2. (a) Write down X(z), the z-transform of x[n] = (1/3)"u[n] - 4^u[-n - 1] ` (b) Draw the pole/zero plot and shade the ROC.
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- Let x(n) be a causal sequence with z-transform X(z) whose pole-zero plot is shown below. Sketch the pole-zero plots and the ROC of the following sequence:1. If a system function has zeros at the origin of the z-plane then the system function is a properrational function. Why? 2. Can you obtain the z-transform of u[n] at z = 0? If you can, what is its value? If you cannot,why not? 3. What is the preferred method for obtaining the inverse z-transform of rational functions?Describe this method. 4. Explain why a one-sided z-transform is able to determine response to a LCCDE with initialconditions while the two-sided z-transform cannot. 5. Can any system with a rational system function be decomposed into a product of an allpasssystem and a minimum-phase system? If yes, then explain how?Evaluate the inverse z-transform of of the functions given below by the three methods described in the text: E(z)=0.5z/(z-1)(z-0.6)
- y''-4y=e-2t Find the general solution of the second order, constant coefficient, non-homogeneous linear differential equation given below by "Laplace transform"?1. Explain in not less than 7 sentences with mathematical equations the relationship of z-transform and convolution sum. 2.Some transforms in the z-transform pairs are exactly the same. How would did they differ in other parts? What would be its implication when solving z-transforms?c) Obtain Z-transform of (1-e-Ts )/s(1/s+1) d) The z-transform X (z) of a function X (k), is given by;X (z) =z(0.1065z +0.09021)/(z-1)(z2-1.5z+0.6967) Determine an expression for X (k), by finding the inverse z-transform function of X(z)
- Q.1 Find the z-transform f(z) and sketch the pole-zero plot with the ROC for each of the following sequences?Given an RC circuit in Figure 1 is modelled by the differential equation written as, ????+1???(?)=1???(?).If the input, ?(?)=0.8?−??(?) with initial condition ?(0−)=3?, by using the Laplace transform approach determine,a) overall output response, ?(?)b) zero input responsec) zero state responsethe following statements given, write if each statement is true or false. a. If the complex frequency domain function, F(s), is derived using a “One-sided” Laplace Transform, the inverse of that function, f(t), is only valid for t ≥ 0. b. When using an ideal transformer, if voltage increases from primary side to secondary side, current also increases from primary side to the secondary side. c. Initial value theorem states that the initial value of a time function f(t) can be obtained from its Laplace Transform F(s) by multiplying the transform by s and letting s approach zero. ..
- a) Based on your understanding, describe the use of Laplace-transform and z-transform in linear time invariant analysis.b) Find Laplace transform and sketch the associated ROC of following function,i. ?(?)=(?+1)?−??(?)ii. ?(?)=?−2?[?(?)−?(?−4)].c) Find z-transform and sketch the associated ROC of following function,i) ?[?]=(23)??[?]+(16)??[?]ii) ?[?]=?3??[?]Simplify given expression using SOP form. K(A,B,C,D) = Σ(0,1,3,7,2,8,10) d(A,B,C,D) = Σ(5,6,11,15)What will be the z transform of the DT Dirac delta function? Show detailed calculation