Properties of ROC of Laplace Transform | Dirichlet’s conditions of Fourier series | Fourier series and Fourier transform

Properties of ROC of Laplace Transform

    
    1. The ROC of X(s) consists of strips parallel to the ># axis in the s-plane.  

    2. The ROC does not contain any poles.  

    3. If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s plane.  

    4. It x(t) is a right sided signal, that is x(t) = 0 for t<t0<∞ then the ROC is of the form Re(s)> max , where max equals the maximum real part of any of the poles of X(s).  

    5. If x(t) is a left sided, that is x(t) = 0 for t>t1> -∞, then the ROC is of the form Re(s)<   min , where  min equals the minimum real part of any of the poles of X(s).  

    6. If x(t) is a two sided signal, than the ROC is of the form  1<Re(s)<  2.

Dirichlet’s conditions of Fourier series

    (i)The function x(t) should be single valued within the interval T0  

    (ii) The function x(t) should have atmost a finite number of discontinuities in the interval T0

    (iii) The function x(t) should have finite number of maxima and minima in the interval T0  

    (iv) The function should have absolutely integrable. 

Difference between Fourier series and Fourier transform

S.NO	Fourier Series	Fourier Transform
1	Fourier series is calculated  for periodic signals.	Fourier Transform is calculated for  non-periodic as well as periodic  signals.
2	Expands the signals in time  domain.	Represents the signal in frequency  domain
3	Three types of Fourier series  such as trigonometric, Polar  and Complex Exponential	Fourier transform has no such types.