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Proof of the sampling theorem

The above discussion has already shown the sampling theorem in an informal and intuitive way that could easily be refined into a formal proof. However, the original proof of the sampling theorem, which will be given here, provides the interesting observation that the samples of a signal with period T s provide Fourier series coefficients for the original signal spectrum on ( - π / T s , π / T s ) .

Let x be a ( - π / T s , π / T s ) bandlimited signal and x s be its samples with sampling period T s . We can represent x in terms of its spectrum X using the inverse continuous time Fourier transfrom and the fact that x is bandlimited. The result is

x ( t ) = 1 2 π - π / T s π / T s X ( ω ) e j ω t d ω

This representation of x may then be sampled with sampling period T s to produce

x s ( n ) = x s ( n T s ) = 1 2 π - π / T s π / T s X ( ω ) e j ω n T s d ω

Noticing that this indicates that x s ( n ) is the n th continuous time Fourier series coefficient for X ( ω ) on the interval ( - π / T s , π / T s ) , it is shown that the samples determine the original spectrum X ( ω ) and, by extension, the original signal itself.

Perfect reconstruction

Another way to show the sampling theorem is to derive the reconstruction formula that gives the original signal x ˜ = x from its samples x s with sampling period T s , provided x is bandlimited to ( - π / T s , π / T s ) . This is done in the module on perfect reconstruction. However, the result, known as the Whittaker-Shannon reconstruction formula, will be stated here. If the requisite conditions hold, then the perfect reconstruction is given by

x ( t ) = n = - x s ( n ) sinc ( t / T s - n )

where the sinc function is defined as

sinc ( t ) = sin ( π t ) π t .

From this, it is clear that the set

sinc ( t / T s - n ) | n Z

forms an orthogonal basis for the set of ( - π / T s , π / T s ) bandlimited signals, where the coefficients of a ( - π / T s , π / T s ) signal in this basis are its samples with sampling period T s .

Practical implications

Discrete time processing of continuous time signals

The Nyquist-Shannon Sampling Theorem and the Whittaker-Shannon Reconstruction formula enable discrete time processing of continuous time signals. Because any linear time invariant filter performs a multiplication in the frequency domain, the result of applying a linear time invariant filter to a bandlimited signal is an output signal with the same bandlimit. Since sampling a bandlimited continuous time signal above the Nyquist rate produces a discrete time signal with a spectrum of the same form as the original spectrum, a discrete time filter could modify the samples spectrum and perfectly reconstruct the output to produce the same result as a continuous time filter. This allows the use of digital computing power and flexibility to be leveraged in continuous time signal processing as well. This is more thouroughly described in the final module of this chapter.


The properties of human physiology and psychology often inform design choices in technologies meant for interactin with people. For instance, digital devices dealing with sound use sampling rates related to the frequency range of human vocalizations and the frequency range of human auditory sensativity. Because most of the sounds in human speech concentrate most of their signal energy between 5 Hz and 4 kHz, most telephone systems discard frequencies above 4 kHz and sample at a rate of 8 kHz. Discarding the frequencies greater than or equal to 4 kHz through use of an anti-aliasing filter is important to avoid aliasing, which would negatively impact the quality of the output sound as is described in a later module. Similarly, human hearing is sensitive to frequencies between 20 Hz and 20 kHz. Therefore, sampling rates for general audio waveforms placed on CDs were chosen to be greater than 40 kHz, and all frequency content greater than or equal to some level is discarded. The particular value that was chosen, 44.1 kHz, was selected for other reasons, but the sampling theorem and the range of human hearing provided a lower bound for the range of choices.

Sampling theorem summary

The Nyquist-Shannon Sampling Theorem states that a signal bandlimited to ( - π / T s , π / T s ) can be reconstructed exactly from its samples with sampling period T s . The Whittaker-Shannon interpolation formula, which will be further described in the section on perfect reconstruction, provides the reconstruction of the unique ( - π / T s , π / T s ) bandlimited continuous time signal that samples to a given discrete time signal with sampling period T s . This enables discrete time processing of continuous time signals, which has many powerful applications.

