Phase Shift and Time Shift - Fourier Transform

In summary, the conversation discusses the relationship between phase shift and time shift in Fourier Transforms. The equation x(t-t0) → e^(jwt0)X(jw) is mentioned as a means of simplifying the problem, but the confusion lies in making the simplification toward the end. It is suggested that the correct equation may be x(t-t0) → exp(-jωt0)X(jω), with a negative sign in the exponent.
  • #1
johnstamos
5
0

Homework Statement



I'm trying to relate phase shift and time shift Fourier Transformers

Homework Equations



x(t-t_0) → e^(jwt0)X(jw)

The Attempt at a Solution



I've attached a picture of my work. I'm a bit confused as to how I would be able to make that simplification towards the end.

http://imageshack.us/photo/my-images/694/photofeb1585114pm.jpg/

Regards
 
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  • #2
johnstamos said:

Homework Statement



I'm trying to relate phase shift and time shift Fourier Transformers

Homework Equations



x(t-t_0) → e^(jwt0)X(jw)

The Attempt at a Solution



I've attached a picture of my work. I'm a bit confused as to how I would be able to make that simplification towards the end.

http://imageshack.us/photo/my-images/694/photofeb1585114pm.jpg/

Regards

How about x(t-t0) → exp(-jωt0)X(jω)? Note the - sign in the exponent.
 

1. What is the Fourier Transform?

The Fourier Transform is a mathematical tool used to analyze and understand the frequency components of a signal. It converts a signal from its original form in the time domain to its representation in the frequency domain. This allows for easier analysis and manipulation of the signal's frequency components.

2. What is Phase Shift?

Phase shift refers to the delay or advancement in the timing of a signal's frequency components. It is measured in degrees and can occur when a signal is passed through a system, such as a filter or amplifier. Phase shift can also be intentionally introduced to a signal for various purposes, such as in audio effects or communication systems.

3. What is Time Shift?

Time shift is the change in the timing of a signal. It can occur when a signal is delayed or advanced in time, either intentionally or unintentionally. Time shift can also be introduced to a signal through various processes, such as filtering or modulation.

4. How are Phase Shift and Time Shift related to the Fourier Transform?

The Fourier Transform allows us to analyze the frequency components of a signal, including any phase or time shifts. It provides a representation of the signal in the frequency domain, which can help us understand and manipulate these shifts. Additionally, the Fourier Transform can be used to identify and correct for any phase or time shifts that may occur in a signal.

5. What are some practical applications of Phase Shift and Time Shift in the real world?

Phase shift and time shift are commonly used in various fields, such as signal processing, telecommunications, audio engineering, and image processing. They are used to design and optimize filters, correct for distortions in signals, and create various effects in audio and visual media. These concepts are also important in understanding the behavior of electronic circuits and systems.

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