Eksamen Invulvraestel Kopiereg voorbehou. Exam Fill in paper Copyright reserved. Linear Systems ELI November 2010

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1 Departement Elektriese, Elektroniese en Rekenaar-Ingenieurswese Eksamen Invulvraestel Kopiereg voorbehou Lineêre Stelsels ELI0 4 November 010 Department of Electrical, Electronic and Computer Engineering Exam Fill in paper Copyright reserved Linear Systems ELI0 4 November 010 Student se besonderhede: Student's details: Naam en Van Name and Surname Tel. nr. gedurende eksamen Tel no. during examination Studentenommer: Student number: Tel. nr. na eksamen Tel. no. after examination Eksameninligting: Examination information: Maksimum punte: Maximum marks: 90 Duur van vraestel: Duration of paper: Totale aantal bladsye (hierdie blad ingeslui: Total numer of pages (including this page): Volpunte: Full marks: 90 3 ure / 3 hours Oopboek / toeboek: Open / closed book: 15 Punt: Mark: Toeboek Closed book BELANGRIK IMPORTANT 1. Die eksamenregulasies van die Universiteit van Pretoria geld. The examination regulations of the University of Pretoria apply.. Hierdie vraestel mag slegs in Afrikaans of Engels beantwoord word. This question paper may only be answered in Afrikaans or English. 3. Geen sakrekenaar mag gebruik word nie. No calculators of any kind are allowed. Interne Eksaminator(e): Internal examiner(s): Mrs. M. Ballot Eksterne Eksaminator: External Examiner: Prof L. Linde EECExam-

2 Vraag 1: Basiese begrippe [15] Question 1: Basic concepts a) Die frekwensie domein van die oordragsfunksie van ʼn lineêre tyd-onveranderlike (LTI) stelsel word gegee deur H (ω). As die inset x = δ ( t π ), gee ʼn uitdrukking vir die frekwensie domein van die uitset, Y (ω) i.t.v. H (ω). Gee ook ʼn vereenvoudigde uitdrukking vir die tyd-domein van die uitset, y ( i.t.v. h (. [] The frequency domain of the transfer function of a linear time-invariant (LTI) system is H (ω). With an input of x = δ ( t π ), provide an expression for the frequency-domain of the output, Y (ω), in terms of H (ω). Also provide a simplified expression for the time-domain, y ( in terms of h (. h t = e vir t 3t b) Die tyddomein oordragfunksie van ʼn stelsel word gegee deur ( ) 0 en h = 0 vir t < 0. Bepaal en skets die uitset as die stelsel aangedryf word met ʼn sein t 1 = Π x. [5] 3 The time domain transfer function is given by ( ) = t h t e vir t 0 and h = 0 for t < 0. Determine and draw the t 1 output of the system when the signal x = Π is applied to it. EECExam van/of 15

3 c) 3t Bepaal die Laplace transvorm van h( = (1 e ) vir t 0 en h = 0 vir t < 0. Wat is die konvergensie gebied van die Laplace transvorm? [3] 3t Determine the Laplace transform of h( = (1 e ) vir t 0 andh = 0 for t < 0. What is the area of convergence of the Laplace transform? 3t d) Bepaal die Fourier transvorm van h( = (1 e ) vir t 0 en h = 0 vir t < 0. [3] 3t Determine the Fourier transform of h( = (1 e ) vir t 0 andh = 0 for t < 0. e) Gee 'n tyd-domein uitdrukking vir die respons/uitset van die stelsel soos in vrae c) d) op 'n inset x = cos3t vir t R. [] Provide a time-domain expression for the response/output of the system as in questions c)-d) when the input is x = cos3t vir t R. EECExam 3 van/of 15

