Orthogonal Transforms - PowerPoint PPT Presentation

orthogonal transforms n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Orthogonal Transforms PowerPoint Presentation
Download Presentation
Orthogonal Transforms

play fullscreen
1 / 67
Orthogonal Transforms
140 Views
Download Presentation
drucilla-terris
Download Presentation

Orthogonal Transforms

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

  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. Note that these are arbitrary functions, we do not assume sinusoids

  6. Illustrate it for Walsh and RM

  7. Mean Square Error

  8. Mean Square Error

  9. Important result

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

  11. Unitary Transforms

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

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

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

  15. Properties of Unitary Transform transform Covariance matrix

  16. Example of arbitrary basis functions being rectangular waves

  17. This determining first function determines next functions

  18. 0 1

  19. Small error with just 3 coefficients

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

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

  22. Orthogonality and separability

  23. Orthogonal and separable Image Transforms

  24. Extending general transforms to 2-dimensions

  25. Forward transform inverse transform separable

  26. Fourier Transforms in new notations We emphasize generality Matrices

  27. Fourier Transform separable

  28. Extension of Fourier Transform to two dimensions

  29. Discrete Fourier Transform (DFT) New notation

  30. 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

  31. Fast Factorization Algorithms are general and there is many of them

  32. 1-dim. DFT (cont.) • Calculation of DFT : Fast Fourier Transform Algorithm (FFT) • Decimation-in-time algorithm Derivation of decimation in time

  33. Decimation in Time versus Decismation in Frequency

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

  35. 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-frequency algorithm (cont.) • Derivation of Decimation-in-frequency algorithm

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

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

  38. Use of Fourier Transforms for fast convolution

  39. Calculations for circular matrix

  40. By multiplying