# what is the maning of e^jw in fourier transform?

1. ## what is the maning of e^jw in fourier transform?

Hi
In fourier transform i don't understand the meaning of e number?
We know e number is =2.71828
But in fourier e^iQ = cosQ+ isinQ , (I understan that it means a complex number if wrong please correct it)
Is this e 2.71828 or is it only a sembol.Is it not "e" number?

I can not solve what the meaning of (e sembol or number) in fourier transform?

Thanks friends.

2. ## Re: what is the maning of e^jw in fourier transform?

Hello
it is all true.
e is real number e=2.71828 it is Euler number
e^iQ = cosQ+ isinQ is Euler formula (see http://en.wikipedia.org/wiki/Euler)
you use it if you want calculate e^iQ for example on simply calc.
But this is basic math you may study it before work with FFT.

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4. ## Re: what is the maning of e^jw in fourier transform?

thanks friend
I passed signals and systems lesson last year but i memorized everything.I know that i still don't understand the meaning of multiplying a signal with e^jw and then taking its integral.( or i can't imagine it.)
Ok.
e is 2.71828 and also e is cosQ+ isinQ.

But one of them is numeric value but the other is complex value or complex represantation.I confused here.

We convert our signal from time-domain to frequency domain means we convert our signal numeric value to complex value or complex represantaion.Is this true?
If true so what is the meaning of e is 2.71828...It is numeric value.

http://en.wikipedia.org/wiki/Euler%27s_identity

But still i can't clarify the meaning of "e^jw=cosw+jsinw=2.71828^jw"

Can you please give me more information about fourier transform(or the meaning of multiplying a signal with e^jw(Isn't this we carry our signal to vectorel plane?)).Still I can't imagine its mechanism.

thanks again
Good works

5. ## Re: what is the maning of e^jw in fourier transform?

Taylor series of some common functions include :
e^x=e^a[1+(x-a)+1/2(x-a)^2+1/6(x-a)^3+....]
sinx=sina+cosa(x-a)-1/2sina(x-a)^1/2-......
you can compute this series :wink:
or the same sites.

hope it helps .

6. ## Re: what is the maning of e^jw in fourier transform?

Sorry amiralib , i can't find my questions' answers in taylor series formula.
I want to add one more question to my last post, an explanantion:
"For example think a cup , it is shapeless.How can we calculate its volume?We can't use scale but we can use water.Here water represent 3D but scalet is 2D."

So can we give this example about converting a signal one domain too another domain?Aer these true?

Numeric values has 1D but complex number has 2 dimensions.So Does multiplying a signal with e^jw mean we carry it to complex plane?If so how is that possible?As i said i missed something so i can't figure out it.

Thanks
(I am engineering student not a science , so they don't shows us the details(mechanism) .They only give us what we need.)

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7. ## what is the maning of e^jw in fourier transform?

Be careful!
How do you get from
e^iQ = cosQ+ isinQ
to
e is 2.71828 and also e is cosQ+ isinQ.
Then how did you loose "^iQ"? :|

8. ## Re: what is the maning of e^jw in fourier transform?

It must be e^iQ = cosQ+ isinQ=2.71828^iQ.Is this true?In fourier transform e is really 2.71828 or before i said only a represantation of a complex number or something else?

Can you please give answer to my questions(or correct my explanantions).I need some datails.

Thanks

9. ## Re: what is the maning of e^jw in fourier transform?

Originally Posted by serimc
It must be e^iQ = cosQ+ isinQ=2.71828^iQ.Is this true?In fourier transform e is really 2.71828 or before i said only a represantation of a complex number or something else?

Can you please give answer to my questions(or correct my explanantions).I need some datails.

Thanks
(1) e = 2.71
(2) e^iQ=CosQ+iSinQ

from (1) and (2) => 2.71^iQ=CosQ+iSinQ

about the 'e' in the fourier transform. well its the same e = 2.71.

about the complex numbers. a complex number can be represented as:

a) a+ib
b) Re^iQ

and they are both the same number only with different representation.

mayyan

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10. ## Re: what is the maning of e^jw in fourier transform?

Ok
thanks mayyan.
So what is the meaning of multiplying a func. with 2.71^iQ=CosQ+iSinQ?( I know we use it for frequency represantation of a signal.But i don't know its meaning.What is the magic behind this calculation)

Thanks again

11. ## Re: what is the maning of e^jw in fourier transform?

i will start with an analogy. a point in a 3 dimentional space can be represented as:

p= ( a , b , c )

now if we take the 3 orthonormal vectors:

x=(1 , 0 ,0) y=(0 , 1 ,0) z=(0 , 0 ,1)

and:

<x,p> , <y,p> , <z,p>

you will get the projection of the point on the 3 axises. fourier is doing the same thing but now the vectors that built the space are sin and cos with different freq Qn.

