# 0.8 Appendix: bilinear forms for linear convolution

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## Appendix: bilinear forms for linear convolution

The following is a collection of bilinear forms for linear convolution.In each case $\left({D}_{n},{D}_{n},{F}_{n}\right)$ describes a bilinear form for $n$ point linear convolution. That is

$y={F}_{n}\left\{{D}_{n},h,*,{D}_{n},x\right\}$

computes the linear convolution of the $n$ point sequences $h$ and $x$ .

For each ${D}_{n}$ we give Matlab programs that compute ${D}_{n}x$ and ${D}_{n}^{t}x$ , and for each ${F}_{n}$ we give a Matlab program that computes ${F}_{n}^{t}x$ . When the matrix exchange algorithm is employed in the design of circular convolutionalgorithms, these are the relevant operations.

## 2 point linear convolution

${D}_{2}$ can be implemented with 1 addition, ${D}_{2}^{t}$ with two additions.

${D}_{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 1& 1\end{array}\right]$
${F}_{2}=\left[\begin{array}{ccc}1& 0& 0\\ -1& -1& 1\\ 0& 1& 0\end{array}\right]$
function y = D2(x) y = zeros(3,1);y(1) = x(1); y(2) = x(2);y(3) = x(1) + x(2); function y = D2t(x) y = zeros(2,1);y(1) = x(1)+x(3); y(2) = x(2)+x(3); function y = F2t(x) y = zeros(3,1);y(1) = x(1)-x(2); y(2) = -x(2)+x(3);y(3) = x(2);

## 3 point linear convolution

${D}_{3}$ can be implemented in 7 additions, ${D}_{3}^{t}$ in 9 additions.

${D}_{3}=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 1\\ 1& -1& 1\\ 1& 2& 4\\ 0& 0& 1\end{array}\right]$
${F}_{3}=\frac{1}{6}\left[\begin{array}{ccccc}6& 0& 0& 0& 0\\ -3& 6& -2& -1& 12\\ -6& 3& 3& 0& -6\\ 3& -3& -1& 1& -12\\ 0& 0& 0& 0& 6\end{array}\right]$
function y = D3(x) y = zeros(5,1);a = x(2)+x(3); b = x(3)-x(2);y(1) = x(1); y(2) = x(1)+a;y(3) = x(1)+b; y(4) = a+a+b+y(2);y(5) = x(3); function y = D3t(x) y = zeros(3,1);y(1) = x(2)+x(3)+x(4); a = x(4)+x(4);y(2) = x(2)-x(3)+a; y(3) = y(1)+x(4)+a;y(1) = y(1)+x(1); y(3) = y(3)+x(5); function y = F3t(x) y = zeros(5,1);y(1) = 6*x(1)-3*x(2)-6*x(3)+3*x(4); y(2) = 6*x(2)+3*x(3)-3*x(4);y(3) = -2*x(2)+3*x(3)-x(4); y(4) = -x(2)+x(4);y(5) = 12*x(2)-6*x(3)-12*x(4)+6*x(5); y = y/6;

## 4 point linear convolution

${D}_{4}={D}_{2}\otimes {D}_{2}$
${F}_{4}=\left[\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& -1& 1& 0& 0& 0& 0& 0& 0\\ -1& 1& 0& -1& 0& 0& 1& 0& 0\\ 1& 1& -1& 1& 1& -1& -1& -1& 1\\ 0& -1& 0& 1& -1& 0& 0& 1& 0\\ 0& 0& 0& -1& -1& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\end{array}\right]$
function y = F4t(x) y = zeros(7,1);y(1) = x(1)-x(2)-x(3)+x(4); y(2) = -x(2)+x(3)+x(4)-x(5);y(3) = x(2)-x(4); y(4) = -x(3)+x(4)+x(5)-x(6);y(5) = x(4)-x(5)-x(6)+x(7); y(6) = -x(4)+x(6);y(7) = x(3)-x(4); y(8) = -x(4)+x(5);y(9) = x(4);

