School of Information Technology and Electrical Engineering EXAMINATION. ELEC3004 Signals, Systems & Control

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1 This exam paper must not be removed from the venue Venue Seat Number Student Number Family Name First Name School of Information Technology and Electrical Engineering EXAMINATION Semester One Final Examinations, 204 This paper is for St Lucia Campus students. Examination Duration: Reading Time: Exam Conditions: This is a Central Examination 80 minutes 0 minutes For Examiner Use Only Question Mark This is a Closed Book Examination - specified materials permitted During reading time - write only on the rough paper provided This examination paper will be released to the Library Materials Permitted In The Exam Venue: (No electronic aids are permitted e.g. laptops, phones) Any unmarked paper dictionary is permitted An unmarked Bilingual dictionary is permitted Calculators - Any calculator permitted - unrestricted One A4 sheet of handwritten or typed notes double sided is permitted Materials To Be Supplied To Students: x 6 Page Answer Booklet x cm x cm Graph Paper Rough Paper Instructions To Students: Please answer ALL questions. Thank you. Page of 4 Total

2 This exam has THREE (3) Sections for a total of 00 Points Section : Linear Signals & Systems % Section 2: Signal Processing % Section 3: Digital Control % Please answer ALL questions. PLEASE RECORD ALL ANSWERS IN THE ANSWER BOOKLET Any material not in Answer Booklet(s) will not be seen. In particular, the exam paper will not be graded or reviewed. Section : Linear Signals & Systems Please Record Answers in the Answer Book (Total: 25%). Whatchamacallit? (5%) Express the signals in the figure below by a single expression valid for all t. a) b) 4 t t -3 5 t 4 2. A Fast and E-Z Fourier (5%) Given a discrete-time unit impulse response with the following difference equation: a) What is its Z-transform? (i.e., what is Y(z))? b) What is its frequency (or Fourier) response? (i.e., what is Y(ω)) [hint: for partial credit, you may leave it in terms of the phasor ] Page 2 of 4

3 3. Is Digital Really Better? (5%) a) Briefly give TWO advantages of digital signals over analogue signals? b) Briefly give TWO advantages of analogue signals over digital signals? 4. Characteristic Roots and Characteristic Modes (5%) For systems having the following pole-zero plots (on the s-plane), please sketch the corresponding zero-input response. a) b) c) Page 3 of 4

4 5. I Spy LTI? A system has the following input (u) and output (y) relation. (5%) Input: Output: Suppose, for three different (and separate) cases the input is slightly varied as below. CASE I: Input (I): Output (I): CASE II: Input (II): Output (II): CASE III: Input (III): Output (III): Please determine if the conditions can or cannot be determined. If it can be determined, then please state if it is or is not the case (please mark a in the table [in the booklet!!]) [Note: Please treat each case separately. That is Case I is independent of Case II ] For CASE I, is the system: Linear Time-Invariant Causal For CASE II, is the system: Linear Time-Invariant Causal For CASE III, is the system: Linear Time-Invariant Causal Yes No Indeterminate Yes No Indeterminate Yes No Indeterminate Page 4 of 4

5 Section 2: Signal Processing Please Record Answers in the Answer Book (Total: 30%) 6. Battle of the Band(limited) Signals (5%) A signal is bandlimited to B Hz. Show that the signal is bandlimited to nb Hz. [Hint: Start with n=2. Use frequency convolution property and the width property of convolution.] 7. Cogito Ergo Sum (5%) a) Briefly explain (and/or show a simple sketch) what is meant by the terms ergodic and ensemble in the context of multiple stochastic digital samples or signals? b) Does the noise have to be white for the system to remain unbiased? (please briefly explain) [Hint: what happens when pink noise is averaged?] 8. An Analogue to Filtering (5%) Consider a filter given by the following transfer function 4 4 a) What is the order of H(s)? b) What is the cut-off frequency of H(s)? c) Design a low-pass digital filter, H(z), with a sampling frequency of 00 Hz, that has the same cut-off frequency as H(s). Page 5 of 4

