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systems analysis and design using matlab

Questions and Answers of Systems Analysis And Design Using MATLAB

=+• what action the function takes if there is a problem
=+1. What is a ‘class invariant’? Choose only one option.
=+(a) A class whose source code is versioned and therefore cannot be changed.
=+(b) A class whose objects have constant fields.
=+(c) A condition that will always be true for an instance of the class.
=+2. What is meant by the term ‘design by fear’? Choose only one option.(a) Design is scary.(b) You cannot know when to trust the code.(c) You design a system too quickly because of time
=+3. What is ‘Design by Contract’? Choose only one option.
=+(a) Designing code as if there were a contract between an object that sends a message and the object that receives it.
=+(b) Reinforcing the contract between every pair of objects by increasing the amount of error-checking.
=+(c) Protecting your software using a contract with a firewall.
=+(d) Designing a software system under contract.
Solve the following second-order DE with initial conditions y(0) ¼ 1 and dy(t) dt t=0 = 4:
Solve the following second-order DE with initial conditions y(0) ¼ 1 and dy(t)dt jt¼0¼ 4:where u(t) is the unit step function. dy(t) dt +3dy(t) dt +2y(t) = eu(t)
Find the Laplace transform and ROC of the signal x(t) ¼ eatu(t).
Find the Laplace transform and ROC of the signal x(t) ¼ u(t), where u(t) is the unit step function
Find the Laplace transform and ROC of the unit impulse signal x(t) ¼ d(t).
Find the inverse Laplace transform of the following function of the right-sided signal x(t) if X(s) === S (s+1)(s+2)
Solve the following second-order DE using Laplace transform. The initial conditions are y(0) ¼ 1 and dy(t)dt jt¼0¼ 4.where u(t) is the unit step function. dy(t). dy(t) +3 dt dt +2y(t)=eu(t)
Solve the following second-order DE with initial conditions y(-2) ¼ =1 and y(-1) ¼ =-1: y(n) = 0.75y(n-1)- 0.125y(n-2)
Consider the following second-order DE:Find the output signal y(n) if the input signal is u(n) ¼ 2(0:4)n for n 0. Assume zero initial conditions, that is, let y(2) ¼ y(1) ¼ 0. = y(n)
Find the z-transform of the one-sided signal x(n) = anu(n).
Consider the sequence x(n) = an, where 0 Im(z) ROC of X(z) a
Find the z-transform of the following sequence: xin)=2(7)"un) +3()"un)
Find the z-transform of the following sequence x(n) ={(0.6)" n3 (0.6)" n4
Convolve the following two sequences = x(n) [231] and h(n) = [1-232]
Convolve the following two sequences using z-transform: x(n) = (0.5)" u(n) h(n) = (0.7)u(n)
Find the z-transform of the sequence x(n) = nau(n)
Find the initial and the final values of the one-sided signal x(n), if X(z) = z2 z(z -0.7) 1.25z+0.25
Find the inverse z-transform of the following function: X(z) === ROC: |z|> 0.3 (z 0.2)(z 0.3)'
Find the inverse z-transform of the function X(z) ¼ z z0:2 using power series expansion.
Find the inverse z-transform of z3-2.3z2 +0.84z X(z) = z3 1.4z2 +0.63z -0.09
Find the inverse z-transform of the irrational function X(z) = log(1+az1) ROC: [z]>a ==
Find the inverse z-transform of the following function: z X(z) = ROC: z> 0.3 (z -0.2)(z -0.3)'
Consider a SISO system described by the second-order DE:Find the output of this system if the input signal is a unit step function x(n) ¼ u(n)with the initial conditions y(1) ¼ y(0) ¼ 0.
