# How to understand negative frequency?

• yangjun1222
In summary, the conversation discusses negative frequency in communication theory and its origins in the Fourier transform. Negative frequency is a mathematical tool used to simplify expressions and does not exist in reality. It was first introduced by Fourier in 1800 and is not a physical quantity. It is often used in communication methods such as AM and DSB, but some may have difficulty understanding its purpose. There are textbooks available that explain the history and development of communication theory and technology in detail.
yangjun1222
Recently I learned some communication theory, which uses negative frequency, what does negative frequency means? So I checked textbook, found the negative frequency comes from the Fourier transform, which comes from the euler formula, e-jx=cosx+isinx,
or another form cosx=(ejx+e-jx)/2, then any physical quantity can be expressed in negative part and positive part.
Can anyone explain this (negative frequency) further? (In physics, I can not understand this, so this is just a expression).
Some communication theory use this much, like AM, DSB,modulation method.
Does this technical develop after mathematical derivation(like the electromagnetic wave) or mathematical is latter developed after experiments. Are there any good textbook introduce this (communication theory and technology and their history) clearly?

In the equations you show it the OP, the independent variable is x. +x and-x mean waves moving to the left and right (or whichever direction is the x axis). That is not negative frequency, it is negative direction.

Negative frequencies do not exist in reality.
As you have shown, they are invented as a fictive quantity (mathematical tool) to simplify some manipulations/expressions in system or communucation theory only.

EDIT (added): In 1800, it was shown by Fourier how any periodic signal with period T can be expanded to an infinite set of complex exponents with frequencies +-k/T, where k = 0, 1, 2, ... That means: Frequencies can be both positive and negative. If to set T infinite, then we arrive at the Fourier transform.
So, negative frequencies are just mathematical quantities (not physical) similar to the imaginary part of a complex signal. In real world, negative frequencies do not exists and the spectral content on negative frequencies must be added to the spectral content at the positive frequencies.

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## 1. What is negative frequency?

Negative frequency is a concept in mathematics and signal processing that represents a frequency value below the zero frequency point. It is often used to describe frequencies that are lower than the fundamental frequency of a signal.

## 2. How is negative frequency represented?

Negative frequency is typically represented using the letter "f" with a negative sign in front of it, such as -f. It can also be represented using the symbol ω (omega) with a negative value, such as -ω.

## 3. What is the difference between positive and negative frequency?

The main difference between positive and negative frequency is their direction on the frequency axis. Positive frequency values represent frequencies above the zero frequency point, while negative frequency values represent frequencies below the zero frequency point. In other words, positive frequencies increase as you move to the right on the frequency axis, while negative frequencies decrease as you move to the left.

## 4. How is negative frequency used in signal processing?

Negative frequency is often used in signal processing to represent reversed or mirrored signals. It can also be used to represent the imaginary component of a complex signal. In some cases, negative frequencies may also be used to cancel out unwanted frequencies in a signal.

## 5. Why is it important to understand negative frequency?

Understanding negative frequency is important for various applications in mathematics and signal processing. It allows us to analyze and manipulate signals with complex frequency components, and can also help us better understand the behavior of certain systems or phenomena. Additionally, many modern technologies, such as wireless communication and digital signal processing, rely on a thorough understanding of negative frequency.

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