Orthogonal Signal Space

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Orthogonal Signal Space

****IMPORTANT****

Vector Orthogonality:

$\begin{displaymath}<\underline{x},\underline{y}>=\sum_{i=1}^{N}x_iy_i^\ast=0 \end{displaymath}$

if $\underline{x}$ and $\underline{y}$ orthogonal, $\underline{x}\perp\underline{y}$
Magnitude $\vert\vert\underline{x}\vert\vert^2=\sum_{i=1}^{N}\vert x_i\vert^2=<\underline{x},\underline{x}>$ as defined above

$\begin{displaymath}\underline{a}\perp\underline{b}\Rightarrow\vert\vert a+b\vert... ...t^2+\vert\vert b\vert\vert^2 \qquad \mbox{(prove it yourself)} \end{displaymath}$

triangle Inequality $\vert\vert\underline{a}+\underline{b}\vert\vert\le\vert\vert\underline{a}\vert\vert+\vert\vert\underline{b}\vert\vert$
Linearity

$\begin{displaymath}L[a\underline{x}+b\underline{y}]=aL[\underline{x}]+bL[\underline{y}] \end{displaymath}$
Orthogonal expansion
Can always write

$\begin{displaymath}\underline{x}=\sum_{i=1}^{N}a_i\underline{e_i} \qquad \mbox{w... ...hspace{2cm}}r} 0 & i\ne j\\ \alpha_i & i=j \end{array} \right. \end{displaymath}$

$\{\underline{e_i}\}$ is an orthogonal basis Therefore,

$\begin{displaymath}L[\underline{x}]=\sum_{i=1}^{N}a_iL[\underline{e_i}] \end{displaymath}$

Eigenvector: $\underline{x}$ is an eigenvector of system $L[\;]$ if

$\begin{displaymath}L[\underline{x}]=\lambda\underline{x} \end{displaymath}$

where $\lambda$ is a constant.
Gram Schmidt
Can we create personalized orthogonal sets? GET YOURS NOW! CALL 1-800-555,... :)
Aside on independence: $\{\underline{x_i}\}$ $\mbox{Card}\{\underline{x_i}\} =J\le N$ are mutually independent if

$\begin{displaymath}\underline{x_i}\ne \sum^J_{j=1, j\ne i} a_j \underline{x_j} \end{displaymath}$

for all choices $\{a_j\}$
Given J mutually independent vectors $\underline{x_1},\cdots,\underline{x_j}$ we can form mutually orthogonal set $\{\underline{\phi_i}\}$ using GRAM-SCHMIDT ORTHOGONALIZATION
Procedure
Start with $\underline{x}_1$

$\begin{displaymath}\underline{\phi_1}=\frac{\underline{x_1}}{\vert\vert\underline{x_1}\vert\vert} \end{displaymath}$

Then use $\underline{x}_2$ and subtract off projection onto $\underline{\phi}_1$

$\begin{displaymath}\underline{\phi_2} =\underline{x_2}- \frac{<\underline{\phi_1... ...c{\underline{\phi_1}}{\vert\vert\underline{\phi_1}\vert\vert} \end{displaymath}$

etcetera and so on...

$\begin{displaymath}\underline{\phi_k} =\underline{x_k}- \sum_{j=1}^{k-1} \frac{<... ...c{\underline{\phi_j}}{\vert\vert\underline{\phi_j}\vert\vert} \end{displaymath}$
Orthogonality Check

$\begin{displaymath}<\underline{\phi_1},\underline{\phi_2}>= <\underline{x_1},\un... ...ne{x_1},\underline{x_1}>}{<\underline{x_1},\underline{x_1}>}=0 \end{displaymath}$

can show recursively that $\underline{\phi_i}\perp \phi_j$ $i\ne j$
Then we make orthonormal:

$\begin{displaymath}\underline{\psi_i}=\frac{\underline{\phi_i}}{\vert\underline{\phi_i}\vert} \end{displaymath}$

$\begin{displaymath}<\underline{\psi_i},\underline{\psi_j}>= \left\{\begin{array}... ... & {i\ne j} \\ \displaystyle {1} & {i=j} \end{array}\right. \end{displaymath}$
What about Signals?
SIGNAL PICTURE

$\begin{displaymath}\underline{f}=(f_1,f_2,\cdots f_N) \end{displaymath}$

$\begin{displaymath}\Delta<\underline{g},\underline{f}>=(\sum_{i=1}^N g_i f_i^*)\Delta \end{displaymath}$

Shrink $\Delta\rightarrow 0$

$\begin{displaymath}\frac{1}{T}\int_0^T g(t)f^*(t)dt\equiv <g(t),f(t)> \end{displaymath}$

if $g(t)\perp f(t)$ on (a,b) then

$\begin{displaymath}\int_a^b g(t)f^*(t)dt=0 \end{displaymath}$
Example

$\begin{displaymath}\omega_0=\frac{2\pi}{T} \end{displaymath}$

$\begin{displaymath}\underbrace{<e^{j\omega_0 kt}, e^{j\omega_0 mt}>}_{\mbox{orth... ...{array}{r @{\qquad}r} 0 & m\ne k\\ 1 & m=k \end{array} \right. \end{displaymath}$

GRAM SCHMIDT WORKS TOO!
Completeness
Complete set means any signal in the signal space could be expanded as a series, i.e.

$\begin{displaymath}f(t)=\sum_{i=1}^{\infty}a_i \phi_i(t) \end{displaymath}$

where $<\phi_i(t),\phi_j(t)>=\delta_{i,j}$
$\{\phi_i(t)\}$ forms a basis for the signal set if any f(t) can be represented this way
Complex exponentials are a complete orthonormal which is why Fourier series works
DECOMPOSITION

$\begin{displaymath}{f(t)=\sum_{k=-\infty}^{\infty} a_k e^{j\omega_0 kt}} \end{displaymath}$

PROJECTION

$\begin{displaymath}a_k=\frac{1}{T}\int_0^T f(t) e^{-j\omega_0 kt}dt \end{displaymath}$

which form the Fourier series pair when f(t)=f(t+T), $\omega_0 = 2\pi/T$ and if f(t) satisfies Dirichlet (see book)

Next: LTI Systems and Periodic Up: Linear Systems Review Previous: LTI Systems: Time domain

1999-02-01