# FFT, Mathematica, Continuous Fourier Transform

• Mathematica
• Anthony
In summary, the person is asking for someone to help them approximate the inverse Fourier transform of a smooth function. They have tried using the built-in inverse Fourier transform function, but it does not work well. They want to know if someone can help them implement a function in Mathematica to get a plot of the inverse Fourier transform.

#### Anthony

Hi all,

First a warning: my Mathematica skills, and computery-type skills in general, are not very hot. My problem is thus: I have a function which I know:

$$\hat{f}(k)$$

I'd like mathematica to approximate the inverse Fourier transform of this function for me and plot the result. I've tried using the built-in function "NInverseFourierTransform", but it fails to produce meaningful results. My function oscillates quite rapidly, so NIntegrate doesn't work too well.

Now I'm aware that I could approximate the inverse Fourier transform using a discrete Fourier transform and the FFT algorithm - but I'm afraid I don't really know how to go about doing it. I can do the following:

• Get as many sample points of $$\hat{f}(k)$$ as you want. Call them $$\{\hat{f}_n\}$$.
• I can make $$\hat{f}$$ rapidly decreasing, so it's pretty much got compact support.
• My $$\hat{f}$$ is smooth.
I figure if I've got the above properties, there must be some way of approximating the inverse Fourier transform using the built in FFT functions in mathematica. I've tried using InverseFourier
• , where list contains the $$\hat{f}_n$$, and plotting the real part of it, but the answer is gibberish. I've proved lots of rigorous results regarding the function $$f$$, so I know (pretty much) what the plot of the inverse Fourier transform should look like!

If anyone could help me implement the built in mathematica functions to get a plot of this inverse Fourier transform, I'd be immensely grateful.

Thanks,
Ant

All sorted now - I rolled up my sleeves and got stuck into mathematica.

i don't understand how fft algorithm works.
but i have to solve the problem by Mathematica code which i have attached.
can anybody help me to solve this function.
it will be if any explain by a simple function.

thanks,
Happy

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## 1. What is FFT?

FFT stands for Fast Fourier Transform, which is a mathematical algorithm used to convert a signal from its original domain (often time or space) to a representation in the frequency domain. It is a commonly used tool in signal processing, data analysis, and scientific computing.

## 2. What is Mathematica?

Mathematica is a software program used for symbolic mathematical computation, data analysis, and visualization. It is popular among scientists and engineers for its powerful capabilities in solving complex mathematical problems and creating interactive visualizations.

## 3. How does the FFT algorithm work?

The FFT algorithm works by breaking down a signal into smaller components, using a divide-and-conquer approach. It then applies a series of mathematical operations to these components to convert them into the frequency domain. This process is repeated until the entire signal has been transformed.

## 4. What is the difference between FFT and Continuous Fourier Transform?

The FFT is a discrete version of the Continuous Fourier Transform, which is used to transform a continuous signal into its frequency components. The FFT is faster and more efficient, but it can only be applied to signals that are sampled at regular intervals. The Continuous Fourier Transform, on the other hand, can be applied to any continuous signal.

## 5. How is the Continuous Fourier Transform calculated in Mathematica?

In Mathematica, the Continuous Fourier Transform can be calculated using the FourierTransform function. It takes the signal as an input and returns a symbolic expression representing the frequency components of the signal. This expression can then be plotted or further manipulated using other Mathematica functions.