# 4.1 Proof

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Proof of Shannon's sampling theorem

In order to recover the signal $x(t)$ from it's samples exactly, it is necessary to sample $x(t)$ at a rate greater than twice it's highest frequency component.

## Introduction

As mentioned earlier , sampling is the necessary fundament when we want to apply digital signalprocessing on analog signals.

Here we present the proof of the sampling theorem. The proof is divided in two. First we find an expression for the spectrum of the signal resulting fromsampling the original signal $x(t)$ . Next we show that the signal $x(t)$ can be recovered from the samples. Often it is easier using the frequency domain when carrying out a proof,and this is also the case here.

## Key points in the proof

• We find an equation for the spectrum of the sampled signal
• We find a simple method to reconstruct the original signal
• The sampled signal has a periodic spectrum...
• ...and the period is $2\times \pi {F}_{s}$

## Proof part 1 - spectral considerations

By sampling $x(t)$ every ${T}_{s}$ second we obtain ${x}_{s}(n)$ . The inverse fourier transform of this time discrete signal is

${x}_{s}(n)=\frac{1}{2\pi }\int_{-\pi }^{\pi } {X}_{s}(e^{i\omega })e^{i\omega n}\,d \omega$
For convenience we express the equation in terms of the real angular frequency $\Omega$ using $\omega =\Omega {T}_{s}$ .We then obtain
${x}_{s}(n)=\frac{{T}_{s}}{2\pi }\int_{\frac{-\pi }{{T}_{s}}}^{\frac{\pi }{{T}_{s}}} {X}_{s}(e^{i\Omega {T}_{s}})e^{i\Omega {T}_{s}n}\,d \Omega$
The inverse fourier transform of a continuous signal is
$x(t)=\frac{1}{2\pi }\int_{()} \,d \Omega$ X Ω Ω t
From this equation we find an expression for $x(n{T}_{s})()$
$x(n{T}_{s})=\frac{1}{2\pi }\int_{()} \,d \Omega$ X Ω Ω n T s
To account for the difference in region of integration we split the integration in into subintervals of length $\frac{2\pi }{{T}_{s}}$ and then take the sum over the resulting integrals to obtain the complete area.
$x(n{T}_{s})=\frac{1}{2\pi }\sum_{k=()}$ Ω 2 k 1 T s 2 k 1 T s X Ω Ω n T s
Then we change the integration variable, setting $\Omega =\eta +\frac{2\times \pi k}{{T}_{s}}$
$x(n{T}_{s})=\frac{1}{2\pi }\sum_{k=-\infty }^{\infty } \int_{\frac{-\pi }{{T}_{s}}}^{\frac{\pi }{{T}_{s}}} X(i(\eta +\frac{2\times \pi k}{{T}_{s}})())e^{i(\eta +\frac{2\times \pi k}{{T}_{s}})()n{T}_{s}}\,d \eta$
We obtain the final form by observing that $e^{i\times 2\times \pi kn}=1$ , reinserting $\eta =\Omega$ and multiplying by $\frac{{T}_{s}}{{T}_{s}}$
$x(n{T}_{s})=\frac{{T}_{s}}{2\pi }\int_{\frac{-\pi }{{T}_{s}}}^{\frac{\pi }{{T}_{s}}} \sum_{k=-\infty }^{\infty } \frac{1}{{T}_{s}}Xi(\Omega +\frac{2\times \pi k}{{T}_{s}})()e^{i\Omega n{T}_{s}}\,d \Omega$
To make ${x}_{s}(n)=x(n{T}_{s})$ for all values of $n$ , the integrands in and have to agreee, that is
${X}_{s}(e^{i\Omega {T}_{s}})=\frac{1}{{T}_{s}}\sum_{k=-\infty }^{\infty } Xi(\Omega +\frac{2\pi k}{{T}_{s}})()$
This is a central result. We see that the digital spectrum consists of a sum of shifted versions of the original, analog spectrum. Observe the periodicity!

We can also express this relation in terms of the digital angular frequency $\omega =\Omega {T}_{s}$

${X}_{s}(e^{i\omega })=\frac{1}{{T}_{s}}\sum_{k=-\infty }^{\infty } Xi\frac{\omega +2\times \pi k}{{T}_{s}}$
This concludes the first part of the proof. Now we want to find a reconstruction formula, so that we can recover $x(t)$ from ${x}_{s}(n)$ .

