Exam. Notes: A single A4 sheet of paper (double sided; hand-written or computer typed)

Transcription

1 Exam February 8th, 8 Sigals & Systems (5-575-) Prof. R. D Adrea Exam Exam Duratio: 5 Mi Number of Problems: 5 Number of Poits: 5 Permitted aids: Importat: Notes: A sigle A sheet of paper (double sided; had-writte or computer typed) Questios must be aswered o the provided aswer sheet; aswers give i the booklet will ot be cosidered. There exist multiple versios of the exam, where the order of the aswers has bee permuted radomly. Every questio has a uique correct aswer. Every questio is worth oe poit for a correct aswer, ad zero poits otherwise. Givig multiple aswers to a questio will ivalidate the aswer. No egative poits will be give for icorrect aswers. Partial poits (Teilpukte) will ot be awarded. You do ot eed to justify your aswers; your calculatios will ot be graded. Use oly the provided paper for your calculatios; additioal paper is available from the supervisors. Good luck!

2 +//59+ Discretizatio Box : Questios, Cosider the cotiuous-time system: q(t) = [ ] q(t) + [ ] u(t). Questio What is the system descriptio resultig from a exact discretizatio with samplig time T s? [ ] [ ] Ts Ts A q[ + ] = q[] + u[] T s [ ] [ Ts B q[ + ] = q[] + u[] ] [ ] [ ] Ts Ts C q[ + ] = q[] + u[] T s [ ] [ Ts Ts ( + D q[ + ] = q[] + T ] s) u[] T s Questio What is the system descriptio resultig from a forward Euler discretizatio with samplig time T s? [ ] [ ] Ts Ts A q[ + ] = q[] + u[] T s [ ] [ Ts B q[ + ] = q[] + u[] ] [ ] [ ] Ts Ts C q[ + ] = q[] + u[] T s [ ] [ Ts Ts ( + D q[ + ] = q[] + T ] s) u[] T s Questio The cotiuous-time sigal ( ) u(t) = cos t is sampled with T s =. resultig i a discrete-time sequece {u[]}. The sigal {u[]} is recorded for N = cosecutive time steps. How may periods of {u[]} are recorded? A B C D E 5 F 6

3 +//58+ Box : Questios, 5 Cosider the followig portio of the periodic cotiuous-time sigal x(t): x(t).5.5 t Questio What is the fudametal period of the discrete-time sigal {x[]} if x(t) is sampled with T s =.5: A B C D 6 E 8 Questio 5 What is the resultig discrete-time sigal {x[]} if x(t) is sampled with T s =, where the first sample is take at t =? A x[] =.5 + cos ( ). si () D B x[] = cos ( x[] =.5 + cos ( ) ) E x[] = si ( ) C x[] =.5 + si ( ) F x[] = cos ( ). si () Box : Questios 6, 7 Cosider the cotiuous-time sigal: x(t) = cos ( ) 6 5 t + cos (t) ad the discrete-time sigals: ( ) ( ) ( 6 6 x [] = + cos 5, x [] = + cos 5, x [] = cos 5 + ). Questio 6 Which discrete-time sequeces result from samplig x(t) with samplig time T s =, where the first sample is take at t =. A Noe B x [] C x [] D x [] E x [] or x [] F x [] or x [] G x [] or x [] H x [], x [], or x []) Questio 7 Which discrete-time sequeces result from samplig x(t) with samplig time T s =, where the first sample is take at t =.5. A Noe B x [] C x [] D x [] E x [] or x [] F x [] or x [] G x [] or x [] H x [], x [], or x [])

4 +//57+ System Properties Box : Questios 8, 9, Cosider the followig systems: G : y[] = u[ ] G : y[] = u[] + u[ + ] G : y[] = y[ ] y[ ] + u[], causal G : y[] = (y[ ]) + u[], causal Questio 8 Select all liear systems. A G, G C G, G E G, G G G, G, G B G, G D G, G F G, G H G, G, G Questio 9 Select all time-ivariat systems. A G, G C G, G E G, G G G, G, G B G, G D G, G F G, G H G, G, G Questio Select all bouded-iput bouded-output stable systems. A G, G C G, G E G, G G G, G, G B G, G D G, G F G, G H G, G, G Box 5: Questio Cosider the followig represetatio of systems: G : h = {...,,,,,,,...} G : y[] = y[ ] + u[], ati causal z G : H(z) = z.5z +.5 Questio Which systems are guarateed to be stable? A G C G E G, G G G, G, G B G D G, G F G, G

