Download presentation

Presentation is loading. Please wait.

Published byMattie Sander Modified over 2 years ago

1
1 DREAM PLAN IDEA IMPLEMENTATION

2
2

3
3 Introduction to Image Processing Dr. Kourosh Kiani Email: kkiani2004@yahoo.comkkiani2004@yahoo.com Email: Kourosh.kiani@aut.ac.irKourosh.kiani@aut.ac.ir Email: Kourosh.kiani@semnan.ac.irKourosh.kiani@semnan.ac.ir Web: www.kouroshkiani.comwww.kouroshkiani.com Present to: Amirkabir University of Technology (Tehran Polytechnic) & Semnan University

4
4 Lecture 08 Fourier Transform Lecture 08 Fourier Transform

5
5 Development of the Fourier Transform Representation of an Aperiodic Signal In the last lecture we saw how a periodic signal could be represented as a linear combination of cos(nω) and sin(nω). In fact, these results can be extended to develop a representation of aperiodic signals as a linear combination of cos(nω) and sin(nω).

6
6 Continuous Fourier transform T 1 T0T0 2T 0 T 1 -T 0 -2T 0 -3T 0 T0T0 T 1 T0T0

7
7 Since And also since x(t)=0 outside this interval, equation (2) can be rewritten as: Therefore, defining the envelope X(ω) of T 0 a k as: We have that the coefficients a k can be expressed as: Combining (1) and (4), can be expressed in the term X(ω) as:

8
8 Or equivalently, since As, approaches x(t), and consequenetly eq. (7) becomes a representation of x(t). Furthermore, as and the right-hand side of eq (7) passes to an integral. Each term in the summation on the right-hand side of eq. (7) is the area of a rectangle of height and width (here t is regarded as fixed). As this by definition converges to the integral of. Therefore, using the fact that as eq. (7) and (4) become

9
9 Fourier Transform

10
10 Comments

11
11 Example

12
12 Samples of Fourier Transforms of Aperiodic Signals Spectrum 0 f3f5f 0 f 3f5f

13
13 CTFT Properties

14
14

15
15

16
16

17
17

18
18

19
19

20
20

21
21

22
22

23
23

24
24

25
25

26
26 Example: the Fourier Transform of a rectangle function: rect(t) Imaginary Component = 0 F()F()

27
27 Example: the Fourier Transform of a decaying exponential: exp(-at) (t > 0) A complex Lorentzian!

28
Questions? Discussion? Suggestions ?

29
29

Similar presentations

Presentation is loading. Please wait....

OK

Week 1.

Week 1.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google