Zero Padding Theorem

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Zero Padding Theorem

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Zero Padding Theorem

A fundamental tool in practical spectrum analysis is zero padding. This theorem shows that zero padding in the time domain corresponds to ideal interpolation in the frequency domain:

Let $x\ and define $y=\. Then $y\ with $M\. Denote the original frequency index by $k$, where $\ and the new frequency index by $k^\, where $\.



Definition: The ideal bandlimited interpolation of a spectrum $X(\, $x\, to an arbitrary new frequency $\ is defined as

\

Note that this is just the definition of the DFT with $\ replaced by $\. That is, the spectrum is interpolated by projecting onto the new sinusoid exactly as if it were a DFT sinusoid. This makes the most sense when $x$ is assumed to be $N$ samples of a time-limited signal. That is, if the signal really is zero outside of the time interval$[0,N-1]$, then the inner product between it and any sinusoid will be exactly as in the equation above. Thus, for time limited signals, this kind of interpolation is ideal.



Definition: The interpolation operator interpolates a signal by an integer factor$L$. That is,

\

Since $X(\ is initially only defined over the $N$roots of unity, while $X(\ is defined over $M=LN$ roots of unity, we define $X(\ for $\ by ideal bandlimited interpolation.



Theorem: For any $x\

\

Proof: Let $M=LN$ with $L\. Then

\

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