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FFT bounds

A Fast Fourier Transform (FFT) is well established computational / mathematical method for calculating the Discrete Fourier Transform (DFT) of a dataset. That's to say, which simple harmonic frequencies, in which ratios, can be combined to approximate a complex varying signal (i.e. waveform).

FFT is a widely used essential tool in signal analysis - in fields such as communications, audio processing, radar, orbital calculations etc. etc. .

The original DFT was discovered in the early 1800s, but the complexity of the calculations led to the development, mainly in the mid 1900s, of a special versions, called FFT, which greatly simplified the mathematical work involved - sometimes by a factor of more than 1,000.

The number of calculations required - known as the 'bounds' - is still substantial, and a goal is to define the minimum number necessary, and if possible improve the current methods.

The lower bounds are currently unknown. Formally stated : Can they be faster than $${\displaystyle O(N\log N)}$$

- where O is the computational load, and N is the size of the dataset.

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