1. The Fourier Transform
Intro: Marisa, Nava, Compression Scheme project. Relies heavily on Fourier Transform

2. What are transforms?
Intro: Marisa, Nava, Compression Scheme project. Relies heavily on Fourier Transform

3. First – A Building Blocks Analogy
Sample Data in an Array 1 4 5 9
Vector (building blocks). Any building blocks. These are called bases. Represent the same set of information, even though they look different because you can go back and forth between them.
Fourier transform takes a vector to the Fourier basis.

4. Standard Building Blocks1

5. Standard Building Blocks1 1 + 4 5 9 1 4 5 9

6. Standard Building Blocks1 1 + 4 5 9 1 4 5 9

7. Non-Standard Building Blocks
1/4 4 + 4 1 20 18 1 4 5 9

8. Non-Standard Building Blocks
1/4 4 + 4 1 20 18 1 4 5 9

9. Non-Standard Building Blocks1 4 5 9
Transform 4 1
Vector of Original Data
Vector of Coordinates

10. Non-standard Building Blocks #2
1/7 -1 4 12 20
2/5
9/10 7 + 1 10 1 4 5 9

11. Non-standard Building Blocks #2
1/7 -1 4 12 20
2/5
9/10 7 + 1 10 1 4 5 9

12. Non-standard Building Blocks #21 4 5 9
Transform 7 1 10
Vector of Original Data
Vector of Coordinates

13. Standard
Non-standard #1
Non-standard #2 1 4 5 9 1 4 5 9 4 1 1 4 5 9
Transform 1 4 5 9
Transform 7 1 10

14. The Fourier Building Blocks
A specific set of building blocks that produces the Fourier Transform.
Fourier theorem: any periodic function f(x) can be expressed as the sum of a series of sines and cosines.
Graphics: Sines and cosines of varying frequencies, sample them at even frequency.
In other words, the building blocks in this case are vector representations of sines and cosines.
When we add these together with coefficients in front of each sine and cosine (just like we did with our standard building blocks), we are creating an equivalent form of the function f(x). This form is called a Fourier series.

15. 1/√8
1/2
-1/2
-1/√8

16. A Motivating Example
Square Wave 1 -1
Vector of Wave Data

17. Recall: Fourier Transform
Transform a data vector to a vector of coordinates using the set of set of Fourier building blocks.
How do we find the new coordinate vector?
???? 1 -1

18. Dot Products!
Construct the new coordinate vector by taking the dot product of each building block and the data vector.
Dot product – multiply element-wise then add.
What do dot products tell us? On a SUPER HIGH LEVEL - how much of one vector is in another vector (takes in two vectors of equal length and returns a single scalar representing a relationship between them).
In this case, it will tell us how much the sin/cosine building block contributes to the signal. If 0, then that frequency is not used at all.
Order is O(n^2) – n points, n multiplies, n-1 adds.

19. * Dot Product = 0 1/√8 + 1/√8 + 1/√8 + 1/√8 - 1/√8 - 1/√8 - 1/√8 --1
= 0
1/√8 +
1/√8 +
1/√8 +
1/√8 -
1/√8 -
1/√8 -
1/√8 -
1/√8

20. Finding Our Coordinates
1/√8
1/2
-1/2
-1/√8  1 -1
= 0  1 -1
= 1  1 -1
= √2 + 1  1 -1
= 0  1 -1
= 0  1 -1
= 1 1 -1
= √2 + 1   1 -1
= 0

21. Finding Our Coordinates
1/√8
1/2
-1/2
-1/√8  1 -1
= 0  1 -1
= 1  1 -1
= √2 + 1  1 -1
= 0  1 -1
= 0  1 -1
= 1 1 -1
= √2 - 1   1 -1
= 0
Vector of Coordinates For Square Wave from Fourier Building Blocks 1
√2 + 1
√2 - 1

22. Transformed Square Wave1 -1
Fourier Transform 1
√2 + 1
√2 - 1

23. 1
√2 + 1
√2 - 1 + + =

24. Compression Use fewer building blocks.
Choose to keep the most meaningful ones, throw out the rest. 1
√2 + 1
√2 - 1 1
√2 + 1
√2 - 1

25. +  + =

26.  + + =

27. Our Results

28. 0%

29. 50%

30. 80%

31. 90%

32. 95%

33. 99%

34. 99.9%

35. 100%