## 6 point linear convolution

${D}_{6}={D}_{2}\otimes {D}_{3}$
${F}_{6}=\frac{1}{6}\left[\begin{array}{ccccccccccccccc}6& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -3& 6& -2& -1& 12& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -6& 3& 3& 0& -6& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -3& -3& -1& 1& -12& -6& 0& 0& 0& 0& 6& 0& 0& 0& 0\\ 3& -6& 2& 1& -6& 3& -6& 2& 1& -12& -3& 6& -2& -1& 12\\ 6& -3& -3& 0& 6& 6& -3& -3& 0& 6& -6& 3& 3& 0& -6\\ -3& 3& 1& -1& 12& 3& 3& 1& -1& 12& 3& -3& -1& 1& -12\\ 0& 0& 0& 0& -6& -3& 6& -2& -1& 6& 0& 0& 0& 0& 6\\ 0& 0& 0& 0& 0& -6& 3& 3& 0& -6& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 3& -3& -1& 1& -12& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 6& 0& 0& 0& 0& 0\end{array}\right]$
function y = F6t(x)y = zeros(15,1); y(1) = 6*x(1)-3*x(2)-6*x(3)-3*x(4)+3*x(5)+6*x(6)-3*x(7);y(2) = 6*x(2)+3*x(3)-3*x(4)-6*x(5)-3*x(6)+3*x(7); y(3) = -2*x(2)+3*x(3)-x(4)+2*x(5)-3*x(6)+x(7);y(4) = -x(2)+x(4)+x(5)-x(7); y(5) = 12*x(2)-6*x(3)-12*x(4)-6*x(5)+6*x(6)+12*x(7)-6*x(8);y(6) = -6*x(4)+3*x(5)+6*x(6)+3*x(7)-3*x(8)-6*x(9)+3*x(10); y(7) = -6*x(5)-3*x(6)+3*x(7)+6*x(8)+3*x(9)-3*x(10);y(8) = 2*x(5)-3*x(6)+x(7)-2*x(8)+3*x(9)-x(10); y(9) = x(5)-x(7)-x(8)+x(10);y(10) = -12*x(5)+6*x(6)+12*x(7)+6*x(8)-6*x(9)-12*x(10)+6*x(11); y(11) = 6*x(4)-3*x(5)-6*x(6)+3*x(7);y(12) = 6*x(5)+3*x(6)-3*x(7); y(13) = -2*x(5)+3*x(6)-x(7);y(14) = -x(5)+x(7); y(15) = 12*x(5)-6*x(6)-12*x(7)+6*x(8);y = y/6;

## 8 point linear convolution

${D}_{8}={D}_{2}\otimes {D}_{2}\otimes {D}_{2}$
${F}_{8}=\left[\begin{array}{ccccccccccccccccccccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& 1& 0& -1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 1& -1& 1& 1& -1& -1& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& -1& 0& 1& -1& 0& 0& 1& 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 1& -1& -1& -1& 1& 0& 0& 0& 1& 1& -1& 0& 0& 0& 0& 0& 0& -1& -1& 1& 0& 0& 0& 0& 0& 0\\ 1& -1& 0& 1& 1& 0& -1& 0& 0& 1& -1& 0& 1& 0& 0& -1& 0& 0& -1& 1& 0& -1& 0& 0& 1& 0& 0\\ -1& -1& 1& -1& -1& 1& 1& 1& -1& -1& -1& 1& -1& -1& 1& 1& 1& -1& 1& 1& -1& 1& 1& -1& -1& -1& 1\\ 0& 1& 0& -1& 1& 0& 0& -1& 0& 1& 1& 0& -1& 1& 0& 0& -1& 0& 0& -1& 0& 1& -1& 0& 0& 1& 0\\ 0& 0& 0& 1& 1& -1& 0& 0& 0& -1& -1& 1& 1& 1& -1& 0& 0& 0& 0& 0& 0& -1& -1& 1& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 0& 0& -1& 1& 0& -1& -1& 0& 1& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& -1& 1& 1& -1& -1& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 0& 1& -1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$
function y = F8t(x) y = zeros(27,1);y(1) = x(1)-x(2)-x(3)+x(4)-x(5)+x(6)+x(7)-x(8); y(2) = -x(2)+x(3)+x(4)-x(5)+x(6)-x(7)-x(8)+x(9);y(3) = x(2)-x(4)-x(6)+x(8); y(4) = -x(3)+x(4)+x(5)-x(6)+x(7)-x(8)-x(9)+x(10);y(5) = x(4)-x(5)-x(6)+x(7)-x(8)+x(9)+x(10)-x(11); y(6) = -x(4)+x(6)+x(8)-x(10);y(7) = x(3)-x(4)-x(7)+x(8); y(8) = -x(4)+x(5)+x(8)-x(9);y(9) = x(4)-x(8); y(10) = -x(5)+x(6)+x(7)-x(8)+x(9)-x(10)-x(11)+x(12);y(11) = x(6)-x(7)-x(8)+x(9)-x(10)+x(11)+x(12)-x(13); y(12) = -x(6)+x(8)+x(10)-x(12);y(13) = x(7)-x(8)-x(9)+x(10)-x(11)+x(12)+x(13)-x(14); y(14) = -x(8)+x(9)+x(10)-x(11)+x(12)-x(13)-x(14)+x(15);y(15) = x(8)-x(10)-x(12)+x(14); y(16) = -x(7)+x(8)+x(11)-x(12);y(17) = x(8)-x(9)-x(12)+x(13); y(18) = -x(8)+x(12);y(19) = x(5)-x(6)-x(7)+x(8); y(20) = -x(6)+x(7)+x(8)-x(9);y(21) = x(6)-x(8); y(22) = -x(7)+x(8)+x(9)-x(10);y(23) = x(8)-x(9)-x(10)+x(11); y(24) = -x(8)+x(10);y(25) = x(7)-x(8); y(26) = -x(8)+x(9);y(27) = x(8);

