Engineering Tripos Part IB, 2P7: Vector Calculus, 2017-18
Lecturer
Timing and Structure
Weeks 1-3 and 6-8 Michaelmas term, 2 lectures/week; weeks 4-5 Michaelmas term, 1 lecture/week. 14 lectures
Aims
The aims of the course are to:
- Provide the necessary background mathematics to ensure that students are confident in handling partial differential equations in vector form while maintaining a tangible physical appreciation of the manipulations involved.
Objectives
As specific objectives, by the end of the course students should be able to:
- Differentiate and integrate scalar functions of two or more variables including transformations to other co-ordinate systems.
- Manipulate vector differential equations including the gradient, divergence and curl operators while retaining a physical appreciation of the mathematical operations involved.
- Perform line, surface and volume integrals and understand their various physical interpretations.
- Set up conservation statements in both differential and integral form and be able to transform from one to the other using Gauss's theorem.
- Appreciate the physical significance of curl and its relationship to circulation via Stokes's theorem in simple examples.
- Solve common PDE's (particularly the Laplace, Poisson, heat conduction and wave equations) with simple boundary conditions by the method of separation of variables.
Content
The course provides an elementary introduction to vector calculus and aims to familiarise the student with the basic ideas of the differential calculus (the vector gradient, divergence and curl) and the integral calculus (line, surface and volume integrals and the theorems of Gauss and Stokes). The physical interpretation of the mathematical ideas will be stressed throughout via applications which centre on the derivation and manipulation of the common partial differential equations of engineering. The analytical solution of simple partial differential equations by the method of separation of variables will also be discussed.
A knowledge of the following Part IA lecture material on functions of more than one variable will be assumed: representation of curves and surfaces (including parametric representation); partial differentiation; total and perfect differentials; Taylor series; maxima and minima.
The course will then consist of lectures on the following topics:
Vector functions and fields; field lines.
Vector differentiation; differentiation formulae.
The vector gradient and its physical interpretation;
Cylindrical and spherical polar co-ordinate systems.
The divergence and its physical interpretation; solenoidal fields; conservation statements;
Surface integrals; volume integrals; Gauss's divergence theorem; integral-differential transformations. Stokes's theorem.
The curl and its physical interpretation; irrotational fields; scalar potential; line integrals; conservative fields.
Types of PDE and boundary conditions; solution by separation of variables; examples of some common PDE's (Laplace, Poisson, heat conduction, wave equation
Booklists
Please see the Booklist for Part IB Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P8
Ability to apply engineering techniques taking account of a range of commercial and industrial constraints.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 31/05/2017 10:00
Engineering Tripos Part IB, 2P7: Vector Calculus, 2019-20
Lecturer
Timing and Structure
Weeks 1-3 and 6-8 Michaelmas term, 2 lectures/week; weeks 4-5 Michaelmas term, 1 lecture/week. 14 lectures
Aims
The aims of the course are to:
- Provide the necessary background mathematics to ensure that students are confident in handling partial differential equations in vector form while maintaining a tangible physical appreciation of the manipulations involved.
Objectives
As specific objectives, by the end of the course students should be able to:
- Differentiate and integrate scalar functions of two or more variables including transformations to other co-ordinate systems.
- Manipulate vector differential equations including the gradient, divergence and curl operators while retaining a physical appreciation of the mathematical operations involved.
- Perform line, surface and volume integrals and understand their various physical interpretations.
- Set up conservation statements in both differential and integral form and be able to transform from one to the other using Gauss's theorem.
- Appreciate the physical significance of curl and its relationship to circulation via Stokes's theorem in simple examples.
- Solve common PDE's (particularly the Laplace, Poisson, heat conduction and wave equations) with simple boundary conditions by the method of separation of variables.
Content
The course provides an elementary introduction to vector calculus and aims to familiarise the student with the basic ideas of the differential calculus (the vector gradient, divergence and curl) and the integral calculus (line, surface and volume integrals and the theorems of Gauss and Stokes). The physical interpretation of the mathematical ideas will be stressed throughout via applications which centre on the derivation and manipulation of the common partial differential equations of engineering. The analytical solution of simple partial differential equations by the method of separation of variables will also be discussed.
