Fourier Transforms is Inflicting Pain

[Nano-Passion]

So I'm doing an internship this summer and one of the things I have to be acquainted with is Fourier Transforms.

My adviser gave me a simple example with that of sin(x). He said that what the Fourier Transform does is transform a time domain signal into one of a frequency signal. Essentially, he added, the peak of a Fourier Transform is that of the greatest match of the frequency to the function sin(x)--where points things that are closest to x-axis are the worst match.

But since I've started this assignment earlier today, I've noticed that Mathematica does not plot Fourier Transform of functions such as sin(x) that have constant frequencies and periodicity for all x. Mathematica plots things such as sin(x)/x where the frequency isn't constant throughout the x-axis. Upon doing some research on FT, I suspect this is because Fourier Transform works for things that vary, not for functions that have constant periodicity. But I don't know any better so I'm asking for your input.

It would also be helpful if someone can clarify the whole concept of Fourier Transforms and how it relates to statistics and data analysis.

 

[Muphrid]

It's probably not plotting the Fourier transform of $\sin x$ because only two frequencies contribute (1 and -1), and everywhere else the transform is zero.

You can imagine the Fourier transform as decomposing a signal into a sum of many sine and cosine functions. The Fourier amplitude $F(\omega)$ tells you what the amplitude of that frequency sine or cosine wave contributes to the signal.

 

[Nano-Passion]

It's probably not plotting the Fourier transform of $\sin x$ because only two frequencies contribute (1 and -1), and everywhere else the transform is zero.

You can imagine the Fourier transform as decomposing a signal into a sum of many sine and cosine functions. The Fourier amplitude $F(\omega)$ tells you what the amplitude of that frequency sine or cosine wave contributes to the signal.

I had trouble running other Fourier Transforms as well, but I'm not sure why.

I've attached a file with some of the Fourier Transforms that I attempted to plot. It would be greatly appreciated if you can take a look at it!

 

[utkarsh1]

sin(x)/x is called sinc x. Its different than sine. Sinc(x) is most important function in communication systems and filters. because theoretical impulse response for "best" filter is a sinc.

 

[Nano-Passion]

sin(x)/x is called sinc x. Its different than sine. Sinc(x) is most important function in communication systems and filters. because theoretical impulse response for "best" filter is a sinc.

Hm interesting. So does sin(x) give itself to Fourier transform methods or am I doing something wrong?

 

[Mute]

Hm interesting. So does sin(x) give itself to Fourier transform methods or am I doing something wrong?

How are you trying to determine the Fourier transforms in Mathematica?

sin(x) does indeed have a Fourier transform, but it is a special kind of Fourier transform. For example, if you try to calculate the Fourier transform in mathematica via the command

Integrate[Exp[-I*2*Pi*k*x]*Sin[x],{x,-Infinity,Infinity}]

you will not get a result. Mathematica will tell you the integral does not converge. This is because in the regular integral sense the integral indeed does not converge. This is because both Sin(x) and Exp(ikx) oscillate as x tends to infinity, but neither decays, so the integral does not converge. However, if you try to transform via

FourierTransform[Sin[x],x,k]

I believe you should get a result that involves the Dirac Delta function. Did you try this way?

Now, if you tried the same two commands with Sin[x]/x, both should give you the same result, I think. Why is this? It's because Sin[x]/x decays as x -> Infinity, so the integral does converge in the usual sense.

 

[Chaitu662]

Even my internship is on Fourier Transforms.

A part of the work is to find the frequency component of the signal which has the maximum amplitude (power density). Right now, I'm performing an fft in Matlab and searching for the maximum peak.

The fft is done to do just this and nothing else. An fft of about 1024 samples would consume a lot of time and would drain battery power.

Is there any simpler way to do this?

Thanks,
Chaitanya.