## 18 point linear convolution

${D}_{8}={D}_{2}\otimes {D}_{3}\otimes {D}_{3}$

${F}_{18}$ and the program F18t are too big to print.

importance of teaching and learning?
we need to know the human mind and behavior and how it works. this can make us successful in our jobs because we know what to do to please the people or to catch them lying etc.
is a Greek word for soul
carolline
scientific study of the human behavior
carolline
Humans are extremely inconsistent at catching out liars. Human intuition and lie detection methods are prone to all kinds of biases. Always follow the evidence and don't let your emotions influence your rationality.
Leon
hello everyone, i was just wondering i have to pick a theoretical perspective to describe a mental disorder, i have picked Anti social personality disorder and was going to use the biomedical model to explain it. does anyone here regard aspd as a neurobiological disorder? kind regards.
is anti social personality disorder a neurobiological illness?
niamh
rationalisation, displacement and reaction formation
only a perspective of what we have pursued till now as an experience
Priyank
what is the meaning of pysche?
The meaning of psyche is Mind
Jobin
Friends can anyone please tell me, why the study of psyche is so important?
Jobin
Well in Greek, it means "soul, spirit, the mind, or the invisible animating entity which occupies the physical body".
Friends can anyone please tell me, why the study of psyche is so important?
Jobin
Friends please tell me how to counsel a student who does not want to change their lives but are comfortable in the same life situation?
Jobin
I think it's such an important study because it's one if those rare things that can't be understood in the conventional manner. The mind (not the brain, but the soul and spirit), cannot be studied physically, or through the use of sight, touch, smell, taste, or hearing.
Much focus and attention would have to be put into it to understand it, and it can't be fully understood because everyone's different and subjected to different circumstances, and we live an ever-changing world. It's important because it helps to develop spiritual, mental, and emotional intelligence
is humanities a good stream? is there scope for a subject like psychology in the future. is it a good choice?
is humanities a good subject?
amrita
Sure it is.
Jobin
you gonna opt for humanities as long as you don't want to pursue psychology in BSc honours , you'll be able to pursue BA psychology only. So if you want to pursue BSc then go for Science stream only.
Priyank
Morning everyone. How can you help a person who can't stop musterbating?
find a different activity to divert the need to masturbate
vincent
masturbation is a need u cant stop it
Abhilash
Try positive punishment then transition to negative punishment
Atone
It probably has something to do with the environment one grew up in and their genetics that caused their puberty to have this psychological effect on them.
Atone
trying to keep yourself busy with stuffs you love doing the most.like reading or listening to music or sports
Naomi
addiction and attachment issues do go hand in hand
vincent
stop watching pornography if you do; it leads you to fantasizing via masturbation. If you are a Christian, try indulging is spirituality (music and prayrer) before your bed time.
Matovu
don't stop watching anything but instead pay attention to yourself and be conscious enough of what happens inside your brain or how the thought process goes that leads you to masturbating
Priyank
Hello everyone have a nice day
Mary
Hello
Georgescu
internal for example a times when when I watched a pornography movie I end up thinking about it a lot so what I used to stop it is to create something in my mind that can ever exist in real life. I can create a story for about a week in my mind just to forget about it and it actually helps me
Mary
@mary charm story construction is helpful but then it open up on doorways for a new problem or addiction per say since it gives boost to the overexaggeration of reality which is not right, instead try to jot down what do you feel, be it digitised or normal writing. painting it also helps.
Priyank
she**
vincent
Marriage is the only solution guys.You can't do it if you get 'it'
Philip
I agree philip Nyenze. in addition to self control
Diamly
Hello y'all! I'd like to start by saying, Marriage isn't a solution for everyone. We all have differences across the board, mental, spiritual and physical. Personally marriage has fixed it for me but not for the reason you'd think. In fact the complete opposite. I found myself with a partner who's..
Samwell
... spiritual views and mental issues (depression) caused my partner to be very distant physically and it caused so much heartbreak and pain within myself and in turn made my mental health to whirlwind out of control. it took 4 years to get through it but I concentrated on my love for my partner...
Samwell
.... outside of the bedroom because I wanted them completely. through trials and error I think I finally got myself in a good place. I don't need to masturbate because even though the hormones are rushing within me and I want to tackle my partner down, my soulmate shines through brighter than that..
