Sine and cosine transforms

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

General

The core of the Fourier transform can be split by means of Euler's identity in a real and imaginary part:

With as the imaginary unit. The real part is used as the core of the cosine transform and the imaginary part as the core of the sine transform. The cosine function is an even function, the cosine transformation is from the even signal component of the Fourier transform of a real signal. Similarly, the odd sine function reflects the odd signal component of the Fourier transform of a real signal.

Sine Transform

The sine transform for real signals is defined by:

Cosine transform

Cosine transformation for real signals defined by:

Context

Fourier transformation

Can make for real signals from the sine and cosine transformation:

For the special cases of real and even signals, the Fourier transformation changes into the cosine transformation for real and odd signals it goes up to a constant prefactor, the sine transform on.