Laplace transform is an integral transform there are many integral transforms and Laplace transform is one out of them in the previous chapter we had discussion on Fourier transform and like Laplace transform Fourier transform is also an integral transform so what do we mean by integral transform they transform having this form is known as integral transform in this there are three functions the first one is G alpha it is the function of alpha and this is the output the second function is f T it is the function of time T and this is the input and the third function is the function of both alpha and T and this is known as integral kernel we know the Fourier transform of signal XT is equal to integration minus infinity to infinity time domain signal XT multiplied to e power minus J Omega T DT now compare the Fourier transform with the general integral transform you will find both are having the same form in case of Fourier transform you can see X Omega is the output and it is the function of Omega XT is the input and it is the function of time T E power minus J Omega T is the integral kernel and it is function of both Omega and time T the Laplace transform is also having the same form I will give you the Laplace transform after some time but first we will know about his discovery laplace transform is named after its Discoverer pierre-simon laplace he was a French mathematician and an astronomer now we will move to the next point Laplace transform is similar to the Fourier transform or more precisely we can see that Laplace transform is the general case of Fourier transform I will give you the Laplace transform first and then we will understand how Laplace transform is the general case of Fourier transform let's see there is a time domain signal ft and its corresponding Laplace transform its corresponding Laplace transform is represented by capital f inside the bracket s so FS which is the Laplace transform is equal to integration minus infinity to infinity the time domain signal ft multiplied to e power minus s T DT now compare the Laplace transform with the general integral transform you will find FS is the output and it is the function of variable s F T is the input and it is the function of T E power minus s T is the integral kernel and it is the function of both s and T now compare the Laplace transform with the Fourier transform you will find there is one change in place of J Omega we are having s and s is equal to Sigma plus J Omega so here we are having a complex variable in the exponential and by doing this we are increasing the possibilities in this case we were having Omega here Omega is a real variable but here we are having s as is a complex variable so by replacing J Omega by s we are generalizing the Fourier transform and we the drawback of Fourier transform Fourier transform do not exist for non absolutely integrable signals but Laplace transform do exist for non absolutely integrable signals this means Laplace transform do exist for neither energy nor power signals therefore compared to Fourier transform Laplace transform can be applied to a broader range of signals and the reason is we are replacing J Omega by s now let's talk about the complex variable s we are having here it is equal to Sigma plus J Omega where Sigma is known as the damping factor it is known as the damping factor and it tells us about the stability and we know Omega is the angular frequency it is the angular frequency in radians per second you can see the Laplace transform is a complex function and it is the complex function of a complex variable s here s is the complex variable or we can say it is complex frequency on the other hand in case of Fourier transform Fourier transform is a complex function of a real variable Omega is a real variable and s is a complex variable now we will talk about bilateral Laplace transform and unilateral Laplace transform the Laplace transform we have written here is known as bilateral Laplace transform by lateral transform we are integrating from minus infinity to infinity and when we perform the integration from minus infinity to infinity the Laplace transform is known as bilateral Laplace transform but if we have a time domain signal ft which is defined when T is greater than 0 or equal to 0 then the Laplace transform will be equal to integration 0 to infinity ft multiplied to E power minus stdt here we are integrating from 0 to infinity and this Laplace transform is known as uni lateral Laplace transform when we see Laplace transform in general it means the unilateral Laplace transform but in this course in signal hand system we will deal with bilateral Laplace transform and when we deal with bilateral Laplace transform it becomes important to mention region of convergence along with the Laplace transform we will talk about what is region of convergence or in short roc-roc is the region in the S plane s is having two variables Sigma and J Omega and when we include all the possible values of Sigma and J Omega we will get a plane and the plane is known as has plane and in s plane the region in which the Laplace transform is finite is known as ROC and outside roc Laplace transform is infinite so we will talk about piracy in the coming presentations we will understand his properties and we will also calculate ROC in the questions having the calculation of Laplace transform so I hope you know understand what is bilateral Laplace transform and unilateral Laplace transform now we will talk about inverse Laplace transform like inverse Fourier transform we have inverse Laplace transform we can have the time domain signal ft using the Laplace transform FS so f T is equal to 1 over 2 pi J integration from Sigma minus J Omega to Sigma plus J Omega Laplace transform offense multiplied to E power s T D s so this is the inverse Laplace transform but we will not use this formula because we have one simple method to obtain the time domain signal using its corresponding Laplace transform we will use the partial fractions using the partial fractions and then comparing the obtained result with the standard Laplace transforms we can have the time domain signal F T the process will be clear when we will take some examples so I think this is all for the introduction alot of the Laplace transform and if you have endowed you mask in the comment section I will end this lecture here you in the next one you [Music]