##### 1.13 LAPLACE TRANSFORMS

In order to find out the response of linear waveshaping circuits to the various types of inputs discussed above, we usually formulate a differential equation and solve the differential equation using Laplace transforms. The Laplace transform of any function in the time domain *t* is given by:

where, *s* = *σ* + *jω*. The lower limit is taken as zero instead of –∞ because the convergence factor for *e*^{−σ t} will diverge for *t* → −∞; therefore, all the information before *t* = 0 is ignored. A function is said to be Laplace transformable when:

The Laplace transform permits us to go from the time domain to the frequency domain whereas the inverse Laplace transform allows us to go from the frequency domain to the time domain.

#### 1.13.1 Basic Properties of Laplace Transformation

**Property 1**: The Laplace transform of the sum or difference of time functions is equal to the sum or difference of the Laplace transforms of the individual time functions.

**Property 2**: The Laplace transform of the product of a constant and a time function is equal to the product of the constant and the Laplace transform of the time function.

**Property 3**:If *F*_{1}(*s*) and *F*_{2}(*s*) are the Laplace transforms of *f*_{1}(*t*) and *f*_{2}(*t*), respectively, then:

*L*[*a*_{1}*f*_{1}(*t*) + *a*_{2}*f*_{2}(*t*)] = *a*_{1}*F*_{1}(*s*) + *a*_{2}*F*_{2}(*s*)

where *a*_{1} and *a*_{2} are arbitrary constants.

**Property 4: Scaling:**

where *a* is an arbitrary constant.

**Property 5: Time Shifting:** The Laplace transform of a time function *f (t*) delayed by time *t*_{0} is equal to the Laplace transform of *f (t*) multiplied by *e*^{−st0}, i.e.,

**Property 6: Frequency Shifting:** The multiplication of *f* (t) by *e*^{−at} has the effect of replacing *s* by (*s* + *a*) in the Laplace transform

**Property 7: Convolution:** This property states that the convolution of two real functions is equal to the multiplication of their respective functions. If

*L*[*f*_{1} (*t*)] = *F*_{1} (*s*) and *L*[*f*_{2} (*t*)] = *F*_{2} (*s*)

By convolution, it can be written as:

The mathematical expression given in Eq. (1.88) is known as the convolution theorem. The word *convolve* means “to revolve continuously”. The two functions *f*_{1}(*t*) and *f*_{2}(*t*) are multiplied in such a way that one is continually moving with time (say *λ*) relative to the other. Laplace transforms are very useful for solving problems in science and engineering. Some useful Laplace transforms are given in Appendix A.