Calculating c for Approximation
Finding
the component of a vector along another vector.
How do we modify our existing
vector equations to account for the non-unit length of vector w?
Well, since our ideal component along w forms a right triangle with the error vector and v, we can use simple trigonometry again
to show that
. [Equation 5.1]
However,
what we really want to find is that constant c, that shows us by how much we
scale vector w to make vector vw. Substituting into Equation 5.1, we find that
, [Equation 5.2]
and
multiplying both sides by |w|,
. [Equation 5.3]
So
finally, substituting into Equation
5.3
according to the dot product formulas found in Equation 1.1 and Equation 1.2
from the inner
product page,
. [Equation
5.4]
We’ve
gone through a lot of work to get to this formula but you can see that it
represents the constant c in a very
understandable way. The dot product in
the numerator is basically the same simple trigonometry operation we performed
in Equation 5.1—the
projection of one vector on another. The
dot product in the denominator is responsible for normalizing c, in the case when our basis vector is
not of unit length. This equality will
work when attempting to find by how much you would scale any vector w to make it
the perfect component—the best approximation—of
any other vector v. Now, we are finally ready to delve into the
world of signals, armed with our vector knowledge. We will find that there are countless
similarities between what we have learned in the realm of vector construction, and
the practice of building signals from their components.
In
terms of a signal,
. [Equation
9.1]
And
thus, the exact value of c, over the
interval [t1, t2], is given by
, [Equation 9.2]
where Eg is the energy of
the basis signal g(t).
Return to the vector – signal analogy table.
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Discovery Project II. Vaz, Richard F. 2001. WPI.
<http://www.ece.wpi.edu/courses/ee2311/temp/dp2.htm>