# 15.8 Orthonormal basis expansions

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The module looks at decomposing signals through orthonormal basis expansion to provide an alternative representation. The module presents many examples of solving these problems and looks at them in several spaces and dimensions.

## Main idea

When working with signals many times it is helpful to break up a signal into smaller, more manageable parts. Hopefully bynow you have been exposed to the concept of eigenvectors and there use in decomposing a signal into one of its possible basis.By doing this we are able to simplify our calculations of signals and systems through eigenfunctions of LTI systems .

Now we would like to look at an alternative way to represent signals, through the use of orthonormal basis . We can think of orthonormal basis as a set of building blockswe use to construct functions. We will build up the signal/vector as a weighted sum of basis elements.

The complex sinusoids $\frac{1}{\sqrt{T}}e^{i{\omega }_{0}nt}$ for all $()$ n form an orthonormal basis for ${L}^{2}(\left[0 , T\right]())$ .

In our Fourier series equation, $f(t)=\sum_{n=()}$ c n ω 0 n t , the $\{{c}_{n}\}()$ are just another representation of $f(t)$ .

For signals/vectors in a Hilbert Space , the expansion coefficients are easy to find.

## Alternate representation

Recall our definition of a basis : A set of vectors $\{{b}_{i}\}$ in a vector space $S$ is a basis if

1. The ${b}_{i}$ are linearly independent.
2. The ${b}_{i}$ span $S$ . That is, we can find $\{{\alpha }_{i}\}$ , where ${\alpha }_{i}\in \mathbb{C}$ (scalars) such that
$\forall x, x\in S\colon x=\sum {\alpha }_{i}{b}_{i}$
where $x$ is a vector in $S$ , $\alpha$ is a scalar in $\mathbb{C}$ , and $b$ is a vector in $S$ .

Condition 2 in the above definition says we can decompose any vector in terms of the $\{{b}_{i}\}$ . Condition 1 ensures that the decomposition is unique (think about this at home).

The $\{{\alpha }_{i}\}$ provide an alternate representation of $x$ .

Let us look at simple example in ${ℝ}^{2}$ , where we have the following vector: $x=\left(\begin{array}{c}1\\ 2\end{array}\right)()$ Standard Basis: $\{{e}_{0}, {e}_{1}\}()=\{\left(\begin{array}{c}1\\ 0\end{array}\right), \left(\begin{array}{c}0\\ 1\end{array}\right)\}()$ $x={e}_{0}+2{e}_{1}$ Alternate Basis: $\{{h}_{0}, {h}_{1}\}()=\{\left(\begin{array}{c}1\\ 1\end{array}\right), \left(\begin{array}{c}1\\ -1\end{array}\right)\}()$ $x=\frac{3}{2}{h}_{0}+\frac{-1}{2}{h}_{1}$

In general, given a basis $\{{b}_{0}, {b}_{1}\}()$ and a vector $x\in {ℝ}^{2}$ , how do we find the ${\alpha }_{0}$ and ${\alpha }_{1}$ such that

$x={\alpha }_{0}{b}_{0}+{\alpha }_{1}{b}_{1}$

## Finding the coefficients

Now let us address the question posed above about finding ${\alpha }_{i}$ 's in general for ${ℝ}^{2}$ . We start by rewriting [link] so that we can stack our ${b}_{i}$ 's as columns in a 2×2 matrix.

$\begin{pmatrix}x\\ \end{pmatrix}={\alpha }_{0}\begin{pmatrix}{b}_{0}\\ \end{pmatrix}+{\alpha }_{1}\begin{pmatrix}{b}_{1}\\ \end{pmatrix}$
$\begin{pmatrix}x\\ \end{pmatrix}=\begin{pmatrix}⋮ & ⋮\\ {b}_{0} & {b}_{1}\\ ⋮ & ⋮\\ \end{pmatrix}\begin{pmatrix}{\alpha }_{0}\\ {\alpha }_{1}\\ \end{pmatrix}$

Here is a simple example, which shows a little more detail about the above equations.

