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

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

Since this is formally the inverse DTFT of $X_d(e^{j\theta^\prime })$ written in terms of $X(j\theta^\prime /T)$ , the result follows.
$\Box$

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Mathematics of the DFT
    Sampling Theory
       Aliasing of Sampled Signals
          Continuous-Time Aliasing Theorem