Quiz. Good luck! Signals & Systems ( ) Number of Problems: 19. Number of Points: 19. Permitted aids:

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Hugh Samuel Leonard
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1 Quiz November th, Signals & Systems (--) Prof. R. D Andrea Quiz Exam Duration: Min Number of Problems: Number of Points: Permitted aids: Important: None Questions must be answered on the provided answer sheet; answers given in the booklet will not be considered. There exist multiple versions of the exam, where the order of the answers has been permuted randomly. Every question has a unique correct answer. Every question is worth one point for a correct answer, and zero points otherwise. Giving multiple answers to a question will invalidate the answer. No negative points will be given for incorrect answers. Partial points (Teilpunkte) will not be awarded. You do not need to justify your answers; your calculations will not be graded. Use only the provided paper for your calculations; additional paper is available from the supervisors. Good luck!

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3 +//+ Discretization Box : Questions, A continuous-time LTI system G c is discretized using the exact discretization method for a zero-order hold input and a uniformly sampled output, such that G d = SG c H. This is done for a sampling time of T s = s and a sampling time of T s = s resulting in the discrete-time system G d, and G d,, respectively. Similarly, the continuous-time input u c is sampled with the two sampling periods T s = s and T s = s resulting in the two discrete time signals u d, and u d,, which are shown in the plot below.. u c u d, u d, Time (s) When the discrete-time input u d, = {...,,,,,,...} is applied to the discrete-time system G d, with state space description: q[n + ] = A d, q[n] + B d, u[n], y[n] = C d, q[n] + D d, u[n], its output is: y d, = {...,,,,,,...}. Question Given what are C d, and D d,? A d, = [ ] B d, = [ ], A C d, = [ ], D d, = [ ] [ C C d, = ], Dd, = [ ] B C d, = [ ], D d, = [ ] D C d, = [ ], D d, = [ ] Question What could be the output of the discrete time system G d,, which was obtained by discretizing G c using a sampling time of T s = s, when the input u d, = {...,,,,,,...} is applied to it? A y d, = {,,,,,...} B y d, = {,,,,,...} C y d, = {,.,,,,...} D y d, = {.,,,,,...} Question Assume you sample a continuous-time signal with the sampling rate f s = Hz. You obtain a periodical discrete-time signal with the fundamental period N =. What continuoustime frequency might the signal have? A ω = Hz B ω = Hz C ω = Hz D ω = Hz E ω = Hz F ω = Hz G ω = Hz H ω = Hz

4 +//+ System Properties Box : Questions,, Consider the following stable, discrete-time systems with input u and output y: System System description G A y[n] = u[n] + y[n ] G B y[n] = u[n] G C y[n] = u[n] Question Select the answer containing all linear systems: A G A B G B C G C D G A, G B E G A, G C F G B, G C G G A, G B, G C H No systems Question Select the answer containing all time-invariant systems: A G A B G B C G C D G A, G B E G A, G C F G B, G C G G A, G B, G C H No systems Question Select the answer containing all causal systems: A G A B G B C G C D G A, G B E G A, G C F G B, G C G G A, G B, G C H No systems

5 +//+ Box : Questions, Consider a causal LTI system G for which y = G u is given by the difference equation y [n] = αu [n ]. G is used to create four systems G A, G B, G C, and G D in the following manner: u G y u + y G A G B G u + y u + G y G C G G D Remark: The sum operator x + x x is defined as x = x + x. Question Which two systems have a finite impulse response? A G A, G B B G A, G C C G A, G D D G B, G C E G B, G D F G C, G D Question Which system has an impulse response given by {h[n]} = {...,,, α, α, α,...}? A G A B G B C G C D G D E None

