Homework Answers
2. Let y(t)(e')u(t) represent the output of a causal, linear and time-invariant continuous-time system with unit...
2. Let y(t)(e')u(t) represent the output of a causal, linear and time-invariant continuous-time system with unit impulse response h[nu(t) for some input signal z(t). Find r(t) Hint: Use the Laplace transform of y(t) and h(t) to first find the Laplace transform of r(t), and then find r(t) using inverse Laplace transform. 25 points
em 2: Given two sequences x[n] = 8 8[n - 8] and h[n] = (0.7)"u[n] Determine...
em 2: Given two sequences x[n] = 8 8[n - 8] and h[n] = (0.7)"u[n] Determine the z-transform of the convolution of the two sequences using the convolution property of the Z-transform Y(z) = X(z) H(2) Determine the convolution y[n] = x[n] * h[n] by using the inverse z-transform Problem 3: Find the inverse z-transform for the functions below. 4z-1 2-4 z-8 X(Z) = + 2-5 Z - 1 2-05 X(Z) = Z 2z2 + 2.7 z + 2
f(t) F(S) (s > 0) S (s > 0) n! t" ( no) (s > 0)...
f(t) F(S) (s > 0) S (s > 0) n! t" ( no) (s > 0) 5+1 T(a + 1) 1a (a > -1) (s > 0) $4+1 (s > 0) S-a 1. Let f(t) be a function on [0,-). Find the Laplace transform using the definition of the following functions: a. X(t) = 7t2 b. flt) 13t+18 2. Use the table to thexight to find the Laplace transform of the following function. a. f(t)=t-4e2t b. f(t) = (5 +t)2...
Problem#3 (16 points) Consider a system that has R(S) as the input and Y (S) as...
Problem#3 (16 points) Consider a system that has R(S) as the input and Y (S) as the output. The transfer function is given by: Y(S) R(S) 45+12 What are the poles of the system? For r(t) output in the time-domain y(t) For r(t) = t, t output in the time-domain y(t) 1- 2- 1,t 0, use partial fraction expansion and inverse Laplace transform to find the 3- 0, use partial fraction expansion and inverse Laplace transform to find the
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+...
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+t' to prove that a unique solution to the IVP exists for short time, as follows: (a) Let b E R2, A E M2 (R) . Show that any function g : R2 -R2.9(x) = Ax+b is Lipschitz. 1 mark (b) Transform the DE for y into a(t) Az(t) +b(t) for a suitable z, A, b. 2...
Problem 1: Find the Laplace transform X(s) of x(0)-6cos(Sr-3)u(t-3). 10 Problem 2: (a) Find the inverse...
Problem 1: Find the Laplace transform X(s) of x(0)-6cos(Sr-3)u(t-3). 10 Problem 2: (a) Find the inverse Laplace transform h() of H(s)-10s+34 (Hint: use the Laplace transform pair for Decaying Sine or Generic Oscillatory Decay.) (b) Draw the corresponding direct form II block diagram of the system described by H(s) and (c) determine the corresponding differential equation. Problem 3: Using the unilateral Laplace transform, solve the following differential equation with the given initial condition: y)+5y(0) 2u), y(0)1 Problem 4: For the...
Mouzey bighalsledsystems tionne 907 octet Acone s ona 27/0 y the 13. The input-output relationship of an LTI system...
Mouzey bighalsledsystems tionne 907 octet Acone s ona 27/0 y the 13. The input-output relationship of an LTI system is deseribed by the difference squation: n]+0.5y[n-1]-xn], Try to figure out two possible unit impulse responses for such a system. Then state which unit impulse response comresponding to tomer les modules com a stable system. 2, b) x,(2)=z" +62 452 | > 1 14) Find the inverse z-transform of the following signals a) X(E)(-2 XI-2) :-5 c) X2(E)-0.5:)1-0.5 )0. <2 15....
Answer for the last box only please. Thanks Entered Answer Preview Result Y's+5°Y+2 Ys + 5Y...
Answer for the last box only please. Thanks
Entered Answer Preview Result Y's+5°Y+2 Ys + 5Y +2 correct (5/s)*([e^(-s)]-1) correct (5*[e^(-s)]-5-2*s/[s*(s+5)] 5e : - 5 - 28 $(8 + 5) correct u(t-1)"[u(t-1)-(e^{-5* (t-1)]]]-u(t)+(e^(-5*t)]-(2/5) ult – 1)(ult – 1) – е11-1)) – u(t) +e5 incorrect At least one of the answers above is NOT correct. (1 point) In this exercise we will use the Laplace transform to solve the following initial value problem: 1 + 5y = -5, ( 0, 0<t<...
3. A sinusoid eơt s or can be expressed as a sum of exponentials e" and e" with complex ncies s-o...
Formuals:
3. A sinusoid eơt s or can be expressed as a sum of exponentials e" and e" with complex ncies s-o +yoo and s* -ju, Locate in the complex plane the complex frequencies of (10 points) the following signals: (a) e cos2t (e) 2 ut) -2t (b) e 3 (c) cos3t (d) e Complex numbers: - R034 1 n even (reje)" rkejke Trignometric Identities sin 2x=2sinxcosx sin2 x+cos2 x = 1 in 1-cos 2 cos2x=1 + cos2x sin(x±y)-sinxcosy±cos x...
Problem 1: (10 pts) Let Y(S) = 4:0251712s+10 Use Matlab's "tf2zpk" and "residue" commands to: k...
