1 / 19

Understanding the Discrete Fourier Transform (DTFT) for Signal Analysis

Learn about the derivation of the DTFT, convergence conditions, and examples of calculating DT Fourier transforms. Explore properties, including convolution and multiplication, and discover the significance of DTFT in various applications like image processing, audio compression, and scientific analysis. Algorithm walkthrough and mathematical basis included.

cbrennan
Télécharger la présentation

Understanding the Discrete Fourier Transform (DTFT) for Signal Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related