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Mathematics of the DFT
    Derivation of the Discrete Fourier Transform (DFT)
       Geometric Series

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Geometric Series

Recall that for any complex number $ z_1\in{\bf C}$, the signal

$\displaystyle x(n)\isdef z_1^n,\quad n=0,1,2,\ldots, $

defines a geometric sequence, i.e., each term is obtained by multiplying the previous term by the (complex) constant $ z_1$. A geometric series is the sum of a geometric sequence:

$\displaystyle S_N(z_1) \isdef \sum_{n=0}^{N-1}z_1^n = 1 + z_1 + z_1^2 + z_1^3 + \cdots + z_1^{N-1} $

If $ z_1\neq 1$, the sum can be expressed in closed form:

$\displaystyle \zbox {S_N(z_1) = \frac{1-z_1^N}{1-z_1}}$   $\displaystyle \mbox{($z_1\neq 1$)}$



Proof: We have

\begin{eqnarray*} S_N(z_1) &\isdef & 1 + z_1 + z_1^2 + z_1^3 + \cdots + z_1^{N-1... ...z_1^N \\ \,\,\Rightarrow\,\,S_N(z_1) &=& \frac{1-z_1^N}{1-z_1}. \end{eqnarray*}

When $ z_1=1$, $ S_N(1)=N$, by inspection of the definition of $ S_N(z_1)$.

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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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