6 9. A Filtered Whatchamacallit! (5%) What kind of compensator is described by pole-zero plots shown below. Please briefly justify the conclusion using a transfer function and a rough sketch of the amplitude and phase response of the filter. a) b) s-plane Im s-plane Im 4 Re 4 Re 0. A-Great Filter (0%) Design a high-pass filter for removing the 50 Hz mains flicker from an audio signal from a guitar. The filter should have minimum transmission 20 db (i.e., an approximate limit to human hearing). Also, the filter should not attenuate music signals from the note A-Great or higher (or more technically 0 Hz, A 2 or A ) by more 3 db. The filtered audio is then played to a concert musician live on stage. [Note: if you need to you may assume 24-bit, 44.-kHz sampled signals if needed] a) What type of filter should we use (analogue or IIR or FIR)? (Please justify) b) What order does this filter need to be? c) Please sketch the frequency response of this filter. Page 6 of 4

7 Section 3: Digital Control Please Record Answers in the Answer Book (Total: 45%). A Discrete Convolution (5%) What is the convolution (y y 2 )[n] between these discrete signals? a) y [n]=δ[n 3], y 2 [n]=δ[n+4] b) y [n]=cos[2πn], y 2 [n]=δ[n 3] 2. An E-Z Correspondence (0%) In mapping from the s-plane to the z-plane, recall that the duration of a time signal is related to the radius (of the pole location) and the sample rate is related to the angle by. From this we can sketch major features of the s-plane to the z-plane such that they have the same features. For the following poles marked on the s-plane: a) b) c), d) ~, and e) The axis labels (σ and jω) please draw their corresponding locations (and/or terms) on the z-plane. Please also briefly justify your mapping/answer. jω jπ/t? σ?? jπ/t s-plane Corresponding points on the z-plane (e.g. the on the σ-jω maps to points outside the unit circle in the z-plane as marked) Page 7 of 4

8 3. Got LTI? (0%) A system consists of two blocks, with the following input u(t) and output y(t) pairs: Input u (t): Output y (t): τ t τ t Input u 2 (t): Output y 2 (t): τ τ t t a) Please provide a transfer function, H (s) and H (z) for y given u. b) Is the entire system (H H 2 ) LTI? (please briefly explain) c) If it is LTI, what is the order of the system? If it is not LTI, what could be done to easily make it LTI? 4. Steer-by-Wire (0%) A steer-by-wire system is proposed in which a DC motor regulates the hydraulic flow of a power steering system. It has the following continuous time plant The system is connected to a digital controller D(z) by a ZOH process 2 having a period of T=0.sec. Input R(s) D(z) ZOH Angle θ(s) G(z) a) Determine G(z) [Hint: For this part you may leave it in terms of the z-transform, Z{ } (i.e. G(z)=Z{G(s)}).] b) Sketch the impulse response of G(z) Such systems are commercially available (often in very high-end vehicles) and should not be confused with all electric steer-by-wire systems (e.g. Nissan s Q50). 2 Zero Order Hold. Recall that a ZOH is modelled by Page 8 of 4

9 5. One Last Stop on the ELEC3004 Express! (0%) As part of an electric train, you need to implement a control system that stops a DC electric motor as fast as possible. Recall that a DC motor is characterized by Where: v : the voltage at its electrical terminals i : the current i : the time derivative of i ( i.e. ) ω : shaft rotational speed (in rad/sec) R : resistance of the motor winding L : inductance of the motor winding k : motor constant V M Recall that the torque is given by. The mechanical torque model is given by To simplify exam calculations, assume unity values for all constants (with, of course, the appropriate physical units). Thus, R=, L=, k=, J=and b=. Given that the motor has some initial speed when the brake is applied at t=0, ω(0), the task is to stop the motor. To do this, we throw a switch that disconnects the driving voltage and connects the terminals of the motor to a stopping circuit. One basic design for this is a big resistor, R Big, resulting in a) Derive a discrete transfer function from ω(0) to v stopping (z) for this plant, using the Tustin s method. b) What is the slowest sampling rate that will not destabilise the system under unitygain proportional negative feedback? c) Under what conditions can the inductance be ignored, and the motor treated as a single pole system? d) What value of R Big results in the motor velocity stopping the fastest? e) Prof Gordian DuKnot suggests that the best thing to do is to set R Big = 0. That is, short circuit the motor. DuKnot s argument is simple: if more voltage makes it go faster, then less voltage makes it slower. Thus, zero voltage stops it. Is this wise? (please explain your reasoning) END OF EXAMINATION Thank you!!! Is the wonder still there? Page 9 of 4

