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Orthogonal Transforms Download Presentation ## Orthogonal Transforms

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1. Orthogonal Transforms Fourier

2. Review • Introduce the concepts of base functions: • For Reed-Muller, FPRM • For Walsh • Linearly independent matrix • Non-Singular matrix • Examples • Butterflies, Kronecker Products, Matrices • Using matrices to calculate the vector of spectral coefficients from the data vector Our goal is to discuss the best approximation of a function using orthogonal functions

3. Orthogonal Functions

4. Orthogonal Functions

5. Illustrate it for Walsh and RM

6. Mean Square Error

7. Mean Square Error

8. Important result

9. We want to minimize this kinds of errors. • Other error measures are also used.

10. Unitary Transforms

11. Unitary Transforms • Unitary Transformation for 1-Dim. Sequence • Series representation of • Basis vectors : • Energy conservation : Here is the proof

12. Unitary Transformation for 2-Dim. Sequence • Definition : • Basis images : • Orthonormality and completeness properties • Orthonormality : • Completeness :

13. Unitary Transformation for 2-Dim. Sequence • Separable Unitary Transforms • separable transform reduces the number of multiplications and additions from to • Energy conservation

14. Properties of Unitary Transform transform Covariance matrix

15. Example of arbitrary basis functions being rectangular waves

16. This determining first function determines next functions

17. 0 1

18. Small error with just 3 coefficients

19. This slide shows four base functions multiplied by their respective coefficients

20. This slide shows that using only four base functions the approximation is quite good End of example

21. Orthogonality and separability

22. Orthogonal and separable Image Transforms

23. Extending general transforms to 2-dimensions

24. Forward transform inverse transform separable

25. Fourier Transforms in new notations We emphasize generality Matrices

26. Fourier Transform separable

27. Extension of Fourier Transform to two dimensions

28. Discrete Fourier Transform (DFT) New notation

29. Fast Algorithms for Fourier Transform Task for students: Draw the butterfly for these matrices, similarly as we have done it for Walsh and Reed-Muller Transforms 2 Pay attention to regularity of kernels and order of columns corresponding to factorized matrices

30. 1-dim. DFT (cont.) • Calculation of DFT : Fast Fourier Transform Algorithm (FFT) • Decimation-in-time algorithm Derivation of decimation in time

31. Decimation in Time versus Decismation in Frequency

32. Butterfly for Derivation of decimation in time • 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-time algorithm (cont.) Please note recursion

33. 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-frequency algorithm (cont.) • Derivation of Decimation-in-frequency algorithm

34. Decimation in frequency butterfly shows recursion • 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-frequency algorithm (cont.)

35. Conjugate Symmetry of DFT • For a real sequence, the DFT is conjugate symmetry

36. Use of Fourier Transforms for fast convolution

37. Calculations for circular matrix

38. By multiplying