5 +/5/56+ Box 6: Questios,, Cosider the followig portios of the system resposes {y []} ad {y []} to iput {s[]}, where { if s[] =. else y[] y[] Questio A {y []} B {y []} Which of the system resposes could result from a FIR filter? C Noe of them D {y []} ad {y []} Questio the system? Cosider the system respose {y []}. Which differece equatio could represet A y[] = y[ ] + u[] B y[] = u[] + u[ ] + u[ ] C y[] = y[ ] + u[] D y[] = u[] u[ ] E y[] = y[ ] + u[ ] Questio the system? Cosider the system respose {y []}. Which differece equatio could represet A y[] = y[] + u[ ] + u[ ] B y[] = y[ ] y[ ] + u[] C y[] = u[] + u[ ] + u[ ] D y[] = u[] + u[ ] + u[ ] + u[ ] E y[] = u[] + u[ ] + u[ ]

6 +/6/55+ Box 7: Questios 5, 6 The followig impulse resposes of four causal FIR filters are covolved with the sequece u h[] h[] h[] h[] u[] resultig i the followig four output sequeces. yd[] yc[] yb[] ya[]

7 +/7/5+ Questio 5 Which output sequece y correspods to y = h u? A y = y A B y = y B C y = y C D y = y D Questio 6 Which impulse respose h correspods to y B = h u? A h = h B h = h C h = h D h = h Box 8: Questios 7, 8, 9 Cosider the followig state-space descriptios of causal, LTI DT systems, where [ ] A B q[ + ] = Aq[] + Bu[] G : is iterpreted as C D y[] = Cq[] + Du[]. G : G : G : G 5 : [ [ ] ] G : Questio 7 Select the aswer cotaiig all stable systems. A G,G C G 5 E G G G,G 5 B G,G,G 5 D G,G,G,G 5 F G H G Questio 8 Select the aswer cotaiig all systems with a FIR. A G B G C G D G E G 5 F G,G G G, G H G, G 5 I G, G 5 Questio 9 Which is a state-space represetatio of a system with LCCDE y[] = y[ ] + u[] + u[ ]? A G B G C G D G E G 5

8 +/8/5+ Box 9: Questio Cosider the followig portios of impulse resposes {h i []} correspodig to stable, LTI systems G i, where i {,,,, 5, 6}. h[].5 h[] h[].5 h[] h5[].5 h6[] Questio Which system has trasfer fuctio H(z) = z + z z? A G B G C G D G E G 5 F G 6

9 +/9/5+ Box : Questios, Cosider a causal system with the differece equatio y[] = y[ ] y[ ] + u[ ]. Questio Iput u is applied to the system with: { if u[] =. else What is the value y[] for? A B.5 C D.5 E 5 Questio A periodic sigal u is applied to the system with: u[] = cos(). What is the output of the system? A y[] = cos( ) B y[] = cos( ) C y[] = 9 cos( ) D y[] = cos() E y[] = cos() F y[] = 9 cos()

10 +//5+ Frequecy Domai Box : Questios, Cosider a causal LTI system G for which y = G u is give by the frequecy respose: H() G is used to create systems G A ad G B i the followig maer: u G A G G y u G B G G + y The iput u to both systems G A ad G B is: Remark: The sum operator ( ) u[] = cos + ( ) cos + si () x + x x is defied as x = x + x. Questio What is the resultig output y = G A u? A y[] = 9 cos ( ) + 9 si () B y[] = cos ( ) +cos ( ) + si () C y[] = si ( ) + si () D y[] = cos ( ) + cos ( ) + si () E y[] = 9 cos ( ) + 9 si () F y[] = si ( ) + si () Questio What is the resultig output y = G B u? A y[] = cos ( ) +cos ( ) + si () B y[] = si ( ) + si () C y[] = 9 cos ( ) + 9 si () D y[] = cos ( ) + si () E y[] = 9 cos ( ) + 9 si () F y[] = cos ( ) + cos ( ) + si ()