10 0 ELEC 3004 / 732: Systems: Signals & Controls Final Examination 204 The Laplace Transform The Z Transform IIR Filter Pre-warp Bi-linear Transform FIR Filter Coefficients Table : Commonly used Formulae c n = t π F (s) = F (z) = 0 f(t)e st dt f[n]z n n=0 ω a = 2 ( ) t tan ωd t 2 s = 2( z ) t( + z ) π/ t 0 H d (ω) cos(nω t) dω Table 2: Comparison of Fourier representations. Time Domain Discrete Continuous Periodic Discrete Fourier Transform X[k] = N x[n] = N n=0 N k=0 x[n]e j2πkn/n X[k]e j2πkn/n Complex Fourier Series X[k] = T x(t) = T/2 T/2 k= x(t)e j2πkt/t dt X[k]e j2πkt/t Non-periodic Discrete-Time Fourier Transform X(e jω ) = x[n]e jωn x[n] = 2π n= π π X ( e jω) e jωn dω Fourier Transform X(jω) = x(t) = 2π x(t)e jωt dt X(jω)e jωt dω Discrete Continuous Freq. Domain Periodic Non-periodic COPYRIGHT RESERVED Page 0 of 4 TURN OVER

11 ELEC 3004 / 732: Systems: Signals & Controls Final Examination 204 Table 3: Selected Fourier, Laplace and z-transform pairs. Signal Transform ROC D x[n] = δ[n pn] X[k] = N x(t) = δ T [t] = p= x[n] = δ[n] δ(t pt ) p= p= δ(t pt ) cos(ω 0 t) { sin(ω 0 t) when t T 0, x(t) = x(t) = πt sin(ω ct) x(t) = δ(t) x(t) = δ(t t 0 ) x(t) = u(t) x[n] = ω c π sinc ω cn x(t) = δ(t) DT X(e jω ) = FS X[k] = T X(jω) = 2π T k= δ(ω kω 0 ) X(jω) = πδ(ω ω 0 ) + πδ(ω + ω 0 ) X(jω) = jπδ(ω + ω 0 ) jπδ(ω ω 0 ) X(jω) = 2sin(ωT 0) ω { when ω ω c, X(jω) = X(jω) = X(jω) = e jωt 0 X(jω) = πδ(w) + DT X(e jω ) = { jw when ω < ω c, L X(s) = all s (unit step) x(t) = u(t) L X(s) = s (unit ramp) x(t) = t L X(s) = s 2 x(t) = sin(s 0 t) x(t) = cos(s 0 t) x(t) = e s 0t u(t) x[n] = δ[n] x[n] = δ[n m] x[n] = u[n] x[n] = z n 0 u[n] x[n] = z n 0 u[ n ] x[n] = a n u[n] L X(s) = L X(s) = L X(s) = z s 0 (s 2 + s 02 ) s (s 2 + s 02 ) s s 0 Re{s} > Re{s 0 } X(z) = all z z X(z) = z m z X(z) = z z z X(z) = z 0 z z > z 0 z X(z) = z 0 z z < z 0 z X(z) = z z a z < a COPYRIGHT RESERVED Page of 4 TURN OVER

