Page 217 - DSP Integrated Circuits

202 Chapter 5 Finite Word Length Effects

spectrum varies with frequency as a scaling criterion. We therefore need a mea-
sure of the "size" of the signals spectrum. To this purpose Jackson [15, 16] has
developed a scaling procedure where Lp-norms are used as a measure of the Fou-
rier transform.
For a sequence, x(n), with Fourier transform X(eJ°^), the Lp-norm is denned as

for any real p > 1 such that the integral exists. It can be shown that

Of particular interest in this context are three cases: the I/i-norm, Z/2-norm,
and Loo-norm.

.Li-Norm

The Lp-norm of the Fourier transform corresponding to a sinusoidal sequence

exists only forp = 1 and is

The Li-norm, which corresponds to the average absolute value, can in practice
be determined either by numeric integration of the magnitude function or by com-
puting the DFT (FFT) of the impulse response. In general, it is difficult or even
impossible to obtain analytical expressions for the Li-norm.

L2-Norm
The L2-norm of a continuous-time Fourier transform is related to the power con-
tained in the signal, i.e., the rms value. The L2-norm of a discrete-time Fourier
transform has an analogous interpretation. The L2-norm is

The L2-norm is simple to compute by using Parsevals relation [5, 27, 33],
which states that the power can be expressed either in the time domain or in the
frequency domain. We get from Parsevals relation

We frequently need to evaluate L2-norms of frequency responses measured
from the input of a filter to the critical overflow nodes. This can be done by imple-
menting the filter on a general-purpose computer and using an impulse sequence
as input. The L2-norm can be calculated by summing the squares of the signal val-