Fourier Series Elementary Question

[agtsp]

Hi,

Elementary question on Fourier Series. I wondered about the series directly summing Sine and the Cosine Terms. But as I understand that these terms are quadrature to each other, there must be the J term for Sine, since we are dealing with signals in vector spaces. Any comments

agtsp

 

[hshah8970]

Re: Fourier Series clarification

The Fourier series breaks down any periodic function into a sum of sines and cosines. If a function is odd, it can be represented by a sum of sines only; if a function is even, it can be represented by a sum of cosines only. For any function that isn't entirely odd or even, a sum of sines as well as cosines is required.

You could say that sines represent the oddness of a function whereas the cosines represent the evenness.

As for the iota multiplier, it isn't required. Assuming that we're dealing with a real 2D function, the 'j' term with the sine terms will only indicate that the original function had a complex part - which would be false.

 

[agtsp]

Re: Fourier Series clarification

Hi,

I am just not able to understand because I am a beginner. If you could bring down to a basic level, I can appreciate it better. Along the similar lines, why do we then multiply with complex number for Fourier transform calculation if function is just a real number.

 

[hshah8970]

Re: Fourier Series clarification

The Fourier Series and the Fourier Transform aren't the same thing.

We use the Fourier series to represent periodic signals as a sum of sines and cosines; the domain remains unchanged. We use the Fourier transform to represent signals in the frequency-domain - we change the domain from 't' to 'jw' i.e. we deliberately take the function into the complex frequency domain so frequency-analysis becomes simpler.

 

[agtsp]

Re: Fourier Series clarification

HI thanks but I have some clarification in fourier transform equations. Why signal is multiplied by exp(-jw) and not just exp(jw), what is the logic

 

[hshah8970]

Re: Fourier Series clarification

You should look into the derivation of the Fourier transform. You will find that we try to form a Fourier series formula for aperiodic signals. After a certain number of steps, we find that it is convenient to introduce the exponential function and change independent variable to omega.

Google "Fourier transform proof."

 

[milind.a.kulkarni]

Re: Fourier Series clarification

In my view ....when you take Fourier series of a signal it gives a decomposition of signal by means of orthogonal functions mapping for the harmonic frequency component with different coefficient .... I really don't understand where is a question of imaginary part comes in Fourier series ?.... Can you be bit elaborate .....

 

[savan]

Why do u feel J to be necessary in Vector Space and there is nothing like imaginary part is must in Vectors!

You can express even Square wave in terms of Fourier Series

Hi,

Elementary question on Fourier Series. I wondered about the series directly summing Sine and the Cosine Terms. But as I understand that these terms are quadrature to each other, there must be the J term for Sine, since we are dealing with signals in vector spaces. Any comments

agtsp

 

[agtsp]

Re: Fourier Series clarification

HI MILIND,

Since you have told about orthogonal functions, it puts me in doubt because any orthogonal functions cannot be summed up arithmetically, so have add vectorially which means putting a j term. As I understand from you, all the signals including a square wave or arbitrary wave are not complex in nature( to be resolved vectorially). Are they any signals in reality which exists in complex form? And regarding the proof of fourier transform, I understand the derivation, but I am unable to think intutively about exp(-jw) term instead of exp(jw)

 

[agtsp]

Hi savan,

Please follow my thread titled "Fourier Series clarification" and kindly give your views

 

[milind.a.kulkarni]

Let me answer the second question answer first.... Can any signal in reality is in complex form - Now a signal is physical quantity exits in the nature.... where variation of one of the component like amplitude or say peak value etc....varies with a variable like space, time etc.....now I think you got your answer the entities exits physically...can be thought as imaginary.....now when you say vector summation of orthogonal you says that the complex number comes in the picture "j" can you do one thing just try to do this ....... generate the signal with mathematical software like matlab..... with T, a0,a1,a2,a3,b1,b2,b3 are constant
f(x) = a0 + a1*sin(2*pi*x/T) + a1*sin(2*2*pi*x/T)+ a1*sin(3*2*pi*x/T) + b1*cos(2*pi*x/T)+ b2*cos(2*2*pi*x/T)+ b3*cos(3*2*pi*x/T)......
and then you do ....
f1(x) = a0
f2(x) = a1*sin(2*pi*x/T)
f3(x) = a2*sin(2*2*pi*x/T)
and so non and then try to add them....
f(x) = f1(x) + f2(x) +......etc
see what you get is original signal or not for your self....here is a big catch where physical entities connects to the mathematics ..... Now look at this way the expression that I had said do not have any complex term at all.....it deals with all the real term getting accumulated with the constant coefficient ..... but now you plot the individual function with respect to x then the point of occurrence for them ( position at with f2(x) become positive and f3(x) become positive on the same plotting variable or independent variable x ) have different positions at different instant....so In shot it is the mathematical notion of complex entities allows you analyze the signal in the mathematical model... I think this will make you better clear...

Good Luck