ECE-314 Fall 2012 Review Questions for Midterm Examination II
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1 ECE-314 Fall 2012 Review Questions for Midterm Examination II First, make sure you study all the problems and their solutions from homework sets 4-7. Then work on the following additional problems. Problem 1 Consider the system represented by the differential equation y + 5y + 6y = x. (a) Find the impulse response for this system. (b) Find the output when the input is a unit step function. (c) What is the connection between your answers in parts a and b? Be precise while explaining the connection thoroughly. (d) Is this system stable? Justify your answer. Problem 2 Repeat Problem (1) considering the system represented by the differential equation y + 4y + 4y = 2x. Problem 3 Consider the differential equation y + 5y + 6y = x, with initial conditions y(0 ) = 2 and y (0 ) = 1. (a) Find the zero-input solution to this equation. (b) Find the total solution using the system s approach when the input is the pulse u(t) u(t 2). To avoid duplicating effort, make sure to use your answer from Problem 1(b). Problem 4 Consider the system represented by the differential equation y +5y +20y = x. (a) Find the impulse response for this system. Hint: First establish the connection between the impulse response for this system and that for the tandem system y + 5y + 20y = x. (b) Is this system stable? Problem 5 Find the impulse response for the system represented by y + 5y + 20y = 3x. Comment on the form of your answer. Problem 6 Consider the differential equation y + 5y + 6y = tu(t), with initial conditions y(0 ) = 2 and y (0 ) = 1. Find the total solution using the classical approach, namely by forming the total solution as a sum of the particular and homogeneous solutions and applying the derived initial conditions y(0 + ) and y (0 + ). Problem 7 Suppose that a system is represented by a certain differential equation of the form y + ay + cy = x. What is the simplest way to determine if it is stable or unstable? Justify your answer. Would the same approach work for the system y + ay + cy = x? What would you do for the latter case in order to determine its stability? Problem 8 Consider the system described by the following difference equation: y[n] = y[n 1] 0.5y[n 2] + x[n] 2x[n 1] + x[n 2], n 0, with specified initial conditions y[ 1] and y[ 2]. a) Show that the zero-input response y 0x [n] has the form y 0x [n] = C 1 ρ n e jϕn + C 2 ρ n e jϕn, 1
2 where C 1 and C 2 are constants. You must define ρ and ϕ. (b) Obtain a set of equations from which the constants C 1 and C 2 can be determined. Do not attempt to solve for these constants. (c) Suppose that C 1 = 1, and C 2+j2 2 = C1. Evaluate y 0x [n] in its simplest form. Your answer should not involve any imaginary numbers. Problem 9 Consider the system described by the following difference equation: y[n] = ay[n 1] + by[n 2] + x[n] + x[n 1], n 0, (1) with initial conditions y[ 1] = y[ 2] = 0. It is known that the solution to another equation y[n] = ay[n 1] + by[n 2] + 0.5δ[n], n 0, with initial conditions at y[ 1] = y[ 2] = 0, is the sequence z[n] = {(0.5) n ( 1) n }u[n]. Express the solution of system (1) in terms of z[n] and the input x[n]. Problem 10 Consider the difference equation y[n] = y[n 1] 1 4 y[n 2] + (1 2 )n u[n] + 5u[n], n 0. a) Determine the form of the particular solution y p [n]. (Do not attempt to evaluate the unknown constants.) b) Given the initial conditions y[ 1] = y[ 2] = 1, explain (in words only) how the total solution is determined assuming that the particular solution is known. Be very specific about any initial conditions you may need to use. Problem 11 In the following questions circle the most appropriate answer. (i) The zero-initial-condition response of a system can always be written as a convolution if the system is: (a) Linear. (b) Linear and time invariant. (c) Linear, time-invariant, and stable. (d) (b) but only when the input is a causal sequence. (ii) For an LTI system, the zero-state response to an eternal sinusoidal input is an eternal sinusoidal if the system is: (a) Stable. (b) Stable and causal. (c) The above statement is always true. (d) The above statement is never true. 2