A knowledge of the following Part IA lecture material on functions of more than one variable will be assumed: representation of curves and surfaces (including parametric representation); partial differentiation; total and perfect differentials; Taylor series; maxima and minima.
The course will then consist of lectures on the following topics:
Vector functions and fields; field lines.
Vector differentiation; differentiation formulae.
The vector gradient and its physical interpretation;
Cylindrical and spherical polar co-ordinate systems.
The divergence and its physical interpretation; solenoidal fields; conservation statements;
Surface integrals; volume integrals; Gauss's divergence theorem; integral-differential transformations. Stokes's theorem.
The curl and its physical interpretation; irrotational fields; scalar potential; line integrals; conservative fields.
Types of PDE and boundary conditions; solution by separation of variables; examples of some common PDE's (Laplace, Poisson, heat conduction, wave equation
Booklists
Please see the Booklist for Part IB Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
P8
Ability to apply engineering techniques taking account of a range of commercial and industrial constraints.
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
Last modified: 16/05/2019 12:23
Engineering Tripos Part IB, 2P6: Fourier Transforms & Signal and Data Analysis, 2024-25
Course Leader
Lecturer
Timing and Structure
Lent Term: 7 lectures Weeks 1-3, 2 lectures, week 4, 1 lecture
Aims
The aims of the course are to:
- Introduce the Fourier Transform as an extension of Fourier techniques on periodic functions and to see how the Fourier Transform is applied to real problems
- Introduce discrete Fourier methods and to develop skills in analysing discrete data.
Objectives
As specific objectives, by the end of the course students should be able to:
- develop the ability to discuss and manipulate signals in terms of their frequency content.
- relate properties of signals in the time domain to those in the frequency domain.
- be familiar with the difference in behaviour/properties of continuous signals compared to sampled signals, and the basic rules that apply to the latter.
Content
Introduction and preliminaries
- Motivation for signal analysis. Examples of typical datasets.
- Power and energy
- Revision and extension of delta functions
- Revision of Fourier series
The Fourier Transform (FT)
- Mathematical formulation of the FT
- Interpretation of the FT
- The inverse Fourier transform (IFT)
- Some important Fourier transforms
Properties of the Fourier Transform
- Linearity and scaling
- Time and frequency shifts (modulation)
- Duality, Parseval's Theorem, convolution
- Relationship to Laplace transforms
Sampling Theory
- The sampling theorem and aliasing
- The discrete time Fourier transform
- Signal reconstruction and the Nyquist frequency
The Discrete Fourier Transform
- Derivation of DFT and inverse DFT
- Examples of using the DFT
- The spectrogram
Booklists
Please refer to the Booklist for Part IB Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 30/07/2024 08:50
Engineering Tripos Part IB, 2P6: Fourier Transforms & Signal and Data Analysis, 2019-20
Lecturer
Timing and Structure
Lent Term: 7 lectures Weeks 1-3, 2 lectures, week 4, 1 lecture
Aims
The aims of the course are to:
- Introduce the Fourier Transform as an extension of Fourier techniques on periodic functions and to see how the Fourier Transform is applied to real problems
- Introduce discrete Fourier methods and to develop skills in analysing discrete data.
Objectives
As specific objectives, by the end of the course students should be able to:
- develop the ability to discuss and manipulate signals in terms of their frequency content.
- relate properties of signals in the time domain to those in the frequency domain.
- be familiar with the difference in behaviour/properties of continuous signals compared to sampled signals, and the basic rules that apply to the latter.
Content
Introduction and preliminaries
- Motivation for signal analysis. Examples of typical datasets.
