Negative frequency

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The negative frequency occurs as a component in harmonic oscillations when the mathematical description is carried out in the complex number plane with complex numbers . The sign has no physical meaning; it specifies the clockwise or counterclockwise rotation of pointers in the complex number plane.

The term negative frequency plays a role in the field of signal theory and signal processing , especially in technical modulation methods .

definition

Vector diagram for the sinusoidal or cosinusoidal course of an oscillation

A frequency f indicates the number of events within a certain period of time and is usually a positive number. A harmonic oscillation s I ( t ) of frequency f and an amplitude A has a sinusoidal curve over time t :

The expression 2π f is usually combined to give the angular frequency ω. As a phasor diagram in the complex plane, as shown in the figure opposite, the sinusoidal curve over time t can be shown when the phasor r rotates on the imaginary axis . The cosine-shaped curve shown on the real axis is equivalent to this and represents a phase shift of π / 2.

The harmonic oscillation can also be given as a complex oscillation using Euler's identity :

with the real part:

and the imaginary part:

The expressions in brackets represent the oscillation as a sum of a positive and a negative frequency. The expression can be graphically represented in the phasor diagram by two counter-rotating pointers, the direction of rotation of the pointers is determined by the sign of the frequency.

Applications

The distinction between positive and negative frequencies is particularly important in technical modulation methods and signal processing . If a signal is modulated as an information carrier by the amplitude modulation from the baseband position for the purpose of transmission into a higher frequency range, the positive and negative frequencies of the signal result in an upper or lower sideband . These two sidebands both carry the same signal information of the baseband.

One possibility of suppressing one of the two sidebands is to use the Hilbert transformation to form an analytical signal from the real signal . An analytic signal is a complex signal, the negative frequency components of which are equal to 0 and only consists of positive frequencies.

Individual evidence

  1. Treatment of the complex representation of waves (PDF; 728 kB) - Negative frequencies and complex fields
  2. Christof Pölczer: Digital Signal Processing - Complex Pointer Calculation . (pdf) Retrieved on February 5, 2018 (lecture notes).

literature

  • Rudolf Mäusl, Jürgen Göbel: Analog and digital modulation methods - baseband and carrier modulations . 1st edition. Hüthig, 2002, ISBN 3-7785-2886-6 .