Exploring Fourier Transform and Its Relation to f(t-a)

Thread starter spaghetti3451
Start date May 31, 2011
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Fourier Fourier transform Transform

In summary, Fourier Transform is a mathematical operation that decomposes a signal into its constituent frequencies, making it easier to analyze and process. It is used in various fields of science, such as signal processing, image processing, quantum mechanics, and spectroscopy. The function f(t-a) is a time-shifted version of the original signal f(t), and the Fourier Transform of f(t-a) will also be a time-shifted version of the original Fourier Transform. While it has many benefits, such as identifying specific frequencies and filtering out noise, Fourier Transform also has limitations, such as assuming the signal is periodic and continuous, and being computationally intensive for large datasets.

May 31, 2011

spaghetti3451

I have to show that the Fourier transform of f(t-a) is exp(-iwa)*F(w).

Any headstart?

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May 31, 2011

micromass

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Hi failexam!

Apply substitution u=t-a in the integral.

May 31, 2011

spaghetti3451

I see, but then F(w) is not the Fourier transform of f(t), but of f(t-a) !

May 31, 2011

micromass

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I don't see that, can you make the calculation? It'll turn out nicely...

1. What is Fourier Transform?

Fourier Transform is a mathematical operation that decomposes a signal into its constituent frequencies. It is used to convert a signal from its original domain (often time or space) to a representation in the frequency domain.

2. What is the relation between Fourier Transform and f(t-a)?

The function f(t-a) is a time-shifted version of the original signal f(t). This means that the original signal has been shifted by a time interval of "a". The Fourier Transform of f(t-a) will also be a time-shifted version of the original Fourier Transform, with the same amount of shift, but in the opposite direction.

3. How is Fourier Transform used in science?

Fourier Transform is used in various fields of science, such as signal processing, image processing, quantum mechanics, and spectroscopy. It helps in analyzing and understanding the frequency components of a signal, which can provide valuable information about the underlying physical processes.

4. What are the benefits of using Fourier Transform?

Fourier Transform allows us to break down a complex signal into simpler components, making it easier to analyze and process. It also helps in identifying specific frequencies present in a signal, which can provide insights into the underlying dynamics. Additionally, Fourier Transform is a powerful tool for filtering out unwanted noise from a signal.

5. Are there any limitations to the Fourier Transform?

While Fourier Transform is a powerful tool, it has some limitations. It assumes that the signal is periodic, which may not be true for all signals. It also assumes the signal is continuous, which may not be the case for discrete signals. Additionally, it can be computationally intensive for large datasets, making it challenging to analyze in real-time applications.