CT203 - Signals & Systems Lecture - 1:...
Transcript of CT203 - Signals & Systems Lecture - 1:...
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CT203 - Signals & SystemsLecture - 1: Introduction
3-1-0-4
Aditya Tatu
![Page 2: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/2.jpg)
Contents
Modeling Signals & Systems.
Modeling systems using State machines.
Linear & Linear Time-Invariant systems.
Frequency representation of a signal - Fourier Transform.
LTI systems and their Frequency representation.
Sampling and Reconstruction.
Laplace and z-transform.
Lecture - 1: Introduction 2/16
![Page 3: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/3.jpg)
Contents
Modeling Signals & Systems.
Modeling systems using State machines.
Linear & Linear Time-Invariant systems.
Frequency representation of a signal - Fourier Transform.
LTI systems and their Frequency representation.
Sampling and Reconstruction.
Laplace and z-transform.
Lecture - 1: Introduction 2/16
![Page 4: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/4.jpg)
Contents
Modeling Signals & Systems.
Modeling systems using State machines.
Linear & Linear Time-Invariant systems.
Frequency representation of a signal - Fourier Transform.
LTI systems and their Frequency representation.
Sampling and Reconstruction.
Laplace and z-transform.
Lecture - 1: Introduction 2/16
![Page 5: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/5.jpg)
Contents
Modeling Signals & Systems.
Modeling systems using State machines.
Linear & Linear Time-Invariant systems.
Frequency representation of a signal - Fourier Transform.
LTI systems and their Frequency representation.
Sampling and Reconstruction.
Laplace and z-transform.
Lecture - 1: Introduction 2/16
![Page 6: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/6.jpg)
Contents
Modeling Signals & Systems.
Modeling systems using State machines.
Linear & Linear Time-Invariant systems.
Frequency representation of a signal - Fourier Transform.
LTI systems and their Frequency representation.
Sampling and Reconstruction.
Laplace and z-transform.
Lecture - 1: Introduction 2/16
![Page 7: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/7.jpg)
Contents
Modeling Signals & Systems.
Modeling systems using State machines.
Linear & Linear Time-Invariant systems.
Frequency representation of a signal - Fourier Transform.
LTI systems and their Frequency representation.
Sampling and Reconstruction.
Laplace and z-transform.
Lecture - 1: Introduction 2/16
![Page 8: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/8.jpg)
Contents
Modeling Signals & Systems.
Modeling systems using State machines.
Linear & Linear Time-Invariant systems.
Frequency representation of a signal - Fourier Transform.
LTI systems and their Frequency representation.
Sampling and Reconstruction.
Laplace and z-transform.
Lecture - 1: Introduction 2/16
![Page 9: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/9.jpg)
References
Edward Lee, and Pravin Varaiya. Structure and Interpretationof Signals & Systems. Second Ed., Univ. of California atBerkeley, 2011.Available online at: leevaraiya.org
Chi-Tsong Chen, Signals and Systems: A Fresh Look, StonyBrook University, 2009.Available online at:www.ece.sunysb.edu/~ctchen/media/freshlook.pdf
Oppenheim, Alan V., and A. S. Willsky. Signals and Systems.Prentice Hall, 1982.
... and many many more. RC has a good collection - Explore.
Lecture - 1: Introduction 3/16
![Page 10: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/10.jpg)
References
Edward Lee, and Pravin Varaiya. Structure and Interpretationof Signals & Systems. Second Ed., Univ. of California atBerkeley, 2011.Available online at: leevaraiya.org
Chi-Tsong Chen, Signals and Systems: A Fresh Look, StonyBrook University, 2009.Available online at:www.ece.sunysb.edu/~ctchen/media/freshlook.pdf
Oppenheim, Alan V., and A. S. Willsky. Signals and Systems.Prentice Hall, 1982.
... and many many more. RC has a good collection - Explore.
