Purdue University
EAS 657
Geophysical Inverse Theory
Robert L. Nowack
Lecture 3a
Vectors in Hilbert
Space
Given a basis for V, (v1,…vN) and a defined inner product, the Gram Schmidt procedure can be used to construct an orthonormal basis (,… ).
a) Let
b) Find
the “orthogonal projection” of v2
on (drop a
perpendicular), then subtract this part from v2. The
angle between and v2 is obtained from
the inner product as
Now, should be parallel to , where
This gives
Then,
and
If this procedure is continued, then
and
CONVERSE: Given a basis v1,…vN of a vector space V, one can always find an inner product such that v1,…vN are orthonormal. This can be done by modifying the definition of the inner product.
(x,y) = yTQx
Choose Q such that x,y are orthonormal.
If , i=1,N are orthonormal, then
for j = 1,…,N
since for an orthonormal basis, (,) =
then,
The representation of a vector
using an orthonormal basis is called a Generalized
Fourier Series, where
with an inner product such that are orthonormal. Note that care must be taken for infinite dimensional spaces.
Now let’s investigate Signal Spaces L2[a,b]. This is the space of square-integrable functions on the line [a,b]. Thus,
This is an infinite dimensional space with inner product defined as
and an induced norm
Ex) For the set of all periodic signals with period T on L2[0,T], choose the complex exponentials
where
and k = , … -1, 0, 1, …
The are orthogonal since
Let
then,
Thus are orthogonal on L2[0,T]. Let,
k = -¥, …-1,0, + 1,…¥
See Luenberger (1968) p. 61-62. These are orthonormal and can also be shown to be complete in L2[0,T]. Now any function in L2[0,T] can be expressed as
where
and
This is called a Generalized Fourier Series since we are free to choose any orthonormal basis. For
then,
The are the Fourier coefficients called the Discrete spectrum of f(t).
A Fourier series pair for L2[0,T] can be written
where .
If we look at the unbounded interval [-, +], the functions are not square-integrable. But if we relax this requirement and look at basis functions , then
.
is a generalized function called a Delta function, where
and
Then,
where
Thus,
or
The spectrum has now coalesced into a continuous function, . Thus, the Continuous Fourier transform pair on with the assistance of generalized functions (a relaxation of the square integrable condition) can be written
However, we aren’t limited to complex exponentials!
Ex) Consider discrete signals where
Let the discrete function be written
then,
where u[k] are the coefficients and are the basis functions.
Define the inner product to be,
Now, is the basis orthonormal? Yes, since
Let
then,
Thus,
The individual sample points provide one basis for a discrete time series.
Ex) Continuous signals that are bandlimited in frequency forming a subspace of
The sampling theorem states that one can completely reconstruct a continuous bandlimited signal from a discrete sampling of the signal. This can be written as
where are the discrete samples with a sampling period T. This is exact if the highest frequency in the signal is less than .
The “impulse response” of this system is where is a sinc function.
In terms of a generalized Fourier series, then the basis functions are with inner product .
Since , then the basis is orthonormal. The coefficient can be written
Polynomials
The polynomials tk are independent, but not orthogonal on L2[-1,1]
with inner product on the interval [-1, 1].
We can use Gram Schmidt to find an orthonormal basis, then,
where Pk(t) are called Legendre polynomials with
The problem with the polynomials, tk, is that they are not nearly orthogonal and an orthonormal basis is preferred. Using different inner products, then different orthogonal polynomials can be obtained.
Ex) Modify the inner product on L2[-1,1]. Choose
Using Gram Schmidt results in the Chebychev polynomials
where
The first several Chebychev polynomials are,
Parseval’s Theorem
Assume two orthonormal bases for the same vector space V, say and with a specified inner product. Given two vectors, v1, v2, then
This is called the generalized Parseval’s theorem. Note that the ordinary Parseval’s theorem states that power is conserved, or
Power in the time domain = Power in the frequency domain.
Let
and
Then,
This results since the are orthonormal, thus
Now use another basis with same inner product, then
and
and
Now the value of the inner product should be preserved regardless of the basis, then
This is the generalized Parseval’s Theorem.
Ex) Assume an inner product
Use a basis where , then the coefficient are the Fourier series.
and
Then,
For , then
Thus, the squared sum in one period equals the sum of squared Fourier coefficients.