Sine and cosine transformation

The sine and cosine transformations are two variants of the continuous Fourier transformation , which are only defined for real numbers , in contrast to the Fourier transformation, which is defined for complex numbers . They are integral transformations with applications in the field of signal processing . The discrete cosine transformation (DCT) and the discrete sine transformation (DST) are derived from this for time-discrete signal sequences .

General

The core of the Fourier transformation can be split into a real and an imaginary part using Euler's identity :

${\displaystyle \mathrm {e} ^{\mathrm {j} \,x}=\cos \left(x\right)+\mathrm {j} \cdot \sin \left(x\right)}$

with as the imaginary unit . The real part is used as the core of the cosine transformation and the imaginary part as the core of the sine transformation. The cosine function is an even function , the cosine transformation maps the even signal component of the Fourier transform of a real signal. Analogously, the odd sine function maps the odd signal component of the Fourier transform of a real signal. ${\displaystyle \mathrm {j} }$

Sine transformation

The sine transformation is defined for real signals by: ${\displaystyle y(t)}$

${\displaystyle \mathrm {Y_{s}} (f)={\mathcal {SIN}}\{y(t)\}=\int _{-\infty }^{\infty }y(t)\sin(2\pi ft)\,\mathrm {d} t.}$

Cosine transformation

The cosine transformation is defined for real signals by: ${\displaystyle y(t)}$

${\displaystyle \mathrm {Y_{c}} (f)={\mathcal {COS}}\{y(t)\}=\int _{-\infty }^{\infty }y(t)\cos(2\pi ft)\,\mathrm {d} t.}$

context

The Fourier transform

${\displaystyle \mathrm {Y} (f)={\mathcal {F}}\{y(t)\}=\int _{-\infty }^{\infty }y(t)\,e^{-2\pi \mathrm {j} f\cdot t}\,\mathrm {d} t}$

can be formed for real signals from the sine and cosine transformation: ${\displaystyle y(t)}$

${\displaystyle {\mathcal {F}}\{y(t)\}={\mathcal {COS}}\{y(t)\}-\mathrm {j} \cdot {\mathcal {SIN}}\{y(t)\}.}$

For the special cases of real and even signals, the Fourier transformation changes into the cosine transformation; for real and odd signals, it changes into the sine transformation, except for a constant prefactor.

literature

• Fernando Puente León, Uwe Kiencke, Holger Jäkel: Signals and Systems . 5th edition. Oldenbourg, 2011, ISBN 978-3-486-59748-6 .