What is negative frequency(fourier transform)

[ankities]

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 ?

 

[Runei]

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

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

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.

 

[the_emi_guy]

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:


cos(ωt) = \frac{e^{+jωt} + e^{-jωt}}{2}

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.