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Discrete-time Fourier Transform

Discrete-time Fourier Transform. Prof. Siripong Potisuk. Derivation of the Discrete-time Fourier Transform. DTFT Pair. Conditions for Convergence. Examples. Example 1: 1 st Order System, Decay Power. stable system. Calculate the DT Fourier transform of the signal: Therefore:. a =0.8.

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Discrete-time Fourier Transform

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  1. Discrete-time Fourier Transform Prof. Siripong Potisuk

  2. Derivation of the Discrete-time Fourier Transform

  3. DTFT Pair

  4. Conditions for Convergence

  5. Examples

  6. Example 1: 1st Order System, Decay Power stable system • Calculate the DT Fourier transform of the signal: • Therefore: a=0.8

  7. Example 2: Rectangular Pulse N1=2 • Consider the rectangular pulse • and the Fourier transform is

  8. IDTFT

  9. 6) Complex Exponentials

  10. DTFT of Periodic Signals Recall the following DTFT pair: Represent periodic signal x[n]in terms of DTFS:

  11. Properties of DTFT

  12. Properties of DTFT

  13. Convolution Property

  14. Multiplication Property

  15. The Discrete Cosine Transform • In the same family as the Fourier Transform • Converts data to frequency domain. • Represents data via summation of variable frequency cosine waves. • Since it is a discrete version, conducive to problems formatted for computer analysis. • Captures only real components of the function. • Discrete Sine Transform (DST) captures odd (imaginary) components → not as useful. • Discrete Fourier Transform (DFT) captures both odd and even components → computationally intense.

  16. Significance / Where is this used? • Image Processing • Compression - Ex.) JPEG • Scientific Analysis - Ex.) Radio Telescope Data • Audio Processing • Compression - Ex.) MPEG – Layer 3, aka. MP3 • Scientific Computing / High Performance Computing (HPC) • Partial Differential Equation Solvers

  17. Algorithm Walk Through • Mathematical Basis • 1D Version: • Where: • 2D Version: • Where α(u) and α(v) are defined as shown in the 1D case.

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