so like a chorus of angelic voices, but in another sphere, that he said to himself: “Who would have thought that Heaven was so near to hell! Or are they accompanying the fearful Judge to try me?” And now a faint glimmer of light appeared at the same distance as the sounds; and the words of the strain were clearly heard: “In pace, in idipsum, dormiam et requiescam.”[156] “Those words are not for me. They might do at a martyr’s entombment; they cannot at a reprobate’s burial.” The light increased; it was like a dawn glowing into day; it entered the gallery and passed across it, bearing in it, as in a mirror, a vision too distinct to be unreal. First, there came virgins robed and holding lamps; then four who carried between them a form wrapped up in a white linen cloth, with a crown of thorns upon the head; after them the youthful acolyte Tarcisius bearing a censer steaming with perfumed smoke; and, after others of the clergy, the venerable Pontiff himself, attended by Reparatus, and another deacon. Diogenes and his sons, with sorrowful countenances, and many others, among whom he could distinguish Sebastian, closed the procession. As many bore lamps or tapers, the figures seemed to move in an unchanging atmosphere of mildest light. And as they passed before him, they chanted the next verse of the psalm: “Quoniam Tu Domine singulariter in spe constituisti me.”[157] “That,” he exclaimed, rousing himself up, “that is for me.” With this thought he had sprung upon his knees; and by an instinct of grace words which he had before heard came back to him like an echo; words suited to the moment; words which he felt that he must speak. He crept forward, faint and feeble, turned along the gallery through which the funeral procession was passing, and followed it, unobserved, at a distance. It entered a chamber and lighted it up, so that a picture
- 41. The Martyr’s Burial. of the Good Shepherd looked brightly down on him. But he would not pass the threshold, where he stood striking his breast and praying for mercy. The body had been laid upon the ground, and other psalms and hymns were sung, and prayers recited, all in that cheerful tone and joyous mood of hopefulness, with which the Church has always treated of death. At length it was placed in the tomb prepared for it, under an arch. While this was being done, Torquatus drew nigh to one of the spectators, and whispered to him the question: “Whose funeral is this?” “It is the deposition,” he answered, “of the blessed Cæcilia, a blind virgin, who this morning fell into the hands of the soldiers, in this cemetery, and whose soul God took to Himself.” “Then I am her murderer,” he exclaimed, with a hollow moan; and staggering forward to the holy bishop’s feet, fell prostrate before him. It was some time before his feelings could find vent in words; when these came, they were the ones he had resolved to utter:
- 42. “Father, I have sinned before heaven, and against Thee, and I am not worthy to be called Thy child.” The Pontiff raised him up kindly, and pressed him to his bosom, saying, “Welcome back, my son, whoever thou art, to thy Father’s house. But thou art weak and faint, and needest rest.” Some refreshment was immediately procured. But Torquatus would not rest till he had publicly avowed the whole of his guilt, including the day’s crimes; for it was still the evening of the same day. All rejoiced at the prodigal’s return, at the lost sheep’s recovery. Agnes looked up to heaven from her last affectionate glance on the blind virgin’s shroud, and thought that she could almost see her seated at the feet of her Spouse, smiling, with her eyes wide open, as she cast down a handful of flowers on the head of the penitent, the first-fruits of her intercession in heaven. Jesus cures the Blind Man, from a picture in the Cemetery of St. Domitilla. Diogenes and his sons took charge of him. An humble lodging was procured for him, in a Christian cottage near, that he might not be within the reach of temptation, or of vengeance, and he was enrolled in the class of penitents, where years of expiation, shortened by the intercession of confessors—that is, future martyrs—would prepare him for full re- admission to the privileges he had forfeited.[158] C H A P T E R X I X . TWOFOLD REVENGE.
