What is Negative Frequency: Exhaustive Insights

The concept of negative frequency is linked to spinning complex exponentials and it rotates opposite to the positive frequency.

Negative frequency is a vector that has the same mathematical meaning as the imaginary part of a complex signal. Negative frequencies do not exist in the actual world; hence the spectral content of negative frequencies must be added to the spectral content of positive frequencies in order to save energy.

what is negative frequency
The vector (cos t, sin t) rotates counter-clockwise at 1 radian/second, and completes a circle every 2π seconds; Image Source: GustavbUnit circle, marked as public domain, more details on Wikimedia Commons

Let us analyze it further by studying its important properties. Negative frequencies utilize complex numbers to analyze real signals in a mathematical framework. This is similar to the case of a double-sided spectrum. Only by adding its conjugate to a complex number can it be rendered real, e.g. (a+bj) + (a-bj) = 2a. As a result, the sum of the complex number and its complex conjugate can be used to describe a real sinusoid using complex exponentials. As a result, the negative frequency is derived from this. The unit of its expression is cycles per second (hertz) or radians per second where one cycle equates to 2π as seen in the figure above.

Sinusoids

Sinusoids can constitute of positive frequencies, unlike the negative argument of a non-negative parameter.

This can be explained by taking a non-negative parameter ω. Let this parameter be of unit radians/second (rad/s). If we plot the angle vs. time graph for this, which is (– ωt+θ), then we get a negative slope value of -ω. This is termed as the negative frequency. The same function returns indistinguishable results when used as an argument of a sine or a cosine function.

What is Negative Frequency: Exhaustive Insights
A negative frequency causes the sin function (violet) to lead the cos (red) by 1/4 cycle; Image Source: MysidNegative frequency, marked as public domain, more details on Wikimedia Commons

We end up with values cos(ωt-θ) and sin(-ωt+θ) , which return the same results as cos(π-ωt+θ) and sin(ωt-θ+π) respectively. This implies that the sign under the slope is vague. Concurrent viewing of the sine and the cosine graphs can settle this obscurity.

Fourier Transforms

Fourier transform is a very common and widely known application of negative frequency.

The calculation of negative frequency is the amount of frequency ω within the intervals (a,b) over the function of time f(t). The evaluation of this frequency ω as a continuous function over the intervals (-∞,+∞) is termed as the Fourier Transform.

X(\omega)=\int_{a}^{b}x(t)e^{-i\omega t}dt

What is Negative Frequency: Exhaustive Insights
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This implies that two complex sinusoids can be multiplied to get another complex sinusoid and its frequency is determined by adding the two original frequencies of the base complex sinusoids.

Thus, all the frequencies of the time function are reduced by the amount of ω itself when it is positive. The frequency content of function x(t) amounting to ω, is a constant value. The amplitude of this constant signifies the intensity of the original content. And whatever portion of x(t) was at frequency zero is replaced with a sinusoid at frequency ω.

Negative Frequency Dependence

The single frequency complex sinusoid is actually mathematically more basic and simpler in comparison to the double frequency real sinusoid.

The product of two complex sinusoids, one of which has a positive frequency and the other, negative, gives a real sinusoid, making it twice as intricate as a complex one. Complex sinusoids are also preferable since their modulus is constant. To acquire the instantaneous frequency of a complex sinusoid, frequency demodulators simply distinguish the phase of the sinusoid.

It’s no wonder, then, that signal processing experts prefer to turn genuine sinusoids into complex sinusoids by filtering off the negative-frequency component before further processing. It depends on how you obtained your time-domain data and your application whether or not you should be concerned with positive and negative frequencies.

You can disregard half of the spectrum and double only one side if your samples are merely real values. The spectrum is symmetrical for real-sampled data; thus, it doesn’t matter which side you choose. Because the spectrum is asymmetric, you’ll need both sides of the spectrum if your samples are complicated.

