What is the difference between DFT and FFT?

The DFT (Discrete Fourier Transform) is the mathematical definition — O(n^2). The FFT (Fast Fourier Transform) is an algorithm that computes the DFT in O(n log n) by exploiting symmetry and recursion. The result is identical; the FFT is just much faster. For n=1024, FFT is ~100x faster than DFT.

Why must the input length be a power of 2?

The Cooley-Tukey FFT algorithm recursively splits the input into even and odd halves. This requires the length to be a power of 2. For other lengths, zero-pad to the next power of 2, or use mixed-radix FFT (supports lengths that are products of small primes). Most FFT libraries handle non-power-of-2 lengths automatically.

TASK

Use Fast Fourier Transform

FFT computes the discrete Fourier transform in O(n log n) instead of O(n^2). Used in signal processing, polynomial multiplication, and audio analysis.

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The FFT converts a signal from time domain to frequency domain. It decomposes a signal into its frequency components. The DFT is O(n^2); the FFT (Cooley-Tukey) is O(n log n) by recursively splitting even and odd indices. Requirements: input length should be a power of 2 (zero-pad if not). Applications: audio processing, image compression (JPEG), polynomial multiplication, convolution.
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Complex Multiply →
FFT uses complex arithmetic. The twiddle factors are complex roots of unity: e^(-2πi k/n). This tool performs complex multiplication, which is the core operation in the FFT butterfly. Each butterfly combines two complex values with a twiddle factor.
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Polynomial Evaluate →
FFT can multiply polynomials in O(n log n) instead of O(n^2): evaluate both polynomials at the n-th roots of unity (FFT), multiply pointwise, then interpolate (inverse FFT). This is faster than direct coefficient multiplication for large polynomials.

Frequently asked questions

What is the difference between DFT and FFT?: The DFT (Discrete Fourier Transform) is the mathematical definition — O(n^2). The FFT (Fast Fourier Transform) is an algorithm that computes the DFT in O(n log n) by exploiting symmetry and recursion. The result is identical; the FFT is just much faster. For n=1024, FFT is ~100x faster than DFT.
Why must the input length be a power of 2?: The Cooley-Tukey FFT algorithm recursively splits the input into even and odd halves. This requires the length to be a power of 2. For other lengths, zero-pad to the next power of 2, or use mixed-radix FFT (supports lengths that are products of small primes). Most FFT libraries handle non-power-of-2 lengths automatically.