How Does the DC Component Affect the Fourier Transform of a Signal?

[rude man]

Since I've just been up all night here, when I stumbled across sin(w) and I was trying to inverse Fourier transform it, I thought I'd done it before, it should be "elementary!", but now looking at it I tried and found myself integrating (from Euler's Identity):
1/2pi * [(exp(2jwt)-1)/2j] ∞→-∞

which obviously doesn't work or trying to use Duality on the table with no success in getting the: j√(pi/2)*[del(t+1)-del(t-1)] that wolfram generated as the answer.

Anyway, a question for another time perhaps.

Thanks for your assistance!

Look at the link I sent you. It tells you how F{sin(wt)} is derived. They use the time-displaced delta function in the w domain in a clever way.

I don't know what field you hope to go into, but if it's electrical engineering I would de-emphasize the Fourier transform and concentrate on the LaPlace. The Fourier comes into its own in optics and statistical control system design (for which the two-sided LaPlace is equally suitable).

 

[toneboy1]

BTW I like the following site for the Fourier transform:

http://www.thefouriertransform.com/transform/fourier.php

Look at the link I sent you. It tells you how F{sin(wt)} is derived. They use the time-displaced delta function in the w domain in a clever way.

I don't know what field you hope to go into, but if it's electrical engineering I would de-emphasize the Fourier transform and concentrate on the LaPlace. The Fourier comes into its own in optics and statistical control system design (for which the two-sided LaPlace is equally suitable).

Right'o, I will concentrate on LaPlace because I am in Electrical Eng; I have done more LaPlace a couple of years ago, than I ever have Fourier.

In that link you sent me, I assume the Sinc and the Sin in which you refer are one and the same. I did notice that using l'hospital's rule sinc(0) = 1 for division by 0 which makes sense.

[ On what might be a side note: I'm confused because my Transform table says (using κ for tau) P_{of width κ}(t) is transformed to κSinc(κω/2pi) but in examples I can see P_κ(t) is transformed into (2/ω)*sin(κω/2) Which would mean that P_κ(t) is transformed into κSinc(κω/2), so where has the pi in the denominator gone? ]To be clear, we're talking about sin(w) FROM frequency domain to time domain aren't we?
(I was trying to use Duality to multiply Sin(w) by 2.pi to get the: j√(pi/2)*[δ(t+1)-δ(t-1)]
Wolfram generated.)

Cheers

 

[rude man]

Right'o, To be clear, we're talking about sin(w) FROM frequency domain to time domain aren't we?
(I was trying to use Duality to multiply Sin(w) by 2.pi to get the: j√(pi/2)*[δ(t+1)-δ(t-1)]
Wolfram generated.)

Cheers

I goofed again. I was thinking F{sin(wt)}, not F^-1{sin(w)}. Sorry.

But the technique is similar: think the time-delayed delta function.

 

[rude man]

Right'o, [ On what might be a side note: I'm confused because my Transform table says (using κ for tau) P_{of width κ}(t) is transformed to κSinc(κω/2pi) but in examples I can see P_κ(t) is transformed into (2/ω)*sin(κω/2) Which would mean that P_κ(t) is transformed into κSinc(κω/2), so where has the pi in the denominator gone? ]

The Fourier transform of a pulse of height 1 and width 2T is F(w) = (T/π)(sin(wT)/wT.
Some folks call sin(x)/x = sinc(x) but I've also seen different definitions of sinc so I avoid that designation in general.

As an aside: you can't let T → ∞ to derive F(1). You have to invoke the delta function.

 

[toneboy1]

The Fourier transform of a pulse of height 1 and width 2T is F(w) = (T/π)(sin(wT)/wT.
Some folks call sin(x)/x = sinc(x) but I've also seen different definitions of sinc so I avoid that designation in general.

Just experimenting, I noticed if w = 3, width = 2 then
(2/w)*sin(2*w/2) is ALMOST the same as (2.pi/w)*sin(2*w/2.pi),
so you seem right about it being a notational thing, but what I can't understand is that the one with the pi in it seems more accurate because in one instance the number was finite and the other kept going. Can you see why this would be?

I goofed again. I was thinking F{sin(wt)}, not F^-1{sin(w)}. Sorry.

But the technique is similar: think the time-delayed delta function.

H'mm, I'm still not clear, I'll elaborate on where I got stuck:

x(t) = (1/2pi)* ∫ (exp(jwt)-exp(-jwt) /2j)*exp(jwt) dw -∞ to ∞

∴ (1/2pi)* ∫(exp(2jwt)-exp(0) / 2j) dw -∞ to ∞

∴ (1/j4pi)*[exp(2jwt)/2jw - w ] -∞ to ∞

And I'm not sure where to go from here. (especially with your time delay)

Alternatively using duality and linearity I attempted to make

sin(ω) = 2pi*sin(-w)/2pi

inverse transformed into: j.pi/2pi * [δ(1-t)-δ(t-1)] which is wrong according to wolfram.

Cheers for any input!

 

[rude man]

Just experimenting, I noticed if w = 3, width = 2 then
(2/w)*sin(2*w/2) is ALMOST the same as (2.pi/w)*sin(2*w/2.pi),
so you seem right about it being a notational thing, but what I can't understand is that the one with the pi in it seems more accurate because in one instance the number was finite and the other kept going. Can you see why this would be?

Not sure what you're getting at here.

In the below, where on Earth did you get that equation? You're apparently trying to do an inverse transform on sin(wt)?? On a function containing t ?

H'mm, I'm still not clear, I'll elaborate on where I got stuck:

x(t) = (1/2pi)* ∫ (exp(jwt)-exp(-jwt) /2j)*exp(jwt) dw -∞ to ∞

Cheers for any input!

I think I need to go to bed!
Cheers!

 

[toneboy1]

Not sure what you're getting at here.

In the below, where on Earth did you get that equation? You're apparently trying to do an inverse transform on sin(wt)?? On a function containing t ?



I think I need to go to bed!
Cheers!

[I was saying
(2/w)*sin(2*w/2) ≈ (2.pi/w)*sin(2*w/2.pi)
for reasons I cannot reckon]

Woops, good call, it should have been 't'less, I.e:
x(t) = (1/2pi)* ∫ (exp(jw)-exp(-jw) /2j)*exp(jwt) dw -∞ to ∞


Heh, heh, fairenough, Night.

 

[rude man]

[I was saying
(2/w)*sin(2*w/2) ≈ (2.pi/w)*sin(2*w/2.pi)
for reasons I cannot reckon]

Woops, good call, it should have been 't'less, I.e:
x(t) = (1/2pi)* ∫ (exp(jw)-exp(-jw) /2j)*exp(jwt) dw -∞ to ∞


Heh, heh, fairenough, Night.

Why are you trying to determine F^-1{(sin(w)} anyway? Its inverse is not at all a common time function ...

If I had to do it I would study the way sin(wt) was transformed, then try to apply that technique to the inversion integral.