Sine and cosine transforms

In mathematics, the Fourier sine and cosine transforms are integral equations that decompose arbitrary functions into a sum of sine waves representing the odd component of the function plus cosine waves representing the even component of the function. The modern Fourier transform concisely contains both the sine and cosine transforms. Since the sine and cosine transforms use sine and cosine waves instead of complex exponentials and don't require complex numbers or negative frequency, they more closely correspond to Joseph Fourier's original transform equations and are still preferred in some signal processing and statistics applications and may be better suited as an introduction to Fourier analysis.

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^[1] only define the cosine transform for even functions $f_{\text{even}}(t)$ , in which case its sine transform is zero. Since cosine is also even and the integral of an even function from ${-}\infty$ to $\infty$ is twice its integral from $0$ to $\infty$ , then the cosine transform of an even function can be written as: ${\hat {f}}^{c}(\xi )=2\int _{0}^{\infty }f_{\text{even}}(t)\cos(2\pi \xi t)\,dt.$

Similarly, if $f_{\text{odd}}(t)$ is an odd function, then its cosine transform is zero. Because the product of two odd functions is an even function, the product ${\textstyle f_{\text{odd}}(t)\sin(2\pi \xi t)}$ is even and its integral similarly simplifies. Thus, the sine transform of an odd function can be written as: ${\hat {f}}^{s}(\xi )=2\int _{0}^{\infty }f_{\text{odd}}(t)\sin(2\pi \xi t)\,dt.$ In the unique decomposition of functions into an even and odd function $(f{=}f_{\text{even}}{+}f_{\text{odd}}),$ cosines represent the even component and sines represent the odd component of the function.

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^[2] ${\hat {f}}^{c}(\xi )={\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }f(t)\cos(2\pi \xi t)\,dt$ and the sine transform as ${\hat {f}}^{s}(\xi )={\sqrt {\frac {2}{\pi }}}\int _{0}^{\infty }f(t)\sin(2\pi \xi t)\,dt.$ Another convention defines the cosine transform as^[3] $F_{c}(\alpha )={\frac {2}{\pi }}\int _{0}^{\infty }f(x)\cos(\alpha x)\,dx$ 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^[4] $f(t)=\underbrace {\int _{-\infty }^{\infty }{\hat {f}}^{c}(\xi )\cos(2\pi \xi t)\,d\xi } _{{\text{even component of }}f(t)}\,+\underbrace {\int _{-\infty }^{\infty }{\hat {f}}^{s}(\xi )\sin(2\pi \xi t)\,d\xi } _{{\text{odd component of }}f(t)}\,.$

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.

Using the addition formula for cosine, the formula can also be rewritten as Fourier's integral formula:^[5]^[6] $f(t)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x)\cos(2\pi \xi (x-t))\,dx\,d\xi .$

Relation with complex exponentials

The complex exponential form of the Fourier transform used more often today is^[7] ${\begin{aligned}{\hat {f}}(\xi )&=\int _{-\infty }^{\infty }f(t)e^{-2\pi i\xi t}\,dt\\\end{aligned}}\,$ where $i$ is the square root of negative one. By applying Euler's formula ${\textstyle (e^{ix}=\cos x+i\sin x),}$ it can be shown that the Fourier transform's real component is the cosine transform (representing the even component of the original function) and the Fourier transform's imaginary component is the negative of the sine transform (representing the odd component of the original function): ${\begin{aligned}{\hat {f}}(\xi )&=\int _{-\infty }^{\infty }f(t)\left(\cos(2\pi \xi t)-i\,\sin(2\pi \xi t)\right)dt&&{\text{Euler's Formula}}\\&=\left(\int _{-\infty }^{\infty }f(t)\cos(2\pi \xi t)\,dt\right)-i\left(\int _{-\infty }^{\infty }f(t)\sin(2\pi \xi t)\,dt\right)\\&={\hat {f}}^{c}(\xi )-i\,{\hat {f}}^{s}(\xi )\,.\end{aligned}}$

An advantage of the modern Fourier transform is that while the sine and cosine transforms together are required to extract the phase information of a frequency, the modern Fourier transform instead compactly packs both phase and amplitude information inside its complex valued result. Adding a sine wave and a cosine wave of the same frequency results a phase-shifted sine wave of that same frequency but whose amplitude and phase depends on the amplitudes of the sine and cosine wave. But a disadvantage is its requirement on understanding complex numbers, complex exponentials, and negative frequency.

The sine and cosine transforms meanwhile has the advantage that all quantities are real. They may also be convenient when the original function is already even or odd, in which case only the cosine or the sine transform respectively is needed, or if the function is already or easily decomposed into odd and even components.

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^[8] 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

^ Mary L. Boas, Mathematical Methods in the Physical Sciences, 2nd Ed, John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1
^ Nyack, Cuthbert (1996). "Fourier Transform, Cosine and Sine Transforms". cnyack.homestead.com. Archived from the original on 2023-06-07. 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.
^ Edwin Titchmarsh (1948), Introduction to the theory of the Fourier integral, Oxford at the Clarendon Press, p. 1
^ Whittaker, Edmund Taylor; Watson, George Neville (1927-01-02). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (4th ed.). Cambridge, UK: Cambridge University Press. p. 189. ISBN 0-521-06794-4. ISBN 978-0-521-06794-2. {{cite book}}: ISBN / Date incompatibility (help)
^ Valentinuzzi, Max E. (2016-01-25). "Highlights in the History of the Fourier Transform". IEEE Pulse. Archived from the original on 2024-05-15. Retrieved 2024-09-09.
^ 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] Mary L. Boas, Mathematical Methods in the Physical Sciences, 2nd Ed, John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1

[2] Nyack, Cuthbert (1996). "Fourier Transform, Cosine and Sine Transforms". cnyack.homestead.com. Archived from the original on 2023-06-07. Retrieved 2018-10-08.

[3] 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)

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

[5] Edwin Titchmarsh (1948), Introduction to the theory of the Fourier integral, Oxford at the Clarendon Press, p. 1

[Whittaker-Watson_1927-6] Whittaker, Edmund Taylor; Watson, George Neville (1927-01-02). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (4th ed.). Cambridge, UK: Cambridge University Press. p. 189. ISBN 0-521-06794-4. ISBN 978-0-521-06794-2. {{cite book}}: ISBN / Date incompatibility (help)

[7] Valentinuzzi, Max E. (2016-01-25). "Highlights in the History of the Fourier Transform". IEEE Pulse. Archived from the original on 2024-05-15. Retrieved 2024-09-09.

[8] 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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