Fourier transform of a shifted and time-reversed sign

In summary: The Fourier transform of the shifted-time-reversed signal is shown in the attached picture.The Fourier transform of the shifted-time-reversed signal is shown in the attached picture.In summary, the Fourier transform of the shifted-time-reversed signal is shown in the attached picture. The Fourier transform of the shifted-time-reversed signal is based on the knowledge of the Fourier transform of the original signal.
  • #1
Legend101
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Homework Statement


given a continuous-time signal g(t) . Its Fourier transform is G(f) ( see definition in picture / "i" is the imaginary number) . It is required to find the Fourier transform of the shifted-time-reversed signal g(a-t) where a is a real constant .
That is , find the Fourier transform of g(a-t) based on the knowledge of the Fourier transform G(f) of g(t)

Homework Equations


The defition of the Fourier transform is shown in the attached picture

The Attempt at a Solution


There are 2 properties of the Fourier transform : shift property + time scaling.
But I'm not sure how to use them both . I prefer to use the definition of the Fourier transform to find the relationship between the Fourier transform of g(a-t) and the Fourier transform of g(t)[/B]
 

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  • #2
So you know ##\mathcal{F} \{g(t) \} = G(f)##.

Try writing this:

$$g(a - t) = g((-1) (t - a))$$

Now you can use both the time scaling and time shifting properties.
 
  • #3
Zondrina said:
So you know ##\mathcal{F} \{g(t) \} = G(f)##.

Try writing this:

$$g(a - t) = g((-1) (t - a))$$

Now you can use both the time scaling and time shifting properties.
Can you check the workout i did in the picture below ?
 

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  • #4
The time scaling property states:

$$\mathcal{F} \{g(\alpha x) \} = \frac{1}{|\alpha|} \hat G \left( \frac{f}{\alpha} \right)$$

You know ##\mathcal{F} \{g(t) \} = \hat G(f)## and ##g(a - t) = g((-1) (t - a))##.

Let ## u = t - a##. Then ##g((-1) (t - a)) = g((-1)u) = g(\alpha u)## with ##\alpha = -1##.

Then you know ##\mathcal{F} \{g(\alpha u) \} = \frac{1}{|\alpha|} \hat G (\frac{f}{\alpha}) = \hat G (-f)##.

This amounts to saying: Since you know ##\mathcal{F} \{g(t) \} = \hat G(f)##, then you can simply apply a negative sign to the argument and get ##\mathcal{F} \{ g(a - t) \}##.
 
  • #5
Zondrina said:
The time scaling property states:

$$\mathcal{F} \{g(\alpha x) \} = \frac{1}{|\alpha|} \hat G \left( \frac{f}{\alpha} \right)$$

You know ##\mathcal{F} \{g(t) \} = \hat G(f)## and ##g(a - t) = g((-1) (t - a))##.

Let ## u = t - a##. Then ##g((-1) (t - a)) = g((-1)u) = g(\alpha u)## with ##\alpha = -1##.

Then you know ##\mathcal{F} \{g(\alpha u) \} = \frac{1}{|\alpha|} \hat G (\frac{f}{\alpha}) = \hat G (-f)##.

This amounts to saying: Since you know ##\mathcal{F} \{g(t) \} = \hat G(f)##, then you can simply apply a negative sign to the argument and get ##\mathcal{F} \{ g(a - t) \}##.
There has to be an exponential factor . Did you check my solution ?
 
  • #6
The solution is fine, I thought it would be insightful to elaborate on the time-reversal property where ##\alpha = -1##.

shifted-time-reversed signal
 

FAQ: Fourier transform of a shifted and time-reversed sign

What is a Fourier transform?

A Fourier transform is a mathematical operation that decomposes a function into its component frequencies. It is often used in signal processing and analysis to understand the frequency content of a signal.

What does it mean to shift a signal in the Fourier domain?

Shifting a signal in the Fourier domain means applying a time delay to the original signal. This results in a phase shift in the frequency domain, which can affect the overall shape and amplitude of the signal.

Why is the Fourier transform of a time-reversed signal different from the original signal?

When a signal is time-reversed, its frequency content is also reversed. This means that the Fourier transform will show the same frequency components but with opposite signs. This results in a reflection of the original Fourier transform about the y-axis.

How does shifting and time-reversing a signal affect its Fourier transform?

Shifting and time-reversing a signal affects its Fourier transform by introducing phase shifts and sign changes. The overall shape and amplitude of the Fourier transform will also be altered, depending on the amount of shift and reversal applied to the signal.

What are some real-world applications of the Fourier transform of a shifted and time-reversed signal?

The Fourier transform of a shifted and time-reversed signal has various applications in signal processing, audio and image compression, and data analysis. It is also used in fields such as astronomy, physics, and engineering to analyze and interpret signals from different sources.

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