Lecture - 1: Introduction 3/16
![Page 11: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/11.jpg)
References
Edward Lee, and Pravin Varaiya. Structure and Interpretationof Signals & Systems. Second Ed., Univ. of California atBerkeley, 2011.Available online at: leevaraiya.org
Chi-Tsong Chen, Signals and Systems: A Fresh Look, StonyBrook University, 2009.Available online at:www.ece.sunysb.edu/~ctchen/media/freshlook.pdf
Oppenheim, Alan V., and A. S. Willsky. Signals and Systems.Prentice Hall, 1982.
... and many many more. RC has a good collection - Explore.
Lecture - 1: Introduction 3/16
![Page 12: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/12.jpg)
References
Edward Lee, and Pravin Varaiya. Structure and Interpretationof Signals & Systems. Second Ed., Univ. of California atBerkeley, 2011.Available online at: leevaraiya.org
Chi-Tsong Chen, Signals and Systems: A Fresh Look, StonyBrook University, 2009.Available online at:www.ece.sunysb.edu/~ctchen/media/freshlook.pdf
Oppenheim, Alan V., and A. S. Willsky. Signals and Systems.Prentice Hall, 1982.
... and many many more. RC has a good collection - Explore.
Lecture - 1: Introduction 3/16
![Page 13: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/13.jpg)
Evaluation policy
First In-Sem - 30%, Second In-Sem - 30%, End-Sem - 40%.
Attendance policy - Attendance will not be taken, No pop-upquizzes.
Announcements will be sent through courses.daiict.ac.in
mailing list or will be made in class.
Lecture - 1: Introduction 4/16
![Page 14: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/14.jpg)
Evaluation policy
First In-Sem - 30%, Second In-Sem - 30%, End-Sem - 40%.
Attendance policy - Attendance will not be taken, No pop-upquizzes.
Announcements will be sent through courses.daiict.ac.in
mailing list or will be made in class.
Lecture - 1: Introduction 4/16
![Page 15: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/15.jpg)
Evaluation policy
First In-Sem - 30%, Second In-Sem - 30%, End-Sem - 40%.
Attendance policy - Attendance will not be taken, No pop-upquizzes.
Announcements will be sent through courses.daiict.ac.in
mailing list or will be made in class.
Lecture - 1: Introduction 4/16
![Page 16: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/16.jpg)
Myth about Signals and Systems
It is needed only for Communication engineers!
Designing weighing machines:
Figure : Weighing machine
Applications: Radio Tuners, Speech & Speaker recognition,Audio equalizers, etc.
Image processing: Noise removal, Face recognition, Imagecompression, etc.
Lecture - 1: Introduction 5/16
![Page 17: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/17.jpg)
Myth about Signals and Systems
It is needed only for Communication engineers!
Designing weighing machines:
Figure : Weighing machine
Applications: Radio Tuners, Speech & Speaker recognition,Audio equalizers, etc.
Image processing: Noise removal, Face recognition, Imagecompression, etc.
Lecture - 1: Introduction 5/16
![Page 18: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/18.jpg)
Myth about Signals and Systems
It is needed only for Communication engineers!
Designing weighing machines:
Figure : Weighing machine
Applications: Radio Tuners, Speech & Speaker recognition,Audio equalizers, etc.
Image processing: Noise removal, Face recognition, Imagecompression, etc.
Lecture - 1: Introduction 5/16
![Page 19: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/19.jpg)
Myth about Signals and Systems
It is needed only for Communication engineers!
Designing weighing machines:
Figure : Weighing machine
Applications: Radio Tuners, Speech & Speaker recognition,Audio equalizers, etc.
Image processing: Noise removal, Face recognition, Imagecompression, etc.
Lecture - 1: Introduction 5/16
![Page 20: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/20.jpg)
Compute ”Shape” of a leaf outline.
Figure : Leaf outlines
How to multiply n digit numbers efficiently?
Analysing preference votes.
Lecture - 1: Introduction 6/16
![Page 21: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/21.jpg)
Compute ”Shape” of a leaf outline.
Figure : Leaf outlines
How to multiply n digit numbers efficiently?
Analysing preference votes.
Lecture - 1: Introduction 6/16
![Page 22: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/22.jpg)
Compute ”Shape” of a leaf outline.
Figure : Leaf outlines
How to multiply n digit numbers efficiently?
Analysing preference votes.
