To view this presentation, you'll need to enable Flash.

Show me how

After you enable Flash, refresh this webpage and the presentation should play.

Loading...

PPT – Lecture 9: Fourier Transform Properties and Examples PowerPoint presentation | free to download - id: 23a87f-Yjk1Y

The Adobe Flash plugin is needed to view this content

View by Category

Presentations

Products
Sold on our sister site CrystalGraphics.com

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Lecture 9 Fourier Transform Properties and

Examples

- 3. Basis functions (3 lectures) Concept of basis

function. Fourier series representation of time

functions. Fourier transform and its properties.

Examples, transform of simple time functions. - Specific objectives for today
- Properties of a Fourier transform
- Linearity
- Time shifts
- Differentiation and integration
- Convolution in the frequency domain

Lecture 9 Resources

- Core material
- SaS, OW, C4.3, C4.4
- Background material
- MIT Lectures 8 and 9.

Reminder Fourier Transform

- A signal x(t) and its Fourier transform X(jw) are

related by - This is denoted by
- For example (1)
- Remember that the Fourier transform is a density

function, you must integrate it, rather than

summing up the discrete Fourier series components

Linearity of the Fourier Transform

- If
- and
- Then
- This follows directly from the definition of the

Fourier transform (as the integral operator is

linear). It is easily extended to a linear

combination of an arbitrary number of signals

Time Shifting

- If
- Then
- Proof
- Now replacing t by t-t0
- Recognising this as
- A signal which is shifted in time does not have

its Fourier transform magnitude altered, only a

shift in phase.

Example Linearity Time Shift

- Consider the signal (linear sum of two time

shifted steps) - where x1(t) is of width 1, x2(t) is of width 3,

centred on zero. - Using the rectangular pulse example
- Then using the linearity and time shift Fourier

transform properties

Differentiation Integration

- By differentiating both sides of the Fourier

transform synthesis equation - Therefore
- This is important, because it replaces

differentiation in the time domain with

multiplication in the frequency domain. - Integration is similar
- The impulse term represents the dc or average

value that can result from integration

Example Fourier Transform of a Step Signal

- Lets calculate the Fourier transform X(jw)of x(t)

u(t), making use of the knowledge that - and noting that
- Taking Fourier transform of both sides
- using the integration property. Since G(jw) 1
- We can also apply the differentiation property in

reverse

Convolution in the Frequency Domain

- With a bit of work (next slide) it can show that
- Therefore, to apply convolution in the frequency

domain, we just have to multiply the two

functions. - To solve for the differential/convolution

equation using Fourier transforms - Calculate Fourier transforms of x(t) and h(t)
- Multiply H(jw) by X(jw) to obtain Y(jw)
- Calculate the inverse Fourier transform of Y(jw)
- Multiplication in the frequency domain

corresponds to convolution in the time domain and

vice versa.

Proof of Convolution Property

- Taking Fourier transforms gives
- Interchanging the order of integration, we have
- By the time shift property, the bracketed term is

e-jwtH(jw), so

Example 1 Solving an ODE

- Consider the LTI system time impulse response
- to the input signal
- Transforming these signals into the frequency

domain - and the frequency response is
- to convert this to the time domain, express as

partial fractions - Therefore, the time domain response is

b?a

Example 2 Designing a Low Pass Filter

- Lets design a low pass filter
- The impulse response of this filter is the

inverse Fourier transform - which is an ideal low pass filter
- Non-causal (how to build)
- The time-domain oscillations may be undesirable
- How to approximate the frequency selection

characteristics? - Consider the system with impulse response
- Causal and non-oscillatory time domain response

and performs a degree of low pass filtering

Lecture 9 Summary

- The Fourier transform is widely used for

designing filters. You can design systems with

reject high frequency noise and just retain the

low frequency components. This is natural to

describe in the frequency domain. - Important properties of the Fourier transform

are - 1. Linearity and time shifts
- 2. Differentiation
- 3. Convolution
- Some operations are simplified in the frequency

domain, but there are a number of signals for

which the Fourier transform do not exist this

leads naturally onto Laplace transforms

Lecture 9 Exercises

- Theory
- SaS, OW,
- Q4.6
- Q4.13
- Q3.20, 4.20
- Q4.26
- Q4.31
- Q4.32
- Q4.33

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Presentation Removal Request Contact Us Send Us Feedback

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Lecture 9: Fourier Transform Properties and Examples" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!