Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is negative frequency(fourier transform)

  1. Dec 20, 2012 #1
    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 dont know the physical significance of the -ve frequencies used?

    are these -ve frequencies just the mathematical imaginary tool ?
     
  2. jcsd
  3. Dec 20, 2012 #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.
     
  4. Dec 20, 2012 #3
    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.
     
    Last edited: Dec 20, 2012
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook