1 / 12

Understanding Fourier Transforms and System Properties in Signal Processing

This document explores key concepts in signal processing, including the uniqueness of the Fourier transform and its implications for analyzing signals. It discusses the preservation of signal strength through Fourier series as evidenced by Parseval's relations. Furthermore, it examines system properties such as memorylessness, time invariance, linearity, causality, and stability, providing insights into how these characteristics affect signals and systems. Relevant examples clarify the concepts, ensuring a comprehensive understanding of these fundamental principles.

mary
Télécharger la présentation

Understanding Fourier Transforms and System Properties in Signal Processing

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. Solutions Q1: False. The Fourier transform is unique. If two signals have the same Fourier transform, then there is a confusion when trying to find the inverse transform. Violates the assumption that x(t) and y(t) are different

  2. Q1b: Solution b)

  3. Q1c: Solution • c) The signal strength before and after the Fourier transform and Fourier series remains the same. This is evident from Parseval’s Relations:

  4. Q2: Solution

  5. Q3: Solution (Memory & Time Invariance) • System is not memoryless because y(t) depends on x(t-2) for some t • System is not time-invariant, because y1(t) ≠ y2(t) below

  6. Q3: Solution (Linearity) • System is linear

  7. Q3: Solution (Causality) • System is causal. Because the system is not LTI, we cannot use the h(t) = 0 for t < 0 test. However, we can use the following test: A linear non-TI system is causal if x(t) = 0 for t < t0 then y(t) = 0 for t < t0 For our system, the above is true for t0=0 as in the definition. So system is causal • We can also see that the output y(t) depends on current x(t) and past x(t-2) values of the input

  8. Q3: Solution (Stability) • Note that because the system is not LTI, we could not use the test • System is BIBO stable

  9. Q4: Solution

  10. Q5: Solution • We have • Taking the inverse transform, we get

More Related