effect of spikes on the probability density function
 

Hi all,
I have some question about the effect of a spike in a probability density function:

1. Why do a spike in B(u) produces a component with characteristic function that behaves as Aexpiks ? When it is evaluated the component phis(k), it is concluded that phis(k)=iks*phis’(k) , where phis’(k) is an other characteristic function that is not A. The fact that the caracteristic function is proportional to exp(iks) does it mean that the characteristic function has imaginary values? If yes, why is this not mentioned?

2. I did not understand why at pag 205 it is written that if the spike is infinitely narrow phi’s(k) is constant. If the spike has a finite width sigmas, phi’s(k) descreases as k/sigma.

3. I did not understand why the function in figure 6.5 (attached image) has such a charactersitic function. Why is it pointed out the width 2pi/s. How did he obtain that value?

Sorry for all my questions

Thanks in advance

These are all properties of Fourier transform or Fourier representation of signals. And these are often assigned as homework for students to prove.

1. phis(k)=iks*phis’(k) is a property of Fourier transforms. Writing that something is exp(iks) means that it is complex. Why is it not mentioned? It is when you write it as exp(iks). If something is written in English, does it also have to be written in French for everyone to understand it? The conditions for it to be purely real are also mentioned.
2. Another property of Fourier transforms.
3. Another property of Fourier transforms.


These are basic mathematical properties of Fourier transforms. You can prove them yourself. Wikipedia is also a good source.

I'll look into it... thank you (merci:D)

Sorry if I write again here. Unfortunately, also after a deep study of the fourier transform, I am still having a lot of unresolved doubt.

1. I have not understand if in the graph 6.5 appear just the real part (and the immaginary part is simply not plotted) or if the immaginary part is absent. The condition to be purely real is mentioned and it is the simmetry of the probability function, but this probability function is not simmetric. So I imagine that it is present an immaginary part, but I would like a confirmation.

2. It is written that the spike has a finite width sigmas, phi’s(k) descreases as k/sigma. I don't know where this property come from, for example if we study a top-hat function the characteristic function associated has this function sin(k*sigma/2)/(k*sigma/2), so it does not decrease as k/sigma, but as 1/(k*sigma).

Thank you in advance

The top-hat function of parameter h in physical space has its transfer function in the form sin(k*h)/(kh). Plot this function along k, you will see that has an infinite number of zeros (the first is at k=pi/h), vanishing only asymptotically

I know that! Indeed I was taking the top-hat function as example for an other question

To mimic a spike, just consider the top-hat for h going to zero

Thank you. I am happy that I was thinking the same thing. But the book says that a characteristic function of spike decrease as k/sigma.
My doubt is the fact that if you consider the characteristic function of the top-hat function, you have (as you also said before) sin(k*h)/(kh). This means that it does not decrease as k/sigma .

So I cannot understand

If the spike is infinitely narrow that means a Dirac function that has the consequence of containing infinite wavenumbers components. I don’t see the concept of decreasing for the transfer function, for fixed sigma the function increases in k

In this case my problem concerns a narrow spike with a finite width. I attached the part of the book that highlight this concept

phi' isn't the characteristic function of B but of Bs. phi is the characteristic function of B

The spike with finite width cannot be infinite as it would not be an integrable function. In my opinion either a top-hat or a Dirac function is discussed

Yes, but in any case I cannot understand why phi' should decrease as k/sigma.

Yes I think that are discussed both, but I dont understand why if it is of finite width it decreases as k/sigma

Ok so the terminology here can be really confusing if you don't agree on what rates of rates are. The point is that if the spike is infinitely narrow, then phi' is a constant function. If the spike has a finite width, and as this width grows wider, phi' is attenuated (decays faster). The wider the the width of the spike in physical space, the more the attenuation (or narrower phi' looks). That is, "phi's is a characteristic function with a width inversely proportional to the width of the spike." That is, for any given k, the width of phi' goes like 1/sigma.


The characteristic plotted in 6.5 is not the characteristic of the entire B(u) but only of the top-hat/spike, and that's why it's real. It is a bit of a mislabeling if you consider only the figure.

Does phi' go like 1/sigma, because the top hat function can mimic the peak (As also FMDenaro said) ? or they are not related