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Sampling Theorem

Let $ x(t)$ denote any continuous-time signal having a continuous Fourier transform

$\displaystyle X(j\omega)\isdef \int_{-\infty}^\infty x(t) e^{-j\omega t} dt. $

Let

$\displaystyle x_d(n) \isdef x(nT), \quad n=\ldots,-2,-1,0,1,2,\ldots, $

denote the samples of $ x(t)$ at uniform intervals of $ T$ seconds. Then $ x(t)$ can be exactly reconstructed from its samples $ x_d(n)$ if $ X(j\omega)=0$ for all $ \vert\omega\vert\geq\pi/T$.D.3

Proof: From the continuous-time aliasing theorem (§D.2), we have that the discrete-time spectrum $ X_d(e^{j\theta})$ can be written in terms of the continuous-time spectrum $ X(j\omega)$ as

$\displaystyle X_d(e^{j\omega_d T}) = \frac{1}{T} \sum_{m=-\infty}^\infty X[j(\omega_d +m\Omega_s )] $

where $ \omega_d \in(-\pi/T,\pi/T)$ is the ``digital frequency'' variable. If $ X(j\omega)=0$ for all $ \vert\omega\vert\geq\Omega_s /2$, then the above infinite sum reduces to one term, the $ m=0$ term, and we have......

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