Lecture - 1: Introduction 6/16
![Page 23: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/23.jpg)
Signals
What is a signal?
Signals are physical entities that convey information.
Examples:
Sound
Images
Temperature of this room
Height, Weight of a person
Lecture - 1: Introduction 7/16
![Page 24: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/24.jpg)
Signals
What is a signal?
Signals are physical entities that convey information.
Examples:
Sound
Images
Temperature of this room
Height, Weight of a person
Lecture - 1: Introduction 7/16
![Page 25: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/25.jpg)
Signals
What is a signal?
Signals are physical entities that convey information.
Examples:
Sound
Images
Temperature of this room
Height, Weight of a person
Lecture - 1: Introduction 7/16
![Page 26: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/26.jpg)
Signals
What is a signal?
Signals are physical entities that convey information.
Examples:
Sound
Images
Temperature of this room
Height, Weight of a person
Lecture - 1: Introduction 7/16
![Page 27: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/27.jpg)
Signals
What is a signal?
Signals are physical entities that convey information.
Examples:
Sound
Images
Temperature of this room
Height, Weight of a person
Lecture - 1: Introduction 7/16
![Page 28: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/28.jpg)
Signals
What is a signal?
Signals are physical entities that convey information.
Examples:
Sound
Images
Temperature of this room
Height, Weight of a person
Lecture - 1: Introduction 7/16
![Page 29: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/29.jpg)
Signals
What is a signal?
Signals are physical entities that convey information.
Examples:
Sound
Images
Temperature of this room
Height, Weight of a person
Lecture - 1: Introduction 7/16
![Page 30: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/30.jpg)
Examples of signals
Lecture - 1: Introduction 8/16
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Examples of signals
Figure : Guess What’s this?Lecture - 1: Introduction 9/16
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Examples of signals
Figure : Diffusion Tensor Imaging
Lecture - 1: Introduction 10/16
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Systems
Systems are black boxes that take signals as inputs andproduce signals with desired properties.
Given desired properties that a system should have, how dowe go about designing such a system?
Map the signals of interest (both input and output) toappropriate mathematical entities, i.e. Mathematically modelsignals.
Lecture - 1: Introduction 11/16
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Systems
Systems are black boxes that take signals as inputs andproduce signals with desired properties.
Given desired properties that a system should have, how dowe go about designing such a system?
Map the signals of interest (both input and output) toappropriate mathematical entities, i.e. Mathematically modelsignals.
Lecture - 1: Introduction 11/16
![Page 35: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/35.jpg)
Systems
Systems are black boxes that take signals as inputs andproduce signals with desired properties.
Given desired properties that a system should have, how dowe go about designing such a system?
Map the signals of interest (both input and output) toappropriate mathematical entities, i.e. Mathematically modelsignals.
Lecture - 1: Introduction 11/16
![Page 36: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/36.jpg)
Examples
Length: Set of objectsruler/tape−−−−−−→ Set of non-negative real
numbers (R+).
Weight: Set of objectsweighing scale−−−−−−−−→ Set of non-negative real
numbers (R+).
Sound: Set of sound wavesmicrophone−−−−−−−→ Set of finite-energy
functions of one variable (L2(R)).
Images: Set of visible radiations falling on a sensorcamera−−−−→ Set
of finite-energy functions of two variables (L2(Ω)), Ω ⊂ R2.
Lecture - 1: Introduction 12/16
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Examples
Length: Set of objectsruler/tape−−−−−−→ Set of non-negative real
numbers (R+).
Weight: Set of objectsweighing scale−−−−−−−−→ Set of non-negative real
numbers (R+).
Sound: Set of sound wavesmicrophone−−−−−−−→ Set of finite-energy
functions of one variable (L2(R)).
Images: Set of visible radiations falling on a sensorcamera−−−−→ Set
of finite-energy functions of two variables (L2(Ω)), Ω ⊂ R2.
Lecture - 1: Introduction 12/16
![Page 38: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/38.jpg)
Examples
Length: Set of objectsruler/tape−−−−−−→ Set of non-negative real
numbers (R+).