11 +//5+ Box : Questios 5, 6 Cosider the followig DT Fourier magitude plots of four differet causal sequeces. XA().5 XB().5 XC().5 XD().5 Questio 5 Which could be the magitude plot correspodig to the followig causal sequece? A X A () B X B () C X C () D X D () x[] Questio 6 Which could be the magitude plot correspodig to the followig causal sequece? A X A () B X B () C X C () D X D () x[]

12 +//9+ Box : Questios 7, 8 Cosider the followig magitude ad phase respose of a stable LTI system G..5 H() Questio 7 Which is the trasfer fuctio of the LTI system G? A H(z) =.5(z+) z.5 B H(z) =.5(z+) z. C H(z) = (z+) z D H(z) = z+ z E H(z) = z+.7 z. Questio 8 The LTI system is cascaded with a delay elemet G D with trasfer fuctio H D (z) = z. Which plot shows the magitude ad phase respose of the resultig system G casc? u G casc G G D y A.5 H() B.5 H() C.5 H() D.5 H()

13 +//8+ Questio 9 The followig sequece is periodic with period of N = 6 {x[]} = k {,,,,, } where k is a iteger. The correspodig DFS coefficiets are {X[k]} = {, X[],,,, X[5]}. What is the magitude of coefficiet X[]? A X[] = k B X[] = 7 k C X[] = 6 k D X[] = 6 k E X[] = 6 k F X[] = k Box : Questios, Cosider the followig DT sigal x[] = e j 7 + e j 7 for all times. A DFT of the first N = samples of sigal {x[]} is take. Let {X[k]} be the correspodig DFT coefficiet. Questio A X[] = B X[] = What is the value of X[]? C X[] = D X[] = E X[] = 6 F X[] = Questio A X[6] = B X[6] = 6 What is the value of X[6]? C X[6] = D X[6] = E X[6] = F X[6] =

14 +//7+ Box 5: Questio Cosider the followig coefficiets of the DFT calculated with the first N = samples of a real, discrete-time sigal {x[]}: X[k] k Questio Which plot could show the coefficiets of the DFT calculated with the first N = 6 samples of the same sigal {x[]}? A X[k] 5 k C X[k] 5 k B X[k] 5 k D X[k] 5 k Questio Cosider the trasfer fuctio of system G H(z) = z + z z. What is the impulse respose of G? A h = {...,,,,,,,...} B h = {...,,,,,,,...} C h = {...,,,, 9, 7,...} D h = {...,,,,, 8,...} E h = {...,,,,,,,...} F h = {...,,,, 9, 7,...}

15 +/5/6+ Questio fuctio Which of the followig magitude resposes belogs to a system with trasfer H(z) = z 8 z + z +.5? A.5 D.5 B.5 E.5 C.5 F.5

16 +/6/5+ Box 6: Questio 5 Let G be a causal LTI system. trasfer fuctio. G Im The followig plot shows the poles ad zeros of the system s Re Recall that: z p is a pole of H(z) ( o a pole-zero plot), if H(z p ) = z z is a zero of H(z) ( o a pole-zero plot), if H(z z ) =. Questio 5 Which of the followig magitude resposes could belog to system G? A.5 C.5 B.5 D.5

17 +/7/+

18 +/8/+ Filterig Questio 6 Cosider the followig CT low-pass ad high-pass filters. HLP (ω) (db) ω (rad/s) ω (rad/s) Which frequecy trasformatio was used to obtai the trasfer fuctio H HP (s) of the high-pass filter from the trasfer fuctio of the low-pass filter H LP (s)? ( A H HP (s) = H ) ( ) ( s LP s C H HP (s) = H + LP E H ( B H HP (s) = H ) s HP (s) = H ) LP s ( LP s D H HP (s) = H s ) LP F H HP (s) = H LP (s) HHP (ω) (db) Box 7: Questios 7, 8 Cosider the followig cascade of systems: u G G + y G G Available systems for G i are gais, i.e. H i (z) = a, low-pass filters, i.e. H i (z) = H LP (z), ad high-pass filters H i (z) = H HP (z). Note that you oly have to choose the class of system, ot its specificatios. Questio 7 How ca G, G, ad G be chose to desig G to be a bad-stop filter? A G : H (z) = H LP (z) G : H (z) = H HP (z) G : H (z) = B G : H (z) = H LP (z) G : H (z) = G : H (z) = H LP (z) C G : H (z) = H LP (z) G : H (z) = G : H (z) = H HP (z) D G : H (z) = H LP (z) G : H (z) = H HP (z) G : H (z) = E G : H (z) = H HP (z) G : H (z) = G : H (z) = H LP (z) F G : H (z) = H LP (z) G : H (z) = G : H (z) = H HP (z) Questio 8 How ca G, G, ad G be chose to desig G to be a bad-pass filter? A G : H (z) = H LP (z) G : H (z) = G : H (z) = H HP (z) B G : H (z) = H LP (z) G : H (z) = G : H (z) = H HP (z) C G : H (z) = H LP (z) G : H (z) = H HP (z) G : H (z) = D G : H (z) = H LP (z) G : H (z) = G : H (z) = H LP (z) E G : H (z) = H LP (z) G : H (z) = H HP (z) G : H (z) = F G : H (z) = H HP (z) G : H (z) = G : H (z) = H LP (z)

