Fourier transforms

Fourier Transformation By Fahad B. Mostafa

1. Fourier Transform By Fahad B. Mostafa

2. Overview Transforms Formal Understandings Fourier Transform Introduction to Fourier Transformation Why Fourier Transform? Inverse F ourier Transform Dirichlet Conditions for convergence of Fourier Transform Fourier sine and cosine transform A pplications Summary References

3. Transforms in Mathematics Transform: Four Types of transformation in mathematics Translation Rotation Reflection Dilation Fourier Transform (FT) involves a whole new paradigm of viewing signals in a context different from the frequency domain of their occurrence.

4. Background In 1807, Joseph Fourier showed that any periodic signal could be represented by a series of sinusoidal functions In picture: the composition of the first two functions gives the bottom one

5. What is Fourier Transform? Understandings The Fourier Transform is a type of mathematical transform. The Fourier Transform transforms a non-periodic function in time domain into a function in its corresponding frequency domain. It is a tool to break a function into a sinusoidal forms characterized by sine and cosine terms. To understand the conduction of heat, wave propagation, digital signal processing, image processing, filtering, etc.  

6. What is Fourier Transform? Mathematical Definition: defined on piecewise continuous in each finite interval absolutely integrable in The function is called the Fourier Transformation of  

7. Why Fourier Transform?

8. Why Fourier Transform ?

9. Time Domain and Frequency Domain

10. Inverse Fourier Transformation is the inverse F ourier transform of . Inverse Fourier Transform is a method of converting a frequency domain function back to its time domain form. If is the Fourier Transform of and if satisfies Dirichlet’s conditions in every finite interval and if is convergent.  

11. Fourier Transform Because of the property: Fourier Transform takes us to the frequency domain:

12. Mathematical Example Sketch the following wave function in the time domain and calculate it by using Fourier transform.  

13. Fourier sine and cosine transform (Infinite) If is a function in then > Infinite Fourier sine transform > Inverse form of Infinite Fourier sine transform > Infinite Fourier cosine transform > Inverse form of Infinite Fourier cosine transform  

14. Fourier sine and cosine transform (finite ) Finite Fourier sine transform of over where is integer . The function is called inverse finite Fourier sine transform of Finite Fourier cosine transform of over where is integer . The function is called inverse finite Fourier cosine transform of  

15. Using FT to Solve Heat equation represents temperature at any point at time ,  

16. Solving BVP By taking FST with limit both sides At ; ;  

17. Solving BVP Using inversion formula of Fourier sine transform  

18. Applications: Image Filtering

19. Other Applications of the FT Signal analysis Sound filtering Partial differential equations Image processing Conduction of heat Wave propagation.

20. Real life Examples : 1 . A doctor examines a patient's heart beat , is done by ECG ( Electro-Cardio-Gram ) 2 . For a musician, all the ragas played are actually in time domain but frequency is more important for him than time 3 . For a circuit, the input and output signals are functions of time.

21. Summary Transforms: Useful in mathematics (solving DE) Fourier Transform: Lets us easily switch between time-space domain and frequency domain so applicable in many other areas Easy to pick out frequencies Many applications

22. References Concepts and the frequency domain http://www.spd.eee.strath.ac.uk/~interact/fourier/concepts.html JPNM Physics Fourier Transform http://www.med.harvard.edu/JPNM/physics/didactics/improc/intro/fourier2.html Fourier Transforms http://www.britannica.com/ebc/article?tocId=9368037&query=transform&ct = D.G. Zill , Advanced Engineering Mathematics, 2002 http://nptel.ac.in