This document is a solutions manual for a textbook on communication systems. It provides step-by-step solutions to problems from each chapter of the textbook. The problems cover topics such as signal representations using Fourier series and integrals, power calculations for periodic signals, and bandpass signal representations. The solutions demonstrate techniques for analyzing and working with signals commonly encountered in electrical communication systems.
In this presentation we discuss about a particular type of analog communication waves that is wideband frequency modulation. In this slide, its expression is discussed along with graphical visuals. Not forgetting its power and bandwidth as well. We also see the use of bessel function and the block diagrams that help to form this type of waves.
In this presentation we discuss about a particular type of analog communication waves that is wideband frequency modulation. In this slide, its expression is discussed along with graphical visuals. Not forgetting its power and bandwidth as well. We also see the use of bessel function and the block diagrams that help to form this type of waves.
4 matched filters and ambiguity functions for radar signalsSolo Hermelin
Matched filters (Part 1 of 2) maximizes the output signal-to-noise ratio for a known radar signal at a predefined time.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Describes Pulse Compression in Radar Systems.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
Since some figures were not downloaded, I recommend to see this presentation on my website under RADAR Folder, Signal Processing subfolder.
A Refined Skew Matrix Model of the CIM3 in the Up-Mixer Extending the Duality...Ealwan Lee
Presented at RFIT-2022
Session info : Advanced Circuit and System Design
Date : Aug 30, 2022
Place : Busan, Korea
Cite with the shortened URL if you like
https://lnkd.in/gBhJmSRa
[ URL/DOI of the paper/preprint ]
DOI) 10.1109/RFIT54256.2022.9882474
URL) https://ieeexplore.ieee.org/document/9882474
https://www.researchgate.net/publication/363510202_A_Refined_Skew_Matrix_Model_of_the_CIM3_in_the_Up-Mixer_Extending_the_Duality_of_IQ_Imbalance
Following(corrective) work of https://lnkd.in/gwJhqgb8
Solutions Manual Digital & Analog Communication Systems (8th Edition) – Answe...JesseDaisy12
Solutions Manual Digital & Analog Communication Systems (8th Edition)
This is completed solutions manual for Digital and Analog Communication Systems 8th Editor.Edition (United States), by Leon W. Couch, II, Pearson/Prentice Hall, Upper Saddle River, NJ,
2011.
This Solutions Manual for Digital and Analog Communication Systems, 8th Edition (United States) contains complete solutions for the homework problems in the 8th Edition.
If the problem is designed for a MATLAB or MATHCAD computer solution, then the MATHCAD printed solution is shown. (MATHCAD solutions are shown since they clearly display the algorithms used and the output takes up less space.)
Click here to view Example:
(Chapter 1) : Solutions-Manual-Digital-&-Analog-Communication-Systems-8-editor Example
4 matched filters and ambiguity functions for radar signalsSolo Hermelin
Matched filters (Part 1 of 2) maximizes the output signal-to-noise ratio for a known radar signal at a predefined time.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Describes Pulse Compression in Radar Systems.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
Since some figures were not downloaded, I recommend to see this presentation on my website under RADAR Folder, Signal Processing subfolder.
A Refined Skew Matrix Model of the CIM3 in the Up-Mixer Extending the Duality...Ealwan Lee
Presented at RFIT-2022
Session info : Advanced Circuit and System Design
Date : Aug 30, 2022
Place : Busan, Korea
Cite with the shortened URL if you like
https://lnkd.in/gBhJmSRa
[ URL/DOI of the paper/preprint ]
DOI) 10.1109/RFIT54256.2022.9882474
URL) https://ieeexplore.ieee.org/document/9882474
https://www.researchgate.net/publication/363510202_A_Refined_Skew_Matrix_Model_of_the_CIM3_in_the_Up-Mixer_Extending_the_Duality_of_IQ_Imbalance
Following(corrective) work of https://lnkd.in/gwJhqgb8
Solutions Manual Digital & Analog Communication Systems (8th Edition) – Answe...JesseDaisy12
Solutions Manual Digital & Analog Communication Systems (8th Edition)
This is completed solutions manual for Digital and Analog Communication Systems 8th Editor.Edition (United States), by Leon W. Couch, II, Pearson/Prentice Hall, Upper Saddle River, NJ,
2011.
This Solutions Manual for Digital and Analog Communication Systems, 8th Edition (United States) contains complete solutions for the homework problems in the 8th Edition.