Samwell
... gorgeous body.
Samwell
You have to stop thinking about the filling that u can get from musturbation and get a partner if only you are 18 years and above.
Tanam
Unless it's a chronic problem like they are addicted to it and do it all the time, then let them be. Masterbation is a natural thing there's nothing wrong with it and it has many health benefits. As long as it isn't a chronic case they're be fine.
UnhealthyCopingMecha
how natural is it?...this i just a formed habit. it takes 3weeks(21)days to form an habit...if youve done it more than 21days,admit its your habit...same it can take 21days to end a formed habit.if you've wanted to stop this habit,contact me personally i'll guide you through
Philip
hi how to join conversation?
i don't know
Azeneth
How can you say that your bestfriend is true to you?
why is my mom so mean?
Hello Krista! Idk what made you think that ur mom is mean, but trust me, that's just not true. She really loves u and cares for u, but it's just that her way of caring is different than others. Ik it sounds mean, but deep inside her heart, she loves u a lot, just like all moms love their children.
Drazy
perhaps, but if you could elaborate on what exactly she does that is demeaning, because sometimes some lessons that are best learned are through a tough and disciplined manner, most likely she really loves you and wants you to learn the lesson once, so you won't have to find it out the hard way
Eldar
I have been assigned to an emotional disturbed student. She seemed fit first but you notice she interrupts or teases other students.would you work with this student an class to improve the situation?
Saleem
Hello ! I think that she's kind of person that shows her bad side before the good. In other words "if you gonna love me ,first you should accept my bad side" These kind of person have the purest love ,once you get through their bad side , it demand some patience and emotional control but , It worth.
aziz
Farhat
We can't know for sure. Whatever decisions your mom makes at the time are her best choices to fulfill her needs, in a way, so her meanness probably in relation to her own prioritization of needs, upbringing, temperament.
frootloop
Also, it's likely part of who you are, as well. By growing up with her, you're likely more vulnerable to her attacks, and they trigger you easily. These are both things that could potentially change, but not without a lot of work
frootloop
Hope this helped! I'm no therapist or psychologist, so take what I say with a grain of salt
frootloop
Rge only one onbthis planet have the ability to be with u whatever happens
Mohammed
What is the relationship between physical development and motor development
what is memory
the faculty by which the mind stores and remembers information. "I've a great memory for faces" something remembered from the past; a recollection.
Amber
what is the difference between Counsellor and Psychologist?
One talks you through your problem helping you solve them on your own (and vent out). The other writes a prescription for you and diagnoses the medical issue lying beneath (eg depression, anxiety)
Vasu
A psychologist has been trained to look at behavioural aspect of mental problems rather than look at biomechanical perspectives. He is more likely to ask his patient about his past and his present behavior, his feelings and his problems to arrive at the root cause.......Read more
Miraj
psychologists have a better understanding and awareness of behavioural reasons of mental problems than others.
Miraj
Instead of the behavioural approach adopted by psychologists, a counsellor tries to encourage the patient to direct the treatment session. He creates an environment where the patient is able to clearly see through his own problems and the reasons underlying these problems........Read more
Miraj
Thus, without relying anybody else, the patient is able to overcome his problems. In a broader sense, both psychologists and counselors both are specialists who try to solve mental problems of people through adopting different approaches.
Miraj
@Kadir shuaib, I hope you understand l
Miraj
Thanks to bits@Miraj Gillani But I've told that Counselling is applied psychology and counseling only effective when theory is applicable.
Miraj
A memory is a faculty which permits human to situate his actions and his state in the times and space.
Sinaly
@vasu a psychologist can't give you a drug as a prescription, that's the job of a psychiatrist
Argoncor
hi I have dealing with a lady who is severely depressed .. she gets suicidal attacks even at the thought of her mother in law . I am working on realeasing her stress .. any other suggestions?
Sriya
hi sriya.. try engazing her in some activity...like cooking,or any u like...make ppl appriciate her effort...plan such that one or other alawys keep appreciating her efforts...that will bring value to her own existence...but it will definetly take some time so u must have patience. tke cre champion.
ARJUN
ARJUN great idea.
Cindy