## Proof part ii - signal reconstruction

For a bandlimited signal the inverse fourier transform is

$x(t)=\frac{1}{2\pi }\int_{\frac{-\pi }{{T}_{s}}}^{\frac{\pi }{{T}_{s}}} X(i\Omega )e^{i\Omega t}\,d \Omega$
In the interval we are integrating we have: ${X}_{s}(e^{i\Omega {T}_{s}})=\frac{X(i\Omega )}{{T}_{s}}$ . Substituting this relation into we get
$x(t)=\frac{{T}_{s}}{2\pi }\int_{\frac{-\pi }{{T}_{s}}}^{\frac{\pi }{{T}_{s}}} {X}_{s}(e^{i\Omega {T}_{s}})e^{i\Omega t}\,d \Omega$
Using the DTFT relation for ${X}_{s}(e^{i\Omega {T}_{s}})$ we have
$x(t)=\frac{{T}_{s}}{2\pi }\int_{\frac{-\pi }{{T}_{s}}}^{\frac{\pi }{{T}_{s}}} \sum_{n=()} \,d \Omega$ x s n Ω n T s Ω t
Interchanging integration and summation (under the assumption of convergence) leads to
$x(t)=\frac{{T}_{s}}{2\pi }\sum_{n=()}$ x s n Ω T s T s Ω t n T s
Finally we perform the integration and arrive at the important reconstruction formula
$x(t)=\sum_{n=()}$ x s n T s t n T s T s t n T s
(Thanks to R.Loos for pointing out an error in the proof.)

## Summary

${X}_{s}(e^{i\Omega {T}_{s}})=\frac{1}{{T}_{s}}\sum_{k=-\infty }^{\infty } Xi(\Omega +\frac{2\pi k}{{T}_{s}})()$