Problem 1: (10 pts) Let Y(S) = 4:0251712s+10 Use Matlab's "tf2zpk" and "residue" commands to: k (s-- (-2) a) (5 pts) Convert y(s) into pole-zero form, 1.e -p -p,Xs-p.)(-p) "polei zero are com en If pole/zero are complex, combine conjugate pairs into 2nd order terms with real coefficients. b) (5 pts) Find the inverse Laplace transform y(t). Please combine conjugate complex parts so that y(t) is real (not complex).
2. Let y(t)(e')u(t) represent the output of a causal, linear and time-invariant continuous-time system with unit impulse response h[nu(t) for some input signal z(t). Find r(t) Hint: Use the Laplace transform of y(t) and h(t) to first find the Laplace transform of r(t), and then find r(t) using inverse Laplace transform. 25 points
em 2: Given two sequences x[n] = 8 8[n - 8] and h[n] = (0.7)"u[n] Determine the z-transform of the convolution of the two sequences using the convolution property of the Z-transform Y(z) = X(z) H(2) Determine the convolution y[n] = x[n] * h[n] by using the inverse z-transform Problem 3: Find the inverse z-transform for the functions below. 4z-1 2-4 z-8 X(Z) = + 2-5 Z - 1 2-05 X(Z) = Z 2z2 + 2.7 z + 2
f(t) F(S) (s > 0) S (s > 0) n! t" ( no) (s > 0) 5+1 T(a + 1) 1a (a > -1) (s > 0) $4+1 (s > 0) S-a 1. Let f(t) be a function on [0,-). Find the Laplace transform using the definition of the following functions: a. X(t) = 7t2 b. flt) 13t+18 2. Use the table to thexight to find the Laplace transform of the following function. a. f(t)=t-4e2t b. f(t) = (5 +t)2...
Problem#3 (16 points) Consider a system that has R(S) as the input and Y (S) as the output. The transfer function is given by: Y(S) R(S) 45+12 What are the poles of the system? For r(t) output in the time-domain y(t) For r(t) = t, t output in the time-domain y(t) 1- 2- 1,t 0, use partial fraction expansion and inverse Laplace transform to find the 3- 0, use partial fraction expansion and inverse Laplace transform to find the
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+t' to prove that a unique solution to the IVP exists for short time, as follows: (a) Let b E R2, A E M2 (R) . Show that any function g : R2 -R2.9(x) = Ax+b is Lipschitz. 1 mark (b) Transform the DE for y into a(t) Az(t) +b(t) for a suitable z, A, b. 2...
Problem 1: Find the Laplace transform X(s) of x(0)-6cos(Sr-3)u(t-3). 10 Problem 2: (a) Find the inverse Laplace transform h() of H(s)-10s+34 (Hint: use the Laplace transform pair for Decaying Sine or Generic Oscillatory Decay.) (b) Draw the corresponding direct form II block diagram of the system described by H(s) and (c) determine the corresponding differential equation. Problem 3: Using the unilateral Laplace transform, solve the following differential equation with the given initial condition: y)+5y(0) 2u), y(0)1 Problem 4: For the...
Mouzey bighalsledsystems tionne 907 octet Acone s ona 27/0 y the 13. The input-output relationship of an LTI system is deseribed by the difference squation: n]+0.5y[n-1]-xn], Try to figure out two possible unit impulse responses for such a system. Then state which unit impulse response comresponding to tomer les modules com a stable system. 2, b) x,(2)=z" +62 452 | > 1 14) Find the inverse z-transform of the following signals a) X(E)(-2 XI-2) :-5 c) X2(E)-0.5:)1-0.5 )0. <2 15....
Answer for the last box only please. Thanks
Entered Answer Preview Result Y's+5°Y+2 Ys + 5Y +2 correct (5/s)*([e^(-s)]-1) correct (5*[e^(-s)]-5-2*s/[s*(s+5)] 5e : - 5 - 28 $(8 + 5) correct u(t-1)"[u(t-1)-(e^{-5* (t-1)]]]-u(t)+(e^(-5*t)]-(2/5) ult – 1)(ult – 1) – е11-1)) – u(t) +e5 incorrect At least one of the answers above is NOT correct. (1 point) In this exercise we will use the Laplace transform to solve the following initial value problem: 1 + 5y = -5, ( 0, 0<t<...
Formuals:
3. A sinusoid eơt s or can be expressed as a sum of exponentials e" and e" with complex ncies s-o +yoo and s* -ju, Locate in the complex plane the complex frequencies of (10 points) the following signals: (a) e cos2t (e) 2 ut) -2t (b) e 3 (c) cos3t (d) e Complex numbers: - R034 1 n even (reje)" rkejke Trignometric Identities sin 2x=2sinxcosx sin2 x+cos2 x = 1 in 1-cos 2 cos2x=1 + cos2x sin(x±y)-sinxcosy±cos x...
Problem 1: (10 pts) Let Y(S) = 4:0251712s+10 Use Matlab's "tf2zpk" and "residue" commands to: k (s-- (-2) a) (5 pts) Convert y(s) into pole-zero form, 1.e -p -p,Xs-p.)(-p) "polei zero are com en If pole/zero are complex, combine conjugate pairs into 2nd order terms with real coefficients. b) (5 pts) Find the inverse Laplace transform y(t). Please combine conjugate complex parts so that y(t) is real (not complex).
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