The Fourier series representation of an aperiodic signal

[bilalkadri]

Fourier series

I have a confusion about the fourier series representation of an aperiodic signal.Some says that there is no fourier series representation of an apeiodic series but some says that it's an infinite series becuase the aperiodic signal have all the frequency components....so wats the right answer

 

[flatulent]

Re: Fourier series

The fourier series is for periodic signals. The fourier transform is for all other signals.

 

[zorro]

Re: Fourier series

Hi friends,

Under some (rather wide) conditions, a (non periodic) signal defined in a finite interval can be represented in a Fourier series. When this interval is expanded from -Infinity up to Infinity, the Fourier series becomes the Fourier integral (Fourier transform).

Let f(t) a function defined on an interval (a,b). Fourier stated that this funcion has an expression in Fourier series under some conditions, and which are the convergence proprierties of this series. If you expand this series, you obtain f(t) inside the interval (a,b) [except at the points where the series does not converge], and outside this interval you obtain the periodical expansion of f(t); this is obvious because the terms of the series are periodical.

It's a pity that many engineering texts treat Fourier series only for periodical signals. Many math texts treat Fourier series also in the above-mentioned vay.
The use of Fourier series for nonperiodic signals plays a fundamental role, for example, in the deduction (or proof) of the samplng theorem, in image coding, etc.

If I'm not confused, Fourier introduced his series in his treatise on the analytical theory of heat (ca. 1800). The problem he analyzed was the diffusion of heat in a finite bar. This is a separable partial differential equation, whose solution in the spatial part uses the Fourier series on a finite interval (the bar length).

Regards

Z

 

[me2please]

Re: Fourier series

Confirm flatulent's answer.

The solution of a differential equation within bounded domain only concern the representation of solution within the domain of interest. How it will behave outside the domain is irrelevant. If it is extended to infinity as = 0 (period = infinity), the representation is Fourier transform. If it is extended as periodic of the bounded interval, it will be represented as Fourier series.

 

[raj_rohit10]

Re: Fourier series

Hi,
To give a simple solution to this problem any signal can be written as sum of its even components and odd components. This is what is reflected when splitting the signal into its fourier series. Now coming to the case of aperiodic signal as the signal has its even and odd compnents spread the series can be writen as an infinite expantion.

 

[shangwa]

Fourier series

Hi,
4 combinations of signals

Continuous+Aperiodic-Fourier Transform
Continuous+Periodic-Fourier Series
Discrete+Aperiodic-Discrete Time Fourier Transform
Discrete+Period-Discrete Fourier Transform

from Steven's DSP book

 

[a_aziz]

Fourier series

hi.
for this reason you can reffer to book of Signals And Systems from Oppenheim A. V as well as Digital signal processing from him.