$x(t)=\sum_{n=()}$ x s n T s t n T s T s t n T s

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how many disorders are there
interesting question
Mahmoud
it is😅
Nelly
I'm guessing it's in the tens of thousands, maybe hundreds if you include counter interactions and stuff like it. although I'm sure it's almost impossible to know for sure, unless you're very rich and connected to the right people. but as I said: guessing.
Beenie
There are more than 200 classified forms of mental illness. Some of the more common disorders are: clinical depression, bipolar disorder, dementia, schizophrenia and anxiety disorders. Symptoms may include changes in mood, personality, personal habits and/or social withdrawal. that is what I think
Mahmoud
too difficult to number. diagnosing a disorder is just checking off boxes on a compilation of symptoms that might match any particular condition on the DSM
what is psychology of the guest
I don't understand the question can you elaborate?
Edgar
the study of philosophy gives to the sociologist
why women are viewed as far more emotional than men?
because they actually are ,I guess.
Collins
May be they are biologically milder than man. It does not mean they are not equal with man. Man can also be emotional and can are oppressed with traditional norms. For example, man are not to be cry, in actual man are also emotional being and they cannot have the right to show their sentiment.
shine
sollungal bro sollungal...
Balaji
because of she has hormonal fluctuations than men..
bavi
usually women are more emotional than men and they are multi talented people
lavanya
Both men and women are capable of expressing emotions. But women has the highest percentage of doing that. Because our society is conditioned by nature in such a way that men are expected to suppress their emotions and motivate them through their acts or thoughts, which has it's side effects of....
Santos
denial of any emotion which they feel that useless at that point. Women in the other hand were encouraged/not controlled to suppress their emotions and let them out what they feel about it. I feel that's the wonderful superpower of the women.
Santos
y
Mahmoud
Men's emotion comes mostly with memories triggered by senses. That means, they thoughts have the power to decide whether to let go of emotions or not.
Santos
Q
Mark
because women tend to be more agreeable than men.
Edgar
women at a young age are conditioned to be more in tune with emotion than man should be less
David
The question is "Why women Viewed are as far more emotional than men ?" it's not a question whether women are more emotional than men. This is more an issue about the point of view from the observer, his/her assumption what emotional behavior is or what emotional behavior is.
Mehmet
The anwers are answers more to the question " Are women more emotional than men?"
Mehmet
hi are you on what'sap
no not really a fan of social laziness
Robert
huh
Parmizan
is there any way for a Btech E.C.E. graduate to take on MSC. PSYCHOLOGY
devesh
ya you can just by cracking entrance , if in India
Mansi
Any stream in UG can go for M.Sc (Psychology)
classification of traits and how they are measured
what was Freud's first name?
Sigmund
Dentriodite
sigmund
Margie
Sigismund Schlomo Freud
Sigmund Freud
Talal
sigmund
Ankita
Leon Vygotsky and Sergei Rubenstein please tell me contributions of these personalities in in 4 lines
Uzma
sigmund Freud
Harpreet
Lev Vygotsky was the founder of socio-cultural theory
Jo
So psychology was based off of the Greek gods I that right or no?
psychology is the study. anything -ology is the study of a certain field. Psyche is a mortal woman who becomes divine in Greek mythology. The etymology of the word "psyche" in Greek means "spirit" or "soul"
seriously there is no scope of psychology all over the world.i hear those how have psychology degrees they have no careers
psychology degree is the waste of time qnd waste of money.is this true?...
at least you know people's minds
Mahmoud
hummmmm
Blaq
well, i like this subject for helping myself and others as well, not for the scope or anything. and i think we shouldn't liberate everything on the basis of outcomes etc
Natasha
Ye
Isaiah
very useful skill
Isaiah
It is not at all true that people who have psychology degree have no career scope
Aaprajita
If u complete your M.Phil then you can work in hospitals as a clinical psychologist.You can also work in school, colleges and IT companies as a counsellor or clinical psychologist.
Aaprajita
yes
aravinth
people are so stressful and tense, and they need a psychologist
aravinth
yo fr tbh lol
Isaiah
if some one have only master degree in psychology what will he/she do...where he/she utilizes degree
Not if you have set goals and a plan. Look into what careers are available with your degree where you live. Also it helps to have a specialty along with your psychology degree. For example a Bachelor's degree in Psychology of Science in Addictions allows for you to become an addictions counselor.
Jo
Psychology is really a valuable degree. I would like to become an "EDUCATIONAL PSYCHOLOGIST "
there is no proper society of Psychology. so it doesn't get a good exposure in our country. we along with whole student need to discuss and approach to government
Shekhar
Psychology is not only a diverse field but also one that is expected to grow tremendously in the years to come. As a subject dealing with the study of the human psyche or mind, psychology finds applications across all avenues of life, be it family, work, relationships, sports, corporate spaces
Clinical Psychology As is suggestive, this branch of psychology is associated to the understanding, diagnosis and treatment of psychological disorders in humans. Clinical psychologists help people facing difficulties in their life to get through it using different treatment methods and therapies.
These professionals work with hospitals, NGOs and even drug rehabilitation centers. As people become increasingly aware of different psychological disorders with time, the need for experts in clinical psychology is only expected to grow.