4 Vraag : Veelvuldige keuse [6 6] Question : Multiple Choice Voltooi hierdie vraag op Kant van die oranje veelvuldige keuse antwoordblad. Complete this question on Side of the orange multiple choice answer sheet. Vir elk van die volgende stelsels (1-4) dui aan of dit: For each of the following systems (1-4) indicate whether it is: [4x] a) Liniêr, tyd-onveranderlik Linear, time-invariant c) Liniêr, tyd-veranderlik Linear, time-variant d x( dx( dy( 1) y = dt dt dt ωot ) y( = 1+ α x( cosωot and x( = sin 5 st = γ + β λ and x( e ; b) Nie-liniêr, tyd-onveranderlik Non-linear, time-invariant d) Nie-liniêr, tyd-veranderlik Non-linear, time-variant 3) y[ n] = h[ i n] x[ i] [] m = α Waar h[n] die stelsel verteenwoordig gegee deur: h [ n] n u[ n 1] < 1. = α α Where h[n] represent the system with h [ n] n u[ n 1] < 1. 4) Die stroombaan hieronder met The circuit below with i v = v = i t = α t 1 R t h1 = F RC 1 e u( RC + R A Bepaal watter klassifikasie die volgende stelsels die beste beskryf: Determine which classification describes the systems below the best: a) Stabiel, intyds en oorsaaklik Stable, instantaneous and causal c) Stabiel, dinamies en oorsaaklik Stable, dynamic and causal e) Stabiel, intyds en nie-oorsaaklik Stable, instantaneous and non-causal g) Stabiel, dinamies en nie-oorsaaklik Stable, dynamic and non-causal b) Onstabiel, intyds en oorsaaklik Unstable, instantaneous and causal d) Onstabiel, dinamies en oorsaaklik Unstable, dynamic and causal f) Onstabiel, intyds en nie-oorsaaklik Unstable, instantaneous and non-causal h) Onstabiel, dinamies en nie-oorsaaklik Unstable, dynamic and non-causal s 1 s + 3 z = z α α α 3 5) ( ) = s H s e ROC Re{ s} > 0 1 6) H ( z) ROC z > = 4 [] [] EECExam 4 van/of 15

5 7) Om te bepaal watter vraestel u doen, moet u nou nommer f) merk. To determine which paper you are doing you must now mark number f). Vir elk van die volgende seine (8-10) dui aan of dit: For each of the following signals (8-10) indicate whether it is: a) Periodiese Drywingsein Periodic Power Signal b) Non-Periodiese Drywingsein Nie-Periodic Power Signal c) Periodiese Energie-sein Periodic Energy Signal d) Nie-periodiese Energie-sein Non-periodic Energy Signal e) Periodiese Energie en Drywingsein Periodic power and energy signal. f) Nie-periodiese Energie en Drywingsein Non-Periodic power and energy signal. g) Periodiese sein maar nie Energie of Drywingsein nie. Periodic signal but not power or energy signal. h) Nie-periodiese sein maar nie Energie of Drywingsein nie. Non-Periodic signal but not power or energy signal. π 8) Die sein x[ n] = sin( n) [] The signal x[ n] = sin( πn) 9) x = sin(3 + cos(5 [] 10) x = t sin ct [] Vir elk van die volgende seine (10-11) dui aan of dit: For each of the following signals (10-11) indicate whether it is: a) Analoog, Deterministies Analogue, Deterministic b) Digitaal, Deterministies Digital, Deterministic c) Analoog, Willekeurig Analogue, Stochastic d) Digitaal, Willekeurig Digital, Stochastic 11) Die Nasionale lied soos gesing deur die skare op Loftus [] The National Anthem as sang on Loftus by the crowd. 1) Die uitset van die horlosie op 'n rekenaar. [] The output of the clock on a computer. 13) Die aantal bewerkings wat nodig is vir n FFT benadering met N monsters gebruik word is: [] The number of calculations required for a FFT approximation with N samples is: a) log ( N ) b) 10log( N ) c) log( ) 0 N d) 16log ( N ) e) N log(n) f) N g) N + N( N 1) h) N ( N 1) N N j) 0 log N i) log ( ) x = p1 ( t n) benader word in die vorm x a = a0 + am cos( mπ lyk die resultaat m= 14) As die funksie n= soos hier onder getoon. Dit word die... genoem If the function n= x = p1 ( t n) is approximated in the form x a = a0 + am cos( mπ the result will = look like the figure below. It is called the a) Armstrong fenomeen / phenomenon b) Dietrich fenomeen / phenomenon c) Dirac fenomeen / phenomenon d) Gibbs fenomeen / phenomenon e) Fourier fenomeen / phenomenon f) Laplace fenomeen / phenomenon g) Nyquist fenomeen / phenomenon h) Parseval fenomeen / phenomenon m EECExam 5 van/of 15

