A doubt related to infinitesimals in continuous fourier transform.

[dexterdev]

Hi all,
Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.)
Now I have a doubt on CTFT (continuous time Fourier transform).

This is a concept I got from my friend.The Fourier series lists particular frequencies, while the Fourier transform is the frequency density function.

I got the former idea but not the later. How do I interpret a Fourier transform plot. Suppose at frequency F if the magnitude spectrum gives X(F) ,do that mean there is a sinusoid with frequency component F with amplitude proportional to X(F)? Is Fourier transform also a frequency density function, if so how we interpret it?

 

[mathman]

Fourier series is used for periodic functions. Fourier integral is for integrable (in some sense) functions over infinite line. (The description is somewhat simplified, but this should give you an understanding of the distinction).

 

[dexterdev]

Thanks for the reply. In the case of a Fourier series plot, the sticks imply sinusoids with corresponding amplitude. But to represent a sinusoid in Fourier transform plot we require a shifted delta function (defined with area equalling amplitude of the particular sinusoid). So how do I interpret a complicated waveform's Fourier transform plot with no delta functions it. This is where infinitesimals come to play. How to read those plots. If it is a low pass signal bandlimited to W Hz , is it right to say that this signal contains many sinusoids from 0 Hz to W Hz with respective amplitudes?

 

[jasonRF]

I think that Parseval's relation helps here:
$$\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df$$
The left hand side is the total energy in the signal, so you can think of the square of the Fourier transform as an energy density, and indeed the square is usually what is plotted. In particular, $|X(f)|^2 df$ is the amount of energy between $f$ and $f+df$.

jason