Hey Sriya! U can take her 2 sum place where she can inhale some fresh air&feel good.&when she gets suicidal attacks 'bout her mom-in-law,u can try calming her down by saying that her mom-in-law wud also appreciate her works&she wud b sad if she knew that her daughter in law is unhappy.Gud luck,girl!
Drazy
Idk about Drazy's comment, something that worked for me was to defy irrational thought patterns and watch out for them to defuse your own suicidal thoughts. Maybe find a focus to life completely independent of the MIL
frootloop
Sriya she needs to do something that makes her happy. I used to spend time with a depressed lady I ask her if anything helped her. when she had anxiety attacks she said music. Whenever I could feel her starting to have an attack I'd hand her head phones along with her favorite music.
Cindy
I asked her she said her husband she has anchored her entire happiness with her husband. she can't even move without him
Sriya
I apologize Sriya I just joined the conversation, I don't know about her husband ; therefore I am not able to give suggestions.
Cindy
@sriya you need to give her an insight about her own individuality so she could seperate herself mentally from her husband and see for herself what she is and where she is! you need to make her understand this that her husband and she herself are two both seperate individuals, may be meditation....
Priyank
...can give her a new tangent to think and pursue reality as it, I use meditation alot on my clients to give them a fresh perception of reality...
Priyank
Hi
ABUBAKAR
Excellent point Priyank
Cindy
thank you everyone I think I know what to do ..
Sriya
Maybe there's a dependency on the husband, and has a bearing on the lady's relationship with her MIL? Like, her MIL is threatening to her
frootloop
Exactly frootloop. Similar to changing a negative thought to a positive thought.
Cindy
What is the deference between Counseling and Psychology?
ABUBAKAR
I think Cindy and Arjun's answers are great. I'm no specialist, but having something private that she's not at war with and can appreciate (activity, hobby, music, etc) could definitely be helpful
frootloop
hey mates..we r part of d universe and essence of universe is continuous activity..even stagnant water fouls..depresn over d top take self value...hence engazin in some purpose oriented work(activity )along with fellows appriciation(value) is two pronged strategy. but it all takes time.u need to b
ARJUN
smart n patience..as depressions dosnt have a face..!!! love u all
ARJUN
always make her to feel important
Mary
@First know the root cause of the issues she's been traumatized with..talk to her in her own language without any Prejudice let her confide in you then you can get the solutions...as the primary reason for depression is being lonely around your own people and house..an open ear to listen to ur silly
harvey
... stories piled up urges, desires a warm hug that human touch we loose over time... saying is easy but living in it is insane..we all need someone to dump our day with..make her realize the consequences if she does hurt herself or others..as u cannot be with them 24/7 so try instilling the...
harvey
...SELF ESTEEM AND BELIEF which she lost over time to her own people..ask her to do some activities which will earn money online and are fun while doing so...keep herself busy not sit and overthink on things!
harvey
Sriya. Give her a story.where you passed the same. If you dont have your own story,then create one. As you give the fake story you tend get more into her shooes.,such that u feel as if u had once being stressed and u overcame it.Make the story bitter that she even symphathizes with u.
Philip
....this way,your patient trusts you more. They believe if they do whatever u tell them they Must overcome their situations....i once told a patient,"if you feel urine and hold on for atleast 2hrs,u will forget all that"...he believed tht so he always drink alot of water to create the urge to urinat
Philip
...he started focusing on how to create urine in his bladder,and he forgot all tht was disturbing.....i nicknamed him "fish."...and am working a theory on the same case. i will post "The Fish theory" once am done with it...though i might change its title,....
Philip
importance of social psychology in the domain of education
Buddhist perspective on motivation..
what is empirical observations
Empirical observation is a type of observation which is based on experiment not on theories. And this observation is not supported by scientific poof.
Miraj
why we do dreams? please tell me
Saleem
the thoughts which we stop thinking suddenly during day. they try to get out of our unconscious mind and they make a story which we term as dreams.
vritee
thank you
Saleem
family members very good morning beautiful lady and family and my family and one that works
holyghostChristine
pls what's sociology of education
solang
The sociology of education is the study of how public institutions and individual experiences affect education and its outcomes. It is mostly concerned with the public schooling systems of modern industrial societies, including the expansion of higher, further, adult, and continuing education.
Amber
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