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Lecture 14: More on finite-length discrete transforms

Lecture 14: More on finite-length discrete transforms. Instructor: Dr. Gleb V. Tcheslavski Contact: gleb@ee.lamar.edu Office Hours: Room 2030 Class web site: http://ee.lamar.edu/gleb/dsp/index.htm. Smiley face. Orthogonal transforms.

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Lecture 14: More on finite-length discrete transforms

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  1. Lecture 14: More on finite-length discrete transforms Instructor: Dr. Gleb V. Tcheslavski Contact:gleb@ee.lamar.edu Office Hours: Room 2030 Class web site:http://ee.lamar.edu/gleb/dsp/index.htm Smiley face

  2. Orthogonal transforms Frequently, it is beneficial to convert a finite-length time-domain sequence xn into a finite-length sequence in other domain and vice versa: Analysis: (14.2.1) Synthesis (14.2.2) We restrict our discussion to the class of orthogonal transforms. For such transforms, the basis sequences (functions) k,n satisfy the following properties: (14.2.3)

  3. Orthogonal transforms For the orthogonal basis functions, we can verify that (14.2.2) is indeed an inverse of (14.2.1): (14.3.1) An important consequence of orthogonality is the energy conservation property of such transforms that allows to compute the energy of a time-domain sequence in the transform domain (Parseval’s relation): (14.3.2) In some applications, it may be important to use a transform that decorrelates the transform coefficients. In other applications, energy compaction (concentration of most of signal energy in few transform coefficients) is highly desirable.

  4. DFT revisited Basis functions: (14.4.1) Therefore, the N-point DFT pair: (14.4.2) (14.4.3) where (14.4.4) As a result, even for real xn, its DFT Xk is generally complex: (14.4.5)

  5. DFT symmetry relations Real and imaginary parts of the DFT sequence can be found as: (14.5.1) (14.5.2) Assuming that the original time-domain signal is complex: (14.5.3) its DFT can be found as: (14.5.4)

  6. DFT symmetry relations Therefore, real and imaginary parts of the DFT sequence are: (14.6.1) (14.6.2) Here xcs,n and xca,n are circular conjugate-symmetric and circular conjugate-antisymmetric parts of xn: (14.6.3) (14.6.4) (14.6.5) circular shift

  7. DFT symmetry relations Therefore, for complex sequences: (14.7.1) (14.7.2) (14.7.3) (14.7.4) (14.7.5) (14.7.6)

  8. DFT symmetry relations For real sequences: (14.8.1) real (14.8.2) Therefore: (14.8.3) (14.8.4) (14.8.5) (14.8.6) (14.8.7) (14.8.8) (14.8.9)

  9. DFT symmetry relations If N (the length of the sequence) is even, the DFT samples X0 and X(N-2)/2 are real and distinct. The remaining N – 2 samples of DFT are complex: a half of them are distinct and the rest are their complex conjugates. If N (the length of the sequence) is odd, the DFT sample X0 is real and distinct. The remaining N – 1 samples of DFT are complex: a half of them are distinct and the rest are their complex conjugates. It is frequently desired to have an orthogonal transform that represents a real time-domain sequence as another real sequence in the transform domain.

  10. Type 2 DCT There are several (8?) types of Discrete Cosine Transform (DCT), for the development of which Dr. Rao (UT Arlington) is credited… The most commonly used is DCT 2: Basis functions: (14.10.1) The DCT pair: (14.10.2) (14.10.3) where (14.10.4)

  11. Type 2 DCT DCT Properties: 1. Linearity: (14.11.1) 2. Symmetry: (14.11.2) (14.11.3) 3. Energy preservation: DCT of a real sequence is another real sequence. DCT has very good energy compaction properties: most of the signals energy is confined to several first DCT coefficients. Therefore, DCT is used for lossy data compression: JPEG, MPG, mp3…

  12. Computation of type 2 DCT DCT is related to DFT (14.12.1) • Therefore, the N-point DCT can be computed (by utilizing FFT algorithms) as follows: • Extend xn to a length-2N sequence xe,n by zero-padding and compute its 2N-point DFT Xe,k; • Extract the first N samples of Xe,k and multiply each sample by ; • Extract the real part of the above samples and multiply each by 2.

  13. The Haar transform The transform pair: (14.13.1) (14.13.2) Haar transform coefficients: (14.13.3) Time-domain samples: (14.13.4) Normalized Haar transform matrix: (14.13.5) (14.13.6) Where denotes the Kronecker product and Ik is a k x k identity matrix.

  14. The Haar transform Properties: 1. Orthogonality: (14.14.1) which, for the inverse transform, leads to: (14.14.2) 2. Energy conservation: (14.13.3) The Haar transform represents a real time-domain sequence as another real transform-domain sequence. Used in data compression algorithms.

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