- Power and energy
- Revision and extension of delta functions
- Revision of Fourier series
The Fourier Transform (FT)
- Mathematical formulation of the FT
- Interpretation of the FT
- The inverse Fourier transform (IFT)
- Some important Fourier transforms
Properties of the Fourier Transform
- Linearity and scaling
- Time and frequency shifts (modulation)
- Duality, Parseval's Theorem, convolution
- Relationship to Laplace transforms
Sampling Theory
- The sampling theorem and aliasing
- The discrete time Fourier transform
- Signal reconstruction and the Nyquist frequency
The Discrete Fourier Transform
- Derivation of DFT and inverse DFT
- Examples of using the DFT
- The spectrogram
Booklists
Please see the Booklist for Part IB Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 16/05/2019 12:22
Engineering Tripos Part IB, 2P6: Fourier Transforms & Signal and Data Analysis, 2021-22
Course Leader
Lecturer
Timing and Structure
Lent Term: 7 lectures Weeks 1-3, 2 lectures, week 4, 1 lecture
Aims
The aims of the course are to:
- Introduce the Fourier Transform as an extension of Fourier techniques on periodic functions and to see how the Fourier Transform is applied to real problems
- Introduce discrete Fourier methods and to develop skills in analysing discrete data.
Objectives
As specific objectives, by the end of the course students should be able to:
- develop the ability to discuss and manipulate signals in terms of their frequency content.
- relate properties of signals in the time domain to those in the frequency domain.
- be familiar with the difference in behaviour/properties of continuous signals compared to sampled signals, and the basic rules that apply to the latter.
Content
Introduction and preliminaries
- Motivation for signal analysis. Examples of typical datasets.
- Power and energy
- Revision and extension of delta functions
- Revision of Fourier series
The Fourier Transform (FT)
- Mathematical formulation of the FT
- Interpretation of the FT
- The inverse Fourier transform (IFT)
- Some important Fourier transforms
Properties of the Fourier Transform
- Linearity and scaling
- Time and frequency shifts (modulation)
- Duality, Parseval's Theorem, convolution
- Relationship to Laplace transforms
Sampling Theory
- The sampling theorem and aliasing
- The discrete time Fourier transform
- Signal reconstruction and the Nyquist frequency
The Discrete Fourier Transform
- Derivation of DFT and inverse DFT
- Examples of using the DFT
- The spectrogram
Booklists
Please refer to the Booklist for Part IB Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 20/05/2021 07:26
Engineering Tripos Part IB, 2P6: Fourier Transforms & Signal and Data Analysis, 2017-18
Lecturer
Timing and Structure
Lent Term: 7 lectures Weeks 1-3, 2 lectures, week 4, 1 lecture
Aims
The aims of the course are to:
- Introduce the Fourier Transform as an extension of Fourier techniques on periodic functions and to see how the Fourier Transform is applied to real problems
- Introduce discrete Fourier methods and to develop skills in analysing discrete data.
Objectives
As specific objectives, by the end of the course students should be able to:
- develop the ability to discuss and manipulate signals in terms of their frequency content.
- relate properties of signals in the time domain to those in the frequency domain.
- be familiar with the difference in behaviour/properties of continuous signals compared to sampled signals, and the basic rules that apply to the latter.
Content
Introduction and preliminaries
- Motivation for signal analysis. Examples of typical datasets.
- Power and energy
- Revision and extension of delta functions
- Revision of Fourier series
The Fourier Transform (FT)
- Mathematical formulation of the FT
- Interpretation of the FT
- The inverse Fourier transform (IFT)
- Some important Fourier transforms
Properties of the Fourier Transform
- Linearity and scaling
- Time and frequency shifts (modulation)
- Duality, Parseval's Theorem, convolution
- Relationship to Laplace transforms
Sampling Theory
- The sampling theorem and aliasing
- The discrete time Fourier transform
- Signal reconstruction and the Nyquist frequency
The Discrete Fourier Transform
- Derivation of DFT and inverse DFT
- Examples of using the DFT
- The spectrogram
Booklists
Please see the Booklist for Part IB Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 31/05/2017 10:00
Engineering Tripos Part IB, 2P6: Fourier Transforms & Signal and Data Analysis, 2020-21
Course Leader
Lecturer
Timing and Structure
Lent Term: 7 lectures Weeks 1-3, 2 lectures, week 4, 1 lecture
Aims
The aims of the course are to:
- Introduce the Fourier Transform as an extension of Fourier techniques on periodic functions and to see how the Fourier Transform is applied to real problems
- Introduce discrete Fourier methods and to develop skills in analysing discrete data.
