The Laplace transform of a signal x(t), x(t) \stackrel{L}{\longrightarrow} X(s)=\int_{-m}^{m} x(t) d^{-n} d t is a generalization of the continuous-time Fourier transform that is useful for studying CTsignals and system.

When s= ju , i.e. the Laplace transform reduces to the CTFT. Most of the times, the Laplace transform can be represented as a ratio of polynomials in s: \boldsymbol{X}(s)=\frac{N(s)}{D(s)} which is also known as rational transforms. Rational transforms can be completely determined by the roots of the polynomial N(s) and D(s), known as zeros and poles. 1. pole-zero diagram.A pole-zero diagram displays the "poles" and "zeros" of the rational transform byplacing an 'x' at each pole location and an 'o' at each zero location in thecomplex s-plane.Poles and zeros can be found out by using roots function in matlab, i.e. \begin{aligned} & \text{^^20Trangfer^^20functian?^^20} \\ & \mathrm{H}(s)=\frac{s-1}{s^2+3s+2}\end{aligned} Using the method given above, find out the zeros and poles of the following system functions and plot them: \begin{array}{l} \text { (1). } H(s)=\frac{s+5}{s^{2}+2 s+3} \\ \text { (2). } H\left(s+\frac{2 s^{2}+5 s+12}{s^{2}+2 s+1 B}\right. \\ \text { (3). } H(s)=\frac{2 s^{2}+5 s+12}{\left.6^{2}+2 s+1 B\right)(s+2)} \end{array} Graph the ROC of each system function manually on your figures. (You can also graph ROC superimposed on your pole-zero diagram if you would.) EECS 360 Lab 9 Laplace Transform