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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
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