Objectives
As specific objectives, by the end of the course students should be able to:
- develop the ability to discuss and manipulate signals in terms of their frequency content.
- relate properties of signals in the time domain to those in the frequency domain.
- be familiar with the difference in behaviour/properties of continuous signals compared to sampled signals, and the basic rules that apply to the latter.
Content
Introduction and preliminaries
- Motivation for signal analysis. Examples of typical datasets.
- Power and energy
- Revision and extension of delta functions
- Revision of Fourier series
The Fourier Transform (FT)
- Mathematical formulation of the FT
- Interpretation of the FT
- The inverse Fourier transform (IFT)
- Some important Fourier transforms
Properties of the Fourier Transform
- Linearity and scaling
- Time and frequency shifts (modulation)
- Duality, Parseval's Theorem, convolution
- Relationship to Laplace transforms
Sampling Theory
- The sampling theorem and aliasing
- The discrete time Fourier transform
- Signal reconstruction and the Nyquist frequency
The Discrete Fourier Transform
- Derivation of DFT and inverse DFT
- Examples of using the DFT
- The spectrogram
Booklists
Please refer to the Booklist for Part IB Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 26/08/2020 09:25
Engineering Tripos Part IB, 2P6: Fourier Transforms & Signal and Data Analysis, 2022-23
Course Leader
Lecturer
Timing and Structure
Lent Term: 7 lectures Weeks 1-3, 2 lectures, week 4, 1 lecture
Aims
The aims of the course are to:
- Introduce the Fourier Transform as an extension of Fourier techniques on periodic functions and to see how the Fourier Transform is applied to real problems
- Introduce discrete Fourier methods and to develop skills in analysing discrete data.
Objectives
As specific objectives, by the end of the course students should be able to:
- develop the ability to discuss and manipulate signals in terms of their frequency content.
- relate properties of signals in the time domain to those in the frequency domain.
- be familiar with the difference in behaviour/properties of continuous signals compared to sampled signals, and the basic rules that apply to the latter.
Content
Introduction and preliminaries
- Motivation for signal analysis. Examples of typical datasets.
- Power and energy
- Revision and extension of delta functions
- Revision of Fourier series
The Fourier Transform (FT)
- Mathematical formulation of the FT
- Interpretation of the FT
- The inverse Fourier transform (IFT)
- Some important Fourier transforms
Properties of the Fourier Transform
- Linearity and scaling
- Time and frequency shifts (modulation)
- Duality, Parseval's Theorem, convolution
- Relationship to Laplace transforms
Sampling Theory
- The sampling theorem and aliasing
- The discrete time Fourier transform
- Signal reconstruction and the Nyquist frequency
The Discrete Fourier Transform
- Derivation of DFT and inverse DFT
- Examples of using the DFT
- The spectrogram
Booklists
Please refer to the Booklist for Part IB Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 24/05/2022 14:09
Engineering Tripos Part IB, 2P6: Fourier Transforms & Signal and Data Analysis, 2018-19
Lecturer
Timing and Structure
Lent Term: 7 lectures Weeks 1-3, 2 lectures, week 4, 1 lecture
Aims
The aims of the course are to:
- Introduce the Fourier Transform as an extension of Fourier techniques on periodic functions and to see how the Fourier Transform is applied to real problems
- Introduce discrete Fourier methods and to develop skills in analysing discrete data.
Objectives
As specific objectives, by the end of the course students should be able to:
- develop the ability to discuss and manipulate signals in terms of their frequency content.
- relate properties of signals in the time domain to those in the frequency domain.
- be familiar with the difference in behaviour/properties of continuous signals compared to sampled signals, and the basic rules that apply to the latter.
Content
Introduction and preliminaries
- Motivation for signal analysis. Examples of typical datasets.
