Question about the second Dirichlet condition

[DrDolittle]

dirichlet conditions explanation

Dirichlet gave the necessary conditions for existence of a fourier spectra for any signal
the second condition tells that number of maxima and minima in a given period should be finite.This actually fills the void created by the first condition
My doubt is how is the finite number of maxima and minima ensures less discontinuities?

Thanks in advance

 

[cedance]

dirichlet conditions arabic

Dirichlet gave the necessary conditions for existence of a fourier spectra for any signal
the second condition tells that number of maxima and minima in a given period should be finite.This actually fills the void created by the first condition
My doubt is how is the finite number of maxima and minima ensures less discontinuities?

Thanks in advance

more the maximas and minimas.. the curve oscillates more. i mean there are more maximas and minimas which are to be present in the signal... that means that there will be more shifts from teh maximum to the minimum for this signal when compared to the signal with finite maximas and minimas.. but u would know that when there is a sudden change from a high value to low value... then that is interpreted as a very high frequency.. which results in spurious components in the same domain... hence recovering the signal is not possible,, hope tis helps,,,

regards,
arunmit168.

 

[me2please]

fourier transform dirichlet conditions

number of maxima and minima in a given period should be finite

To my knowledge, this condition is an iterpretation of a more general Jordan's condition of bounded variation.

Dirichlet's condition is to have finite number of "jump discontinuities" for piecewise smooth function.

Then, Jordan generalized it to "bounded variations" which includes "jump discontinuities" and "continuosly oscillating" part.

From my understanding, Dirichlet's convergence depends on the property that you can cut the function into small monotonic(either increase or decrease) section, so that the average value at any point [f(x+)+f(x-)]/2 can be calculated. This value is well defined also at jump discontinuities and is the converged value of fourier transform of that point. But for unbounded variations, this average is not well defined for it doesn't matter how small you cut, you cannot guarantee to have a monotonic piece.

Hope it helps. 8)

 

[DrDolittle]

dirichlet conditions in signal

Oh yeah,I have read jordan's contour and residue theorem related to convergece of a function.
Iam thinking of an explanation like this.
when the signal dont have finite number of maxima and minima the signal is said to have a high frequency signal.
when such a high frequency signal is convoluted with a square pulse it suppresses high frequency components which is not a desired eventuality.If we have infinite maxima and minima then the square pulse should be of negligible width which is not feasible.Hope i haven't confused.

Regards
drdolittle