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Magnitude spectrum of a discrete time signal

  1. Nov 2, 2012 #1
    Hello, friends. I have a question about the magnitude spectrum of a discrete time signal.

    Suppose I have a continuous time signal [itex]g(t)[/itex] and its Fourier transform:

    [itex] G(f) = \mathcal{F}[f(t)] = \int_{-\infty}^{\infty} f(t) e^{−2 \pi i f t} dt [/itex]

    where [itex] \mathcal{F} [/itex] represents the Fourier transform. Now suppose the time signal is sampled periodically such that for a given cycle [itex] n [/itex], the discrete signal is [itex] g(n T) [/itex] (where [itex] T [/itex] is the sampling period). I take the discrete time Fourier transform of [itex] g(n T) [/itex]:

    [itex] \mathcal{F}[g(n T)]=T \sum_{n = -\infty}^{\infty} g(nT) e^{−2 \pi i f T n} = \sum_{k = -\infty}^{\infty} G(f+k/T) [/itex]

    where the last step utilizes a standard relation for discrete time Fourier transforms. Using this last relation, I would write the squared magnitude spectrum like this:

    [itex] |F[g(n T)]|^{2} = |\sum_{n = -\infty}^{\infty}G(f+k/T)|^{2} [/itex]

    However, I have seen some fairly authoritative references quote the following relation for the squared magnitude spectrum:

    [itex] |F[g(n T)]|^{2} = \sum_{n = -\infty}^{\infty}|G(f+k/T)|^{2} [/itex]

    which confuses me since the relation [itex] | \sum_{k = -\infty}^{\infty} G(f+k/T)|^{2} = \sum_{k = -\infty}^{\infty}|G(f+k/T)|^{2} [/itex] isn't generally true. Therefore, I assume this relation must only be true for specific kinds of signals.

    Will anyone please tell me what kind of signals satisfy this relation?
     
    Last edited by a moderator: Nov 6, 2012
  2. jcsd
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