If the problem is designed for a MATLAB or MATHCAD computer solution, then the MATHCAD printed solution is shown. (MATHCAD solutions are shown since they clearly display the algorithms used and the output takes up less space.)
Click here to view Example:
(Chapter 1) : Solutions-Manual-Digital-&-Analog-Communication-Systems-8-editor Example
QAM modulation is a combination of Amplitude Shift Keying and Phase Shift Keying, both carrier wave is modulated by changing both its amplitude and phase...
Reference: Mobile payment industry in china 2012-2015 C. Keiko Funahashi
Referenced in presentation, "The Seven Wonders of China's Mobile World"
http://www.slideshare.net/ckeikofunahashi/m-learncon-session-907-ckeikofunahashi
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
1. Solutions Manual
to accompany
Communication
Systems
An Introduction to Signals and Noise in
Electrical Communication
Fourth Edition
A. Bruce Carlson
Rensselaer Polytechnic Institute
Paul B. Crilly
University of Tennessee
Janet C. Rutledge
University of Maryland at Baltimore
3. 2-1
Chapter 2
2.1-1
0
0
0
/2
2 ( )
/2
0
sinc( )
0 otherwise
jj
T
j m n f t j
n T
Ae n mAe
c e dt Ae m n
T
φφ
π φ−
−
=
= = − =
∫
2.1-2
0 0
0
0
/ 4 / 2
0 / 4
0 0 0
( ) 0
2 2 2 2
cos ( )cos sin
2
T T
n
T
c v t
nt nt A n
c A dt A dt
T T T n
π π π
π
=
= + − =∫ ∫
n 0 1 2 3 4 5 6 7
nc 0 2 /A π 0 2 / 3A π 0 2 / 5A π 0 2 / 7A π
arg nc 0 180± ° 0 180± °
2.1-3
0
0
/2
20
0 0 0
( ) / 2
2 2 2
cos sin (cos 1)
( )
T
n
c v t A
At nt A A
c A dt n n
T T T n n
π
π π
π π
= =
= − = − −
∫
n 0 1 2 3 4 5 6
nc 0.5A 0.2A 0 0.02A 0 0.01A 0
arg nc 0 0 0 0
2.1-4
0 / 2
0
0
0 0
2 2
cos 0
T t
c A
T T
π
= =∫ (cont.)
4. 2-2
( ) ( )
[ ]
0
0
/2
/2
0 0
0
0 0 0 0 0 0 0
sin 2 / sin 2 /2 2 2 2
cos cos
4( ) / 4( ) /
/ 2 1
sinc(1 ) sinc(1 )
0 otherwise2
T
T
n
n t T n t Tt nt A
c A dt
T T T T n T n T
A nA
n n
π π π ππ π
π π π π
− +
= = +
− +
= ±
= − + + =
∫
2.1-5
0
0
/ 2
0
0 0
( ) 0
2 2
sin (1 cos )
T
n
c v t
nt A
c j A dt j n
T T n
π
π
π
= =
= − = − −∫
n 1 2 3 4 5
nc 2 /A π 0 2 / 3A π 2 / 5A π
arg nc 90− ° 90− ° 90− °
2.1-6
0 ( ) 0c v t= =
( ) ( )
[ ]
0
0
/2
/2
0 0
0
0 0 0 0 0 0 0