6 Vraag 3: Monstering [3] Question 3: Sampling 1 a) Bepaal skets die amplitude en fase van die frekwensie domein van = sin c( f t ) x m. Merk alle afsnitte, Determine and draw the amplitude and phase of the frequency domain of ( ) amplitudes en waarde duidelik. [6] 1 x = sin c f m t. Indicate all axescrossings, amplitudes and values clearly. EECExam 6 van/of 15

7 n= b) Die sein in vraag 4a) word gemonster deur ʼn reeks impuls funksies, c = δ ( t ) (, met nt s T s 1 Nf =. Gee ʼn uitdrukking vir die resulterende sein, y(, in die tyddomein en bepaal en skets die amplitude en fase van die frekwensie domein weergawe van y(, Y ). N=5 [5] (ω n= The signal in question 4a) is sampled with a series of impulse functions, c = δ ( t ) ( with nt s T s m 1 Nf =. Provide an expression for the resultingtime signal, y(,in the time domain and determine and draw the amplitude and phase of the frequency domain of y(, Y ). N=5. (ω m EECExam 7 van/of 15

8 c) Ontwerp 'n herkonstrueringsfilter vir die monsteringstelsel hierbo. Die filter moet moontlike steurende seine met ongeveer 0dB onderdruk. Spesifiseer die tipe wat u sal gebruik, motiveer u antwoord. Dui alle ontwerpsvergelyking, stroombaandiagramme en komponent keuses aan. U hoef nie die fisiese komponent waardes uit te werk nie, maar verwerk die vergelykings tot op so 'n vlak dat jou graad 7 boetie dit sal kan uitwerk. [1] Design a reconstruction for the sampling system above. The filter must suppress possible interfering signals with around 0dB. Specify the type of filter you will use with a motivation. Show all design equations, circuit diagrams, when you choose component values. You do not need to calculate the physical component values, but process it to a level that your grade 7 brother will be able to calculate it EECExam 8 van/of 15

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10 Vraag 4: [1] Question 4: Jou oom, 'n elektroniese ingenieur uit 'n vorige generasie, wie se kennis van lineêre stelsels en seine ook so bietjie verroes is kom klop by jou aan vir hulp. Daar is nou hierdie nuwe stelsel: OFDVT (MA) (Ortogonale Frekwensie Divisie Veelvuldige Toegang. Albei hoofstroom 4G tegnologieë, WiMax en LTE maak daarvan gebruik. Hy het jou genader om hom bietjie te help om insig te kry. Hy vermoed dat julle saam die ding sal kan uitpluis. Hy het die blok-diagram hieronder saamgebring. x[n] is simbole wat eintlik maar as komplekse getalle beskou kan word. T 0 is die simbool periode. Your uncle, an electronics engineer from 'n previous generation, whose knowledge of linear systems and signal are a bit rusted come to you for some help. There is now 'n new system: OFDMA (Orthogonal Frequency Division Multiple Access). Both main stream 4G technologies, WiMax and LTE use this. You uncle hope that between the two of you, you will be able to understand this new method. He brought the block diagram below with. x[n] are symbols that can be viewed as complex numbers. T 0 is the symbol period. Sender / Transmitter Analogue to Digital Conversion Mapping of bits onto symbols Serial to Parallel Conversion s 1 cosω 0 t s cos ω 0 t s = i + x[ i + 4 p] Π p= 0 T0 u ( t ) t pt 0 s ( t 3 ) cos3ω 0 t s 4 cos 4ω 0 t cosω 0 t r 1 r Parallel to Serial Conversion Mapping of symbols to bits Digital to Analogue Conversion cos ω 0 t u ( t ) r ( t 3 ) cos3ω 0 t r 4 cos 4ω 0 t Ontvanger / Receiver EECExam 10 van/of 15

