Sine and cosine transforms

In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics.^[1]

Definition

The Fourier sine transform of $f (t)$ , sometimes denoted by either ${\hat {f}}^{s}$ or ${\mathcal {F}}_{s}(f)$ , is ${\hat {f}}^{s}(\xi )=\int _{-\infty }^{\infty }f(t)\sin(2\pi \xi t)\,dt.$

If $t$ means time, then $ξ$ is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other.

This transform is necessarily an odd function of frequency, i.e. for all $ξ$ : ${\hat {f}}^{s}(-\xi )=-{\hat {f}}^{s}(\xi ).$

The numerical factors in the Fourier transforms are defined uniquely only by their product. Here, in order that the Fourier inversion formula not have any numerical factor, the factor of 2 appears because the sine function has $L 2$ norm of ${\tfrac {1}{\sqrt {2}}}.$

The Fourier cosine transform of $f (t)$ , sometimes denoted by either ${\hat {f}}^{c}$ or ${\mathcal {F}}_{c}(f)$ , is ${\hat {f}}^{c}(\xi )=\int _{-\infty }^{\infty }f(t)\cos(2\pi \xi t)\,dt.$

It is necessarily an even function of frequency, i.e. for all $ξ$ : ${\hat {f}}^{c}(\xi )={\hat {f}}^{c}(-\xi ).$ Since positive frequencies can fully express the transform, the non-trivial concept of negative frequency needed in the regular Fourier transform can be avoided.

Simplification to avoid negative t

Some authors^[2] only define the cosine transform for even functions of $t$ , in which case its sine transform is zero. Since cosine is also even, a simpler formula can be used, ${\hat {f}}^{c}(\xi )=2\int _{0}^{\infty }f(t)\cos(2\pi \xi t)\,dt.$

Similarly, if $f$ is an odd function, then the cosine transform is zero and the sine transform can be simplified to ${\hat {f}}^{s}(\xi )=2\int _{0}^{\infty }f(t)\sin(2\pi \xi t)\,dt.$

Other conventions

Just like the Fourier transform takes the form of different equations with different constant factors (see Fourier transform § Other conventions), other authors also define the cosine transform as^[3] ${\hat {f}}^{c}(\xi )={\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }f(t)\cos(2\pi \xi t)\,dt.$ and sine as ${\hat {f}}^{s}(\xi )={\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }f(t)\sin(2\pi \xi t)\,dt,$ or, the cosine transform as^[4] $F_{c}(\alpha )={\frac {2}{\pi }}\int _{0}^{\infty }f(x)\cos(\alpha x)\,dx$ and the sine transform as $F_{s}(\alpha )={\frac {2}{\pi }}\int _{0}^{\infty }f(x)\sin(\alpha x)\,dx$ using $\alpha$ as the transformation variable. And while $t$ is typically used to represent the time domain, $x$ is often used alternatively, particularly when representing frequencies in a spatial domain.

Fourier inversion

The original function $f$ can be recovered from its transform under the usual hypotheses, that $f$ and both of its transforms should be absolutely integrable. For more details on the different hypotheses, see Fourier inversion theorem.

The inversion formula is^[5] $f(t)=\int _{-\infty }^{\infty }{\hat {f}}^{c}(\xi )\cos(2\pi \xi t)\,d\xi +\int _{-\infty }^{\infty }{\hat {f}}^{s}(\xi )\sin(2\pi \xi t)\,d\xi ,$

which has the advantage that all quantities are real. Using the addition formula for cosine, this can be rewritten as $f(t)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(\tau )\cos(2\pi \xi (\tau -t))\,d\tau \,d\xi .$

If the original function $f$ is an even function, then the sine transform is zero; if $f$ is an odd function, then the cosine transform is zero. In either case, the inversion formula simplifies.

Relation with complex exponentials

The form of the Fourier transform used more often today is ${\begin{aligned}{\hat {f}}(\xi )&=\int _{-\infty }^{\infty }f(t)e^{-2\pi i\xi t}\,dt\\&=\int _{-\infty }^{\infty }f(t)\left(\cos(2\pi \xi t)-i\,\sin(2\pi \xi t)\right)dt&&{\text{Euler's Formula}}\\&=\left(\int _{-\infty }^{\infty }f(t)\cos(2\pi \xi t)\,dt\right)-i\left(\int _{-\infty }^{\infty }f(t)\sin(2\pi \xi t)\,dt\right)\\&={\hat {f}}^{c}(\xi )-i{\hat {f}}^{s}(\xi )\end{aligned}}$

Numerical evaluation

Using standard methods of numerical evaluation for Fourier integrals, such as Gaussian or tanh-sinh quadrature, is likely to lead to completely incorrect results, as the quadrature sum is (for most integrands of interest) highly ill-conditioned. Special numerical methods which exploit the structure of the oscillation are required, an example of which is Ooura's method for Fourier integrals^[6] This method attempts to evaluate the integrand at locations which asymptotically approach the zeros of the oscillation (either the sine or cosine), quickly reducing the magnitude of positive and negative terms which are summed.

References

Whittaker, Edmund, and James Watson, A Course in Modern Analysis, Fourth Edition, Cambridge Univ. Press, 1927, pp. 189, 211

^ "Highlights in the History of the Fourier Transform". pulse.embs.org. Retrieved 2018-10-08.
^ Mary L. Boas, Mathematical Methods in the Physical Sciences, 2nd Ed, John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1
^ "Fourier Transform, Cosine and Sine Transforms". cnyack.homestead.com. Retrieved 2018-10-08.
^ Coleman, Matthew P. (2013). An Introduction to Partial Differential Equations with MATLAB (Second ed.). Boca Raton. p. 221. ISBN 978-1-4398-9846-8. OCLC 822959644.{{cite book}}: CS1 maint: location missing publisher (link)
^ Poincaré, Henri (1895). Theorie analytique de la propagation de chaleur. Paris: G. Carré. pp. 108ff.
^ Takuya Ooura, Masatake Mori, A robust double exponential formula for Fourier-type integrals, Journal of computational and applied mathematics 112.1-2 (1999): 229-241.

[1] "Highlights in the History of the Fourier Transform". pulse.embs.org. Retrieved 2018-10-08.

[2] Mary L. Boas, Mathematical Methods in the Physical Sciences, 2nd Ed, John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1

[3] "Fourier Transform, Cosine and Sine Transforms". cnyack.homestead.com. Retrieved 2018-10-08.

[4] Coleman, Matthew P. (2013). An Introduction to Partial Differential Equations with MATLAB (Second ed.). Boca Raton. p. 221. ISBN 978-1-4398-9846-8. OCLC 822959644.{{cite book}}: CS1 maint: location missing publisher (link)

[5] Poincaré, Henri (1895). Theorie analytique de la propagation de chaleur. Paris: G. Carré. pp. 108ff.

[6] Takuya Ooura, Masatake Mori, A robust double exponential formula for Fourier-type integrals, Journal of computational and applied mathematics 112.1-2 (1999): 229-241.

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