- Power and energy
- Revision and extension of delta functions
- Revision of Fourier series
The Fourier Transform (FT)
- Mathematical formulation of the FT
- Interpretation of the FT
- The inverse Fourier transform (IFT)
- Some important Fourier transforms
Properties of the Fourier Transform
- Linearity and scaling
- Time and frequency shifts (modulation)
- Duality, Parseval's Theorem, convolution
- Relationship to Laplace transforms
Sampling Theory
- The sampling theorem and aliasing
- The discrete time Fourier transform
- Signal reconstruction and the Nyquist frequency
The Discrete Fourier Transform
- Derivation of DFT and inverse DFT
- Examples of using the DFT
- The spectrogram
Booklists
Please see the Booklist for Part IB Courses for references for this module.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 17/05/2018 15:28
Engineering Tripos Part IB, 2P6: Fourier Transforms & Signal and Data Analysis, 2023-24
Course Leader
Lecturer
Timing and Structure
Lent Term: 7 lectures Weeks 1-3, 2 lectures, week 4, 1 lecture
Aims
The aims of the course are to:
- Introduce the Fourier Transform as an extension of Fourier techniques on periodic functions and to see how the Fourier Transform is applied to real problems
- Introduce discrete Fourier methods and to develop skills in analysing discrete data.
Objectives
As specific objectives, by the end of the course students should be able to:
- develop the ability to discuss and manipulate signals in terms of their frequency content.
- relate properties of signals in the time domain to those in the frequency domain.
- be familiar with the difference in behaviour/properties of continuous signals compared to sampled signals, and the basic rules that apply to the latter.
Content
Introduction and preliminaries
- Motivation for signal analysis. Examples of typical datasets.
- Power and energy
- Revision and extension of delta functions
- Revision of Fourier series
The Fourier Transform (FT)
- Mathematical formulation of the FT
- Interpretation of the FT
- The inverse Fourier transform (IFT)
- Some important Fourier transforms
Properties of the Fourier Transform
- Linearity and scaling
- Time and frequency shifts (modulation)
- Duality, Parseval's Theorem, convolution
- Relationship to Laplace transforms
Sampling Theory
- The sampling theorem and aliasing
- The discrete time Fourier transform
- Signal reconstruction and the Nyquist frequency
The Discrete Fourier Transform
- Derivation of DFT and inverse DFT
- Examples of using the DFT
- The spectrogram
Booklists
Please refer to the Booklist for Part IB Courses for references to this module, this can be found on the associated Moodle course.
Examination Guidelines
Please refer to Form & conduct of the examinations.
UK-SPEC
This syllabus contributes to the following areas of the UK-SPEC standard:
Toggle display of UK-SPEC areas.
GT1
Develop transferable skills that will be of value in a wide range of situations. These are exemplified by the Qualifications and Curriculum Authority Higher Level Key Skills and include problem solving, communication, and working with others, as well as the effective use of general IT facilities and information retrieval skills. They also include planning self-learning and improving performance, as the foundation for lifelong learning/CPD.
IA1
Apply appropriate quantitative science and engineering tools to the analysis of problems.
IA3
Comprehend the broad picture and thus work with an appropriate level of detail.
KU1
Demonstrate knowledge and understanding of essential facts, concepts, theories and principles of their engineering discipline, and its underpinning science and mathematics.
KU2
Have an appreciation of the wider multidisciplinary engineering context and its underlying principles.
E1
Ability to use fundamental knowledge to investigate new and emerging technologies.
E2
Ability to extract data pertinent to an unfamiliar problem, and apply its solution using computer based engineering tools when appropriate.
E3
Ability to apply mathematical and computer based models for solving problems in engineering, and the ability to assess the limitations of particular cases.
E4
Understanding of and ability to apply a systems approach to engineering problems.
P1
A thorough understanding of current practice and its limitations and some appreciation of likely new developments.
P3
Understanding of contexts in which engineering knowledge can be applied (e.g. operations and management, technology, development, etc).
US1
A comprehensive understanding of the scientific principles of own specialisation and related disciplines.
US2
A comprehensive knowledge and understanding of mathematical and computer models relevant to the engineering discipline, and an appreciation of their limitations.
US3
An understanding of concepts from a range of areas including some outside engineering, and the ability to apply them effectively in engineering projects.
US4
An awareness of developing technologies related to own specialisation.
Last modified: 30/05/2023 15:14
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