Lecture 4: A Return to the Geometric Viewpoint

Schedule :: Periodic Signals :: Fourier Spectrum :: Symmetry :: Fundamental Frequency :: Smoothness of the Function :: Fourier Series :: Exponential Series :: Spectra :: Bandwidth :: Energy of Signals :: Geometric Viewpoint

We have seen that functions can be represented as series of orthogonal functions, and have seen examples of orthogonal functions, the trigonometric and the complex exponentials. Historically, these were first examined by Jean Baptiste Fourier, who used these to solve the partial differential equations related to heat flow. At first his methods were considered unconventional by mathematicians. Now the generalization of Fourier's methods form one of the largest and most fruitful areas of mathematics.

Are there any other useful orthogonal functions? Given a set of functions that are not orthogonal, is it possible to make it orthogonal somehow? Both answers are yes.

We begin looking at a set of orthogonal polynomials, defined over the interval $[-1,1]$ . It is easy to check that the set of polynomials $\{1,t,t^2,t^3,\ldots\}$ is not orthogonal. For example,

$\begin{displaymath}\la t,t^3 \ra = \int_{-1}^1 (t)(t^3) dt = \frac{2}{5} \end{displaymath}$

Consider, however, the polynomials

$\begin{displaymath}P_0(t) = 1 \end{displaymath}$

$\begin{displaymath}P_1(t) = t \end{displaymath}$

$\begin{displaymath}P_2(t) = \frac{3}{2}t^2 - \frac{1}{2} \end{displaymath}$

$\begin{displaymath}P_3(t) = \frac{5}{2}t^3 - \frac{3}{2}t \end{displaymath}$

In general,

$\begin{displaymath}P_n = \frac{1}{2^n n!}\frac{d^n}{dt^n}(t^2-1)^n. \end{displaymath}$

These polynomials are known as the Legendre polynomials. It can be shown that

$\begin{displaymath}\int_{-1}^1 P_m(t) P_n(t) dt = \left\{\begin{array}{ll} 0 & m \neq n \frac{2}{2m+1} & m = n \end{array}\right. \end{displaymath}$

For functions that are defined over any finite interval, Legendre polynomials can be used as a functional representation.

Another way that orthogonal functions arise is by means of another inner product. While we have not mentioned it in the past, it is possible to introduce a positive weighting factor into the inner product. Every one of these produces a new inner product, each with properties that may be useful for particular applications. If $w(t)$ is a non-negative function, then we can define an inner product

$\begin{displaymath}\la f,g \ra_w = \int_I w(t) f(t)g(t) dt \end{displaymath}$

where the subscript $_w$

indicates the weighting and

is some interval of integration. For example, for $w(t) = \frac{1}{\sqrt{1-t^2}}$ and

we might define an inner product as

$\begin{displaymath}\la f(t),g(t) \ra_w = \int_{-1}^1 \frac{1}{\sqrt{1-t^2}} f(t) g(t) dt \end{displaymath}$

A set of polynomials that is orthogonal with respect to this norm is the set of Chebyshev polynomials:

$\begin{displaymath}T_0(t) = 1 \end{displaymath}$

$\begin{displaymath}T_n(t) = \cos n\cos^{-1} t \end{displaymath}$

For example,

$\begin{displaymath}T_1(t) = t \end{displaymath}$

$\begin{displaymath}T_2(t) = \cos 2 \cos^{-1} t = 2\cos^2(\cos^{-1} t) -1 = 2t^2 - 1 \end{displaymath}$

$\begin{displaymath}T_3(t) = \cos 3 \cos^{-1} t = 4\cos^3(\cos^{-1} t) - 3 \cos(\cos^{-1} t) = 4 t^3 - 3 t \end{displaymath}$

Notice that these have the ``equal ripple'' property. This makes them useful in function approximation (and in fact, Chebyshev functions are used at the heart of the Remez algorithm).

Another set of interesting orthogonal functions that have been the topic of an incredible amount of research lately are wavelet functions. Unlike the familiar and common trigonometric functions, there are actually several families of wavelets (this is part of what makes it confusing). One type of wavelets is described by two sets of functions: $\phi(t)$ (known as a scaling function), and $\psi(t)$ (known as a wavelet function). In dealing with these functions, instead of looking at different frequencies, we scale and shift these functions. We define

$\begin{displaymath}\phi_{j,k}(t) = 2^{j/2}\phi(2^j t - k) \end{displaymath}$

$\begin{displaymath}\psi_{j,k}(t) = 2^{j/2}\psi(2^j t - k) \end{displaymath}$

Make plots and explain. Then (somewhat miraculously), the following orthogonality properties exist:

$\begin{displaymath}\la \phi_{j,k}(t),\phi_{j,m}(t) \ra = \delta_{k,m} \end{displaymath}$

$\begin{displaymath}\la \psi_{j,k}(t),\psi_{l,m}(t) \ra = \delta_{j,l}\delta_{k,m} \end{displaymath}$

$\begin{displaymath}\la \phi_{j,k}(t),\psi_{l,m}(t) \ra = \delta_{j,l}\delta_{k,m} \end{displaymath}$
These are pretty remarkable! This gives us a whole bunch of functions that we can use to signal representations:

$\begin{displaymath}f(t) = \sum_{k} a_{J,k}\phi_{J,k}(t) + \sum_{j,k} b_{j,k}\psi_{j,k}(t) \end{displaymath}$

This is useful for a variety of things which will become more clear after we talk about Fourier transforms (next chapter). But notice that we can represent any function, not just periodic functions, localizing both frequency information and time information. There are also very efficient algorithms that are faster than the Fast Fourier Transform (FFT) to compute a discrete wavelet transform.

Citation: admin. (2006, May 23). Lecture 4: A Return to the Geometric Viewpoint. Retrieved November 23, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Signals_and_Systems/4_12node12.html.