Show that Fourier transform does not violate causality

[coushal]

Homework Statement

Show that Fourier transform does not violate causality. In other words, let $$\hat{E}(\omega)$$ be Fourier transform of function $${E}(t)$$. Show that $$E(t_1)$$, as evaluated from inverse Fourier transform formula using $$\hat{E}(\omega)$$, does not depend on $$E(t_2)$$ for
$$t_2>t_1$$

Homework Equations

The Attempt at a Solution

I don't quite get the meaning of ``does not depend''. Obviously, for sufficiently ``nice'' $$E(t)$$, the inverse Fourier transform formula gives exactly $$E(t)$$ (if the transforms are appropriately normalized). So if one fixes the value $$E(t_1)$$ and perturbs $$E$$ for other $$t$$ (no matter in the past or in the future), the inverse Fourier transform will still return the function which has value $$E(t_1)$$ at point $$t_1$$.

So, it looks like I don't understand the problem. Explanation is greatly appreciated.

 

[mighty2000]

Thank you for your question. The concept of causality in the context of Fourier transforms can be confusing, so let me try to explain it in more detail.

Causality refers to the principle that the cause of an effect must precede the effect in time. In other words, a system cannot respond to a stimulus before it has occurred. This applies to the Fourier transform as well.

Let us consider a function E(t) that represents some physical quantity, such as the electric field at a point in space. The Fourier transform of this function, denoted by ̂E(ω), represents the frequency components of E(t). Now, if we take the inverse Fourier transform of ̂E(ω), we should get back the original function E(t).

In other words, if we know the frequency components of E(t), we should be able to reconstruct the function E(t) at any point in time. However, this reconstruction should not depend on any future values of E(t). This is what is meant by "E(t1), as evaluated from inverse Fourier transform formula using ̂E(ω), does not depend on E(t2) for t2>t1".

To understand this concept better, let us consider an example. Suppose we have a function E(t) that is zero for all negative values of t and has a peak at t=0. In other words, the function E(t) represents a pulse of energy that is released at t=0. The Fourier transform of this function, ̂E(ω), will have non-zero values only for positive frequencies.

Now, if we take the inverse Fourier transform of ̂E(ω), we should get back the same function E(t). However, if we change the value of E(t) at some point in the future, say t=1, the inverse Fourier transform will still give us the same function E(t) with a peak at t=0. This is because the inverse Fourier transform does not depend on any future values of E(t).

In summary, the Fourier transform does not violate causality because the inverse Fourier transform does not depend on any future values of the function E(t). It only depends on the frequency components of E(t) which are present at the time of the transform.

I hope this explanation helps you understand the concept better. If you have any further questions, please do not hesitate to ask.[Your