Continuous-Time Aliasing Theorem

Let denote any continuous-time signal having a Fourier Transform (FT)

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

Let

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

denote the samples of

at uniform intervals of

seconds, and denote its Discrete-Time Fourier Transform (DTFT) by

$\displaystyle X_d(e^{j\theta})\isdef \sum_{n=-\infty}^\infty x_d(n) e^{-j\theta n}.$

Then the spectrum

of the sampled signal

is related to the spectrum

of the original continuous-time signal

$\displaystyle X_d(e^{j\theta}) = \frac{1}{T} \sum_{m=-\infty}^\infty X\left[j\left(\frac{\theta}{T} + m\frac{2\pi}{T}\right)\right].$

The terms in the above sum for $m\neq 0$ are called aliasing terms. They are said to alias into the base band $[-\pi/T,\pi/T]$ . Note that the summation of a spectrum with aliasing components involves addition of complex numbers; therefore, aliasing components can be removed only if both their amplitude and phase are known.

Proof: Writing as an inverse FT gives

$\displaystyle x(t) = \frac{1}{2\pi}\int_{-\infty}^\infty X(j\omega) e^{j\omega t} d\omega.$

Writing

as an inverse DTFT gives

$\displaystyle x_d(n) = \frac{1}{2\pi}\int_{-\pi}^\pi X_d(e^{j\theta}) e^{j \theta t} d\theta$

where $\theta \isdef 2\pi \omega_d T$ denotes the normalized discrete-time frequency variable.

The inverse FT can be broken up into a sum of finite integrals, each of length $\Omega_s \isdef 2\pi f_s= 2\pi/T$ , as follows:

$\begin{eqnarray*} x(t) &=& \frac{1}{2\pi}\int_{-\infty}^\infty X(j\omega) e^{j\o... ...y X\left(j\omega + j m\Omega_s \right) e^{j\Omega_s m t} d\omega \end{eqnarray*}$

Let us now sample this representation for at to obtain $x_d(n) \isdef x(nT)$ , and we have

$\begin{eqnarray*} x_d(n) &=& \frac{1}{2\pi}\int_{-\Omega_s /2}^{\Omega_s /2} e^{... ..._{m=-\infty}^\infty X\left(j\omega + j m\Omega_s \right) d\omega \end{eqnarray*}$

since and are integers. Normalizing frequency as $\theta^\prime = \omega T$ yields

$\displaystyle x_d(n) = \frac{1}{2\pi}\int_{-\pi}{\pi} e^{j\theta^\prime n} \f... ...t(\frac{\theta^\prime }{T} + m\frac{2\pi}{T}\right) \right] d\theta^\prime .$