Fourier transform of function times periodic function

In summary, a Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. It is calculated by integrating the function over all time or space, multiplied by a complex exponential function. The Fourier transform of a function times a periodic function represents the spectral density of the original function and has applications in fields such as signal processing, image processing, and audio analysis. However, it has limitations in that it can only be used for continuous functions with finite energy and assumes periodicity.
  • #1
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Suppose I have a function of the type:

h(t) = g(t)f(t)

where g(t) is a periodic function. Are there any nice properties relating to the Fourier transform of such a product?

Edit: If not then what about if g(t) is taken as the complex exponential?
 
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  • #2
Not sure about your first question, but if
$$
h(t) \Leftrightarrow H(f)
$$
is a transform pair, then
$$
h(t) e^{-2 \pi i f_0 t} \Leftrightarrow H(f - f_0)
$$
and its called frequency shifting (taken from Numerical Recipes in C).
 

FAQ: Fourier transform of function times periodic function

What is a Fourier transform?

A Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. It takes a function in the time or spatial domain and converts it into a function in the frequency domain.

How is a Fourier transform calculated?

The Fourier transform of a function is calculated by integrating the function over all time or space, multiplied by a complex exponential function. This integration is typically done using mathematical software or tables.

What does the Fourier transform of a function times a periodic function represent?

The Fourier transform of a function times a periodic function represents the spectral density of the original function, taking into account the periodicity of the second function. It can be used to analyze signals that have periodic components.

What are some applications of the Fourier transform of a function times a periodic function?

The Fourier transform of a function times a periodic function has many applications in various fields, including signal processing, image processing, and audio analysis. It is also used in quantum mechanics and electromagnetic theory.

Are there any limitations to using the Fourier transform of a function times a periodic function?

While the Fourier transform is a powerful tool, it does have some limitations. It can only be used for functions that are continuous and have finite energy. It also assumes that the function is periodic, which may not always be the case in real-world applications.

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