CT203 - Signals & SystemsLecture - 1: Introduction

3-1-0-4

Aditya Tatu

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Compute ”Shape” of a leaf outline.

Figure : Leaf outlines

How to multiply n digit numbers efficiently?

Analysing preference votes.

Lecture - 1: Introduction 6/16

Compute ”Shape” of a leaf outline.

Figure : Leaf outlines

How to multiply n digit numbers efficiently?

Analysing preference votes.

Lecture - 1: Introduction 6/16

Compute ”Shape” of a leaf outline.

Figure : Leaf outlines

How to multiply n digit numbers efficiently?

Analysing preference votes.

Lecture - 1: Introduction 6/16

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

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

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

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

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

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

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

Examples of signals

Lecture - 1: Introduction 8/16

Examples of signals

Figure : Guess What’s this?Lecture - 1: Introduction 9/16

Examples of signals

Figure : Diffusion Tensor Imaging

Lecture - 1: Introduction 10/16

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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