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12. ## Re: what is the maning of e^jw in fourier transform?

thanks this analogy , i need such expression.
So can we say that this multiplying operation is not numeric multiplying.(Like 3*4=12)

This must be different because this multiplication and then integration returns me a vector not a numeric value.Is this true?I wonder the magic behind this multiplication.

Also i want to ask one more question , i gave a analogy:
""For example think a cup , it is shapeless.How can we calculate its volume?We can't use scale but we can use water.Here water represent 3D but scalet is 2D."

So can we give this example about converting a signal one domain to another domain?"Is this right?

Thanks for all help.
Good works...

13. ## Re: what is the maning of e^jw in fourier transform?

hi all
assume f(t)=A0+A1 sin(wt)+...An sin(nwt)+...

∫sin(mwt) *sin(n(wt) dt=½∫[cos(n+m)wt +cos(n-m)wt] dt

for a priod (wt=0 to pi) ∫[cos(n+m)wt =0
∫cos(n-m)wt dt =0 n≠m
=1/τ n=m
it is because this term is priodic function

so ∫sin(nwt) f(t) dt =1/τ An wt=0 to pi
above equation known as Orthogonal transformation.
be happy

14. ## Re: what is the maning of e^jw in fourier transform?

<f(t),g(t)> = <f(t)|g(t)> = (f(t),g(t)) = inner product

15. ## Re: what is the maning of e^jw in fourier transform?

hi
Same as Previous post
∫exp(jwt) exp(-jwt) is an orthogonal transformation.
for f(t)=A0+A1 exp(-jwt)+...+An exp(-jnwt)+...
Each of this term exp(jnw) are the fundamental or natural frquency of wave form.
exp(jwt) is a sampler in frquecy w. And by exp(jwα) f(α) ,we measure the energy of wave form in frquency w for t=α.
by integration for all of the time (∫exp(jwt) f(t) dt t=-∞ to+∞) we calculat total energy of wave form in frquency w .

what is j at all?

In 2d calculation domain
summation role (x1,y1)+(x2,y2)=(x1+x2 ,y1+y2)
and product role (x1,y1)*(x2,y2)=(x1 x2-y1 y2,x1y2+x2y1)
we define a transfomation from 2D to 1D as below
f(x,y)=x

we define j=(0,1) so
j^2=(0,1)*(0,1)=(-1,0)
f(j^2)=f(-1,0)=-1

Im
|
| (x,y)
|
|______________ real

16. ## what is the maning of e^jw in fourier transform?

i Is Imaginary Number And We Use This To Simplify Our Notation (a,b)*(c,d) = (ac-bd,ad+bc)
(a,b) Is Complex Number.
I Think This Is Revoloution In Mathematics To Use Hegel's TRIAD And Removing Contradiction And Opposition In Real Numbers

17. ## Re: what is the maning of e^jw in fourier transform?

ok
I have one more and last question.
why 2.71828^iQ id equal to cosQ+ isinQ.I mean why 2.71828.why not 10 or 53 but 2.71828.
There is no relationship between 2.71828 and cos-sin.But they are two part of an equation and 2.71828^iQ is equal to cosQ+ isinQ.

One of my friend said that you can declare whatever number you want and it will be your space.I think he mean i can declare 10^iQ=cosQ+ isinQ.
Is this explanation true?

18. ## Re: what is the maning of e^jw in fourier transform?

hello check this :
http://mathworld.wolfram.com/EulerFormula.html
equations 4>6 is a prove to euler and so is 7>12

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19. ## Re: what is the maning of e^jw in fourier transform?

hi

exp(x) =1+ x/1! + (x²/2! )+.....x^n /n!+....

cos(x)=1-x^2+x^4/4!-x^6/6!+......
sin(x)=x-x^3/3!+x^5/5!-

exp(jw)=1+jw/1!-w^2/2!-jw^3/3!......
exp(-jw)=1-jw/1!-w^2/2!+jw^3/3!......

exp(jw)+exp(-jw)= 2-w^2/2!+w^4/4!...= 2cos(w)

exp(jw)-exp(-jw)=2jw/1!- ....= 2j sin(w)

20. ## Re: what is the maning of e^jw in fourier transform?

Hi,

You are confused this much just because you might have read a wrong/advanced book for your course.

Fast fourier transform and its applications := Oran E Brigham.

After reading this you may be able to understand any other book on DSP.

Basic Engineering mathematics you will get in the book by Kreysig.
For the title and all you can search in www.amazon.com

You may also read about othogonal representation of signals (Say, Grahm schmit orthogonalisation procedure) just to get the clear picture.

The numbers like pi (3.14), e (2.7) are actually sum of some special infinite series.
Infact sin and cos are also sum of some infinite series.

Euler showed that how these three (e,sin,cos) are related.

So you cannot write 10^iQ=cosQ+isinQ.

Cheers..

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