19 +/9/+ Box 8: Questios 9, A approach to realize a secod order ifiite impulse respose filter with trasfer fuctio has the followig form. H(z) = b + b z + b z + a z + a z () u g + g y G d G d g + g G d G d g + g 5 Remarks: The delay elemets G d have trasfer fuctio H d (z) = z ad the gai operator g x x is defied as x = g x with g R. Questio 9 realized? How should the gais be chose, such that secod order IIR respose filter () is A g =, g = a, g = a, g = b, g = b, g 5 = b B g =, g = a, g = a, g = b, g = b, g 5 = b C g = b, g = b, g = b, g =, g = a, g 5 = a D g = b, g = b, g = b, g =, g = a, g 5 = a Questio The form show above eeds four delay uits to realize a secod order IIR respose filter with trasfer fuctio (). Which statemet is correct? A Five is the miimum umber of delay elemets eeded to realize a geeral secod order IIR filter. Four elemets were oly sufficiet above because of a = i (). B Four is the miimum umber of delay elemets eeded to realize a geeral secod order IIR filter. C Three is the miimum umber of delay elemets eeded to realize a geeral secod order IIR filter. D Two is the miimum umber of delay elemets eeded to realize a geeral secod order IIR filter. E Oe is the miimum umber of delay elemets eeded to realize a geeral secod order IIR filter.

20 +//+ Box 9: Questios,,, Cosider the sequece {u[]}: u[] It is filtered usig four differet causal discrete-time filters G i, i {,,, } resultig i the sequeces y i = G i u: y[] y[] y[] y[]

21 +//+ Questio Which filter could have the followig magitude respose?.5 A G B G C G D G Questio Which could be the gai of the filter G at =? A H ( = ) = B H ( = ) = C H ( = ) = E D H ( = ) =.98 H ( = ) = Questio Select the aswer cotaiig all filter(s) which could have a FIR shorter tha samples. A G C G E G, G G G, G I G, G B G D G F G, G H G, G Questio Which filter could be a media filter? A G B G C G D G Questio 5 Cosider a cotiuous-time, causal system with trasfer fuctio H(s) = βs + αs + s + with α > ad β >. The system is discretized usig the biliear trasform ad samplig time T s. What is the phase of the discrete-time trasfer fuctio H(z), at z = e j? A H(z) = B H(z) = C H(z) = D H(z) = E H(z) = F H(z) = Box : Questio 6 Cosider the followig system descriptios: G : h = {...,,, /, /,,...} G : y[] = y[ ] + u[] G : y[] = u[ ] G : y[] = u[ + ] + u[] + u[ ] Questio 6 Select all systems whose phase-respose is zero. A G C G E G, G G G, G I G, G B G D G F G, G H G, G J G, G