11 a) Jou oom verstaan nie lekker van die "ortogonale frekwensies" nie. Dui aan dat z cos n t z z m m = cos mω t ortogonaal is oor 'n periode van o 0 π = ω Your uncle don't get this thing about "orthogonal frequencies". Show that = cos mω t are orthogonal, over a period of o 0 n ( = ω en T (met n m) [4] 0 π = ω T (with n m) 0 z = cos nω t and n o o ω ω ω 4ω b) Neem u = s cos t + s cos t + s cos3 t s cos t, as die inset van die ontvanger. Bepaal r ( ) funksies vir slegs die eerste simbool periode, dit is wanneer p=0. [] Take r ( ) 3 t ( 1 o o 3 o + 4 t 3 s1 cosωot + s cos ω ot + s3( cos3ω ot + s4 cos 4 o u =, as the input to the receiver. Detemine for only the first simbol period. i.e. when p=0. ω t o EECExam 11 van/of 15

12 c) As x[1]=1, x[]=-1, x[3]=1 en x[4]=-1, bepaal 'n uitdrukking vir en teken die frekwensie vlak van u( vir die eerste simbool periode. Teken dit op die reele vlak. Dui alle waardes en as-afsnitte duidelik aan. (Wenk: teken dit eers in potlood) [6] If x[1]=1, x[]=-1, x[3]=1 and x[4]=-1, determine an expression for and draw the frequency domain of u( for the first symbol period. Draw it on the real plane. Indicate all values and axis-crossings clearly. (Hint: first draw it in pencil) EECExam 1 van/of 15

13 Vraag 5: Stelsel versyfering [14] Question 5: System digitisation U en n vriend word aangestel as tutors vir die nuwe ELI groep. U opdrag is om ʼn stelsel te ontwerp wat digitaal die konsep van ʼn oordragsfunksie en impulsrespons te illustreer op ʼn rekenaar. You and a friend are appointed as tutors for the new ELI group. You assignment is to develop a system that will digitally illustrate the concept of the transfer function on a computer. Die stroombaan wat u moet illustreer is: The circuit that you need to simulate is: R 1 + F ( ) RA H s = RCs + 1 Om berekening te vereenvoudig neem To simplify calculations take: RF 1 + R a = RC A ; b = RC a) Bepaal die kontinue-tyd oordragsfunksie van die stelsel in die tyd-domein i.e. die impulsresponse. [3] Determine the continuous-time transfer function of the system in the time-domain. i.e. the impulse response. EECExam 13 van/of 15

14 b) Bepaal die differensiaal vergelyking wat die stelsel hierbo beskryf. Die stelsel word versyfer, teen n monsteringsperiode van T. Stel n verskilvergelyking op om die stelsel as n diskrete tyd benadering van die werklike stelsel voor te stel s deur gebruik te maak van die differensiaalvergelyking. [5] Determine the differensial equation that will describe the system above. The system is to be digitised using a sampling period oft. Derive a difference equation to represent the system as a discrete time approximation of the s actual system using the differential equation. c) Bepaal die z-transvorm van die oordragsfunksie vir die verskilvergelyking hierbo. [] Determine the z-transform of the transfer-function for the difference equation above. EECExam 14 van/of 15

15 d) Aan watter kriteria moet a, b en T s voldoen om te verseker dat die stelsel stabiel is en n goeie benadering sal gee? [] Which criteria do a, b and T s adhere to, to ensure that the system is stable and provide a good approximation? e) Bepaal die frekwensie domein voorstelling van die stelsel. [] Determine the frequency domain representation of the system EECExam 15 van/of 15