Question

What increase in speed can be expected in using a fast Fourier transform algorithm rather than direct arithmetic to compute the Fourier transform of an image of size 1024 × 1024?

Answer 1

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Q) What increase in speed can be expected in using a fast Fourier transform algorithm rather than direct arithmetic to compute the Fourier transform of an image of size 1024 × 1024?

Answer: -

Before I answer your question lets know what is FFT (Fast fourier transform) algorithm ?

The Fast Fourier Transform (FFT) is an algorithm that determines discrete Fourier Transform of an input significantly more faster than computing it directly using arithmetics.

So as per defination of Fast Fourier Transform we can say that a fast Fourier transform algorithm is more in speed rather than direct arithmetic to compute the Fourier transform of an image.

The FFT is just a faster implementation of the discrete Fourier transform .The FFT algorithm reduces an n-point Fourier transform to about

(n/2) log₂ (n)  complex multiplications.

Calculated directly, a DFT on 1,024 (i.e., 2¹⁰) data points would require

n² = 1,024 × 1,024 = 2²⁰ = 1,048,576 multiplications.

The Fast Fourier Transform algorithm reduces this to about

(n/2) log₂ (n) = 512 × 10 = 5,120 multiplications, for a factor-of-200 improvement.

You can see the expected increase in speed by using a fast Fourier transform algorithm rather than direct arithmetic to compute the Fourier transform of an image of size 1024 × 1024

But the increase in speed comes at the cost of versatility.

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