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Let (t) 2eeu(t). (Recall that u(t) is the unit step function.) Use the Laplace trans- form...
Laplace Transform Problem 3. (15 points) Given f(t) = 4e-2tu(t) + 29u(-t) a) Using the Laplace...
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...
16. Given f(t) = 2e-tu(t) + 4u(-t) a) Using the Unilateral Laplace Transform table and the...
16. Given f(t) = 2e-tu(t) + 4u(-t) a) Using the Unilateral Laplace Transform table and the procedure described in class and the text, determine the Bilinear Laplace Transform Fb (s) and sketch the region of convergence (ROC) in the s-plane showing poles. State the ROC as an inequality. b) Another function is added so that fa(t) = 2e-u(t) + 4u(-t) + 4e -0.5t u(t). Find the Bilinear Laplace Transform and sketch the region of convergence in s-plane also showing poles.
3. Let the Laplace transforms of signals (t) and y(t) be X(s) and Y(s) with appropriate...
3. Let the Laplace transforms of signals (t) and y(t) be X(s) and Y(s) with appropriate regions of convergence, respectively (a) Show that the Laplace transform of x(t) * y(t) is X(s)Y (s). What is the region of convergence? (b) Show that the Laplace transform of tx(t) is -dX(s)/ds with the same region of x(t) convergence as tn-1 1 for Re{sa} > 0. -at e (c) Show that the Laplace transform of 'u(t) is n 1)! (sa)" 1 for Refsa}...
4. Laplace Transform. (15 pts) Find the Laplace Transform of the following signals and sketch the...
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)
1. Laplace Transform. (10 pts) Find the Laplace Transform of the following signals and sketch the...
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)
2. (40 pts) Determine the Laplace transform (i.e. calculate it yourself without use of a Table),...
2. (40 pts) Determine the Laplace transform (i.e. calculate it yourself without use of a Table), identify the associated region of convergence, and sketch the pole-zero for the following time-domain signals: a. x)-ecos(120t)-u() b. x(t)-2[u(t)-u(t-5)
Use Definition 7.1.1. DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t ≥...
Use Definition 7.1.1.
DEFINITION 7.1.1 Laplace Transform Let f be a function defined for
t ≥ 0. Then the integral ℒ{f(t)} = ∞ e−stf(t) dt 0 is said to be
the Laplace transform of f, provided that the integral converges.
Find ℒ{f(t)}. (Write your answer as a function of s.) ℒ{f(t)} = (s
> 0)
Use Definition 7.1.1. DEFINITION 7.1.1 Laplace Transform et f be a function defined for t2 0. Then the integral is said to be the Laplace...
Please answer the A,b, and C in clear handwriting. Consider the following cases involving sinusoids: (a)...
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...
A continuous-time LTI system has unit impulse response h(t). The Laplace transform of h(t), also called...
A continuous-time LTI system has unit impulse response h(t). The
Laplace transform of h(t), also called the “transfer function” of
the LTI system, is
.
For each of the following cases, determine the region of
convergence (ROC) for H(s) and the corresponding h(t), and
determine whether the Fourier transform of h(t) exists.
(a) The LTI system is causal but not stable.
(b) The LTI system is stable but not causal.
(c) The LTI system is neither stable nor causal
8...
USE DEFINITION 1 TO DETERMINE THE LAPLACE TRANSFORM OF THE FOLLOWING FUNCTION. f(t)= e sin(t) Laplace...
USE DEFINITION 1 TO DETERMINE THE LAPLACE TRANSFORM OF THE FOLLOWING FUNCTION. f(t)= e sin(t) Laplace Transform Definition 1. Let f(t)be a function on [0,00). The Laplace transform of f is the function defined by the integral The domain of F(s) is all the values of " for which the integral in (1) exists.' The Laplace transform of fis denoted by both and ${/}. QUESTION 2. (3PTS) USE TABLE 7.1 AND 7.2 TO DETERMINE THE LAPLACE TRANSFORM OF THE GIVEN...
Laplace Transform
Problem 3. (15 points) Given f(t) = 4e-2tu(t) + 29u(-t) a) Using the Laplace Transform table 9.2 find the bilinear Laplace transform, F($) 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...
16. Given f(t) = 2e-tu(t) + 4u(-t) a) Using the Unilateral Laplace Transform table and the procedure described in class and the text, determine the Bilinear Laplace Transform Fb (s) and sketch the region of convergence (ROC) in the s-plane showing poles. State the ROC as an inequality. b) Another function is added so that fa(t) = 2e-u(t) + 4u(-t) + 4e -0.5t u(t). Find the Bilinear Laplace Transform and sketch the region of convergence in s-plane also showing poles.
3. Let the Laplace transforms of signals (t) and y(t) be X(s) and Y(s) with appropriate regions of convergence, respectively (a) Show that the Laplace transform of x(t) * y(t) is X(s)Y (s). What is the region of convergence? (b) Show that the Laplace transform of tx(t) is -dX(s)/ds with the same region of x(t) convergence as tn-1 1 for Re{sa} > 0. -at e (c) Show that the Laplace transform of 'u(t) is n 1)! (sa)" 1 for Refsa}...
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)
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)
2. (40 pts) Determine the Laplace transform (i.e. calculate it yourself without use of a Table), identify the associated region of convergence, and sketch the pole-zero for the following time-domain signals: a. x)-ecos(120t)-u() b. x(t)-2[u(t)-u(t-5)
Use Definition 7.1.1.
DEFINITION 7.1.1 Laplace Transform Let f be a function defined for
t ≥ 0. Then the integral ℒ{f(t)} = ∞ e−stf(t) dt 0 is said to be
the Laplace transform of f, provided that the integral converges.
Find ℒ{f(t)}. (Write your answer as a function of s.) ℒ{f(t)} = (s
> 0)
Use Definition 7.1.1. DEFINITION 7.1.1 Laplace Transform et f be a function defined for t2 0. Then the integral is said to be the Laplace...
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...
A continuous-time LTI system has unit impulse response h(t). The
Laplace transform of h(t), also called the “transfer function” of
the LTI system, is
.
For each of the following cases, determine the region of
convergence (ROC) for H(s) and the corresponding h(t), and
determine whether the Fourier transform of h(t) exists.
(a) The LTI system is causal but not stable.
(b) The LTI system is stable but not causal.
(c) The LTI system is neither stable nor causal
8...
USE DEFINITION 1 TO DETERMINE THE LAPLACE TRANSFORM OF THE FOLLOWING FUNCTION. f(t)= e sin(t) Laplace Transform Definition 1. Let f(t)be a function on [0,00). The Laplace transform of f is the function defined by the integral The domain of F(s) is all the values of " for which the integral in (1) exists.' The Laplace transform of fis denoted by both and ${/}. QUESTION 2. (3PTS) USE TABLE 7.1 AND 7.2 TO DETERMINE THE LAPLACE TRANSFORM OF THE GIVEN...
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