### Fixed-Point FFTs and NFFTs

Recall (*e.g.*, from Eq.(6.1)) that the inverse DFT requires a division by that the forward DFT does not. In fixed-point arithmetic (Appendix G), and when is a power of 2, dividing by may be carried out by right-shifting 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 (

*i.e.*, is implemented by

*half*as many right shifts as dividing by ). 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 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 is not a power of , the number of right-shifts in the forward and reverse transform must differ by one (because is odd instead of even).

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Radix 2 FFT