- 43. EBASTIAN’S visit to the cemetery had been not merely to take thither for sepulture the relics of the first martyr, but also to consult with Marcellinus about his safety. His life was too valuable to the Church to be sacrificed so early, and Sebastian knew how eagerly it was sought. Torquatus now confirmed this, by communicating Fulvius’s designs, and the motive of his attendance at the December ordination. The usual papal residence was no longer safe; and a bold idea had been adopted by the courageous soldier,—the “Protector of the Christians,” as his acts tell us he had been authoritatively called. It was to lodge the Pontiff where no one could suspect him to be, and where no search would be dreamt of, in the very palace of the Cæsars.[159] Efficiently disguised, the holy Bishop left the cemetery, and, escorted by Sebastian and Quadratus, was safely housed in the apartments of Irene, a Christian lady of rank, who lived in a remote part of the Palatine, in which her husband held a household office. Early next morning Sebastian was with Pancratius. “My dear boy,” he said, “you must leave Rome instantly, and go into Campania. I have horses ready for you and Quadratus; and there is no time to be lost.” “And why, Sebastian?” replied the youth, with sorrowful face and tearful eye. “Have I done something wrong, or are you doubtful of my fortitude?” “Neither, I assure you. But you have promised to be guided by me in all things, and I never considered your obedience more necessary than now.” “Tell me why, good Sebastian, I pray.” “It must be a secret as yet.” “What, another secret?” “Call it the same, to be revealed at the same time. But I can tell you what I want you to do, and that I think will satisfy you. Corvinus has got orders to seize on Chromatius and all his community, yet young in the faith, as the wretched example of Torquatus has shown us; and, what is worse, to put your old master Cassianus, at Fundi, to a cruel death. I want you to hasten before his messenger (perhaps he may go himself), and put them on their guard.” Pancratius looked up brightly again; he saw that Sebastian trusted him. “Your wish is enough reason for me,” said he, smiling; “but I would go the
- 44. world’s end to save my good Cassianus, or any other fellow-Christians.” He was soon ready, took an affectionate leave of his mother; and before Rome had fully shaken off sleep, he and Quadratus, each with well- furnished saddle-bags on their powerful steeds, were trotting across the campagna of Rome, to reach the less-frequented, and safer, track of the Latin way. Corvinus having resolved to keep the hostile expedition in his own hands, as honorable, lucrative, and pleasant, it was delayed a couple of days, both that he might feel more comfortable about his shoulders, and that he might make proper preparations. He had a chariot hired, and engaged a body of Numidian runners, who could keep up with a carriage at full speed. But he was thus two days behind our Christians, though he, of course, travelled by the shorter and more beaten Appian road. When Pancratius arrived at the Villa of Statues, he found the little community already excited, by the rumors, which had reached it, of the edict’s publication. He was welcomed most warmly by all; and Sebastian’s letter of advice was received with deep respect. Prayer and deliberation succeeded its perusal, and various resolutions were taken. Marcus and Marcellianus, with their father Tranquillinus, had already gone to Rome for the ordination. Nicostratus, Zoë, and others followed them now. Chromatius, who was not destined for the crown of martyrdom, though commemorated, by the Church, with his son, on the 11th of August, found shelter for a time in Fabiola’s villa, for which letters had been procured from its mistress, without her knowing the reason why; for he wished to remain in the neighborhood a little while longer. In fine, the villa ad Statuas was left in charge of a few faithful servants, fully to be depended upon. When the two messengers had given themselves and their horses a good rest, they travelled, by the same road as Torquatus had lately trodden, to Fundi, where they put up at an obscure inn out of the town, on the Roman road. Pancratius soon found out his old master, who embraced him most affectionately. He told him his errand, and entreated him to fly, or at least conceal himself. “No,” said the good man, “it must not be. I am already old, and I am weary of my unprofitable profession. I and my servant are the only two Christians in the town. The best families have, indeed, sent their children to my school, because they knew it would be kept as moral as paganism will
- 45. permit; but I have not a friend among my scholars, by reason of this very strictness. And they want even the natural refinement of Roman heathens. They are rude provincials; and I believe there are some among the elder ones who would not scruple to take my life, if they could do so with impunity.” “What a wretched existence indeed, Cassianus, you must be leading! Have you made no impression on them?” “Little or none, dear Pancratius. And how can I, while I am obliged to make them read those dangerous books, full of fables, which Roman and Greek literature contain? No, I have done little by my words; perhaps my death may do more for them.” Pancratius found all expostulation vain, and would have almost joined him in his resolution to die; only he had promised Sebastian not to expose his life during the journey. He, however, determined to remain about the town till he saw the end. Corvinus arrived with his men at the villa of Chromatius; and early in the morning rushed suddenly through the gates, and to the house. He found it empty. He searched it through and through, but discovered neither a person, a book, nor a symbol of Christianity. He was confounded and annoyed. He looked about; and having found a servant working in the garden, asked him where his master was. “Master no tell slave where he go,” was the reply, in a latinity corresponding to such a rude phraseology. “You are trifling with me. Which way did he and his companions go?” “Through yonder gate.” “And then?” “Look that way,” answered the servant. “You see gate? very well; you see no more. Me work here, me see gate, me see no more.” “When did they go? at least you can answer that.” “After the two come from Rome.” “What two? Always two, it seems.” “One good youth, very handsome, sing so sweet. The other very big, very strong, oh, very. See that young tree pulled up by the roots? He do that as easy as me pull my spade out of the ground.”