Questions & Answers

what's a psy.d degree
Robin Reply
Psychology Doctorate. The main difference between PsyD and PhD Psychology is that PsyD mainly focuses on clinical sessions and PhD focuses on research.
so most of the phd holders become professors teaching what they have learnt
Hi, I have a question. What factors influence social facilitation and social inhibition in groups?
Amritpal Reply
it really depends on that person on how the can task in life either they can work better with other people or you do worse working with people.
the task
they task
Do you know of any studies done to show social facilitation and social inhibition
social facilitation is often influenced by the person's perception of the situation and their appraisal of the task. for example, a track athlete who views their competition as a challenge rather than a threat, will tend to run the fastest they have ever ran under a large crowd.
Another condition is if the person has mastered the ability to perform that task. A track athlete who trains everyday, will perform above average under a large crowd, however, someone who does not run, will tend to do worse than their actual ability under a large crowd
social inhibition can be explained through evolutionary biology or attachment personality theory. some people develop an avoidance personality trait, which when they feel under pressure, they tend to leave the group or avoid a gathering due to high lev of stress
Does anyone know how I can transfer my higher national diploma credits in usa?
Qwanta Reply
Oh wow. I never realised there was a chat on this app!
Amritpal Reply
Then welcome dearest
Thank you! ❤️
thank you
Hello Ana, good afternoon from here
Philip Reply
hello im new student here.
Junalyn Reply
o sorry madm
I m also student.
what is easier option to psychology?
I'm new too
As in the the specialisation or ?
good morning
understanding human nature is what we are going to talk about.
okay, go ahead
“ Every child is left to evaluate his experiences for himself, and to take care of his own personal development outside the classroom. There is no tradition for the acquisition of a true knowledge of the human psyche. The science of human nature thus finds itself today in the position that chemis
well, I think every individual is different and to a great extent we can't say confidently that we understand the human nature in its entirety
lack determines what we will become in life?.
Not 100% but it plays a huge role
Hi! I wasn't aware there was a chat feature on this app 😊
👍Philip Anyaegbu
yes Ana I'm here
alternatively we are different, we have basic similar qualities
Yes true Bridget.
hello there
who we are how we think what we do insight
@ bridget. yes that's true
Ana ioanidis? do you speak Greek?
Qwanta no I dont speak Greek my husband does
are you a psychologist? or your husband?
I need some reliable informations about my studies...
not a psychologist
I ask questions
so what is the cause why a person experiences a psychological disorder if it not of their own?
Junalyn Reply
it's partly genetics,culture, environment and substance abuse. generally these are the common factors which can lead to most of the disorders
r u proficcer
u r psychology subject
not a professor but a clinical psychologist
can i ask you, i keep getting headache, and had a history of hypothyroid ans diagnosed as having mild anxiety disorder. i want to finish. my undergraduate psychology thesis.. but always unable to have good ambition and energy to focus on writing the case and the theories... whats the best remedy
had a headache and have a history
297 according to the DSM 5
how many disorders are there
Janareon Reply
interesting question
it is😅
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.
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
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
Lirilong Reply
I don't understand the question can you elaborate?
the study of philosophy gives to the sociologist
Aman Reply
why women are viewed as far more emotional than men?
Wario Reply
because they actually are ,I guess.
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.
sollungal bro sollungal...
because of she has hormonal fluctuations than men..
usually women are more emotional than men and they are multi talented people
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....
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.
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.
because women tend to be more agreeable than men.
women at a young age are conditioned to be more in tune with emotion than man should be less
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.
The anwers are answers more to the question " Are women more emotional than men?"
hi are you on what'sap
Wunuji Reply
no not really a fan of social laziness
is there any way for a Btech E.C.E. graduate to take on MSC. PSYCHOLOGY
ya you can just by cracking entrance , if in India
Any stream in UG can go for M.Sc (Psychology)
classification of traits and how they are measured
chinedu Reply
what was Freud's first name?
Robin Reply
Sigismund Schlomo Freud
Sigmund Freud
Leon Vygotsky and Sergei Rubenstein please tell me contributions of these personalities in in 4 lines
sigmund Freud
Lev Vygotsky was the founder of socio-cultural theory
So psychology was based off of the Greek gods I that right or no?
Tanya Reply
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"
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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