sin 2 / sin 2 /2 2 2 2
sin sin
4( )/ 4( )/
/ 2 1
sinc(1 ) sinc(1 )
0 otherwise2
T
T
n
n t T n t Tt nt A
c j A dt j
T T T T n T n T
jA nA
j n n
π π π ππ π
π π π π
− +
= − = − −
− +
= ±
= − − − + =
∫
m
2.1-7
]
0 0
0 0
0
/ 2
0 / 2
0
1
( ) ( )
T T
jn t jn t
n
T
c v t e dt v t e dt
T
ω ω− −= +
∫ ∫
0 0
0 0 0 0
0
0
0
/ 2
/ 2
0
/ 2 0
/ 2
0
where ( ) ( /2)
( )
T T
jn t jn jn T
T
T
jn tjn
v t e dt v T e e d
e v t e dt
ω ω λ ω
ωπ
λ λ− − −
−
= +
= −
∫ ∫
∫
since 1 for even , 0 for evenjn
ne n c nπ
= =
5. 2-3
2.1-8
2 2 2 2 2 2
0 0 0 0 0 0 0 0
1
0
2
2 2 2 2
2
2 2 2 2 2 2
2 2 sinc 2 sinc2 2 sinc3
1
where 4
1 1 1 3
1 2sinc 2sinc 2sinc 0.23
16 4 2 4
2 1 1 3 5 3 7
1 2sinc 2sinc 2sinc 2sinc 2sinc 2sinc
16 4 2 4 4 2 4
n
n
P c c Af Af f Af f Af f
f
A
f P A
A
f P
τ τ τ τ τ τ τ
τ
τ
τ
∞
=
= + = + + + +
=
> = + + + =
> = + + + + + +
∑ L
2
2
2 2 2
0.24
1 1 1
1 2sinc 2sinc 0.21
2 16 4 2
A
A
f P A
τ
=
> = + + =
2.1-9
0 0
0
2
2 2
/ 2 / 2
/ 2 0
0 0 0 0
2 2 2
2 2 2
0 02 2 2
0 even
2
odd
n
41 2 4 1
a) 1 1
3
4 4 4
2 2 2 0.332 so / 99.6%
9 25
8 8 8
b) ( ) cos cos3 cos5
9 25
n
T T
T
n
c
n
t t
P dt dt
T T T T
P P P
v t t t
π
π π π
ω ω ω
π π π
−
=
= − = − =
′ ′= + + = =
′ = + +
∫ ∫
0t
2.1-10
( )
0
0
2 2 2
/ 2 2
/ 2
0
0 even
2
odd
1 2 2 2
a) 1 1 2 0.933 so / 93.3%
3 5
n
T
T
n
c j
n
n
P dt P P P
T
π
π π π−
= −
′ ′= = = + + = =
∫
(cont.)
6. 2-4
( ) ( ) ( )
( ) ( ) ( )
0 0 0
0 0 0
4 4 4
b) ( ) cos 90 cos 3 90 cos 5 90
3 5
4 4 4
sin sin 3 sin 5
3 5
v t t t t
t t t
ω ω ω
π π π
ω ω ω
π π π
′ = − ° + − ° + − °
= + +
2.1-11
0
2
0
0 0
1/2 01 1
1/2 03
T
n
nt
P dt c
n nT T π
=
= = =
≠
∫
4 4
4 4 4
odd
2 2 1 1 1 1
2 2
1 3 5 3n
P
nπ π
∞
= = + + + =
∑ L
2 2
2 2 2
1 1 1 4 1 1
Thus,
1 2 3 2 3 4 6
π π
+ + + = − =
L
2.1-12
0
2
/ 2
20
0 0
0 even2 4 1
1
(2/ ) odd3
T
n
nt
P dt c
n nT T π
= − = =
∫
2 2
2 2 2 2
1
1 1 1 2 1 1 1 1
2
2 2 4 4 1 2 3 3n
P
nπ π
∞
=
= + = + + + + =
∑ L
4 4
4 4 4 4
1 1 1 1
Thus,
1 3 5 2 2 3 96
π π
+ + + = =
⋅
L
2.2-1
( )
( )
( )
( )
[ ]
/ 2
0
( ) 2 cos cos2
sin 2 sin 2
2 22 sinc( 1/2) sinc( 1/2)
22 2 2 2
t
V f A ftdt
f f
A
A f f
f f
τ
π π
τ τ
π π
τ τ
π
π
τ
τ τ
π π
τ
τ τ
π π
+
+
=
−
= + = − + +
−
∫
(cont.)