Find the DTFT of the five-point discrete signal: x(n) [2-3 7-3 2]
Find the inverse DTFT of X( jv) given by = X(jw)-e-for-
Find the z-transform of the sequence x(n1,n2)= a b 1, 20 otherwise
Find the z-transform of the 2-D sequence x(n1,n2) = { 10 otherwise
Find the inverse DSFT of X( jv1, jv2) given bywhere D is the dashed area shown in Figure 3.12. 1 X(jon, jw2) 0 if (1,02) ED otherwise
Find the eigenvalues and eigenvectors of the 2 * 2 matrix A: A= [ -2 2 -24 12
Find the eigenvalues and eigenvectors of the 3 * 3 matrix A: 2.6 1.3 -2.5] A: 0.8 5.4 0.8 1.4 -5 -1
Find the characteristic polynomial of the following 3 * 3 matrix: -2 4 A = 1 6 23 3 1 -1 -5
Find the modal matrix of 2.6 1.3 -2.5" A= 0.8 5.4 -5 0.8 1.4 -1
Diagonalize 3 * 3 matrix A given by 2.6 1.3 -2.5 A=08 5.4 -5 0.8 1.4 -1
Diagonalize matrix A: A = 5 2 UNT 1 0 40 -1 1 6
Diagonalize the 2 * 2 matrix A: A ^= [44] -4
Check the following symmetric matrices for their definiteness: (a) A = [5 ] -2 5 1 (b) B = [2] (c) C = 1 -2 -6 -4 -4 -6
Find the SVD of matrix A given by 2 46 A = -26 -4
Consider the 256 256 cameraman image. If we decompose the image into its SVD components, each component is an image of size 256 256. The ith component of the image is s2i uivi and the whole image
Find the p ¼ 1, p ¼ 2, p¼1, and Frobenius norm of the following matrix: A = 4 -6
Figure 3.18 shows 2-D scatter data obtained by plotting two neighboring pixels x ¼ f (i, j) and y ¼ f (i þ 1, j) of the LENA image. The estimated covariance matrix of this data set is R = [0.2529
In this example, the gray-scale cameraman image is used to obtain the principal components (eigenimages). These eigenimages are used as basis functions for compressing the image. The image is
Show that matrix A satisfies its characteristic polynomial A= 1
Find the matrix polynomial |f(A) A8-8A6 +245 +2A3-3A2 + A+7/ if
Find f (A) = eAt if 0 A= ^= [ 2 - 3 ] 1 -3
Find f (A) = Ak if [0.75 0.5 -0.75 A= 0.5 3 -3 0.5 2 -2
Find SeTdT if
Find eAt if -3 -1 A = 0 -4
Find Ak, if 0.1 0.4 -0.1 A= -0.15 0.6 -0.05 -0.2 0.4 0.2
Let x 2 R3 and f (x) ¼ x31þ 3x21þ 2x2 7x2x3 þ 4. Find af(x) ax
Let x 2 Rn and f (x) ¼ cTx, where c is an n * 1 column vector, then f(x) = cx=c1x1 + C2x2 ++ Coxn
Let x 2 R3 and f (x) ¼ 13x21þ 8x22 4x23 3x1x2 þ 5x1x3 þ 6x2x3 af(x) Find fix and fix f(x)
Let x 2 Rn, A be an n n symmetric matrix, b be an n 1 column vector, and c a scalar. Define f (x) ¼ 12 xTAx þ bTx þc. Find af(x) and a f(x) ax ax
Let x 2 R3 andthen f(x)= xx2+3x2+4x3-5 X1X2 X1 X2 X22X1X2X3 +
3.1 (a) LetFind x(n) * h(n).(b) Consider the LTI system with impulse responseUsing z-transform, find the response y(n) of this system to the input x(n) = d(n)+28(n-1)+38(n-2) and h(n)=8(n)
3.2 Determine the ROCs and z-transforms of each of the following sequences:(a) x(n) ¼ (0:3)nu(n) þ 2nu(n 1)(b) y(n) ¼ (0:2)nu(n 2)(c) z(n) ¼ u(n) þ d(n) þ 3nu(n)
3.3 Evaluate the inverse z-transform of the following function: 2-2+ (a) X(z) = ROC: < < 2-22 +2 = (b) X(2) log (1-2z), ROC: z] > 2
3.4 Determine the z-transform and its ROC for each of the following 2-D sequences: (a) x(n,n2)=()u(n)u(n2) (b) y(n1,n2)=()u(n)u(2)
3.5 Find the eigenvalues and eigenvectors of matrix A given below: A = 3 2
3.6 Find the eigenvalues and eigenvectors of the following matrices: 2 0 A -- B = [6%]. 0 1 3 a and C= 0 30 b a 300
3.7 Let A be a 4 4 matrix with eigenvalues 1, 2, 3, and 4. Find (if possible)the following quantities:(a) det (AT)(b) Trace (A1)(c) det (A 8I)
3.8 Compute the eigenvalues and eigenvectors of the matrixCan matrix A be diagonalized? If yes, find the transformation to diagonalize matrix A. A = [0 2 2 202 22 0
3.9 Compute the eigenvalues and eigenvectors of the matrixFind, if possible, the matrix M such that M1AM is a diagonal matrix. A = 1 1 1 021 00 0 3
3.10 Show that the eigenvalues of A þ aI are related to the eigenvalues of A by A(A+al) (A) +
3.11 Find the SVD of the following matrices: -1 0 A B = 3 -2 0 and C=2 -3. -2 4 1 -3 5 3 4
