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Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Fourier Theorems for the DFT
       Fourier Theorems
          Downsampling Theorem (Aliasing Theorem)

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Downsampling Theorem (Aliasing Theorem)



Theorem: For all $ x\in{\bf C}^N$,

$\displaystyle \zbox {\hbox{\sc Downsample}_L(x) \;\longleftrightarrow\;\frac{1}{L}\hbox{\sc Alias}_L(X).} $



Proof: Let $ k^\prime \in[0,M-1]$ denote the frequency index in the aliased spectrum, and let $ Y(k^\prime )\isdef \hbox{\sc Alias}_{L,k^\prime }(X)$. Then $ Y$ is length $ M=N/L$, where $ L$ is the downsampling factor. We have

\begin{eqnarray*} Y(k^\prime ) &\isdef & \hbox{\sc Alias}_{L,k^\prime }(X) \isd... ...n) e^{-j2\pi k^\prime n/N} \sum_{l=0}^{L-1}e^{-j2\pi l n M/N}. \end{eqnarray*}

Since $ M/N=1/L$, the sum over $ l$ becomes

$\displaystyle \sum_{l=0}^{L-1}\left[e^{-j2\pi n/L}\right]^l = \frac{1-e^{-j2\p... ...right) \\ [5pt] 0, & n\neq 0 \left(\mbox{mod}\;L\right) \\ \end{array}\right. $

using the closed form expression for a geometric series derived in §6.1. We see that the sum over $ L$ effectively samples $ x$ every $ L$ samples. This can be expressed in the previous formula by defining $ m\isdeftext n/L$ which ranges only over the nonzero samples:

\begin{eqnarray*} \hbox{\sc Alias}_{L,k^\prime }(X) &=& \sum_{n=0}^{N-1}x(n) e^{... ... & L\cdot \hbox{\sc DFT}_{k^\prime }(\hbox{\sc Downsample}_L(x)) \end{eqnarray*}

Since the above derivation also works in reverse, the theorem is proved.

An illustration of aliasing in the frequency domain is shown in Fig.7.12.


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Next: Illustration of the Downsampling/Aliasing Theorem in Matlab
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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