6 +//+ Box : Questions, Consider the causal LTI system with difference equation y[n] =.y[n ] + u[n] + u[n ]. Question What is the impulse response h of the system? A None of the options B h = {...,,,,,...} C h = {...,,,,.,.,.,...} D h = {...,,,.,.,.,...} E h = {...,,,.,.,.,...} F h = {...,,,,,...} Question What is the output {y[n]} of the system if the constant input signal u[n] = is applied for all times n? A {y[n]} = {} B {y[n]} = {} C {y[n]} = {} D {y[n]} = {. n } E {y[n]} = {. n } F {y[n]} = {. n } Question Consider a system G with impulse response h = {...,,,,,,...}. What is the system s output y = Gu given the input u = {...,,,,,...}? A y = {...,,,,,,...} B y = {...,,,,,,...} C y = {...,,,,,,...} D y = {...,,,,,,...}

7 +//+ Frequency Domain. Magnitude and Phase Response Box : Question Let G be a causal LTI system. The following plot shows the poles and zero of the system s transfer function. Recall that: z p is a pole of H(z) ( on a pole-zero plot), if H(z p ) = z z is a zero of H(z) ( on a pole-zero plot), if H(z z ) =. G Im Re Question Which of the following magnitude responses could belong to system G? A H(Ω). C H(Ω). B H(Ω). D H(Ω).

8 +//+ Box : Question Consider the magnitude and phase responses of a causal LTI systems G. H(Ω). H(Ω) Question Which of the following input sequences u to system G yields the output sequence {y[n]} = Gu = { ( )} cos n? A {u[n]} = { sin ( { ( n)} B {u[n]} = cos { ( n)} C {u[n]} = sin n)} D {u[n]} = { cos ( { ( n)} E {u[n]} = cos n)} F {u[n]} = { cos ( n)} G {u[n]} = { cos ( { ( n)} H {u[n]} = sin n)} I {u[n]} = { sin ( n)}. DFS Box : Question Consider the following three periodic DT signals, shown on the range n =,..., : xa[n] xb[n] xc[n] n n n Question Which signal has a Fourier series representation with period N =? A x A B x B C x C D x A, x B E x A, x C F x B, x C G x A, x B, x C

9 +//+. DFT Box : Questions,,, Given the following four DT sequences which are all periodic with N = : x[n] x[n] x[n] x[n] n n and the following magnitude and phase plots of DFT coefficients calculated with the first N = samples: / / / / / / XA[k] XB[k] XC[k] XD[k] k XA[k] XB[k] XC[k] XD[k] / / / / / / / / / / / / / / / / / / k

10 +//+ Question Which are the DFT coefficients of sequence x? A X A B X B C X C D X D E None Question Which are the DFT coefficients of sequence x? A X A B X B C X C D X D E None Question Which are the DFT coefficients of sequence x? A X A B X B C X C D X D E None Question Which are the DFT coefficients of sequence x? A X A B X B C X C D X D E None Box : Question Consider the magnitude of the DFT coefficients {X[k]} computed with a DFT length N = : X[k] k Question Which of the following continuous-time signals relates to the DFT coefficients {X[k]} when sampled with T s =? A None of the options B x(t) = cos(t) C x(t) = cos(.t) D x(t) = cos(.t) E x(t) = cos(t) F x(t) = cos(.t)

11 +//+ Answer sheet: student number please encode your student number, and write your first and last name below. Firstname and lastname: How to select answer B : (Unwanted answer clearly removed) (Desired answer clear) (Cannot distinguish B and C) Discretization Answers must be given exclusively on this sheet; answers given on the other sheets will not be counted. Frequency Domain Q: A B C D Q: A B C D Q: A B C D E F G H System Properties Q: A B C D E F G H Q: A B C D E F G H Q: A B C D E F G H Q: A B C D E F Q: A B C D E Q: A B C D E F Q: A B C D E F Q: A B C D. Magnitude and Phase Response Q: A B C D Q: A B C D E F G H I. DFS Q: A B C D E F G. DFT Q: A B C D E Q: A B C D E Q: A B C D E Q: A B C D E Q: A B C D E F

Final Exam January 31, Solutions

Final Exam January 31, Solutions Final Exam January 31, 014 Signals & Systems (151-0575-01) Prof. R. D Andrea & P. Reist Solutions Exam Duration: Number of Problems: Total Points: Permitted aids: Important: 150 minutes 7 problems 50 points