Counseling Psychology Counseling psychology is a branch of psychology that helps solve people’s personal and interpersonal issues. These issues can be problematic but are different from serious mental health issues. Counseling psychologists or counselors help people deal with such issues when they
fail to do it on their own.Career counseling, guidance counseling, marital counseling and rehabilitation counseling are among the many applications of counseling psychology. These professionals either manage their own counseling set-ups or work with therapycenters, career centers, NGOs as well as
schools and universities. Industrial or Organizational Psychology Organizational psychologists are professionals who apply the principles of psychology in organizations or workplaces. They analyze the issues of the workplace at individual as well as organizational level, and work towards resolving
them to enhance the efficiency of the employees. Those specializing in organizational psychology can choose to work as – • Human resource development specialists • HR managers • Organizational consultants With increasing pressure and stress levels at work places, the need for emotional understandi
understanding and gauging social intelligence has become stronger. Child Psychology (Development psychology) Development psychology is a branch of psychology that is dedicated to studying the psychological development of human beings over the course of their lifetime. Child psychology is one of
the more popular variants of development psychology, and deals with emotional, cognitive, social and psychomotor development in infants and children. Child psychologists work with schools, child therapy centers and also NGOs. They also play big roles in special education centers for children.
Sports Psychology This branch of psychology deals with the study of elements that influence an athlete’s performance. These elements or factors could be emotional, cognitive, psychological or motivational. Sports psychologists can go on to work with sports coaching centers, leagues, academies or
sports teams. Sports psychologists aid sportspersons to stay in their best mental forms, thus improving their performances on the ground. With the frenzy regarding sports in India, sports psychology is a promising career to pursue.
Forensic Psychology Forensic psychology offers a unique opportunity to apply psychology to the benefit of legal organizations, especially pertaining to specific contents or witness testimony. Professionals in this field utilize their skills to look deep into the psyche of criminals and figure out
the intent behind the crime. This helps define the quantum and nature of sentence to be rendered. With the onset of new legal era, forensic psychology is expected to influence policy making.
SO,THESE ARE THE CAREER SCOPE IN THE FIELD OF PSYCHOLOGY👥
psychology and the jobs in psychology are the ever growing fields. there are number of jobs available. but, yes, one needs the right qualification, such as, MPhil, PhD, NET, etc.
Paul
No. Psychology has a great scope all over world
Talal
Because of 90% people has stress or anxiety or any other mental problem. So they have a need of Psychologist.
Talal
when a student simultaneously wants to attend a party and to study for the exam next morning is an example of ?
it is an example of conflict
Atta
which one?
Natasha
approach-avoidance approach-approach double - approach avoidance avoidance - avoidance
Natasha
when a person have two choices but he cant decision about one of those choices
Atta
and this is approach avoidence conflict
Atta
thank u so much
Natasha
incase if you are motivated to give the exam and equally motivated to go for the party, it will be approach approach conflict.
Ankita
Good day! I had a friend suddenly become unusual. Before he do not talk a lot but now he talk too much and most of the time inappropriately. He treat himself like he knows a lot of things. He thinks that he is more genius than ancient scientist. He saw something unusual and creepy to imagine.
Den
He acknowledge himself that he could see and use magic. He used to have hand gesture as if trying to show us magic. He speaks that he only understand oftentimes unlike before. He always told us that no one could understand him and that no one could surpass his intelligent.
Den
He is so active now, oftentimes misbehave. He wants to do something that makes him feel busy than relaxing until he gets tired. He is not a boy we know before. And there are lot more. We are all do confused and some suggest that it matters on psychological disorder.
Den
Hoping this message would be given concern. can you me figure it out what it is?
Den
you can consult a psychiatrist for this, they will look into his symptoms deeply. a detailed casehistory regarding his symptoms are required.
Ankita
question. where do you draw the line between hoarding an just not wanting to be wasteful?
Probably the anxiety and guilt complex that arises when you think of the item(s) in question. ;¬)
I'm not a hoarder I did have some collections,hot wheels that we're still in packages some from 1987 cards most sports an a mo vie collection around 3500 of them my house burned down 2 years ago along with things I collected since I was 8 years old the things I lost that bothered me was pictures my
Robert
diploma sentimental value the rest is just materialistic crap that can be replaced
Robert
s.s.venkateshwar Andaman portblair India
Is men and women see colour and black and white dreams? s.s.venkateshwar
hoarding and wanting is the same, especially we do not need it,back to use ur things wisely
Agnes
I believe (but don't quote me on it) that a lot of people dream or did dream in black & white as a result (or due to the influence of) growing up with a black & white TV. Personally, having little memory of black & white televisions; I have vivid dreams in full colour. Hope this helps @anonymoususer
Leigh
what is anxiety to you? what outward signs of anxiety can you see? that for a certainly it's anxiety
Robert
hi can I you help me please
NAVEED
hello what do you need
Parmizan
hello
Parmizan