7. 2-5
2.2-2
( )
( )
( )
( )
[ ]
/ 2
0
2 2
2 2
2
( ) 2 sin cos2
sin 2 sin 2
2 22 sinc( 1) sinc( 1)
22 2 2 2
t
V f j A ftdt
f f
A
j A j f f
f f
τ
π π
τ τ
π π
τ τ
π
π
τ
τ τ
π π
τ
τ τ
π π
+
+
= −
−
= − − = − − − +
−
∫
2.2-3
2 2
20
2
( ) 2 cos 2sin 1 1 sinc
( ) 2
t A
V f A A tdt A f
τ τ ωτ
ω τ τ
τ ωτ
= − = − + =
∫
2.2-4
20
2
( ) 2 sin (sin cos )
( )
(sinc2 cos2 )
t A
V f j A tdt j
A
j f f
f
τ τ
ω ωτ ωτ ωτ
τ ωτ
τ π τ
π
= − = − −
= − −
∫
2.2-5
2
2
2
1
( ) sinc2
2 2
1 1 1
sinc2
2 2 4 2
f
v t Wt
W W
f
Wt dt df df
W W W W
∞ ∞ ∞
−∞ −∞ −∞
= ↔ Π
= Π = =
∫ ∫ ∫
8. 2-6
2.2-6
( )
2 2 2
2
2 20 0
2
2 arctan
2 (2 )
W
bt A A A W
E Ae dt E df
b b f b b
π
π π
∞
−
′= = = =
+∫ ∫
50% / 22 2
arctan
84% 2 /
W bE W
W bE b
ππ
ππ
=′
= =
=
2.2-7
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
j t
j t
v t w t dt v t W f e df dt
W f v t e dt df W f V f df
ω
ω
∞ ∞ ∞
−∞ −∞ −∞
∞ ∞ ∞
− −
−∞ −∞ −∞
=
= = −
∫ ∫ ∫
∫ ∫ ∫
22
( ) *( ) when ( ) is real, so ( ) ( ) *( ) ( )V f V f v t v t dt V f V f df V f df
∞ ∞ ∞
−∞ −∞ −∞
− = = =∫ ∫ ∫
2.2-8
2 2 2 ( )
( ) ( ) ( ) ( )
Let ( ) ( ) so ( ) ( ) and ( ) ( )
Hence ( ) ( ) ( ) ( )
j ft j ft j f t
w t e dt w t e dt w t e dt W f
z t w t Z f W f W f Z f
v t z t dt V f Z f df
π π π
∗ ∗
∞ ∞ ∞
∗ − − − ∗
−∞ −∞ −∞
∗ ∗ ∗
∞ ∞
−∞ −∞
= = =
= = − = −
= −
∫ ∫ ∫
∫ ∫
2.2-9
1
sinc so sinc
2 2
( ) sinc ( ) for
2 2
t f
A Af At
A A A
t f
v t V f A
τ τ
τ τ
Π ↔ ↔ Π
= ↔ = Π =
2.2-10
[ ]
[ ]
[ ]
cos sinc( 1/2) sinc( 1/2)
2
( )
so sinc( 1/2) sinc( 1/2) cos cos
2
Let and 2 ( ) sinc(2 1/2) sinc(2 1/2)
t t B
B f f
B f f f f
t t B B
B A W z t AW Wt Wt
π τ
τ τ
τ τ
τ π π
τ τ
τ τ τ τ
τ
Π ↔ − + +
− −
− + + ↔ Π = Π
= = ⇒ = − + +
2.2-11
[ ]
[ ]
[ ]
2
sin sinc( 1) sinc( 1)
2
2 ( ) 2
so sinc( 1) sinc( 1) sin sin
2
Let and 2 ( ) sinc(2 1) sinc(2 1)
t t B
B j f f
B f f f f
j t t B B
B jA W z t AW Wt Wt
π τ
τ τ
τ τ
τ π π
τ τ
τ τ τ τ
τ
Π ↔ − − + +
− −
− − + + ↔ Π = − Π
= − = ⇒ = − + +
9. 2-7
2.2-12
( )
( )
( )
2
2 2 2 2 2 2
2
2
2
22 2 0 2 2
2
2 30 2 2
2 4 /
(2 ) (2 ) (2 )
1 /
2
2
1 1
Thus,
2 2 4
b t a t
a t
b a a
e e
b f a f a f
a a df
e dt df
a a f a f
dx
a a aa x
π
π
π π
π π π
π
π π
π π
π
− −
∞ ∞ ∞−
−∞ −∞
∞
↔ ⇒ ↔ =
+ + +
= = =
+ +
= =
+
∫ ∫ ∫
∫
2.3-1
( ) ( ) ( ) where v( ) ( / ) sinc
so Z( ) ( ) ( ) 2 sinc cos2j T j T
z t v t T v t T t A t A f
f V f e V f e A f fTω ω
τ τ τ
τ τ π−
= − + + = Π ↔
= + =
2.3-2
2 2
( ) ( 2 ) 2 ( ) ( 2 ) where v( ) ( / ) sinc
( ) ( ) ( ) ( ) 2 (sinc )(1 cos4 )j T j T
z t v t T v t v t T t a t A f
Z f V f e V f V f e A f fTω ω
τ τ τ
τ τ π−
= − + + + = Π ↔
= + + = +
2.3-3
2 2
