This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum). For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.

The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform of a Gaussian function is another Gaussian function. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The imaginary part of ĝ( ω) is negated because a negative sign exponent has been used in the Fourier transform, which is the default as derived from the Fourier series, but the sign does not matter for a transform that is not going to be reversed.įunctions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The Fourier transform decomposes a function into eigenfunctions for the group of translations. delay) in the time domain is interpreted as complex phase shifts in the frequency domain. The bottom row shows a delayed unit pulse as a function of time ( g( t)) and its Fourier transform as a function of frequency ( ĝ( ω)). The top row shows a unit pulse as a function of time ( f( t)) and its Fourier transform as a function of frequency ( f̂( ω)). The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.

The term Fourier transform refers to both this complex-valued function and the mathematical operation. The output of the transform is a complex-valued function of frequency. In physics, engineering and mathematics, the Fourier transform ( FT) is an integral transform that converts a function into a form that describes the frequencies present in the original function. A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed. The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. An example application of the Fourier transform is determining the constituent pitches in a musical waveform.

Light intensity equation exponent