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Explain Gibbs phenomenon .Also explain conditions necessary for the convergence of Fourier Series.

Mumbai University > EXTC > Sem 4 > Signals and Systems

Marks : 7M

Year : Dec 2014

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1. The exponential form of Fourier series of a continuous time periodic signals x(t) is given by,

The above equation is frequency domain representation of the signal x(t) as a sum of infinite series with each term in the series representing a harmonic frequency component.

2. When the signal x(t) is reconstructed with only N number of terms of the infinite series, the reconstructed signal exhibits oscillations.

3. Consider a periodic square pulse signal shown in fig along with the reconstructed signal using N-terms of Fourier series.

4. It can be observed that the reconstructed signal exhibits oscillations and the oscillations are compressed towards points of discontinuities with increasing value of N. Also it can be observed that, at the points of discontinuity, the Fourier series converges to average value of the signal on either side of discontinuity.

5. This phenomenon is called as Gibbs phenomenon and the oscillations are called Gibbs oscillations.

6. Conditions for ROC of Fourier series :

Condition 1 = Over any period, x (t) must be absolutely integrable $\int_0^T | x(t) | dt \lt \infty$ .

Condition 2 = In any finite interval, x (t) is of bounded variation; that is no more than a finite number of maxima and minima during any single period of the signal.

Condition 3 = In any finite interval of time, there are only a finite number of discontinuities. Further, each of these discontinuities are finite.