- 46. “The very two,” exclaimed Corvinus, thoroughly enraged. “Again that dastardly boy has marred my plans and destroyed my hopes. He shall suffer well for it.” As soon as he was a little rested, he resumed his journey, and determined to vent all his fury on his old master; unless, indeed, he whom he considered his evil genius should have been there before him. He was engaged during his journey, in plotting vengeance upon master and fellow- student; and he was delighted to find, that one at least was at Fundi, when he arrived. He showed the governor his order for the arrest and punishment of Cassianus, as a most dangerous Christian; but that officer, a humane man, remarked that the commission superseded ordinary jurisdiction in the matter, and gave Corvinus full power to act. He offered him the assistance of an executioner, and other requisites; but they were declined. Corvinus had brought an abundant supply of strength and cruelty, in his own body- guard. He took, however, a public officer with him. He proceeded to the school-house when filled with scholars; shut the doors, and reproached Cassianus, who advanced with open hand and countenance to greet him, as a conspirator against the state and a perfidious Christian. A shout arose from the boyish mob; and by its tone, and by the look which he cast around, Corvinus learnt there were many present like himself—young bears’ cubs, with full-grown hyenas’ hearts within them. “Boys!” he shouted out, “do you love your master Cassianus? He was once mine too, and I owe him many a grudge.” A yell of execration broke out from the benches. “Then I have good news for you; here is permission from the divine emperor Maximian for you to do what you like to him.” A shower of books, writing tablets, and other school missiles, was directed against the master, who stood unmoved, with his arms folded, before his persecutor. Then came a rush from all sides, with menacing attitudes of a brutal onslaught. “Stop, stop,” cried out Corvinus, “we must go more systematically to work than this.” He had reverted in thought to the recollection of his own sweet school- boy days; that time which most look back on from hearts teeming with softer feelings than the contemplation of present things can suggest. He indulged in the reminiscence of that early season in which others find but
- 47. the picture of unselfish, joyous, happy hours; and he sought in the recollection what would most have gratified him then, that he might bestow it as a boon on the hopeful youths around him. But he could think of nothing that would have been such a treat to him, as to pay back to his master every stroke of correction, and write in blood upon him every word of reproach that he had received. Delightful thought, now to be fulfilled! It is far from our intention to harrow the feelings of our gentle readers by descriptions of the cruel and fiendish torments inflicted by the heathen persecutors on our Christian forefathers. Few are more horrible, yet few better authenticated, than the torture practised on the martyr Cassianus. Placed, bound, in the midst of his ferocious young tigers, he was left to be the lingering victim of their feeble cruelty. Some, as the Christian poet Prudentius tells us, cut their tasks upon him with the steel points used in engraving writing on wax-covered tablets; others exercised the ingenuity of a precocious brutality, by inflicting every possible torment on his lacerated body. Loss of blood, and acute pain, at length exhausted him, and he fell on the floor without power to rise. A shout of exultation followed, new insults were inflicted, and the troop of youthful demons broke loose, to tell the story of their sport at their respective homes. To give Christians decent burial never entered into the minds of their persecutors; and Corvinus, who had glutted his eyes with the spectacle of his vengeance, and had urged on the first efforts at cruelty of his ready instruments, left the expiring man where he lay, to die unnoticed. His faithful servant, however, raised him up, and laid him on his bed, and sent a token, as he had preconcerted, to Pancratius, who was soon at his side, while his companion looked after preparations for their departure. The youth was horrified at what he beheld, and at the recital of his old master’s exquisite torture, as he was edified by the account of his patience. For not a word of reproach had escaped him, and prayer alone had occupied his thoughts and tongue. Cassianus recognized his dear pupil, smiled upon him, pressed his hand in his own, but could not speak. After lingering till morning he placidly expired. The last rites of Christian sepulture were modestly paid to him on the spot, for the house was his; and Pancratius hurried from the scene, with a heavy heart and a no slight rising of its indignation, against the heartless savage who had devised and witnessed, without remorse, such a tragedy. He was mistaken, however. No sooner was his revenge fulfilled than Corvinus felt all the disgrace and shame of what he had done; he feared it