12 2 ELEC 3004 / 732: Systems: Signals & Controls Final Examination 204 Table 4: Properties of the Discrete-time Fourier Transform. Property Time domain Frequency domain Linearity ax [n] + bx 2 [n] ax (e jω ) + bx 2 (e jω ) Differentiation (frequency) nx[n] j dx(ejω ) dω Time-shift x[n n 0 ] e jωn 0 X(e jω ) Frequency-shift e jω0n x[n] X(e j(ω ω0) ) Convolution x [n] x 2 [n] X (e jω )X 2 (e jω ) Modulation x [n]x 2 [n] 2π (e jω ) X 2 (e jω ) Time-reversal x[ n] X(e jω ) Conjugation x [n] X (e jω ) Symmetry (real) Im{x[n]} = 0 X(e jω ) = X (e jω ) Symmetry (imag) Re{x[n]} = 0 X(e jω ) = X (e jω ) Parseval x[n] 2 = π X(e jω ) 2 dω 2π n= π Table 5: Properties of the Fourier series. Property Time domain Frequency domain Linearity a x (t) + b x 2 (t) ax [k] + bx 2 [k] d x(t) j2πk Differentiation dt T X[k] (time) Time-shift x(t t 0 ) e j2πkt0/t X[k] Frequency-shift e j2πk 0t/T x(t) X[k k 0 ] Convolution x (t) x 2 (t) T X [k]x 2 [k] Modulation x (t) x 2 (t) X [k] X 2 [k] Time-reversal x( t) X[ k] Conjugation x (t) X [ k] Symmetry (real) Im{ x(t)} = 0 X[k] = X [ k] Symmetry (imag) Re{ x(t)} = 0 X[k] = X [ k] T/2 Parseval x(t) 2 dt = X[k] 2 T T/2 k= COPYRIGHT RESERVED Page 2 of 4 TURN OVER

13 3 ELEC 3004 / 732: Systems: Signals & Controls Final Examination 204 Table 6: Properties of the Fourier transform. Property Time domain Frequency domain Linearity a x (t) + b x 2 (t) ax (jω) + bx 2 (jω) Duality X(jt) 2πx( ω) Differentiation dx(t) dt jωx(jω) Integration t x(τ) dτ X(jω) + πx(j0)δ(ω) jω Time-shift x(t t 0 ) e jωt 0 X(jω) Frequency-shift e jω0t x(t) X(j(ω ω 0 )) Convolution x (t) x 2 (t) X (jω)x 2 (jω) Modulation x (t)x 2 (t) 2π (jω) X 2 (jω) Time-reversal x( t) X( jω) Conjugation x (t) X ( jω) Symmetry (real) Im{x(t)} = 0 X(jω) = X ( jω) Symmetry (imag) Re{x(t)} = 0 X(jω) = X ( ) ( jω) jω Scaling x(at) a X a Parseval x(t) 2 dt = 2π X(jω) 2 dω Table 7: Properties of the z-transform. Property Time domain z-domain ROC Linearity ax [n] + bx 2 [n] ax (z) + bx 2 (z) R x R x2 Time-shift x[n n 0 ] z n 0 X(z) Scaling in z z0 n x[n] X(z/z 0 ) R x z 0 R x in z R x Time-reversal x[ n] X(/z) /R x Differentiation nx[n] z dx(z) dz Conjugation x [n] X (z ) R x Symmetry (real) Im{x[n]} = 0 X(z) = X (z ) Symmetry (imag) Re{X[n]} = 0 X(z) = X (z ) Convolution x [n] x 2 [n] X (z)x 2 (z) R x R x2 Initial value x[n] = 0, n < 0 x[0] = lim X(z) z z = 0 or z = may have been added or removed from the ROC. COPYRIGHT RESERVED Page 3 of 4 TURN OVER

14 4 ELEC 3004 / 732: Systems: Signals & Controls Final Examination 204 Table 8: Commonly used window functions. 0 Rectangular: w rect [n] = { when 0 n M, Bartlett (triangular): 2n/M when 0 n M/2, w bart [n] = 2 2n/M when M/2 n M, Hanning: { w hann [n] = 2 2 cos (2πn/M) when 0 n M, Hamming: w hamm [n] = { cos (2πn/M) when 0 n M, Blackman: cos (2πn/M) when 0 n M, w black [n] = cos (4πn/M) 20 log 0 W rect (e jω ) 20 log 0 W bart (e jω ) 20 log 0 W hann (e jω ) 20 log 0 W hamm (e jω ) 20 log 0 W black (e jω ) ω ( π) ω ( π) ω ( π) ω ( π) ω ( π) Type of Window Peak Side-Lobe Amplitude (Relative; db) Approximate Width of Main Lobe Peak Approximation Error, 20 log 0 δ (db) Rectangular 3 4π/(M + ) 2 Bartlett 25 8π/M 25 Hanning 3 8π/M 44 Hamming 4 8π/M 53 Blackman 57 2π/M 74 COPYRIGHT RESERVED Page 4 of 4 END