List of Fourier-related Transforms - Continuous Transforms

Continuous Transforms

Applied to functions of continuous arguments, Fourier-related transforms include:

• Two-sided Laplace transform
• Mellin transform, another closely related integral transform
• Laplace transform
• Fourier transform, with special cases:
  • Fourier series
    • When the input function/waveform is periodic, the Fourier transform output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called Fourier series coefficients. The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients.
    • When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued. But a discrete subset of its values is sufficient to reconstruct/represent the portion that was analyzed. The same discrete set is obtained by treating the duration of the segment as one period of a periodic function and computing the Fourier series coefficients.
  • Sine and cosine transforms: When the input function has odd or even symmetry around the origin, the Fourier transform reduces to a sine or cosine transform.
• Hartley transform
• Short-time Fourier transform (or short-term Fourier transform) (STFT)
• Chirplet transform
• Fractional Fourier transform (FRFT)
• Hankel transform: related to the Fourier Transform of radial functions.