22 +//9+ 5 System Idetificatio Box : Questios 7, 8 The objective of the followig MATLAB code is to estimate the trasfer fuctio coefficiets a, b ad b of the trasfer fuctio H(z) = b + b z + a z give estimated frequecy resposes Ĥ() for frequecies icludig = ad = saved i MATLAB vectors H.mat ad Omega.mat, respectively. %% First order trasfer fuctio estimatio % load estimated frequecy resposes at tested frequecies load( H.mat ); % H is a xl matrix with the estimated freq resp 5 load( Omega.mat ); % Omega is a xl matrix with the correspodig freq 6 7 % umber of frequecies tested 8 L=legth(Omega); 9 % assert that eough frequecies were tested assert(...); % assert(cod) throws a error if cod is false. % matrices for least-squares problem F = zeros(*l, ); 5 G = zeros(*l, ); 6 7 for l=:l- 8 R = abs(h(l+)); % the magitude of the estimated freq resp 9 Phi = phase(h(l+)); % the phase of the estimated freq resp Om = Omega(l+); % the freq correspodig to the estimated freq resp % populate F ad G F(*l+,:) =...; F(*l+,:) = [-R*si(Phi-Om),, -si(om)]; 5 G(*l+) = R*cos(Phi); 6 G(*l+) = R*si(Phi); 7 ed 8 9 % ow estimate the trasfer fuctio coefficiets usig least-squares Theta = F\G; % defie the trasfer fuctio; % remember, a = is ot i the Theta parameter vector a = [; Theta()]. ; 5 b = Theta(:). ; Questio 7 Which optio correctly completes lie? A assert(*l>6); B assert(*l>5); C D assert(*l>); assert(*l>); E assert(*l>);

23 +//8+ Questio 8 Which optio correctly completes lie? A F(*l+,:) = [-R*cos(Om),, cos(phi-om)]; B F(*l+,:) = [-R*cos(Phi-Om),, cos(phi-om)]; C F(*l+,:) = [R*cos(Phi-Om),, cos(om)]; D F(*l+,:) = [-R*cos(Phi-Om),, cos(om)]; E F(*l+,:) = [-R*cos(Phi-Om),, cos(om)]; Questio 9 To determie the frequecy respose H() of a stable LTI system at frequecy =, a siusoidal iput {u e[]} = {A cos( )} is applied to the system ad the output sequece {y m []} is measured. The first N T samples are discarded ad the followig N samples are used to calculate the DFT coefficiets Y m [l] = N T +N N T +N y m []e j N l ad U e [l] = u e []e j N l, =N T =N T where N T = ad N =. Which is the frequecy respose estimate at frequecy =? A B C Ĥ( = ) = Ym[5] U e[5] Ĥ( = ) = Ym[5] U e[5] Ĥ( = ) = Ym[5] U e[5] D E F Ĥ( = ) = Ym[] U e[] Ĥ( = ) = Ym[5] U e[5] Ĥ( = ) = Ym[5] U e[5] Questio 5 You coducted three experimets to idetify a ukow trasfer fuctio ad obtaied the followig magitude respose estimates: Ĥ( = ) =.5, Ĥ( = ) =.9, Ĥ( = ) =.96. Which of the followig trasfer fuctios fits your experimets best? A H(z) = z+ z.5 B H(z) = z.5 z +.5 C H(z) = D H(z) = z z+.75 z z+

24 Corrected Trigoometric Fuctios cos ω si ω ω (rad) ta ω 6 6 ω (rad)

25 Corrected

26 Corrected Aswer sheet: studet umber please ecode your studet umber, ad write your first ad last ame below. Firstame ad lastame: How to select aswer B : (Uwated aswer clearly removed) (Desired aswer clear) (Caot distiguish B ad C) Discretizatio Aswers must be give exclusively o this sheet; aswers give o the other sheets will ot be couted. Q7: A B C D E F G H Q: A B C D Q: A B C D Q: A B C D E F Q: A B C D E Q5: A B C D E F Q6: A B C D E F G H Q7: A B C D E F G H System Properties Q8: A B C D E F G H Q9: 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 Q: A B C D Q: A B C D E Q: A B C D E Q5: A B C D Q6: A B C D Q8: A B C D E F G H I Q9: A B C D E Q: A B C D E F Q: A B C D E Q: A B C D E F Frequecy Domai Q: A B C D E F Q: A B C D E F Q5: A B C D Q6: A B C D Q7: A B C D E Q8: A B C D Q9: A B C D E F Q: A B C D E F Q: A B C D E F Q: A B C D Q: A B C D E F Q: A B C D E F

27 Corrected Q5: A B C D Filterig Q6: A B C D E F Q7: A B C D E F Q8: A B C D E F Q9: A B C D Q: A B C D E Q: A B C D Q: A B C D E Q: A B C D E F G H I Q: A B C D Q5: A B C D E F Q6: A B C D E F G H I J 5 System Idetificatio Q7: A B C D E Q8: A B C D E Q9: A B C D E F Q5: A B C D