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Lesson 3

Lesson 3. Discrete-time Fourier Analysis. Lesson 2 Recap. Basis Signals. Scaled and Shifted Unit Impulse:. Complex Exponentials:. Convolution is Matrix-Vector Multiplication. Vector x multiplies with a matrix A, whose rows are folded and shifted h.

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Lesson 3

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  1. Lesson 3 Discrete-time Fourier Analysis

  2. Lesson 2 Recap

  3. Basis Signals Scaled and Shifted Unit Impulse: Complex Exponentials:

  4. Convolution is Matrix-Vector Multiplication. • Vector x multiplies with a matrix A, whose rows are folded and shifted h. • We can think of a LTI as a matrix, the input signal as a vector and the output signal as another vector.

  5. Complex exponentials are eigenvectors of the convolution matrix. • What is an eigenvector of a matrix? A x = λ x x is an eigenvector of A λ is an eigenvalue corresponding to x

  6. Represent an input signal as an superposition of the complex exponentials

  7. How to calculate Ak? Discrete-time Fourier Transform (DTFT)

  8. Two Important Properties • Periodicity • Symmetry Implication: We only need to consider

  9. Matlab Implementation

  10. Properties of DTFT • Linearity

  11. Properties of DTFT • Time Shifting

  12. Properties of DTFT • Frequency shifting

  13. Properties of DTFT • Conjugation

  14. Properties of DTFT • Folding

  15. Properties of DTFT • Convolution

  16. Properties of DTFT • Multiplication

  17. Properties of DTFT • Energy (Parseval’s Theorem)

  18. Frequency-domain Representation of LTI • Complex exponentials are the eigenvectors of LTI. • The Fourier Transform of the impulse response gives the eigenvalues.

  19. Frequency-domain representation of LTI Phase response Magnitude (gain) response

  20. An Example

  21. Sampling of Analog Signals Continuous-Time Fourier Transform (CTFT)

  22. Sampling of Analog Signals Sampling Interval Sampling Frequency (sam / sec)

  23. Poisson Summation Formula

  24. Poisson Summation Formula

  25. Poisson Summation Formula • Dirac Comb

  26. Sampling of Analog Signals Aliasing Formula

  27. Sampling of Analog Signals Signal Bandwidth Nyquist rate Harry Nyquist

  28. Claude Elwood Shannon

  29. Example

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