$\begin{pmatrix}x(0)\\ x(1)\\ \end{pmatrix}={\alpha }_{0}\begin{pmatrix}{b}_{0}(0)\\ {b}_{0}(1)\\ \end{pmatrix}+{\alpha }_{1}\begin{pmatrix}{b}_{1}(0)\\ {b}_{1}(1)\\ \end{pmatrix}=\begin{pmatrix}{\alpha }_{0}{b}_{0}(0)+{\alpha }_{1}{b}_{1}(0)\\ {\alpha }_{0}{b}_{0}(1)+{\alpha }_{1}{b}_{1}(1)\\ \end{pmatrix}()$
$\begin{pmatrix}x(0)\\ x(1)\\ \end{pmatrix}=\begin{pmatrix}{b}_{0}(0) & {b}_{1}(0)\\ {b}_{0}(1) & {b}_{1}(1)\\ \end{pmatrix}\begin{pmatrix}{\alpha }_{0}\\ {\alpha }_{1}\\ \end{pmatrix}$

## Simplifying our equation

To make notation simpler, we define the following two itemsfrom the above equations:

• Basis Matrix : $B=\begin{pmatrix}⋮ & ⋮\\ {b}_{0} & {b}_{1}\\ ⋮ & ⋮\\ \end{pmatrix}()$
• Coefficient Vector : $\alpha =\begin{pmatrix}{\alpha }_{0}\\ {\alpha }_{1}\\ \end{pmatrix}()$
This gives us the following, concise equation:
$x=B\alpha$
which is equivalent to $x=\sum_{i=0}^{1} {\alpha }_{i}{b}_{i}$ .

Given a standard basis, $\{\begin{pmatrix}1\\ 0\\ \end{pmatrix}, \begin{pmatrix}0\\ 1\\ \end{pmatrix}\}$ , then we have the following basis matrix: $B=\begin{pmatrix}0 & 1\\ 1 & 0\\ \end{pmatrix}$

To get the ${\alpha }_{i}$ 's, we solve for the coefficient vector in [link]

$\alpha =B^{(-1)}x$
Where $B^{(-1)}$ is the inverse matrix of $B$ .

## Examples

Let us look at the standard basis first and try to calculate $\alpha$ from it. $B=\begin{pmatrix}1 & 0\\ 0 & 1\\ \end{pmatrix}=I$ Where $I$ is the identity matrix . In order to solve for $\alpha$ let us find the inverse of $B$ first (which is obviously very trivial in this case): $B^{(-1)}=\begin{pmatrix}1 & 0\\ 0 & 1\\ \end{pmatrix}$ Therefore we get, $\alpha =B^{(-1)}x=x$

Let us look at a ever-so-slightly more complicated basis of $\{\begin{pmatrix}1\\ 1\\ \end{pmatrix}, \begin{pmatrix}1\\ -1\\ \end{pmatrix}\}=\{{h}_{0}, {h}_{1}\}$ Then our basis matrix and inverse basis matrix becomes: $B=\begin{pmatrix}1 & 1\\ 1 & -1\\ \end{pmatrix}$ $B^{(-1)}=\begin{pmatrix}\frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{-1}{2}\\ \end{pmatrix}$ and for this example it is given that $x=\begin{pmatrix}3\\ 2\\ \end{pmatrix}$ Now we solve for $\alpha$ $\alpha =B^{(-1)}x=\begin{pmatrix}\frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{-1}{2}\\ \end{pmatrix}\begin{pmatrix}3\\ 2\\ \end{pmatrix}=\begin{pmatrix}2.5\\ 0.5\\ \end{pmatrix}$ and we get $x=2.5{h}_{0}+0.5{h}_{1}$

Now we are given the following basis matrix and $x$ : $\{{b}_{0}, {b}_{1}\}=\{\begin{pmatrix}1\\ 2\\ \end{pmatrix}, \begin{pmatrix}3\\ 0\\ \end{pmatrix}\}$ $x=\begin{pmatrix}3\\ 2\\ \end{pmatrix}$ For this problem, make a sketch of the bases and then represent $x$ in terms of ${b}_{0}$ and ${b}_{1}$ .

In order to represent $x$ in terms of ${b}_{0}$ and ${b}_{1}$ we will follow the same steps we used in the above example. $B=\begin{pmatrix}1 & 2\\ 3 & 0\\ \end{pmatrix}$ $B^{(-1)}=\begin{pmatrix}0 & \frac{1}{2}\\ \frac{1}{3} & \frac{-1}{6}\\ \end{pmatrix}$ $\alpha =B^{(-1)}x=\begin{pmatrix}1\\ \frac{2}{3}\\ \end{pmatrix}$ And now we can write $x$ in terms of ${b}_{0}$ and ${b}_{1}$ . $x={b}_{0}+\frac{2}{3}{b}_{1}$ And we can easily substitute in our known values of ${b}_{0}$ and ${b}_{1}$ to verify our results.

