Abstract:
As with the general theme of this workshop, the goal
is to refine approximations of signals by coarsely
quantized expansions over an overcomplete generating
family of signal vectors. In this talk, we present
the most abstract idea that can be extracted from
the method of SigmaDelta modulation to perform the
quantization operation. The first principle
implicitly introduced by SigmaDelta modulation is
to perform a change of generating family, in the
same sense as "change of basis". The new family is
constructed by taking each individual vector of the
original family and subtracting to it a linear
combination of the other generating vectors, to
obtain a residual vector of reduced norm. The second
principle is to quantize the expansion coefficients
of the input signal with respect to the original
generating family, while observing the expansion
coefficients of the quantization error signal with
respect to the generating family of residual
vectors. Given the small norm of these vectors, the
primary concern is to ensure that the latter
coefficients remain bounded with reasonable bounds.
SigmaDelta modulation, the way it is commonly
known, results from these abstract principles when
the quantization control algorithm is restricted to
be causal (onepass) and timeinvariant. This
automatically implies the use of a dynamical system.
Based on these abstract principles, we give an
overview of the various aspects of SigmaDelta
modulation (often ignoring each other) under a
unified signalprocessing framework. We will show
from a topdown presentation what is the position of
the various research directions in SigmaDelta
modulation with respect to each other. We will
include research in the context of finite and
infinite dimensional signal spaces, onedimensional
and multidimensional signal index spaces, lowpass,
bandpass and multichannel signal expansions. With
respect to the second principle, we perform a
unified classification of all existing dynamical
system architectures of SigmaDelta modulation.
[LECTURE SLIDES]
