# Magnitude spectrum of a discrete time signal

1. Nov 2, 2012

### 3mileisland

Hello, friends. I have a question about the magnitude spectrum of a discrete time signal.

Suppose I have a continuous time signal $g(t)$ and its Fourier transform:

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

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

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

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:

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

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

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

which confuses me since the relation $| \sum_{k = -\infty}^{\infty} G(f+k/T)|^{2} = \sum_{k = -\infty}^{\infty}|G(f+k/T)|^{2}$ 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