A change of basis simply looks at $x$ from a "different perspective." $B^{(-1)}$ transforms $x$ from the standard basis to our new basis, $\{{b}_{0}, {b}_{1}\}$ . Notice that this is a totally mechanical procedure.

## Extending the dimension and space

We can also extend all these ideas past just ${ℝ}^{2}$ and look at them in ${ℝ}^{n}$ and ${ℂ}^{n}$ . This procedure extends naturally to higher (>2) dimensions. Given a basis $\{{b}_{0}, {b}_{1}, \dots , {b}_{n-1}\}$ for ${ℝ}^{n}$ , we want to find $\{{\alpha }_{0}, {\alpha }_{1}, \dots , {\alpha }_{n-1}\}$ such that

$x={\alpha }_{0}{b}_{0}+{\alpha }_{1}{b}_{1}+\dots +{\alpha }_{n-1}{b}_{n-1}$
Again, we will set up a basis matrix $B=\begin{pmatrix}{b}_{0} & {b}_{1} & {b}_{2} & \dots & {b}_{n-1}\\ \end{pmatrix}()$ where the columns equal the basis vectors and it will alwaysbe an n×n matrix (although the above matrix does not appear to be square since we left terms in vector notation).We can then proceed to rewrite [link] $x=\begin{pmatrix}{b}_{0} & {b}_{1} & \dots & {b}_{n-1}\\ \end{pmatrix}\begin{pmatrix}{\alpha }_{0}\\ ⋮\\ {\alpha }_{n-1}\\ \end{pmatrix}=B\alpha$ and $\alpha =B^{(-1)}x$