Weight: Set of objectsweighing scale−−−−−−−−→ Set of non-negative real
numbers (R+).
Sound: Set of sound wavesmicrophone−−−−−−−→ Set of finite-energy
functions of one variable (L2(R)).
Images: Set of visible radiations falling on a sensorcamera−−−−→ Set
of finite-energy functions of two variables (L2(Ω)), Ω ⊂ R2.
Lecture - 1: Introduction 12/16
![Page 39: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/39.jpg)
Examples
Length: Set of objectsruler/tape−−−−−−→ Set of non-negative real
numbers (R+).
Weight: Set of objectsweighing scale−−−−−−−−→ Set of non-negative real
numbers (R+).
Sound: Set of sound wavesmicrophone−−−−−−−→ Set of finite-energy
functions of one variable (L2(R)).
Images: Set of visible radiations falling on a sensorcamera−−−−→ Set
of finite-energy functions of two variables (L2(Ω)), Ω ⊂ R2.
Lecture - 1: Introduction 12/16
![Page 40: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/40.jpg)
Homomorphism
The set of signals are mapped to a mathematical set (set ofreal numbers, set of functions of one variable, etc.), such that
essential properties of the signal are preserved -homomorphism, and
in some cases we also want the mapping to be invertible.
These mappings are realized using transducers, for example, aweighing scale, microphone, loudspeaker, etc.
Length
In the physical world, it is possible to join () two objects tocreate a longer object, Ok Om.
There should be a corresponding operation, addition (+) onthe chosen set R+.
Let S denote the set of physical objects, and L denote the
mapping L : Sruler/tape−−−−−→ R+. Observe that
L(Ok Om) = L(Ok) + L(Om).
Lecture - 1: Introduction 13/16
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Homomorphism
The set of signals are mapped to a mathematical set (set ofreal numbers, set of functions of one variable, etc.), such that
essential properties of the signal are preserved -homomorphism, and
in some cases we also want the mapping to be invertible.
These mappings are realized using transducers, for example, aweighing scale, microphone, loudspeaker, etc.
Length
In the physical world, it is possible to join () two objects tocreate a longer object, Ok Om.
There should be a corresponding operation, addition (+) onthe chosen set R+.
Let S denote the set of physical objects, and L denote the
mapping L : Sruler/tape−−−−−→ R+. Observe that
L(Ok Om) = L(Ok) + L(Om).
Lecture - 1: Introduction 13/16
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Homomorphism
The set of signals are mapped to a mathematical set (set ofreal numbers, set of functions of one variable, etc.), such that
essential properties of the signal are preserved -homomorphism, and
in some cases we also want the mapping to be invertible.
These mappings are realized using transducers, for example, aweighing scale, microphone, loudspeaker, etc.
Length
In the physical world, it is possible to join () two objects tocreate a longer object, Ok Om.
There should be a corresponding operation, addition (+) onthe chosen set R+.
Let S denote the set of physical objects, and L denote the
mapping L : Sruler/tape−−−−−→ R+. Observe that
L(Ok Om) = L(Ok) + L(Om).
Lecture - 1: Introduction 13/16
![Page 43: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/43.jpg)
Homomorphism
The set of signals are mapped to a mathematical set (set ofreal numbers, set of functions of one variable, etc.), such that
essential properties of the signal are preserved -homomorphism, and
in some cases we also want the mapping to be invertible.
These mappings are realized using transducers, for example, aweighing scale, microphone, loudspeaker, etc.
Length
In the physical world, it is possible to join () two objects tocreate a longer object, Ok Om.
There should be a corresponding operation, addition (+) onthe chosen set R+.
Let S denote the set of physical objects, and L denote the
mapping L : Sruler/tape−−−−−→ R+. Observe that
L(Ok Om) = L(Ok) + L(Om).
Lecture - 1: Introduction 13/16
![Page 44: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/44.jpg)
Homomorphism
The set of signals are mapped to a mathematical set (set ofreal numbers, set of functions of one variable, etc.), such that
essential properties of the signal are preserved -homomorphism, and
in some cases we also want the mapping to be invertible.
These mappings are realized using transducers, for example, aweighing scale, microphone, loudspeaker, etc.
