Page 216 - DSP Integrated Circuits

5.5 Scaling of Signal Levels 201

5.5.2 FFT Scaling

In order to scale the signal levels in the FFT we first scale the inputs to the butter-
fly operations. We will later discuss scaling of the signals that are internal to the
butterfly operations. We will use the safe scaling technique just discussed.
We have for the complex input values to the butterfly (see Figure 5.14)

and for the two output values, X and Y, we have

In order to assure that the input sig-
nals to the succeeding stage of butterflies
also are properly scaled, the outputs of
each butterfly are divided by 2, as illus-
trated in Figure 5.14.
We also need to scale the input sig-
nal to the FFT by dividing by 2 so that
the signal level to the butterflies in the
first stage becomes properly scaled. Fur-
ther, the scaling of the butterflies in the
last stage is not necessary. However, from an implementation point or view it
may be better to retain the scaling and multiply the outputs with a factor two.
M
The result is that the output of the scaled FFT will be divided by the factor 2 = N,
where M = log2(AO stages.
It is easy to show that this scaling policy is conservative and corresponds to
the safe scaling just discussed. Assume that the input is a sinusoidal sequence.
The DFT is

Hence, a sinusoidal input sequence with amplitude A will appear in the frequency
domain with amplitude A. The FFT is therefore safely scaled.

5.5.3 Lp-Norms
In the next few sections, we will discuss more efficient scaling policies based on the
observation that it is the frequency properties of a signal that are of major interest
in many DSP algorithms—for example, frequency selective filters. This implies
that the spectral properties of the input signal are characteristic in these applica-
tions. It is therefore natural to try to exploit knowledge of how the input signal