Fast Fourier Transform and its inverse

In summary, the conversation discusses the presence of \(i\) in the FFTs of different equations, including the NLS and KdV equations. It is confirmed that the FFT for the KdV equation does not need to remove the \(i\).
  • #1
Dustinsfl
2,281
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Does every FFT have \(i\) in it?

Given \(u_t = -(u_{xxx} + 6uu_x)\).

\(f'''(x) = \mathcal{F}^{-1}\left[(ik)^3\mathcal{F}(f(x))\right]\)
\(f'(x) = \mathcal{F}^{-1}\left[(ik)\mathcal{F}(f(x))\right]\)

The only equation I have used the pseudo-spectral method on was the NLS which is
\(u_t = i(u_{xx} + |u|^2u)\). In this case, I know I will have \(i\) in the FFT.

Are my transforms for the KdV correct or do I need to remove \(i\)?
 
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  • #2
Yes, your transforms are correct and you do not need to remove the \(i\).
 

FAQ: Fast Fourier Transform and its inverse

What is the Fast Fourier Transform (FFT)?

The Fast Fourier Transform is an algorithm used to efficiently compute the Discrete Fourier Transform (DFT) of a discrete signal or sequence. It is widely used in various fields such as signal processing, image processing, and data compression.

How does the FFT work?

The FFT works by breaking down a signal into its individual frequency components. It starts by dividing the signal into smaller segments and then combines the results to calculate the overall DFT. This process reduces the number of calculations needed, making it much faster than traditional methods for computing DFT.

What is the difference between FFT and its inverse?

The FFT calculates the frequency components of a signal, while its inverse, the Inverse Fast Fourier Transform (IFFT), reconstructs the original signal from its frequency components. The FFT is used to compress and analyze signals, while the IFFT is used to decompress and synthesize signals.

What are the applications of FFT?

The FFT has a wide range of applications, including audio and video processing, image and signal compression, speech recognition, and solving differential equations. It is also used in various scientific fields such as astronomy, physics, and engineering for data analysis and visualization.

Are there any limitations of FFT?

While FFT is a powerful and widely used algorithm, it does have some limitations. It requires the input signal to be discrete and periodic, and the number of data points must be a power of 2 for the most efficient computation. Additionally, FFT is not suitable for signals with sharp edges or discontinuities.

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