If you have an over abundance of what you and your family need or use and your basically collecting it just to watch it grow. its hoarding in my opinion. if I have not touched it or thought about it or needed it 6 months then it's in the garbage or it's given away.
Natasha
Psychology is really a valuable degree. I would like to become an "EDUCATIONAL PSYCHOLOGIST "
KORLAPATI
is it possible!?
KORLAPATI
is there any scope for psychology?
KORLAPATI
there is scope of psychology
Oyedokun
Psychology has a great scope all over world
psychology and the jobs in psychology are the ever growing fields. there are number of jobs available. but, yes, one needs the right qualification, such as, MPhil, PhD, NET, etc.
Always thinking of someone do they think of me?
Maybe you just miss that someone.. 😁
Mary
they could very well just be very heavily imprinted on your mind but there's a lot we can't say we understand yet. try to find out.
Elliot
could someone who is empathic develop PTSD from experiencing other people's memories and feelings.
Yvette
Yvette only if they don't let go but something as strong as ptsd can only happen if it relates to your own feelings thus affecting in a amplified way
Dhruv
Yvette
Are you familiar with pet's behavior suchlike dog's. They pick up the Aura energy of those who are the most important or most influential people in their lives. Sometimes we as spiritual beings do the same thing.
Leonard
what are the causal factors of this disorder
Heart break 💔
Abwooli
yes
Shah
.
allen
s
allen
mentally or really?
danur
mental breakdowns,bc of internal and external happenings,external traumatas
Agnes
and/or
Agnes
what are symtom s of a hoarder
what are the symtoms of a hoarder
Anita
please bare with me. first time in. what are the symtoms of a hoarder
Anita
Inability to throw away possessions and has severe anxiety when attempting to discard items. Distressed, overwhelmed and/or embarrassed by possessions. Indecisive about where to keep possessions. Having a suspicion about other people touching items.
Stephanie
There is also an obsessive quality to thoughts and actions. Such as having a fear of running out of an item that might be needed in the future as well as checking the trash for accidentally discarded items.
Stephanie
Why do you think it is that even after assertions/fact proving research that these findings take so long to work their way into any main kind of diagnostic assessment or tool? I have terrible health as a result of myriad psychophysiological factors & physiological factors.
Why do you think it is that even after assertions/fact proving research that these findings take so long to work their way into any main kind of diagnostic assessment or tool?
Leigh
Medicine practices itself work on guesses since their is so much data for every symptom and their multiple combinations so doctors assume the most probable decisions and treat like a hit and trial
Dhruv
And brain being one of the strangest and mysterious things of all makes it really tough and time consuming
Dhruv
Unfortunately, I think it's money, money, money! Consider, your particular situation. More money can be made with a broad diagnosis because..."we think you're...so take two of these daily." if that doesn't work then delve back into that BROAD diagnosis and try again.
Clifford
Then, if and when the side effects of the treatment begins to affect body in a negative way. Well, let's just say these days there isxa pill for everything! Not saying all physicians are only after profit but in our capitalistic society....
Clifford
Yeah mental health especially is a minefield. I've spent most of my life trying to find the right combination of treatments etc. It just seems that there there is research out there now attesting to x y or z for example & the gp is still using a to z in his diagnosis if that makes sense.
Leigh
It leaves you without much hope for getting better when the variables are too much for even your GP to fully comprehend. What chance do we then have. Mental health services are even more lacking.
Leigh
Facts...dont equal profit in America.
Clifford
Clifford I do agree with what you're saying to a certain extent. My GP had me on medication for pain that it later transpired was more than 4 x the maximum dose for pain. I have had to spend weeks reducing this medication to avoid triggering my epilepsy & withdrawals. I just wanted help.
Leigh
You almost have to fight for your health now.
Leigh
I'm in the UK to avoid confusion
Leigh
To say that they're all their just for money rather than the health of the patient would be bad case of generalization and I live in India and our country has a lot of these money hungry situations and many extremely greedy ones too.
Dhruv
There *
Dhruv
Dhruv...my apologies for the assumption. I can only fairly make that assumption based on what I have experienced. Im actually impressed wit the UK's healthcare system! I think America should take notes.
Clifford
@Leigh,...well that sucks! But unfortunately that is a reality for so many people these days. I also believe the best a doc can give me is an educated guess. No man or woman designed the human body, therefore, how can we know exactly what is wrong and how to fix it.
Clifford
Yeah of course no need to apologize we're all speaking from experience only am just saying facts oftentimes lie outside our experience
Dhruv
I personally believe meds are not always the best course...considering human being have thrived long before pills!
Clifford
I completely agree Dhruv! Great point! Do you think Leigh would have the same problems in India? What is healthcare like there?
Clifford
I also believe that doctors are too quick to assume a medication is the only solution, however in my personal experience it is a combination of medication, alternative therapies & counselling / psychiatric help that really helps.
Leigh
Similarly, we are so complex that I agree there is not one size fits all solution, however I think there is a general lack of awareness by GP's. How can one be referred to the right service if the GP isn't fully aware of certain *flags that warrant further investigation.
Leigh
Personally, I have been waiting to see the pain clinic for example, for a year and a half now.
Leigh
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