Decide whether or not you care about the direction of propagation when modelling a system in MATLAB. You can merely utilise cosines and look at a single-sided spectrum for applications like simple real-valued filters. When modelling something where propagation and/or reflection are important (such as a radar system), you should use complex exponentials. In such situation, you’re concerned about both ends of the political spectrum.

What is the physical significance of Negative Frequency?

Forward travelling waves are represented by negative frequencies, whereas backward travelling waves are represented by positive frequencies.

Sinusoids are waves, and the direction of wave propagation is determined by the sign of the frequency, which is based on standard convention. Wave propagation is defined by physicists as positive frequencies moving forward. But Fourier Transform breaks the signal into complex exponentials, hence negative frequency does not have any useful significance for sinusoids. These are spirals that are spinning in the complex plane.

F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt

The concept of negative frequency originates from the fact that spirals can rotate either clockwise or counterclockwise. This can also be conceived as the forward or backward phase angle in time. Real signals comprise of two equal but complex exponentials that revolve in opposite directions. The phase of both the spirals decides whether the complex exponentials will nullify each other to produce a purely real sine wave, a totally imaginary sine wave or a strictly real cosine wave.

 The knowledge of the true signal allows us to ignore the opposite side of the spectrum, irrespective of the requirement of dual sign frequencies to create a real signal. Complex signals in general though, require understanding of both sides of the frequency spectrum.

Are Negative Frequencies same as imaginary frequencies?

Negative frequencies do not exist, despite its advantages in the mathematical description of the spectrum of a signal.

The existence of negative frequencies cannot be determined through conventional mathematics. Although, their significance can be witnessed in complex numbers. Negative frequencies aren’t required in the “actual” world since a real signal always contains equal amounts of positive and negative frequencies. As a result, the fact that we cannot measure imaginary components instantly negates the question of whether a negative frequency is measurable. Hence, negative frequencies and the imaginary numbers can be equally considered as mathematical constructs.

A sinusoidal signal in the complex plane can be represented by a phasor. To represent a genuine periodic signal, each positive frequency complex sinusoid should be added to a negative frequency complex sinusoid of equal amplitude, such that the imaginary parts cancel out, leaving only the real signal. Perhaps a better way to explain this is to say that a phasor’s counterclockwise circular motion should be accompanied by an equal and opposite clockwise circular motion.

Negative Frequency Evidences

  • A previously unknown resonant emission component from solitons has now been discovered and analyzed. A soliton is a localized “lump” of light that is the outcome of wave effects in a nonlinear medium and can generate low-intensity, positive frequency resonant radiation in its wake under specific conditions. Eleonora Rubino of the University of Insubria in Como, Italy, and partners discovered that this resonant emission has a negative frequency equivalent, identified experimentally in two separate systems. The Physical Review Letters has published this work in one of its issues.
  • Event Horizon witnesses the transformation of positive-frequency waves into negative-frequency ones through the concept of Hawking radiation of black holes. When and where the flow exceeds the wave velocity in black-hole analogies, horizons emerge for waves travelling in a medium against the current.
  • Propagation of the incident waves on the water bodies, against the motion of the current, causes them to become steeper. As a result, waves can be created close to the crest, possibly with additional vorticity generation, and geometric cusps can form as a result of nonlinear dynamics. The flow then sweeps these crests waves away. Negative frequencies are clearly witnessed here.

About Esha Chakraborty

What is Negative Frequency: Exhaustive InsightsI have a background in Aerospace Engineering, currently working towards the application of Robotics in the Defense and the Space Science Industry. I am a continuous learner and my passion for creative arts keeps me inclined towards designing novel engineering concepts.
With robots substituting almost all human actions in the future, I like to bring to my readers the foundational aspects of the subject in an easy yet informative manner. I also like to keep updated with the advancements in the aerospace industry simultaneously.

Connect with me with LinkedIn - http://linkedin.com/in/eshachakraborty93

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