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Mathematics of the DFT
    Fast Fourier Transform (FFT) Algorithms
       Mixed-Radix Cooley-Tukey FFT
          Decimation in Time

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Decimation in Time

The DFT is defined by

$\displaystyle X(k) = \sum_{n=0}^{N-1} x(n) W_N^{kn}, \quad k=0,1,2,\ldots,N-1, $

where $ x(n)$ is the input signal amplitude at time $ n$, and

$\displaystyle W_N \isdef e^{-j\frac{2\pi}{N}}.\quad \hbox{(primitive $N$th root of unity)} $

Note that $ W_N^N=1$.

When $ N$ is even, the DFT summation can be split into sums over the odd and even indexes of the input signal:

$\displaystyle X(\omega_k)$ $\displaystyle \isdef$ $\displaystyle \hbox{\sc DFT}_{{N,k}}\{x\} \isdef \sum_{n=0}^{N-1} x(n) e^{-j\omega_k n T}, \quad \omega_k \isdef \frac{2\pi k}{NT}$  
  $\displaystyle =$ $\displaystyle \sum_{{\stackrel{n=0}{\vspace{2pt}\mbox{\tiny$n$\ even}}}}^{N-2} ... ...stackrel{n=0}{\vspace{2pt}\mbox{\tiny$n$\ odd}}}}^{N-1} x(n) e^{-j\omega_k n T}$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^{\frac{N}{2}-1} x(2n) e^{-j2\pi \frac{k}{N/2} n} + e^{-j2\pi\frac{k}{N}}\sum_{n=0}^{\frac{N}{2}-1} x(2n+1) e^{-j2\pi \frac{k}{N/2} n},$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^{\frac{N}{2}-1} x_e(n) W_{N/2}^{kn} + W_N^k \sum_{n=0}^{\frac{N}{2}-1} x_o(n) W_{N/2}^{kn}$  
  $\displaystyle \isdef$ $\displaystyle \hbox{\sc DFT}_{{\frac{N}{2},k}}\{\hbox{\sc Downsample}_2(x)\}$  
    $\displaystyle \mathop{\quad} +\;W_N^k\cdot\hbox{\sc DFT}_{{\frac{N}{2},k}}\{\hbox{\sc Downsample}_2[\hbox{\sc Shift}_1(x)]\}, \protect$ (A.1)

where $ x_e(n)\isdef x(2n)$ and $ x_o(n)\isdef x(2n+1)$ denote the even- and odd-indexed samples from $ x$. Thus, the length $ N$ DFT is computable using two length $ N/2$ DFTs. The complex factors $ W_N^k=e^{-j\omega_k}=\exp(-j2\pi k/N)$ are called twiddle factors. The splitting into sums over even and odd time indexes is called decimation in time. (For decimation in frequency, the inverse DFT of the spectrum $ X(\omega_k)$ is split into sums over even and odd bin numbers $ k$.)

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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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