Why is Fourier Transform of a Real Function Complex?

[LunaFly]

Homework Statement

Find the Fourier transform F(w) of the function f(x) = [e^-2x (x>0), 0 (x ≤ 0)]. Plot approximate curves using CAS by replacing infinite limit with finite limit.

Homework Equations

F(w) = 1/√(2π)*∫ f(x)*e^-iwxdx, with limits of integration (-∞,∞).

The Attempt at a Solution

I solved the integral and found the Fourier transform of f(x) to be:

F(w) = 1/√(2π) * (2+iw)/(4+w²).

I am pretty confident in my solution (but feel free to correct me if I'm wrong!). Where I have an issue is this.. F(w) is complex, so how do I plot it? Do I only plot the real part? How do the real and complex parts of F(w) relate to f(x)?

I may have this question because I am still having a hard time fundamentally understanding the relationship between F(w) and f(x), so any information on this topic would be welcome. Describing F(w) as a function that determines the coefficient (contribution) of e^iwx in f(x) makes some sense, (as explained in the below link):

http://math.stackexchange.com/questions/1002/fourier-transform-for-dummies (answer #2, with plot of sines).

However I still am confused how a complex-valued F(w) is tied in.

Thanks!

 

[vela]

Homework Statement

Find the Fourier transform F(w) of the function f(x) = [e^-2x (x>0), 0 (x ≤ 0)]. Plot approximate curves using CAS by replacing infinite limit with finite limit.

Homework Equations

F(w) = 1/√(2π)*∫ f(x)*e^-iwxdx, with limits of integration (-∞,∞).

The Attempt at a Solution

I solved the integral and found the Fourier transform of f(x) to be:

F(w) = 1/√(2π) * (2+iw)/(4+w²).

I am pretty confident in my solution (but feel free to correct me if I'm wrong!). Where I have an issue is this.. F(w) is complex, so how do I plot it? Do I only plot the real part? How do the real and complex parts of F(w) relate to f(x)?

It would probably be more useful to plot the magnitude and phase of $F(\omega)$.

I may have this question because I am still having a hard time fundamentally understanding the relationship between F(w) and f(x), so any information on this topic would be welcome. Describing F(w) as a function that determines the coefficient (contribution) of e^iwx in f(x) makes some sense, (as explained in the below link):

http://math.stackexchange.com/questions/1002/fourier-transform-for-dummies (answer #2, with plot of sines).

However I still am confused how a complex-valued F(w) is tied in.

Thanks!

The expression for a sine wave of frequency $\omega$ is $A \sin(\omega x + \phi)$. You need to specify two numbers, $A$ and $\phi$, to describe the wave. $|F(\omega)|$ corresponds to $A$, and the complex phase of $F(\omega)$ basically corresponds to $\phi$.

 

[Buzz Bloom]

Hi LunaFly:

Here is a hint.

e^-iwx = cos wx + i sin wx

Regards,
Buzz

 

[LunaFly]

It would probably be more useful to plot the magnitude and phase of $F(\omega)$.

The expression for a sine wave of frequency $\omega$ is $A \sin(\omega x + \phi)$. You need to specify two numbers, $A$ and $\phi$, to describe the wave. $|F(\omega)|$ corresponds to $A$, and the complex phase of $F(\omega)$ basically corresponds to $\phi$.

Thank you Vela for the insight. So the Fourier transform F(w) is more of a phasor relating the "amount" of oscillatory function e^-iwt of frequency w present in the function f(x) than a constant coefficient.. Interesting!

 

[LunaFly]

Hi LunaFly:

Here is a hint.

e^-iwx = cos wx + i sin wx

Regards,
Buzz

Thanks for the hint Buzz. I am guessing you are pointing to the fact that when applying a Fourier transform to a function f(x), we are introducing a function with non-real terms, namely e^-iwx. Or you may be hinting at the fact that the inverse Fourier transform is a continuous linear combination of complex terms weighted by F(w) that is equal to f(x), meaning the weight function F(w) must have complex terms in order to equal a real function f(x). Maybe? Thanks regardless.

 

[Buzz Bloom]

Hi LunaFly:

Actually I had a different thought than either of the two ideas in your post. Here is a second hint.

Any real function f(x) over [-∞,+∞] can be separated into two parts: f(x) = s(x) + a(x)
s(x) is symmetric, i.e., s(-x) = s(x)
a(x) is anti-symmetric, i.e., a(-x) = -a(x).
Now consider the transform: F(w) = S(w) + A(w).

Now combine that idea with my first hint.
Good luck.

Regards,
Buzz