Length
In the physical world, it is possible to join () two objects tocreate a longer object, Ok Om.
There should be a corresponding operation, addition (+) onthe chosen set R+.
Let S denote the set of physical objects, and L denote the
mapping L : Sruler/tape−−−−−→ R+. Observe that
L(Ok Om) = L(Ok) + L(Om).
Lecture - 1: Introduction 13/16
![Page 45: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/45.jpg)
Homomorphism
The set of signals are mapped to a mathematical set (set ofreal numbers, set of functions of one variable, etc.), such that
essential properties of the signal are preserved -homomorphism, and
in some cases we also want the mapping to be invertible.
These mappings are realized using transducers, for example, aweighing scale, microphone, loudspeaker, etc.
Length
In the physical world, it is possible to join () two objects tocreate a longer object, Ok Om.
There should be a corresponding operation, addition (+) onthe chosen set R+.
Let S denote the set of physical objects, and L denote the
mapping L : Sruler/tape−−−−−→ R+. Observe that
L(Ok Om) = L(Ok) + L(Om).
Lecture - 1: Introduction 13/16
![Page 46: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/46.jpg)
Homomorphism
The set of signals are mapped to a mathematical set (set ofreal numbers, set of functions of one variable, etc.), such that
essential properties of the signal are preserved -homomorphism, and
in some cases we also want the mapping to be invertible.
These mappings are realized using transducers, for example, aweighing scale, microphone, loudspeaker, etc.
Length
In the physical world, it is possible to join () two objects tocreate a longer object, Ok Om.
There should be a corresponding operation, addition (+) onthe chosen set R+.
Let S denote the set of physical objects, and L denote the
mapping L : Sruler/tape−−−−−→ R+. Observe that
L(Ok Om) = L(Ok) + L(Om).
Lecture - 1: Introduction 13/16
![Page 47: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/47.jpg)
Homomorphism
The set of signals are mapped to a mathematical set (set ofreal numbers, set of functions of one variable, etc.), such that
essential properties of the signal are preserved -homomorphism, and
in some cases we also want the mapping to be invertible.
These mappings are realized using transducers, for example, aweighing scale, microphone, loudspeaker, etc.
Length
In the physical world, it is possible to join () two objects tocreate a longer object, Ok Om.
There should be a corresponding operation, addition (+) onthe chosen set R+.
Let S denote the set of physical objects, and L denote the
mapping L : Sruler/tape−−−−−→ R+. Observe that
L(Ok Om) = L(Ok) + L(Om).
Lecture - 1: Introduction 13/16
![Page 48: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/48.jpg)
Homomorphism
Sound
In the physical world, it is possible to mix () two soundstogether, and to turn up/down (l) volume of a sound. Let
M : Smic−−→ L2(R) represent the mapping from the set of
sounds to the set of functions. The corresponding operations on L2(R) are point-wise
addition (+1)of functions and scalar multiplication (·1) by areal number to a function, defined as
(f +1 g)(t) :=f (t) + g(t), ∀f , g ∈ L2(R), ∀t ∈ R(a ·1 f )(t) :=a · f (t), ∀f ∈ L2(R), ∀a ∈ R+, ∀t ∈ R
Note here that +, · are the usual addition and multiplicationbetween real numbers.
Then we know that M(sk sm) = M(sk) +1 M(sm) andM(a l sk) = a ·1 M(sk).
Lecture - 1: Introduction 14/16
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Homomorphism
Sound
In the physical world, it is possible to mix () two soundstogether, and to turn up/down (l) volume of a sound. Let
M : Smic−−→ L2(R) represent the mapping from the set of
sounds to the set of functions. The corresponding operations on L2(R) are point-wise
addition (+1)of functions and scalar multiplication (·1) by areal number to a function, defined as
(f +1 g)(t) :=f (t) + g(t), ∀f , g ∈ L2(R), ∀t ∈ R(a ·1 f )(t) :=a · f (t), ∀f ∈ L2(R), ∀a ∈ R+, ∀t ∈ R
Note here that +, · are the usual addition and multiplicationbetween real numbers.