( ) ( 2 ) 2 ( ) ( 2 ) where ( ) ( / ) sinc
( ) ( ) 2 ( ) ( ) 2 (sinc )(cos4 1)j T j T
z t v t T v t v t T v t a t A f
Z f V f e V f V f e A f fTω ω
τ τ τ
τ τ π−
= − − + + = Π ↔
= − + = −
2.3-4
/ 2
/ 2
( ) ( )
2
( ) 2 sinc2 ( ) sincj T j T
t T t T
v t A B A
T T
V f AT fTe B A T fTeω ω− −
− −
= Π + − Π
= + −
10. 2-8
2.3-5
2 2
2 2
( ) ( )
4 2
( ) 4 sinc4 2( ) sinc2j T j T
t T t T
v t A B A
T T
V f AT fTe B A T fTeω ω− −
− −
= Π + − Π
= + −
2.3-6
/ /
1
Let ( ) ( ) ( ) ( / )
1
Then ( ) [ ( / )] ( / ) so ( ) ( ) ( / )d dj t a j t a
d d
w t v at W f V f a
a
z t v a t t a w t t a Z f W f e V f a e
a
ω ω− −
= ↔ =
= − = − = =
2.3-7
2 ( )
( ) ( ) ( ) ( )c c cj t j t j f f tj t
cv t e v t e e dt v t e dt V f fω ω πω
∞ ∞ − −−
−∞ −∞
= = = − ∫ ∫F
2.3-8
[ ]
( ) ( / )cos with 2 /
( ) sinc( ) sinc( ) sinc( 1/2) sinc( 1/2)
2 2 2
c c c
c c
v t A t t f
A A A
V f f f f f f f
τ ω ω π π τ
τ τ τ
τ τ τ τ
= Π = =
= − + + = − + +
2.3-9
[ ]
/ 2 /2
( ) ( / )cos( /2) with 2 2 /
( ) sinc( ) sinc( )
2 2
sinc( 1) sinc( 1)
2
c c c
j j
c c
v t A t t f
e e
V f A f f A f f
A
j f f
π π
τ ω π ω π π τ
τ τ τ τ
τ
τ τ
−
= Π − = =
= − + +
= − − − +
2.3-10
2
2 2 2 2
2
( ) ()cos ( )
1 (2 )
1 1
( ) ( ) ( )
2 2 1 4 ( ) 1 4 ( )
t
c
c c
c c
A
z t v t t v t Ae
f
A A
Z f V f f V f f
f f f f
ω
π
π π
−
= = ↔
+
= − + + = +
+ − + +
2.3-11
/ 2 / 2
( ) ()cos( /2) ( ) for 0
1 2
/ 2 / 2
( ) ( ) ( )
2 2 1 2 ( ) 1 2 ( )
/ 2 / 2
2 ( ) 2 ( )
t
c
j j
c c
c c
c c
A
z t v t t v t Ae t
j f
e e jA jA
Z f V f f V f f
j f f j f f
A A
j f f j f f
π π
ω π
π
π π
π π
−
−
= − = ≥ ↔
+
−
= − + + = +
+ − + +
= −
− − − +
11. 2-9
2.3-12
( )
2
2
( ) ( ) ( ) 2 sinc2
sin2 2
( ) 2 (2 ) cos2 2 sin2
2 (2 )
1
( ) ( ) sinc2 cos2
2
A t
v t t z t z t A f
d d f A
Z f A f f f
df df f f
d jA
V f Z f f f
j df f
τ
τ τ
π τ
πτ π τ πτ π τ
π τ π τ
τ π τ
π π
= = Π ↔
= = −
−
= = −
−
2.3-13
2 2
22 2 2 2
2
( ) ( ) ( )
(2 )
1 2 2
( )
2 (2 ) (2 )
b t Ab
z t tv t v t Ae
b f
d Ab j Abf
Z f
j df b f b f
π
π π π
−
= = ↔
+
= =
− + +
2.3-14
( ) [ ]
2
2 3
( ) ( ) ( ) for 0
2
1 2
( )
22 2
t A
z t t v t v t Ae t
b j f
d A A
Z f
df b j fj f b j f
π
ππ π
−
= = ≥ ↔
+
= = +− +
2.3-15
2 2
2 2
2 2
2 2
( ) ( / )
2 ( ) ( / )
( ) ( / )
( ) ( / )
1
( ) ( )
2
( ) ( ) 2
1
( ) ( )
2
Both results are equivalent to
bt f b
bt f b
bt f b
bt f b
v t e V f e
b
d j f
a v t b te e
dt b
d f
b te V f e
j df jb
bte jf e
π π
π π
π π
π π
π
π
π
− −
− −
− −
− −
= ↔ =
= − ↔
↔ =
−
↔ −
2.4-1
2
0
2
0
( ) 0 0
0 2
2
2 2
t
y t t
At
A d t
A d A t
λ λ
λ λ
= <
= = < <
= = >
∫
∫
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