Problem 1 (20 points) Given a signal x(t) = e-지디 1) Plot the signal x(t) in...
Laplace Transform Problem 3. (15 points) Given f(t) = 4e-2tu(t) + 29u(-t) a) Using the Laplace Transform table 9.2 find the bilinear Laplace transform, F($) and sketch the region of convergence (ROC) in the s-plane showing all poles. State the ROC as an inequality. b) Another function is added so that fa(t) = 4e-2tu(t) + 7u(-t) – 10e-10t u(-t). Find the Bilinear Laplace Transform of fa(t) and sketch the region of convergence in s-plane also showing all the poles. State...
3. Given a signal x(t) = e-stuſt – 1), calculate its Laplace transform X(s) with ROC.
4. Laplace Transform. (15 pts) Find the Laplace Transform of the following signals and sketch the corresponding pole-zero plot for each signal. In the plot, indicate the regions of convergence (ROC). Write X(s) as a single fraction in the form of DO (a) (5 pts)-(t-e*ta(t) + e-8tu(t). Show that X(s) =は,,늚. with ROC of Re(s) >-6. (b) (5 pts)-(t) = M(-t) +Au(-t). (c) (5 pts)-(t) 6(t)-a(-t). (s+6) (s+8)
Please answer all questions with math detail 3. (21 points) Laplace Transform (a) (15 points) Find the Laplace transforms of the following signals and determine their region of convergence sinwot)-iu i. f(t) -i, e-2(t-3 2<t otherwise (b) (6 points) The Laplace transform of a causal signal x(t) is given by X (s) = s2 , ROC: Re{s) > -1 Which of the following Fourier transforms can be obtained from X(s) without actu- ally determining the signal x(t)? In each case,...
1. Laplace Transform. (10 pts) Find the Laplace Transform of the following signals and sketch the corresponding pole-zero plot for each signal. In the plot, indicate the regions of convergence (ROC). Write X(s) as a single fraction in the forin of (a) (2 pts) z(t) = e-Mu(t) + e-6tu(t). Show that X(s)-AD10 (b) (4 pts)-(t) = e4ta(-t) + e8ta(-t). (c) (4 pts) (t)-(t)-u(-t) . with ROC of Re(s) >-4. (s+4)(8+6)
Please answer the A,b, and C in clear handwriting. Consider the following cases involving sinusoids: (a) 3.3 Find the Laplace transform of r(t) = sin(2n)[u(I)-a(1-1)I and its region of convergence. Carefully plot yt). Determine the region of convergence of Y(s) (b) A very smooth pulse, called the raised cosine, is x(t) = 1-cos( 2π 1), O 1 1, and zero elsewhere. Find its Laplace transform and its corresponding region of convergence. (c) Indicate three possible approaches to finding the Laplace...
Laplace Transform 5. Given a causal LTI system with pole-zero cancellation such as H(s)= S+1 what is the region of convergence and why. (5+1)(3+2) i. ROC = undefined ii. ROC = Re(s) > 0 iii. ROC = Re(s) >-2 iv. ROC = Re(s) >-1
Signal and Systems Problem 2 (3 points) Given the unilateral Laplace transform of the impulse response for a causal system H(s) Determine h(t) the impulse response? Hint synthetic division! (s+10) 40 t-10 Problem 3 a) (2 points) What is the initial value of time function f(t) corresponding to the one sided Laplace Transform F(s) = (i.e. f(t) is causal) s(s+10)(2+4) lim f(t) = 0 40 lim f(t) = 1. t-0 10 x 4 lim f(t) = 0 t-0 lim f(t)...
Let (t) 2eeu(t). (Recall that u(t) is the unit step function.) Use the Laplace trans- form integral to compute (not look up from a table) this signal's Laplace transform X(s) and find its region of convergence (ROC). Draw a sketch that shows the pole(s) and ROC of X(s)
part c) Figure 1 5 ma Pag c) The zero-pole diagram of the Laplace transform of y(t), Y(s), is shown in fig. 2 jw Figure 2 [Please turn over] Page 2 of 9 Determine and justify the region of convergence (ROC) ifit is known that Y(Go), that is the Fourier transform of y(t), exists. 5 marks/ Figure 1 5 ma Pag c) The zero-pole diagram of the Laplace transform of y(t), Y(s), is shown in fig. 2 jw Figure 2...