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Gives various Fourier transform properties

Introduction

This module will look at some of the basic properties of the Discrete-Time Fourier Transform (DTFT).

We will be discussing these properties for aperiodic, discrete-time signals but understand that very similarproperties hold for continuous-time signals and periodic signals as well.

Discussion of fourier transform properties

Linearity

The combined addition and scalar multiplication properties in the table above demonstrate the basic property oflinearity. What you should see is that if one takes the Fourier transform of a linear combination of signals then itwill be the same as the linear combination of the Fourier transforms of each of the individual signals. This is crucialwhen using a table of transforms to find the transform of a more complicated signal.

We will begin with the following signal:

z n a f 1 n b f 2 n
Now, after we take the Fourier transform, shown in the equation below, notice that the linear combination of theterms is unaffected by the transform.
Z ω a F 1 ω b F 2 ω

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Symmetry

Symmetry is a property that can make life quite easy when solving problems involving Fourier transforms. Basicallywhat this property says is that since a rectangular function in time is a sinc function in frequency, then a sincfunction in time will be a rectangular function in frequency. This is a direct result of the similaritybetween the forward DTFT and the inverse DTFT. The only difference is the scaling by 2 and a frequency reversal.

Time scaling

This property deals with the effect on the frequency-domain representation of a signal if the time variable isaltered. The most important concept to understand for the time scaling property is that signals that are narrow intime will be broad in frequency and vice versa . The simplest example of this is a delta function, a unit pulse with a very small duration, in time that becomes an infinite-length constant function in frequency.

The table above shows this idea for the general transformation from the time-domain to the frequency-domainof a signal. You should be able to easily notice that these equations show the relationship mentioned previously: if thetime variable is increased then the frequency range will be decreased.

Time shifting

Time shifting shows that a shift in time is equivalent to a linear phase shift in frequency. Since the frequencycontent depends only on the shape of a signal, which is unchanged in a time shift, then only the phase spectrum will be altered. This property is proven below:

We will begin by letting z n f n η . Now let us take the Fourier transform with the previousexpression substituted in for z n .

Z ω n f n η ω n
Now let us make a simple change of variables, where σ n η . Through the calculations below, you can see that only the variable in the exponential are altered thusonly changing the phase in the frequency domain.
Z ω η f σ ω σ η n ω η σ f σ ω σ ω η F ω

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Convolution

Convolution is one of the big reasons for converting signals to the frequency domain, since convolution in time becomesmultiplication in frequency. This property is also another excellent example of symmetry between time and frequency.It also shows that there may be little to gain by changing to the frequency domain when multiplication in time isinvolved.

We will introduce the convolution integral here, but if you have not seen this before or need to refresh your memory,then look at the discrete-time convolution module for a more in depth explanation and derivation.

y n f 1 n f 2 n η f 1 η f 2 n η

Time differentiation

Since LTI systems can be represented in terms of differential equations, it is apparent with this property that convertingto the frequency domain may allow us to convert these complicated differential equations to simpler equationsinvolving multiplication and addition. This is often looked at in more detail during the study of the Z Transform .

Parseval's relation

n f n 2 ω F ω 2
Parseval's relation tells us that the energy of a signal is equal to the energy of its Fourier transform.

Modulation (frequency shift)

Modulation is absolutely imperative to communications applications. Being able to shift a signal to a differentfrequency, allows us to take advantage of different parts of the electromagnetic spectrum is what allows us to transmittelevision, radio and other applications through the same space without significant interference.

The proof of the frequency shift property is very similar to that of the time shift ; however, here we would use the inverse Fourier transform in place of the Fourier transform. Since we wentthrough the steps in the previous, time-shift proof, below we will just show the initial and final step to this proof:

z t 1 2 ω F ω φ ω t
Now we would simply reduce this equation through anotherchange of variables and simplify the terms. Then we will prove the property expressed in the table above:
z t f t φ t

Properties demonstration

An interactive example demonstration of the properties is included below:

Interactive Signal Processing Laboratory Virtual Instrument created using NI's Labview.

Summary table of dtft properties

Discrete-time Fourier transform properties and relations.
Discrete-time fourier transform properties
Sequence Domain Frequency Domain
Linearity a 1 s 1 n a 2 s 2 n a 1 S 1 2 f a 2 S 2 2 f
Conjugate Symmetry s n real S 2 f S 2 f
Even Symmetry s n s n S 2 f S 2 f
Odd Symmetry s n s n S 2 f S 2 f
Time Delay s n n 0 2 f n 0 S 2 f
Multiplication by n n s n 1 2 f S 2 f
Sum n s n S 2 0
Value at Origin s 0 f 1 2 1 2 S 2 f
Parseval's Theorem n s n 2 f 1 2 1 2 S 2 f 2
Complex Modulation 2 f 0 n s n S 2 f f 0
Amplitude Modulation s n 2 f 0 n S 2 f f 0 S 2 f f 0 2
s n 2 f 0 n S 2 f f 0 S 2 f f 0 2