Then we know that M(sk sm) = M(sk) +1 M(sm) andM(a l sk) = a ·1 M(sk).
Lecture - 1: Introduction 14/16
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Homomorphism
Sound
In the physical world, it is possible to mix () two soundstogether, and to turn up/down (l) volume of a sound. Let
M : Smic−−→ L2(R) represent the mapping from the set of
sounds to the set of functions. The corresponding operations on L2(R) are point-wise
addition (+1)of functions and scalar multiplication (·1) by areal number to a function, defined as
(f +1 g)(t) :=f (t) + g(t), ∀f , g ∈ L2(R), ∀t ∈ R(a ·1 f )(t) :=a · f (t), ∀f ∈ L2(R), ∀a ∈ R+, ∀t ∈ R
Note here that +, · are the usual addition and multiplicationbetween real numbers.
Then we know that M(sk sm) = M(sk) +1 M(sm) andM(a l sk) = a ·1 M(sk).
Lecture - 1: Introduction 14/16
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Homomorphism
Sound
In the physical world, it is possible to mix () two soundstogether, and to turn up/down (l) volume of a sound. Let
M : Smic−−→ L2(R) represent the mapping from the set of
sounds to the set of functions. The corresponding operations on L2(R) are point-wise
addition (+1)of functions and scalar multiplication (·1) by areal number to a function, defined as
(f +1 g)(t) :=f (t) + g(t), ∀f , g ∈ L2(R), ∀t ∈ R(a ·1 f )(t) :=a · f (t), ∀f ∈ L2(R), ∀a ∈ R+, ∀t ∈ R
Note here that +, · are the usual addition and multiplicationbetween real numbers.
Then we know that M(sk sm) = M(sk) +1 M(sm) andM(a l sk) = a ·1 M(sk).
Lecture - 1: Introduction 14/16
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Mathematical Structure
The operations defined on the set will have certain properties(called axioms henceforth), for example, the additionoperation defined above is commutative as well as associative.
The set, operations and the axioms give rise to what is calleda Mathematical structure.
Some popular mathematical structures are groups, rings,fields, vector spaces. You will learn them and study theirproperties in SC-116.
Lecture - 1: Introduction 15/16
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Mathematical Structure
The operations defined on the set will have certain properties(called axioms henceforth), for example, the additionoperation defined above is commutative as well as associative.
The set, operations and the axioms give rise to what is calleda Mathematical structure.
Some popular mathematical structures are groups, rings,fields, vector spaces. You will learn them and study theirproperties in SC-116.
Lecture - 1: Introduction 15/16
![Page 54: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/54.jpg)
Mathematical Structure
The operations defined on the set will have certain properties(called axioms henceforth), for example, the additionoperation defined above is commutative as well as associative.
The set, operations and the axioms give rise to what is calleda Mathematical structure.
Some popular mathematical structures are groups, rings,fields, vector spaces. You will learn them and study theirproperties in SC-116.
Lecture - 1: Introduction 15/16
![Page 55: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/55.jpg)
Assignment - 1
Install GNU Octave:http://www.gnu.org/software/octave/.
Download the octave files: record.m, playaudio.m, fromthe course webpage.
Test out the homomorphism for sound via experiments.
Lecture - 1: Introduction 16/16
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Assignment - 1
Install GNU Octave:http://www.gnu.org/software/octave/.
Download the octave files: record.m, playaudio.m, fromthe course webpage.
Test out the homomorphism for sound via experiments.
Lecture - 1: Introduction 16/16
![Page 57: CT203 - Signals & Systems Lecture - 1: Introductioncourses.daiict.ac.in/pluginfile.php/22296/mod_resource/content/1/Lecture1.pdfModeling Signals & Systems. Modeling systems using State](https://reader034.fdocuments.us/reader034/viewer/2022052612/5f1252f9b9828b34400b8769/html5/thumbnails/57.jpg)
Assignment - 1
Install GNU Octave:http://www.gnu.org/software/octave/.
Download the octave files: record.m, playaudio.m, fromthe course webpage.
Test out the homomorphism for sound via experiments.
Lecture - 1: Introduction 16/16