3.12 Consider the symmetric square matrix(a) Find eigenvalues of A þ 3I.(b) For what value ofa, the matrix A þ aI is singular. A a
3.13 Determine whether the following matrices are (a) positive definite, (b) positive semi-definite, (c) negative definite, and (d) negative semi-definite: A = 2 140 0 1 B 1 -4 -1 and C c = [6 7 6 2
3.14 Consider the following square matrix:(a) Compute Ak and eAt using the Cayley–Hamilton theorem.(b) Find the square roots of the matrix, that is, find all matrices B such that B2 ¼ A using the
3.15 Consider the following square matrix:a) Compute A7 and A73 using the Cayley–Hamilton theorem.(b) Compute eAt using the Cayley–Hamilton theorem. 2 2 0 A=002 002
3.16 Assume that the following matrix is nonsingular:Compute the natural logarithm of the matrix C; that is, identify the matrix B ¼ ln (C) that satisfies the equation C ¼ eB. Is the assumption of
3.17 Given the 3 * 3 matrixExpress the matrix polynomial f (A) ¼ A4 þ 2A3 þ A2 A þ 3I as a linear combination of the matrices A2, A, and I only. A = -3 2 1 0 4 -2 1 5
3.18 Let the 2* 2 matrix A be defined asCompute Ak and eAt . A= [2]
3.19 Let the 3 * 3 matrix A be defined asCompute Ak and eAt . 030 0-3 A-3 0 0 -3
3.20 Let the 3 * 3 matrix A be defined asCompute Ak and eAt . A = -1 0 0 -1
Consider the following SISO system:Obtain a state-space representation of this system in (a) controllable and (b)observable canonical forms. dy(t) dt +5 y(t) dy (t) dr +2 +8y9. dt du(t) = +7 dt +>
Find the transfer function corresponding to the state equations given by X(t)] X2(1) = [x (t] + 5 u(t) and y(t) [45] [x(t)] (4.59) -6 X2(1) X2(t)
Find the state-transition matrix of the following dynamic system using the two methods described above FA- x(t) -1 x(t)
Find the output of the dynamic system described by the state equation:With the input and initial conditions given by, -1 1 x(t) = x(t)+ u(t) -1 -1 y(t) = [1-1]x(t)
Consider the following SISO discrete-time LTI systemObtain a state-space representation of this system in (a) controllable and (b)observable canonical forms y(k+2)+0.75y(k+1)+ 0.25y(k) =
Find the transfer function corresponding to the state equation given by [x (k+1)] [x2(k+1) 0 1 [x(k) + u(k) -0.6 -0.5x2(k), y(k)= [1 -2][X1(k)]
Find the state-transition matrix of the following dynamic system using the two methods presented previously: x(k+1)= -0.25 1.5 -0.25] x(k)
Find the output of the following dynamic system if u(k)= [1k0 lok
4.1 Find the state-space representation of the following dynamical systems in controllable and observable canonical forms: (a) dy(t) dy(t) dr3 +2- dy(t) dz +3 dr + y(t) = 4u(t) (b) y(k +2)+y(k + 1) +
4.2 Find the state-space representation of the following dynamical systems: dy(t) dy(t) dr (a) +3dy(1) du(t) dr +6- +2y(t): == +8u(t) dt dt (b) y(k+3)+0.5y(k+2)+y(k+1)+0.89y(k) = 2.4u(k)
4.3 Obtain the response y(t) of the following system, where u(t) is the unit step function: x(0) [A] y = [1_0] [X] y(1
4.4 Obtain the response y(k) of the following system, when u(k) ¼ (1)k for k 0: [x(k+1) [x2(k+1)] x(0) = [7 -0.25 ] [ x ] + [ ] 4 (k) = [8] y(k) = [1 1] [x1(k)]
4.5 Show that the continuous-time state-transition matrix satisfies the following properties: 4(t) = et (b) (-1)=e=(e)-1=Q(1) (p) (11)(1)(1) ()() = (7)(11) = (1+ 1) (5) 1 = (0) (e)
4.6 Show that the discrete-time state-transition matrix satisfies the following properties: (a) (0) = 1 4(k) = A* (b) q(-k) =A= (A)-1=Q(k) (c) (k1+k)=4(k1)4(k2) = q(k2)4(k1) (d) (k3-k2)4(k2 - k)=4(k3
4.7 The state-transition matrix of a time-varying system is the solution to the following partial differential equation: a(1, T) t A(t)p(t, T), with boundary condition (T,T) = I Find the
4.8 An LTI system is described by the following differential equations:Let x1(0) ¼ 1, x2(0) ¼ 1, and u(t) ¼ et; derive expressions for x1(t) and x2(t).(a) Find a nonsingular transformation matrix

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