Purpose of fourier series and fourier transform

In summary, the Fourier transform is used to transform a signal from the time domain to the frequency domain. The Fourier integral is used to understand the effects of limited bandwidth on an input signal.
  • #1
Jncik
103
0
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?

[tex] X(j\omega)= \int_{-\infty}^{+\infty} x(t) e^{-j \omega t} dt[/tex]

we have a function in the frequency domain, so in the x-axis we will have values for [tex]\omega[/tex] 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 :)
 
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  • #2
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.
 
  • #3
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 cn represents how much the nth harmonic contributes and its relative phase, G(ω) represents the amplitude and phase of the frequency component ω.
 
  • #4
rude man said:
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.
 
  • #5
vela said:
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
 
Last edited by a moderator:

What is the purpose of Fourier series?

The purpose of Fourier series is to decompose a periodic function into a sum of simpler, sinusoidal functions. This allows us to analyze and understand the behavior of the original function in terms of its individual frequency components.

What is the difference between Fourier series and Fourier transform?

Fourier series is used for periodic functions, while Fourier transform is used for non-periodic functions. Fourier transform also provides a continuous frequency spectrum, while Fourier series only provides a discrete frequency spectrum.

How is Fourier transform used in signal processing?

Fourier transform is used in signal processing to analyze and manipulate signals in the frequency domain. This allows us to filter out unwanted frequencies, compress or expand signals, and extract useful information from the signal.

Why is the Fourier transform important in mathematics and engineering?

The Fourier transform is important in mathematics and engineering because it provides a powerful tool for analyzing and understanding complex functions and signals. It is also widely used in fields such as image and sound processing, communications, and control systems.

What are some real-world applications of Fourier series and Fourier transform?

Some real-world applications of Fourier series and Fourier transform include audio and image compression, noise reduction in audio and image signals, analysis of electric circuits, and solving differential equations in physics and engineering.

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