- 48. should be known to his father, who had always esteemed Cassianus; he feared the anger of the parents, whose children he had that day effectually demoralized, and fleshed to little less than parricide. He ordered his horses to be harnessed, but was told they must have some more hours’ rest. This increased his displeasure; remorse tormented him, and he sat down to drink, and so drown care and pass time. At length he started on his journey, and after baiting for an hour or two, pushed on through the night. The road was heavy from continued rain, and ran along the side of the great canal which drains the Pontine marshes, and between two rows of trees. Corvinus had drunk again at his halt, and was heated with wine, vexation, and remorse. The dragging pace of his jaded steeds provoked him, and he kept lashing them furiously on. While they were thus excited they heard the tramp of horses coming fast on behind, and dashed forward at an uncontrollable speed. The attendants were soon left at a distance, and the frightened horses passed between the trees on to the narrow path by the canal, and galloped forward, rocking the chariot from side to side at a reckless rate. The horsemen behind hearing the violent rush of hoofs and wheels, and the shout of the followers, clapped spurs to their horses, and pushed gallantly forward. They had passed the runners some way when they heard a crash and a plunge. The wheel had struck the trunk of a tree, the chariot had turned over, and its half-drunken driver had been tossed head over heels into the water. In a moment Pancratius was off his horse and by the side of the canal, together with his companion. By the faint light of the rising moon, and by the sound of his voice, the youth recognized Corvinus struggling in the muddy stream. The side was not deep, but the high clayey bank was wet and slimy, and every time he attempted to climb it his foot slipped, and he fell back into the deep water in the middle. He was, in fact, already becoming benumbed and exhausted by his wintry bath. “It would serve him right to leave him there,” muttered the rough centurion. “Hush, Quadratus! how can you say so? give me hold of your hand. So!” said the youth, leaning over the bank and seizing his enemy by his arm, just as he was relaxing his hold on a withered shrub, and falling back fainting into the stream. It would have been his last plunge. They pulled him out and laid him on the road, a pitiable figure for his greatest foe. They chafed his
- 49. temples and hands, and he had begun to revive when his attendants came up. To their care they consigned him, together with his purse, which had fallen from his belt as they drew him from the canal. But Pancratius took possession of his own pen-knife, which dropped out with it, and which Corvinus carried about him, as evidence to convict him of having cut down the edict. The servants pretended to Corvinus, when he had regained consciousness, that they had drawn him out of the water, but that his purse must have been lost in it, and lay still buried in the deep mud. They bore him to a neighboring cottage, while the carriage was being repaired, and had a good carouse with his money while he slept. The Anchor and Fish, emblematic of Christianity, found in the Catacombs. Two acts of revenge had been thus accomplished in one day,—the pagan and the Christian. C H A P T E R X X . THE PUBLIC WORKS. F, before the edict, the Thermæ of Dioclesian were being erected by the labor and sweat of Christian prisoners, it will not appear surprising, that their number and their sufferings should have greatly increased, with the growing intensity of a most savage persecution. That emperor himself was expected for the inauguration of his favorite building, and hands were doubled on the work to expedite its completion. Chains of supposed culprits arrived each day from the port of Luna, from Sardinia, and even from the Crimea, or Chersonesus, where they had been engaged in quarries or mines; and were put to labor in the harder departments of the building art. To transport materials, to saw and cut stone and marble, to mix the mortar, and
- 50. to build up the walls, were the duties allotted to the religious culprits, many of whom were men little accustomed to such menial toil. The only recompense which they received for their labor, was that of the mules and oxen which shared their occupation. Little better, if better, than a stable to sleep in, food sufficient in quantity to keep up their strength, clothing enough to guard them from the inclemency of the season, this was all they had to expect. Fetters on their ankles, heavy chains to prevent their escape, increased their sufferings; and task-masters, acceptable in proportion as they were unreasonable, watched every gang with lash or stick in hand, ever ready to add pain to toil, whether it were to vent their own wanton cruelty upon unresisting objects, or to please their crueller masters. But the Christians of Rome took peculiar care of these blessed confessors, who were particularly venerated by them. Their deacons visited them, by bribing their guards; and young men would boldly venture among them, and distribute more nourishing food, or warmer clothing to them, or give them the means of conciliating their keepers, so as to obtain better treatment at their hands. They would then also recommend themselves to their prayers, as they kissed the chains and the bruises, which these holy confessors bore for Christ. This assemblage of men, convicted of serving faithfully their divine Master, was useful for another purpose. Like the stew in which the luxurious Lucullus kept his lampreys ready fattened for a banquet; like the cages in which rare birds, the pens in which well-fed cattle, were preserved for the sacrifice, or the feast of an imperial anniversary; like the dens near the amphitheatre, in which ferocious beasts were fed for exhibition at the public games; just so were the public works the preserves, from which at any time could be drawn the materials for a sanguinary hecatomb, or a gratification of the popular appetite for cruel spectacles, on any occasion of festivity; public stores of food for those fierce animals, whenever the Roman people wished to share in their savage propensities. Such an occasion was now approaching. The persecution had lingered. No person of note had been yet captured; the failures of the first day had not
