21-09-2012, 04:03 PM
Building blocks for signals: Vector spaces
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An informal example
The ingredients are denoted by ik. The mixture for the cake batter consists
of certain amounts of each ingredient. The ingredients for a cake might be, for
example: 500 ml white
our, 300 ml granulated sugar, 2 eggs, 20 ml vegetable oil,
200 ml water, and 10 ml baking powder. (Hint: don't try to make this at home!).
Assuming that the quantities in the table above are placed in the correct units
(\normalized"), this recipe could be written as follows:
c = 500i1 + 300i3 + 10i7 + 2i9 + 200i1120i12:
The set of all possible cakes forms a \vector space". If the set of ingredients is
able to make every element in the space (i.e. every cake), the said of ingredients
is complete. Notice that a complete cake space is not necessarily able to make
everything else: we could not, for example, make every possible shampoo with the
set of ingredients to make cakes. (We don't have, for example, any aloe in our list
above, or even any FD&C Red # 21.)
Several interesting questions now arise. Given a cake, is it possible to determine
the quantities of each ingredient that goes into it? (This is the analysis question.)
Suppose that we only want to use a certain subset of the ingredients (say, we have
run out of ingredients and don't want to run to the store). What is the best
approximation to a desired cake that we can make? What is the error between the
desired cake and the cake we actually get? (This is the approximation question.)
Obviously, some of these questions don't make a lot of sense when applied to cakes.
However, these kinds of things will be very applicable when it comes to analyzing
signals, which may also be built up from a set of building blocks.
Vector spaces
We brie
y review the concept of a vector space. A vector space V has the following
key property: If v;w 2 V then av + bw 2 V for any scalars a and b. That is, linear
combinations of vectors give vectors.
Most of your background with vectors has been for vectors in Rn. But: the
signals that we deal with are also elements of a vector space, since linear
combinations of signals also gives a signal. This is a very important and powerful
idea.
Recall that in vector spaces we deal with concepts like the length of a vector, the
angle between vectors, and the idea of orthogonal vectors. All of these concepts
carry over, by suitable denitions, to vector spaces of signals.
This powerful idea captures most of the signicant and interesting notions in
signal processing, controls, and communications. This is really the reason why the
study of linear algebra is so important.
Function spaces
One of the neat things about all of this is that we can do the same techniques for
innite-dimensional spaces, which includes spaces of functions. Now our ingredients
are not vectors in the usual sense, but functions. Suppose we have a set of ingredient
functions called i1(t); i2(t); : : : ; in(t). We want to nd a representation of some other
function f(t)