
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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