Digital Signal Processing Reference
In-Depth Information
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Chapter 3
Signals and Hilbert Spaces
In the 17th century, algebra and geometry started to interact in a fruitful syn-
ergy which continues to the present day. Descartes's original idea of trans-
lating geometric constructs into algebraic form spurred a new line of attack
in mathematics; soon, a series of astonishing results was produced for a
number of problems which had long defied geometrical solutions (such as,
famously, the trisection of the angle). It also spearheaded the notion of vec-
tor space, in which a geometrical point could be represented as an
n
-tuple
of coordinates; this, in turn, readily evolved into the theory of linear alge-
bra. Later, the concept proved useful in the opposite direction: many al-
gebraic problems could benefit from our innate geometrical intuition once
they were cast in vector form; from the easy three-dimensional visualiza-
tion of concepts such as distance and orthogonality, more complex alge-
braic constructs could be brought within the realm of intuition. The fi-
nal leap of imagination came with the realization that the concept of vec-
tor space could be applied to much more abstract entities such as infinite-
dimensional objects and functions. In so doing, however, spatial intuition
could be of limited help and for this reason, the notion of vector space had
to be formalized in much more rigorous terms; we will see that the defini-
tion of Hilbert space is one such formalization.
Most of the signal processing theory which in this topic can be usefully
cast in terms of vector notation and the advantages of this approach are ex-
actly what we have just delineated. Firstly of all, all the standard machinery
of linear algebra becomes immediately available and applicable; this greatly
simplifies the formalism used in the mathematical proofs which will follow
and, at the same time, it fosters a good intuition with respect to the under-
lying principles which are being put in place. Furthermore, the vector nota-
tion creates a frame of thought which seamlessly links the more abstract re-
