Fourier transform basic

[neptune]

can you tell me the limits of Impulse..you know there is no width of impulse

 

[MrAl]

Hi,

Yes, that's right, sort of. But it does have a known area (unit area) so you can start by integrating over some small range and let the interval get smaller as 1/T gets larger and thus end up with a FT. In other words, integrate from t0 to delta_t and multiply by 1/delta_t and let delta_t get smaller and smaller (a rectangle delta_t wide and 1/delta_t high always has unit area), the limit is the transform of the impulse.

This question however was supposed to be one of those two or three second questions, where you think about it for a little while and then realize that you can not find a Fourier Series for an impulse or any other non periodic signal, but you can find a Fourier Transform. That was the whole point. The Fourier Transform is defined for non periodic signals so that's the difference.

It's also interesting if you try to find the Fourier Series for a unit step. It becomes impossible to find a good time to integrate over. On the other hand there is a Fourier Transform for the unit step.
There is a way to approximate though, if you use a long enough pulse (relative to any time constant of your circuit) you can find the series for that and use it with some limitations Actually we could do this and you'd see how the 'phase' we had been talking about fits into the picture.


The transform gives us a continuous spectrum while the series gives us a line spectrum.

I havent needed to find a Fourier Transform for anything useful since maybe the mid 1990's. I end up using Laplace transforms for everything.

 

[neptune]

The transform gives us a continuous spectrum while the series gives us a line spectrum.

if i am not wrong a continous spectrum contains all of the line spectrum ?
ok, if FT of unit pulse u(T) is a sinc fuction, like Tsinc(T) then its graph will be a continous spectrum .. and this graph will show all possible frequencies that closely defines this unit pulse u(T) .. but only a single frequency that defines it completely.

This man builds two graph to show FT
https://demonstrations.wolfram.com/RectangularPulseAndItsFourierTransform/

 

[MrAl]

Hi,


That's perhaps a good example to work with. Since you picked it we can use that. Note we dont need any other examples or any other stuff on the web now
This is good enough, work with that for now and i think you'll be at least a little happy.

Taking his example, you can see he posted two plots. One is the amplitude and the other is the phase. This is what we were talking about earlier. All you have to do now is reproduce his graphs. First find the amplitude, then find the phase, plot both. It may not show you how the phase fits in however, but we'll get to that in a minute.

Make one very important note here first though, his result is not 100 percent accurate, as are many things on the web due to typo's and copy and paste mistakes. We all do this at some point so it's not nice to be too critical, but in the past i have looked up much more complex stuff only to find that i had to go through the whole derivation myself anyway because the web posted results were not typographically correct. What a shame, it's the downside of the web and it's hard to do anything about.

The inaccuracy being talked about here is of course the lower case 'T' in his resulting transform ('t'). How could we have a frequency transform with lower case T in it? Answer: we dont That lower case 't' needs to be changed to either lower case J or lower case I (the imaginary operator of course) and then everything is good to go.
So the true transform does not have:
e^(-t*2*pi*f*t0)
in it, it has instead:
e^(-j*2*pi*f*t0)
in it, and this is a simple replacement of 't' with 'j'.
Reexamining the new result, we see we just have the transform of a pulse 'delayed' by a time value of t0.

Finding the amplitude and the phase you should be able to come up with the same plots he got after fixing the little 't' typo.

This is a good example of finding the amplitude and phase but we should also do a complete transform/reconstruction so you can see how this works using a Fourier Series, unless you've done that already. Once you play around with that a little it becomes very apparent why the amplitude/phase is really just another form of the same thing.
Amplitude/phase representation is often important in electronic/electrical circuits though so it's good to get a feel for this.

 

[neptune]

hi,
the point i was making in my previous post about transform is that it builds continous spectrum , where main lobe contains frequencies which can define this Pulse almost accurately. and other side lobes not so accurately, have i got this right ?

One more question that his sinc function is wrongly built, because sinc function has dyeing oscillation which gooes from -ve to +ve amplitude ?

 

[MrAl]

Hi,


The central lobe may approximate, but it would depend on the pulse width i think. Are you looking to approximate the impulse function this way?

Why do you say his sinc function is not right?

 

[neptune]

I thought Sinc function looks like this

 

[MrAl]

Hi again,


Oh ok i understand your question better now.

What you have graphed there indeed looks like the *sinc* function (and a pretty good hand drawing too i might add).
What you need to graph next is the *amplitude* function, which is not exactly the same as the sinc function. That's what he has graphed.

 

[neptune]

What you need to graph next is the *amplitude* function, which is not exactly the same as the sinc function. That's what he has graphed.

what the heck is *Amplitude* function ?
i have a pathetic hand writing

 

[MrAl]

Hi,

The amplitude function is the square root of the sum of the squares of the real part and imaginary part:
Ampl=sqrt(real^2+imag^2)

Simple inspection shows that if we only have either just real or just imag part then the function boils down to a simple absolute value function:
Ampl=|real| or Ampl=|imag|

If you care to try this you should get matching results with the link.

 

[neptune]

Hi,
I have an idea to transfer a periodic wave over any media digitally. instead of transffering continously we can transfer there Fourier series component.
it will have 3 components A0 , Ak & Bk.
For one tone we will have 3 components and for other tone we will have other components.
at receiving end we can reconstruct these waves with pure quality.

 

[MrAl]

Hi,

You might want to look into the Jpeg standard for image compression. The image information is transformed into frequency components in two dimensions, and that is the data that is actually stored, not the picture raw 'signal' itself. On decompression, the frequency components are transformed back into the 'time' components and that gets displayed as a very close approximation of the original picture.

 

[neptune]

nice,
Q1 - but can we use this technology in Radio transmission ?
Q2 - is there any other better and new technology then JPEG compression ?

 

[MrAl]

Hi,

I suppose you could if you felt like going through all that trouble. You'd have to figure out if it has any benefit in the long run.

There is wavelet compression which is 'supposed' to be better, and there is always fractal compression which im not sure ever made it off the ground too far yet.

Jpeg uses the discrete cosine transform in two dimensions for the conversions, although it's often implemented as a FFT to make it faster with some required matrix transforms and a few constants.

 

[neptune]

ok i will study wavelet compression and fractal compression.
I just googled and i see that Digital Radio already exist LoL

one more Question that can we use any orthogonal periodic functions in Fourier transform. instead of sine and cosine

 

[MrAl]

Hi,

Yes the wavelet is used in modern image processing. I've never used it myself actually, but did do a custom FFT to DCT for a Jpeg decoder long time ago.

If you use something other than sine and cosine then you dont have a Fourier anymore
You can use powers of x too, but im sure there are a ton of other ways to encode a signal.