What is negative frequency(fourier transform)

In summary, in Fourier transforms of normal baseband signals, spectral components are replicated on both positive and negative sides of the frequency axis. The negative frequencies are a mathematical tool and have no physical significance. However, they are used to simplify the Fourier transform and can also be used to describe modulation in terms of a pair of orthogonal modulation components.
  • #1
ankities
9
0
in Fourier transforms of normal baseband sigal , spectral components are replicated on both +ve and -ve sides of frequency axis.

i know that both -ve and +ve frequency components contribute to the total power of the signal

but i don't know the physical significance of the -ve frequencies used?

are these -ve frequencies just the mathematical imaginary tool ?
 
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  • #2
There is no physical reality of the negative frequency. It is, as you say, a mathmatical tool.

However, the negative frequencies only emerge because we wish to simplify the Fourier transform.

It is perfectly possible to have a Fourier Transform without any imaginary and negative components.

If you look into Fourier Series (from which Fourier Transforms are developed), you will see that it can be represented as

[itex]f(t) = a_{0} + \sum_{n}a_{n} cos(n\omega t + \theta_{n}) + \sum_{m}b_{m} sin(m\omega t + \theta_{m})[/itex]

It is only because we wish to simplify this that we make use of eulers identity that

e±iθ = cos(θ) ± i sin(θ)

When substituting this you will get "negative frequencies" when deriving all the formulas.
 
  • #3
Runei said:
There is no physical reality of the negative frequency. It is, as you say, a mathmatical tool.

Perhaps it would be more precise to say that there is no physical reality of complex frequencies in general, and that any real signals consist of a sum of positive and negative complex frequencies:


[itex]cos(ωt) = \frac{e^{+jωt} + e^{-jωt}}{2}[/itex]

On the other hand there is a physical reality to complex frequencies when we use them to describe modulation.

Specifically, they represent a *pair* of real modulation signals.

There are two independent degrees of freedom when modulating a sinusoid; phase/amplitude in polar coordinates, I/Q in rectangular coordinates. We can incorporate both of these independent signals into our single complex frequency expression.

In other words just as complex numbers can represent a pair of real numbers on an Argand diagram, complex frequencies can represent a pair of orthogonal modulation components of a real sinusoid.
 
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What is negative frequency?

Negative frequency is a concept in Fourier analysis, which is a mathematical tool used to break down a complex signal into simpler components. In this context, negative frequency refers to the direction of rotation of a complex signal in the frequency domain. It represents the frequency components that rotate in the opposite direction to the positive frequencies.

How is negative frequency represented in the Fourier transform?

In the Fourier transform, negative frequency is represented by a negative value on the x-axis of the frequency domain. This is because the Fourier transform uses complex numbers to represent the magnitude and phase of a signal at different frequencies, and negative numbers on the x-axis correspond to the opposite direction of rotation.

What is the significance of negative frequency in Fourier analysis?

Negative frequency is important in Fourier analysis because it allows us to fully represent and analyze periodic signals that have both clockwise and counterclockwise rotation in the frequency domain. It enables us to accurately capture the phase information of a signal, which is crucial in many applications such as digital signal processing, image processing, and communication systems.

Can negative frequency have a physical meaning?

No, negative frequency does not have a physical meaning. It is a mathematical concept used to simplify the representation and analysis of signals. In the physical world, frequencies only have positive values and represent the number of cycles or oscillations per unit time.

How can negative frequency be interpreted in the time domain?

In the time domain, negative frequency can be interpreted as a time-reversed version of the original signal. This means that the signal has the same shape and magnitude but is flipped in time. This interpretation is useful in understanding how the Fourier transform converts a signal from the time domain to the frequency domain.

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