Purpose of fourier series and fourier transform

[Jncik]

Hi I'm trying to understand what we mean when we say that the Fourier transform is used to transform a signal from the time domain to the frequency domain and what we actually have in the frequency domain.

In Fourier series we are actually using a different representation of the signal in terms of the sum of harmonically related sinusoids which is very important in signal and system analysis

but in Fourier transform, we use it mostly for aperiodic signals right?

but what exactly does this function represent?

$$X(j\omega)= \int_{-\infty}^{+\infty} x(t) e^{-j \omega t} dt$$

we have a function in the frequency domain, so in the x-axis we will have values for $$\omega$$ while on the y-axis this function, which represents what exactly? and why is it so important?

can you please explain in the most simple words? thanks :)

 

[rude man]

First, it should be X(w), not X(jw). X is always real.

The Fourier integral does for a single pulse what the Fourier series do to a periodic function going from -∞ to +∞. By graphing X(w) vs. w you get a picture of the relative frequency components in x(t), just as the coefficients of the harmonics in a Fourier series give you the magnitude of each freq. component. In the case of the Fourier integral, though, you have a CONTINUOUS spectrum, whereas the series give a LINE spectrum (harmonics of the basic frequency 1/period only).

The Fourier integral is especially useful if you want to know the effects of limited bandwidth on an input signal. The inverse F.I. let's you do that elegantly by just replacing the infinity limits by + and -w0, where w0 is the radian bandwidth.

 

[vela]

It's the same idea. Compare the exponential form of the Fourier series to the Fourier integral:
\begin{align*}
f(t) &= \sum_{n=-\infty}^\infty c_n e^{in\omega t} \\
g(t) &= \int_{-\infty}^\infty G(\omega) e^{i\omega t} \, d\omega
\end{align*}(Depending on your convention, there could be factors of 2 pi floating around.) Both are a sum of exponentials. For periodic signals, you only need the frequency components corresponding to multiples of the fundamental frequency. For an aperiodic signal, you need all of the frequencies. Just as c_n represents how much the nth harmonic contributes and its relative phase, G(ω) represents the amplitude and phase of the frequency component ω.

 

[vela]

First, it should be X(w), not X(jw). X is always real.

I've seen some engineering texts where its written as X(jω).

X is generally not real as it contains both phase and amplitude information.

 

[rude man]

I've seen some engineering texts where its written as X(jω).

X is generally not real as it contains both phase and amplitude information.

X is always real. Fin d'histoire.

Or maybe not. I may have to eat crow on this one. Will check it out.

Anyway, for a rectangular pulse, for example, X is real, and writing X(jw) is clearly in error.

OK, I concede defeat. X is in general complex. It's real only if f(t) is even.

However, it is STILL wrong to write X(jw).

http://s-mat-pcs.oulu.fi/~ssa/ESignals/sig2_2.htm