differences between general anxiety disorder and phobia
refer to D.S.M.
Rick
huge differences I would say between phobias vs a disorder
Rick
coming from someone who suffers from PTSD. i would say anxiety is different then phobias. although my phobia is snakes with causes a huge panic attack.
Betty
hi im.from Colombia i install this program but i dont underdtand this... is kind of a chat?
Gabriela
i just wanna learn psychology in english coz im psychologist already but is very interesting ... i just seee all this time this conversations and i dont understand
Gabriela
u can find people who know both Spanish n English
Anmol
phobia is a type of anxiety. It's an irrational, unreasonable fear of an object or situation. Common phobias may be fear of the heights, dark, fear of certain animals such as snakes or spiders, or fear of blood -- some people get frightened of having their blood drawn when they go to see their docto
Betty
Anmol Joy, thank you so much, and Betty thanks for your post.. blesses bye
Gabriela
np
Betty
Hello what causes over thinking pls ?
Binta
Anxiety and overthinking tend to be evil partners. One of the horrible hallmarks of any type of anxiety disorder is the tendency to overthink everything. The anxious brain is hypervigilant, always on the lookout for anything it perceives to be dangerous or worrisome. ... Because anxiety causes me to
Betty
hello what are the things to do or to learn to make mind controlled and powerful
hari
I agree
Richard
I saw the posting about phobias being irrational and unreasonable, I don't completely agree with that
Richard
unless you're referring to learned phobia
Richard
overthinking causes bad consequences in brain
Abid
the problem with the world is that the intelligent people are full of doubts, while the stupid people are full of confidence.
Captain
how can web help someone who fears spiders?
ummm..(?) place a decoy spider on it and expose ur self.. until you can endure the sight of real spider.
Princeed
How does selection relate to population divergence
Is this the only app edition? If so does it contain recent studies such as from 2015?
I am going to look for it right now.
Melissa
lookbit up at google play store just time in i mind or whatever the series would be considerd.
Melissa
did you find anything out?
Zed
i have a question for you dear humans. What is the purpose of your life ?
Akando
Akando Evans are u not human
hari
or u dont know
hari
You're absolutely right.
Akando
I'm sorry but I am more than human. My life's purpose is to teach and lightworker
Rebecca
Only app edition of the one we are currently on if I'm not mistaken
Rebecca
Rebecca ? If you assume to be more than human, then i simply ask you. What is it like being just human ?
Akando
we are multi dementional beings having human experiences
Rebecca
ooooooo my life purpose is to learn and gain knowlege and gain something too
hari
multidimensional ? Interesting. Do elaborate and exemplify please.
Akando
can i ask u one thing Rebecca Lopez
hari
Rebecca ? If you assume to be more than human, then i simply ask you. What is it like being human ?
Akando
Humans do have lot of responsibilities and some productive work to do Alien Akando... They can't spend whole time for some useless explanations..
sunny
Take a minute, and observe the ironical paradox in your message. I will wait.
Akando
do any one believe that humans are stronger and powerfull than any creature in the all dimensions i do . do any one believe?
hari
Yes I do believe humans possess immeasurable power within, I'm Len Allen
Len
is this edition good for graduate study?
MsGTG
nice tomeet u len allen do u believe humans have some natural powrrs
hari
i want yo know is it posdiable ot not
hari
possible
hari
I believe we all possess natural powers that are in fact supernatural supported by theory that infanete power can be dirived from the unknown
Captain
does telekinesis possible to learn or not
hari
telekinesis is possible to learn with a clear mind and being zen with the planet orbital sun and the third eye
Captain
what is meant by being zen with the planet orbital sun and the third eye
hari
in reality you must focus on the object for 10 mins and try to focus how you want to shape it and bend it with your mind
Captain
and tai chi hpnoticesv tele kinesis is these are possible to human being and controlling others mind is posdible if possible.' what things should we get mastry before learning these please say
hari
where are you from?
Captain
hari
india
hari
because the knowledge is out there if you seek it, it is there
Captain
I only ask because India is a country that knows alot of the mysteries you seek
Captain
reallly what u said i did not understand once say again please i want to know about it
hari
where arr u from
hari
hari
share ur knowledge
hari
I'm in the United States
Captain
ok what r the bacis we need to learn before to learn telekiness etc
hari
ok please share knowledge captain nemo
hari
Guys, can anyone please suggest books to read statistics and research methodology for psychology? These are tricky topics for me so kindly share books which are easy to understand, thanks!
Himshikha
hi sharma did u get any philosophy book
hari
Nope!
Himshikha
are u a ug student ?
vishav
Yes!
Himshikha
I need it for preparation of entrance of MA Applied Psychology
Himshikha
oooo nice group all the best if u got please share me
hari
ohh! I thought u need it for ug exams ..still i'll ask my professor for the book.
vishav
Thanks so much! Are you an undergraduate student?
Himshikha
sss how u know
hari
I was asking Vishav. Are you too?
Himshikha
sss
hari
i m also looking for admission in MA PSYCHOLOGY but in our state there is no subjective test for enterance..there is common aptitude test
vishav
Oh I see, you could share the books you read in your UG
Himshikha
yeah ! himshikha just completed my last semester
vishav
Oh great! What did you refer to for stats?
Himshikha
honesty ! I studied from YouTube
vishav
That's even better! Could you share the channel?
Himshikha
I just need to clear my concepts
Himshikha
vishav
All the basic statistical methods used in Psychology
Himshikha
I'm too afraid of things and I over think about them too much ,which lags me behind and suppresses my confidence level how to overcome this
Binta
I started giving time for myself and learning not to give a Crap
Dana
You should live the moment and just do it pretty much. Your fears will just hold u back
Dana
does meditating helps me
Rajesh
be confident with yourself. you are beautiful and perfect.