Questions & Answers

does psychology deal with love?
Mohammed Reply
Maybe, i think
edem
I definitely would say yes
Clara
how so
Isaiah
*triarchic
Meredith
there are so many different reasons why you can fall in love with someone, many of them develope subconsciously -> psychology
Clara
love messes with the brain, a lot, ergo I believe that Psychology does indeed deal with love
what is synapse
Katie Reply
In the central nervous system, a synapse is a small gap at the end of a neuron that allows a signal to pass from one neuron to the next. synapse are found where nerve cells connect with other nerve cells
Najeem
can you do auto book auto
Mariah Reply
WHT u mean?
usef
yes
MD
heyy, may i join the conversation please?
edem Reply
who is the father of psychology
Richy Reply
aristatil
Ramadevi
and please, how would you guys, describe the study of psychology at college ?
edem
psychologist student?
Aspen
i mean not yet but am about to start college so wanna know how is it(college in general and psychology course) please
edem
Psychology is the study of mind and behaviour. So if you will take psychology as a subject so you will get to know how your everything (physical, mental, social, spiritual aspects) effects your behaviour
sakina
With this brief knowledge you can help people to cope up with their problems and only you can guide them correctly
sakina
And if you go for further specialisations you can study hypnosis, face reading, body language etc
sakina
Thanks a lot🙏🏾 And ik some of the stuffs u said but i am also going to write thesis, right ?
edem
ok no prob, thanks a lot🙏🏾✨
edem
cerebellum
Khan
hae everyone, hope you are well this evning my question is what is the difference between drive and motivation
Michael
good question
Rainee
drive is more like an impulse or urge and i think they both go together (drive and motivation) even if there is a slight difference
edem
@ Michael Drive is delivered to be innate without the use of an external stimuli, motivation normally evolves an outside stimuli which may include praise, appreciate, or reward.
Reginald
*believed...sorry for typo
Reginald
@Reginald, can't the motivation come from the inner self?
edem
Good question, please give an example.
Reginald
can we say desire of success for example
edem
Wilhelm Wundt is the father of psychology
ipau
Wilhem Wundt thank you for the road that you opened.
Qwanta
You mean who is the father of having a great educated argumentative guess? nothing is more wrong than this question. The question is you should ask yourselfs is, how sure are you abour their scientific studying? one's percieved assimilated approach to judging another person and saying they are
Roger
the biggest problem with scientific research and data is that ya you could get the same result 1000 times then it could go the other way 1000 times, but we would never know that and we did, we would still say ya but the proof is there. The only thing science proves is that humanity has
Roger
no facts about human behavior in the scientific context, but more in the trial and error.. sorry to tell you, but so far no one has proven Father of anything, thats up to you and i, judgement is bias, science is good enough lazy
Roger
cognitive development is the growing and development of the brain.
Jessy Reply
Anyone knows about Techno-fascism?
Hussein Reply
Ecofascism is a theoretical political model in which an authoritarian government would require individuals to sacrifice their own interests to the "organic whole of nature". The term is also used as a rhetorical pejorative to undermine the environmental movement.
ipau
what's the big difference between prejudice and discrimination?
Danice Reply
A prejudiced person may not act on their attitude.  Therefore, someone can be prejudiced towards a certain group but not discriminate against them.  Also, prejudice includes all three components of an attitude (affective, behavioral and cognitive), whereas discrimination just involves behavior
Nancy Lee
hi
basher
hello
Rahul
what is all about cognitive development?
Kamohelo
cognitive development is the growing and development of the brain
Jessy
how do you control a variable when using spss whilst running a pearsons correlation analysis?
Jessie Reply
it dependa on your study. according to what you want to say and explain your result
Pouran
why does it say her and she
Jayla Reply
stages of cognitive development
brivia Reply
sensory preoperatinal concrete formal
Rajendra
what is psychology
Chethani Reply
the study of insecurities and the effect on the host .
Sera
Psychology is the scientific study of behavior & mental processes
Angela
psychology is science about learning human behaviour
Zhamshid
behaviorosm
Khan
In thinking about the case of Candace described earlier, do you think that Candace benefitted or suffered as a result of consistently being passed on to the next grade?
Nene Reply
what is reward
Angeles Reply
reward is a technique to change behaviour
Rajendra
Reward is a way to promote a specific behaviour or to teach someone/ something to behave a specific way or perform a specific task.
Johan
a reward is something that is usually associated with desirable behavior. The child got a reward for winning the game. A reinforcer is different in that a reinforcer is anything that increases behavior, even if it is increasing an undesirable behavior.
Meredith
reward is earned effort realized
freweini
why heroine addicted people smoke heroine in a dirty and polluted place ?
Najeem Reply
?
Sera
?
Sera
G
ky
/i don't know
Rajendra
🤔
Hammam
I believe they don't wanna be seen
Ruphine
Rajendra Singh I'm asking from those who know thanks for your comment
Najeem
Ruphine it's not a scientific answer
Najeem
Ok
Ruphine
more my personal opinions and experiences than a real answer but more of an answer than the question marks
Jehsika
what's the data or fact that actually say that heroine smokers host in dirty and polluted place.
amaan
Generally the more time an addict is using and the harder they use the more their life and everything around them crumbles. it's quite common to find neat respectable clean users but over time the addiction increases and their health declines along with their ability to keep up day to day duties.
Jehsika
I'll see if I can find some links for you to have a read.
Jehsika
hi. as heroine affedted exactly on the brain, it should be because the chages heroine makes on frintal lobe and sensory and motor parts. after a while the cannot behave in a normal way. in a way heroine ruines the neurones
Pouran
The things said in the thread are some of the most horridly presumptuous & unwarrantedly pretentious statements I've read in my entire life. It's distastefully disrespect, to say the least.
Kaytee
Pouran Hi! thanks for answering I'm agree with your opinion and may the answer be the same
Najeem
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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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