Sine and cosine transforms explained

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, sometimes denoted by either

{\hat f} ^{s} or { l F} _{s} (f) , is$$^s(\backslash xi)\; =\; \backslash int\_^\backslash infty\; f(t)\backslash sin(2\backslash pi\; \backslash xi\; t)\; \backslash ,dt.$$

If means time, then 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 :$$^s(-\backslash xi)\; =\; -\; ^s(\backslash xi).$$

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 norm of

\tfrac{ 1 }{\sqrt 2 }.

The Fourier cosine transform of, sometimes denoted by either

{\hat f} ^{c} or { l F} _{c} (f) , is$$^c(\backslash xi)\; =\; \backslash int\_^\backslash infty\; f(t)\backslash cos(2\backslash pi\; \backslash xi\; t)\; \backslash ,dt.$$

It is necessarily an even function of frequency, i.e. for all :$$^c(\backslash xi)\; =\; ^c(-\backslash xi).$$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, in which case its sine transform is zero. Since cosine is also even, a simpler formula can be used, $$^c(\backslash xi)=2\; \backslash int\_0^\backslash infty\; f(t)\backslash cos(2\backslash pi\; \backslash xi\; t)\; \backslash ,dt.$$

Similarly, if is an odd function, then the cosine transform is zero and the sine transform can be simplified to $$^s(\backslash xi)\; =2\backslash int\_0^\backslash infty\; f(t)\backslash sin(2\backslash pi\; \backslash xi\; t)\; \backslash ,dt.$$

Other conventions

Just like the Fourier transform takes the form of different equations with different constant factors (see), other authors also define the cosine transform as^[3] $$^c(\backslash xi)=\backslash sqrt\; \backslash int\_0^\backslash infty\; f(t)\backslash cos(2\backslash pi\; \backslash xi\; t)\; \backslash ,dt.$$and sine as$$^s(\backslash xi)\; =\backslash sqrt\; \backslash int\_0^\backslash infty\; f(t)\backslash sin(2\backslash pi\; \backslash xi\; t)\; \backslash ,dt,$$or, the cosine transform as^[4] $$F\_c(\backslash alpha)\; =\; \backslash frac\; \backslash int\_0^\backslash infty\; f(x)\; \backslash cos(\backslash alpha\; x)\; \backslash ,\; dx$$ and the sine transform as $$F\_s(\backslash alpha)\; =\; \backslash frac\; \backslash int\_0^\backslash infty\; f(x)\; \backslash sin(\backslash alpha\; x)\; \backslash ,\; dx$$ using

\alpha as the transformation variable. And while is typically used to represent the time domain, is often used alternatively, particularly when representing frequencies in a spatial domain.

Fourier inversion

The original function can be recovered from its transform under the usual hypotheses, that 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] $$f(t)\; =\; \backslash int\; \_^\backslash infty\; ^c(\backslash xi)\; \backslash cos\; (2\backslash pi\; \backslash xi\; t)\; \backslash ,d\backslash xi\; +\; \backslash int\; \_^\backslash infty\; ^s(\backslash xi)\; \backslash sin\; (2\backslash pi\; \backslash xi\; t)\; \backslash ,d\backslash xi,$$

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

If the original function is an even function, then the sine transform is zero; if 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

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.

See also

• Discrete cosine transform
• Discrete sine transform

References

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

1. Web site: Highlights in the History of the Fourier Transform. pulse.embs.org. en-US. 2018-10-08.
2. [Mary L. Boas]
3. Web site: Fourier Transform, Cosine and Sine Transforms. cnyack.homestead.com. 2018-10-08.
4. Book: Coleman, Matthew P. . An Introduction to Partial Differential Equations with MATLAB . 2013 . 978-1-4398-9846-8 . Second . Boca Raton . 221 . 822959644.
5. Book: Poincaré, Henri. Henri Poincaré

   . Henri Poincaré. Theorie analytique de la propagation de chaleur. 1895. G. Carré. Paris. 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.