janneth
yes. at least, take some quiet time and enjoy yourself. believe that you can do anything.
janneth
Meditating can help too :)
Dana
3,9,6 Tesla
Keith
find the origin of problem...why you get scared..
confidence level just doesn't increase on its own. first you have to accomplish something, anything big or small doesn't matter. after that your confidence increases, then try doing things that gives you fear, not everything together, in small parts.
Akarui
it may take time but this confidence is alot more true than the confidence some people fake to hide their insecurity.
Akarui
people in glass houses shouldn't throw stones, sure it's an old proverb
Keith
remember the great fllood
Keith
?
Keith
truth will set you free, don't quote me ... somebody else far wiser knew the power of voice & voices
Keith
check out Niney the observer ... Mix up song
Keith
is there anyone addicted to mast.?
Do anyone have solution to get rid of it?
Su
Mast?
Dana
anyone watched Dr Jordan Peterson lectures?
Himshikha
there's many , search a short clip of his interview on channel 4 uk
Keith
okay
Himshikha
jordan Peterson is fab! I'm in the middle of putting together a comparative models presentation and we chose the subject of fairy tales to demonstrate this .. in particular carl jungs analytical theory and petersons lectures have been invaluable ... Peterson is well worth watching! .. love him!!
Kimmie
Hello, this is the first time I join this, please give me some advice for this. Thank yiu very much!
throw it out there
Keith
music expression of truth tales
Keith
***youtu.be/uYpgsPB-Bkw
Keith
I'm new and I have no idea how these chats work 😂
Dana
start with the being or beginning? no pressure ... I've not clue either
Keith
I still haven't used this app yet to be honest
Dana
😉
Vanessa
haha
Keith
ok are people waking up?
Keith
same here but am still learning
sekele
overcoming anxiety, is it possible?
Definitely!
Shuaijun
quelling your caring to social issues
Keith
emotional intelligence
Keith
think it's David Coleman book
Keith
social change without going to far left & equally too far right
Keith
anxiety is a feeling causse to us it problem to co trol it but if u controled u r mind u can do it
hari
increasing your levels of confidence by different means like trying out small tasks n completing it nicely....making friends and joining social gatherings can reduce ua anxiety level
Anmol
what you need to translate in hindi
Ravi
😂
Anmol
idk. i don't need 2 translate anything.
Vanessa
can someone tell me what is extremes of intelligence
Pralay
why?.. you don't know the answer it what 😂
Pralay
hormones and vitamin levels can affect our thoughts and behavior though. Right?
yes
Himshikha
everything we ate affects us.
Whatever
If thoughts as we perceive them are neither matter or energy, what are they?
neural connections
Himshikha
they all are absolutely a form of energy
virus
such a nice question!!
Anmol
maybe it's an undiscovered form of matter and energy called mattergy......I'm not sure though.
Eric
bro thought as I believe it is .....is nothing but nerve impulses and a very complicated one too
mubii
is it not energy? for example worries can make one tired?
Krish
thoughts I agree that they are neural impulses with energy, an energy that connects u to human feelings, emotions, an energy that has a power to emphasize ultimately the one which forms a network from one thought to other. which results in attachment or clashes between individual....
Anmol
wen u see a person and like him at once , can u guarantee that there isn't the power of thoughts uniting u? don't u think there is some connections of thoughts getting formed at the very moment.. I have these doubts of thoughts with energy
Anmol
but energy have two kinds one is positive other is negative
virus
Do you use energy in the context of an analogy? Because energy is not what you seem to get at here. It is body language, mindset and empathy that you seem to describe.
aemilius
Aemilius then what is energy? Isn't all matter energy? If the whole universe which I am a part of, is energy, and my perceived thoughts are part of me then what are they? it took us a really long time to be able to measure the cosmos, for a much longer time we couldn't.
Dale
I'm ok with stepping outside the boundaries of empiricism, which forces us to ignore the very subject of this conversation, and say subjective thoughts must have some substance we haven't been able to isolate and measure yet.
Dale
Energy is a force that moves something. It can't be destroyed or created.
aemilius
It's a bunch of connections from the things we saw, we heard, we tasted.... Your brain mix it up like a bowl of soup.
Whatever
wow, glad I have been able to understand
ruth
how can I teach a dog to dance with a particular tone
dance with him daily
Aayush
sit with him normally nd put the song on and and start with that song.. and change the song again and sit normally.. and after 5 more mins do the same 5 time a day u wll see it urself after 15 day..
mahliqa
Teach him steps and give his favorite food pieces after every step done by him.
Cholistan
heya Can someone suggest me in my stuff to find out the courses and colleges of psychology?
Lata
It depends upon from where, you would like to do ( the place ) . In genral every university offers courses in Psychology , you may Google and search some colleges in your localities .
Ravi
give him treats for success
Dani
How can someone overcome the fear of not been able to speak in public?
Can an introvert become extrovert
kabiru
not completely....infact no individual is completely an introvert or extrovert....there is term ambivert by Jung...which means there is both level of extroversion and introversion in an in individual....u can increase ua level of extroversion by various ways
Anmol
social anxiety can be reduced by 1st forming small group of friends and then increasing ua group members participanting in functions, parties, trying to give some stage performances can help u.
Anmol
What are the ways
kabiru
use a personal "stress ball" I have a smooth rock I hold in my hand and rub with my thumb, it word for me. everyone is different and it will take a few tries to figure out what works for you.
Dawn
Plx what are the concept of motivations
There are three major components to motivation: activation, persistence, and intensity. Activation involves the decision to initiate a behavior, such as enrolling in a psychology class
Michael
thanks Michael
Iliyas
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