3 (iii) Let y h [n] be the homogeneous solution of a second-order linear difference equation with certain specified initial conditions y[ 1] and y[ 2]. Suppose that y h [n] = 3 cos(0.2πn)u[n], then it must be true that (a) The roots of the characteristic equation are real. (b) The roots of the characteristic equation are complex but with negative real parts. (c) The roots of the characteristic equation are complex conjugates of each other. (d) The roots of the characteristic equation are on the unit circle. (e) c & d. Problem 12 Obtain an analytical solution to the convolution z[n] = x[n] y[n], where x[n] = (0.3) n u[n] and { 1, n = 0,..., 8, y[n] = 0, otherwise. Note: You must specify z[n] for all values of n. Problem 13 For the signals x[n] and y[n] shown below, let z[n] = x[n] y[n]. Compute z[4]. x[n] = δ[n 2] + 2δ[n 3] + δ[n 4] and y[n] = 2δ[n] + 2δ[n 1] + δ[n 2] Problem 14 It is known that if the input 2δ[n 1] is applied to a certain linear time-invariant system, the output is 2.33(0.7) n u[n 2]. Use this information to write an expression for the output y[n] due to an arbitrary input x[n]. (Do not to attempt to evaluate or simplify the expression since x[n] is not explicitly given.) Hint: Determine the impulse response first. Problem 15 A linear time-invariant systems has an impulse response a) Is this system causal? b) Is it stable? h[n] = 0.88u[n] n u[n + 1]. 3
4 c) What is the simplest way to determine if a system represented by an mth-order difference equation is stable or unstable? Justify your answer. Problem 16 Repeat problem 15 for h(t) = e t u(t) + e 0.6t u(t 1). a) Is this system causal? b) Is it stable? Problem 17 A certain system is described by the following difference equation: y[n] = 1.5y[n 1] + y[n 2] 3x[n] + 0.5x[n 1], n 0. with y[ 1] = y[ 2] = 0. Determine if the system is stable. Show all your work. Problem 18 Consider the following difference equation: y(n) = y(n 1) 0.25y(n 2) + 3(0.5) n u(n), n 0, with initial conditions y( 1) = 1 and y( 2) = 3. a) Determine the homogeneous solution y h (n). b) Determine the form of the impulse response h(n) and explain how you would go about evaluating its constants. You must set up the equations but do not solve for the constants. c) How can you calculate the particular solution y p (n)? Do not carry out any algebraic calculations. Problem 19. A discrete-time system has a frequency response (DTFT of its impulse response) H(e jθ ) as given below. If the input given by H(e jθ ) = sin 2 (θ/2) e 2jθ, 0 θ 2π. x[n] = A{1 + m cos(0.2πn)} is applied to the filter, determine the filter output. Show that you can write the output in the form A {1 + m cos(0.2πn)}? How does m compare to m? Comment on your answer. Problem 20 An LTI system has an impulse response h(t). It is known that H(jω) = h(t)e jωt dt = 1/(jω + 1). Calculate the output of the system when the applied input is: (a) x 1 (t) = 2e j200πt (b) x 2 (t) = cos(400πt). 4
5 Problem 21 Calculate the discrete-time Fourier series representation of periodic signal x(n) = sin(nπ/4). Be specific about the period of x. Problem 22 Show that the DTFT of a real signal is symmetric about the line Ω = π. Problem 23 A real signal x has the DTFT X(e jω ) whose restriction to the period [0, π] is defined as follows: X(e jω ) = 1 when 0 Ω 0.5 or 2 Ω π; X(e jω ) = 0 when 0.5 Ω 2. Find x. Is x a causal signal? Problem 24 State the applicability of each of the FS, DTFS, and DTFT to signals. In each case state the inversion formula. 5
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