# Initial value theorem

• Initial value theorem:

• Final value theorem:

, if all poles of  are in the left half-plane.

The final value theorem is useful because it gives the long-term behaviour without having to perform partial fraction decompositions or other difficult algebra. If a function’s poles are in the right-hand plane (e.g.  or  ) , this formula is not useful.

### Proof of theLaplacetransform of a function’s derivative:

In signal and system we use directly the formula oflaplacetransform but for differentiation equation we need some another formula which is derived below:

It is often convenient to use the differentiation property of theLaplacetransform to find the transform of a function’s derivative. This can be derived from the basic expression for aLaplacetransform as follows:

yielding

and in the bilateral case,

The general result

where fn is the n-th derivative of f.

#### Fourier transform

In signal and system, fourier transform is same as laplacetransform .But it does not have ROC(region of convergence). The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument s = iω or s = 2πfi :

The above relation is valid as stated if and only if the region of convergence (ROC) of F(s) contains the imaginary axis, σ = 0. For example, the function f(t) = cos(ω0t) has a Laplace transform F(s) = s/(s2 + ω02) whose ROC is Re(s) > 0. As s = iω is a pole of F(s), substituting s = iω in F(s) does not yield the Fourier transform of f(t)u(t), which is proportional to the Dirac delta-function δ(ω-ω0).

However, a relation of the form

holds under much weaker conditions..

#### Z-transform

The unilateral or one-sided Z-transform is simply theLaplacetransform of an ideally sampled signal with the substitution of

where  is the sampling period (in units of time e.g., seconds) and

is the sampling rate (in samples per second or hertz).

Let

be a sampling impulse train (also called a Dirac comb) and

be the continuous-time representation of the sampled

are the discrete samples of  .

The Laplace transform of the sampled signal  is

This is precisely the definition of the unilateral Z-transform of the discrete function

with the substitution of

Comparing the last two equations, we find the relationship between the unilateral Z-transform and theLaplacetransform of the sampled signal: