FFT vs Filon's Rule: Accuracy Comparison

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In summary, FFT is preferred over Filon's rule because it is faster and more versatile, even though it may not always be as accurate.
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Homework Statement


Suppose we can calculate a quantity f(t) and we need its Fourier transform F(w). Looks to me that accuracywise Filon's rule, e.g. approximating the computed f(t) by splines and analytically integrating piecewise should be more accurate than an FFT, at least for a smooth f(t); especially for large enough w(>w_Nyquist) the FFT would show aliasing, while Filon's rule might lose accuracy due to the summation of a large number of terms of possibly different sign. Why is FFT always used?
 
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Homework Equations N/AThe Attempt at a Solution FFT is used because it is more computationally efficient than Filon's rule. While Filon's rule requires repeated integrations of the approximated spline, FFT is just a sequence of fast discrete Fourier transforms that can be implemented using matrix multiplication algorithms. Additionally, FFT can be used to calculate the Fourier transform of a function of any length and resolution, whereas Filon's rule is only applicable to evenly spaced data points.
 

FAQ: FFT vs Filon's Rule: Accuracy Comparison

1. What is the difference between FFT and Filon's rule?

FFT (Fast Fourier Transform) is an algorithm used to quickly compute the discrete Fourier transform of a signal or sequence. Filon's rule is a numerical integration technique used to approximate the value of a definite integral.

2. Which method is more accurate, FFT or Filon's rule?

This depends on the specific application and the accuracy requirements. In general, FFT is more accurate for computing the Fourier transform of a signal, while Filon's rule may be more accurate for numerical integration tasks.

3. Can FFT and Filon's rule be used interchangeably?

No, FFT and Filon's rule are used for different purposes and cannot be used interchangeably. FFT is used for signal processing and digital signal analysis, while Filon's rule is a numerical integration method.

4. Are there any drawbacks to using FFT or Filon's rule?

Both methods have their limitations. FFT can be computationally intensive and may not be suitable for certain types of signals. Filon's rule may not be accurate for highly oscillatory functions or functions with singularities.

5. When should I use FFT and when should I use Filon's rule?

FFT is typically used for analyzing signals and computing the Fourier transform, while Filon's rule is useful for numerical integration tasks. It is important to understand the specific requirements of your problem and choose the method that is most appropriate for your application.

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