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## Orthogonal Transforms

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**Orthogonal Transforms**Fourier**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**Note that these are arbitrary functions, we do not assume**sinusoids**We want to minimize this kinds of errors.**• Other error measures are also used.**Unitary Transforms**• Unitary Transformation for 1-Dim. Sequence • Series representation of • Basis vectors : • Energy conservation : Here is the proof**Unitary Transformation for 2-Dim. Sequence**• Definition : • Basis images : • Orthonormality and completeness properties • Orthonormality : • Completeness :**Unitary Transformation for 2-Dim. Sequence**• Separable Unitary Transforms • separable transform reduces the number of multiplications and additions from to • Energy conservation**Properties of Unitary Transform**transform Covariance matrix**Example of arbitrary basis functions being rectangular waves****0**1**This slide shows four base functions multiplied by their**respective coefficients**This slide shows that using only four base functions the**approximation is quite good End of example**Forward transform**inverse transform separable**Fourier Transforms in new notations**We emphasize generality Matrices**Fourier Transform**separable**Discrete Fourier Transform (DFT)**New notation**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**Fast Factorization Algorithms are general and there is many**of them**1-dim. DFT (cont.)**• Calculation of DFT : Fast Fourier Transform Algorithm (FFT) • Decimation-in-time algorithm Derivation of decimation in time**Butterfly for Derivation of decimation in time**• 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-time algorithm (cont.) Please note recursion**1-dim. DFT (cont.)**• FFT (cont.) • Decimation-in-frequency algorithm (cont.) • Derivation of Decimation-in-frequency algorithm**Decimation in frequency butterfly shows recursion**• 1-dim. DFT (cont.) • FFT (cont.) • Decimation-in-frequency algorithm (cont.)**Conjugate Symmetry of DFT**• For a real sequence, the DFT is conjugate symmetry