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Mixed-Radix Cooley-Tukey FFT

When the desired DFT length $ N$ can be expressed as a product of smaller integers, the Cooley-Tukey decomposition provides what is called a mixed radix Cooley-Tukey FFT algorithm.A.2

Two basic varieties of Cooley-Tukey FFT are decimation in time (DIT) and its Fourier dual, decimation in frequency (DIF). The next section illustrates decimation in time.

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$.)

Radix 2 FFT

When $ N$ is a power of $ 2$, say $ N=2^K$ where $ K>1$ is an integer, then the above DIT decomposition can be performed $ K-1$ times, until each DFT is length $ 2$. A length $ 2$ DFT requires no multiplies. The overall result is called a radix 2 FFT. A different radix 2 FFT is derived by performing decimation in frequency.

A split radix FFT is theoretically more efficient than a pure radix 2 algorithm [73,31] because it minimizes real arithmetic operations. The term ``split radix'' refers to a DIT decomposition that combines portions of one radix 2 and two radix 4 FFTs [22].A.3On modern general-purpose processors, however, computation time is often not minimized by minimizing the arithmetic operation count (see §A.7 below).

Radix 2 FFT Complexity is N Log N

Putting together the length $ N$ DFT from the $ N/2$ length-$ 2$ DFTs in a radix-2 FFT, the only multiplies needed are those used to combine two small DFTs to make a DFT twice as long, as in Eq.$ \,$(A.1). Since there are approximately $ N$ (complex) multiplies needed for each stage of the DIT decomposition, and only $ \lg N$ stages of DIT (where $ \lg N$ denotes the log-base-2 of $ N$), we see that the total number of multiplies for a length $ N$ DFT is reduced from $ {\cal O}(N^2)$ to $ {\cal O}(N\lg N)$, where $ {\cal O}(x)$ means ``on the order of $ x$''. More precisely, a complexity of $ {\cal O}(N\lg N)$ means that given any implementation of a length-$ N$ radix-2 FFT, there exist a constant $ C$ and integer $ M$ such that the computational complexity $ {\cal C}(N)$ satisfies

$\displaystyle {\cal C}(N) \leq C N \lg N

for all $ N>M$. In summary, the complexity of the radix-2 FFT is said to be ``N log N'', or $ {\cal O}(N\lg N)$.

Fixed-Point FFTs and NFFTs

Recall (e.g., from Eq.$ \,$(6.1)) that the inverse DFT requires a division by $ N$ that the forward DFT does not. In fixed-point arithmetic (Appendix G), and when $ N$ is a power of 2, dividing by $ N$ may be carried out by right-shifting $ \log_2(N)$ places in the binary word. Fixed-point implementations of the inverse Fast Fourier Transforms (FFT) (Appendix A) typically right-shift one place after each Butterfly stage. However, superior overall numerical performance may be obtained by right-shifting after every other butterfly stage [8], which corresponds to dividing both the forward and inverse FFT by $ \sqrt{N}$ (i.e., $ \sqrt{N}$ is implemented by half as many right shifts as dividing by $ N$). Thus, in fixed-point, numerical performance can be improved, no extra work is required, and the normalization work (right-shifting) is spread equally between the forward and reverse transform, instead of concentrating all $ N$ right-shifts in the inverse transform. The NDFT is therefore quite attractive for fixed-point implementations.

Because signal amplitude can grow by a factor of 2 from one butterfly stage to the next, an extra guard bit is needed for each pair of subsequent NDFT butterfly stages. Also note that if the DFT length $ N=2^K$ is not a power of $ 4$, the number of right-shifts in the forward and reverse transform must differ by one (because $ K=\log_2(N)$ is odd instead of even).

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