- 51. been fully repaired; and something more wholesale was expected. The people demanded more sport; and an approaching imperial birthday justified their gratification. The wild beasts, which Sebastian and Pancratius had heard, yet roared for their lawful prey. “Christianos ad leones” might seem to have been interpreted by them, as meaning “that the Christians of right belonged to them.” One afternoon, towards the end of December, Corvinus proceeded to the Baths of Dioclesian, accompanied by Catulus, who had an eye for proper combatants in the amphitheatre, such as a good dealer would have for cattle at a fair. He called for Rabirius, the superintendent of the convict department, and said to him: “Rabirius, I am come by order of the emperor, to select a sufficient number of the wicked Christians under your charge, for the honor of fighting in the amphitheatre, on occasion of the coming festival.” “Really,” answered the officer, “I have none to spare. I am obliged to finish the work in a given time, and I cannot do so, if I am left short of hands.” “I cannot help that; others will be got to replace those that are taken from you. You must walk Catulus and myself through your works, and let us choose those that will suit us.” Rabirius, grumbling at this unreasonable demand, submitted nevertheless to it, and took them into a vast area, just vaulted over. It was entered by a circular vestibule lighted from above, like the Pantheon. This led into one of the shorter arms of a cruciform hall of noble dimensions, into which opened a number of lesser, though still handsome, chambers. At each angle of the hall, where the arms intersected one another, a huge granite pillar of one block had to be erected. Two were already in their places, one was girt with ropes delivered round capstans, ready to be raised on the morrow. A number of men were actively employed in making final preparations. Catulus nudged Corvinus, and pointed, with his thumb, to two fine youths, who, stripped slave-fashion to their waists, were specimens of manly athletic forms. “I must have those two, Rabirius,” said the willing purveyor to wild beasts; “they will do charmingly. I am sure they are Christians, they work so cheerfully.”
- 52. “I cannot possibly spare them at present. They are worth six men, or a pair of horses, at least, to me. Wait till the heavy work is over, and then they are at your service.” “What are their names, that I may take a note of them? And mind, keep them up in good condition.” “They are called Largus and Smaragdus; they are young men of excellent family, but work like plebeians, and will go with you nothing loth.” “They shall have their wish,” said Corvinus, with great glee. And so they had later. As they went through the works, however, they picked out a number of captives, for many of whom Rabirius made resistance, but generally in vain. At length they came near one of those chambers which flanked the eastern side of the longer arm of the hall. In one of them they saw a number of convicts (if we must use the term) resting after their labor. The centre of the group was an old man, most venerable in appearance, with a long white beard streaming on his breast, mild in aspect, gentle in word, cheerful in his feeble action. It was the confessor Saturninus, now in his eightieth year, yet loaded with two heavy chains. At each side were the more youthful laborers, Cyriacus and Sisinnius, of whom it is recorded, that, in addition to their own task-work, one on each side, they bore up his bonds. Indeed, we are told that their particular delight was, over and above their own assigned portion of toil, to help their weaker brethren, and perform their work for them.[160] But their time was not yet come; for both of them, before they received their crowns, were ordained deacons in the next pontificate. Several other captives lay on the ground, about the old man’s feet, as he, seated on a block of marble, was talking to them, with a sweet gravity, which riveted their attention, and seemed to make them forget their sufferings. What was he saying to them? Was he requiting Cyriacus for his extraordinary charity, by telling him that, in commemoration of it, a portion of the immense pile which they were toiling to raise, would be dedicated to God, under his invocation, become a title, and close its line of titulars by an illustrious name?[161] Or was he recounting another more glorious vision, how this smaller oratory was to be superseded and absorbed by a glorious temple in honor of the Queen of Angels, which should comprise the entire of that superb hall, with its vestibule, under the directing skill of the
- 53. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookultra.com