Solutions. Number of Problems: 4. None. Use only the prepared sheets for your solutions. Additional paper is available from the supervisors.

Transcription

1 Quiz November 4th, 23 Sigals & Systems (5-575-) P. Reist & Prof. R. D Adrea Solutios Exam Duratio: 4 miutes Number of Problems: 4 Permitted aids: Noe. Use oly the prepared sheets for your solutios. Additioal paper is available from the supervisors.

2 Quiz Sigals & Systems Problem

3 Quiz Sigals & Systems Problem poits A system is govered by the differece equatio y[] = 2x[] x[ 3]. Determie aalytically whether the system: a) is liear; (3 poits) b) is time-ivariat. (3 poits) Now, cosider the impulse respose of a differet system, which is liear ad time-ivariat: h[] = 2 u[ + ]. c) Is the system causal? Justify your aswer. (2 poits) d) Is the system bouded-iput bouded-output (2 poits) (BIBO) stable? Justify your aswer. Solutio a) Let y [] = 2x [] x [ 3], y 2 [] = 2x 2 [] x 2 [ 3], be the system outputs to arbitrary iputs x [] ad x 2 []. The, with x 3 [] := α x [] + α 2 x 2 [], we have: y 3 [] = 2x 3 [] x 3 [ 3] = 2α x [] + 2α 2 x 2 [] α x [ 3] α 2 x 2 [ 3] = α (2x [] x [ 3]) + α 2 (2x 2 [] x 2 [ 3]) = α y [] + α 2 y 2 []. Because the system satisfies the superpositio property, we coclude that the system is liear.

4 Quiz Sigals & Systems Problem Gradig: 2 poits for the correct use of the superpositio property. poit for the correct coclusio. b) Let x 4 [] := x [ + ], the y 4 [] = 2x 4 [] x 4 [ 3] = 2x [ + ] x [ + 3] = y [ + ]. Therefore, a time-shift by ay iteger of the iput correspods to a time-shift i the output, i.e. the system is time-ivariat. Gradig: 2 poits for approach. poit for correct coclusio. c) A discrete-time LTI system is causal if ad oly if h[] =, <. For the give impulse respose we have h[ ] = 2. The system is therefore ot causal. Gradig: poit for the correct use of the causality coditio. poit for the correct coclusio. d) A discrete-time LTI system is BIBO stable if ad oly if its impulse respose is absolutely summable, that is h[k] <. = = For the give impulse respose, we have 2 u[k + ] = = 2, which is ot absolutely summable (ad thus ot BIBO stable). Gradig: poit for the correct use of the stability coditio. poit for the correct coclusio.

5 Quiz Sigals & Systems Problem 2 poits This problem cosists of five multiple-choice questios. Each questio is worth 2 poits if aswered correctly ad poits if aswered icorrectly. Write your aswers i the Your Aswer box at the ed of each questio. If you chage your mid, cross out your aswer ad write the ew oe ext to the box. No poits will be awarded for crossed out aswers ) A cotiuous-time sigal x(t) = si(6πt) is uiformly sampled with samplig time T secods to produce the discrete-time sigal x[] = x(t ) show below. x[] Based o this plot, what is the value of T? (a) /6 (b) /2 (c) /24 (d) /36 Your Aswer:

6 Quiz Sigals & Systems Problem 2 2) The plot below shows the discrete-time sigal x[] = cos(ω ). x[] Which of the followig plots is x [] = cos((ω + 2π))? Plot (a) x[] Plot (b) x[] Plot (c) x[] Plot (d) x[] Your Aswer:

7 Quiz Sigals & Systems 3) Cosider the LTI system with iput x[] ad output y[] show i the figure below: x[] H(Ω) y[] The system s magitude respose H(Ω) ad phase respose H(Ω) are show i the followig plots: H(Ω) 2 H(Ω) π ل 2 ل 2 Ω π π ل 4 ل 4 Ω π Suppose that the iput is x[] = cos( ل ) for all time. What is the output y[] for all time? (a) y[] = cos( ل 5 ل 4 ) (b) y[] = 2 cos( ل ) (c) y[] = 2 cos( ل ل 4 ) (d) y[] = cos( ل 5 ) Your Aswer:

8 Quiz Sigals & Systems Problem 2 4) The plots below show the impulse respose h[] of a LTI system ad the iput x[] to that system. h[] x[] Which of the plots below is the output y[] of the system? Plot (a) y[] Plot (b) y[] Plot (c) y[] Plot (d) y[] Your Aswer:

9 Quiz Sigals & Systems 5) Cosider the followig cotiuous-time siusoids: x(t) x2(t) x3(t) t t t x4(t) t The siusoids are uiformly sampled at 5 Hz. statemets is true? Which of the followig (a) Oly x (t) ca be sampled without aliasig. (b) Oly x (t) ad x 2 (t) ca be sampled without aliasig. (c) Oly x (t), x 2 (t) ad x 3 (t) ca be sampled without aliasig. (d) All the sigals ca be sampled without aliasig. Your Aswer:

10 Quiz Sigals & Systems Problem 2 Solutio 2 ) The correct aswer is (d) (2 poits). Notice that: Twelve samples occur per period of the siusoid The frequecy of the siusoid si(2π(3)t) is 3 Hz, therefore three periods per secod Three periods per secod twelve samples per period = 36 samples per secod Samplig frequecy of 36 Hz correspods to a samplig time of /36 secods. Thus (d) is the aswer. 2) The correct aswer is (b) (2 poits). Rememberig that is per defiitio a iteger ( Z), we have that: cos((ω + 2π)) = cos(ω + 2π) = cos(ω ) The same fuctio as show i the itroductory plot. aswer. Thus (b) is the 3) The correct aswer is (c) (2 poits). Notice that: The iput siusoid x[] = cos( ل ) has frequecy Ω = ل H(Ω) = 2 for Ω = Ω, meaig siusoidal iputs at this frequecy have a gai of 2 H(Ω) = ل 4 for Ω = Ω, meaig siusoidal iputs at this frequecy are phase shifted by ل 4

11 Quiz Sigals & Systems Therefore, optio (c) is the correct aswer 4) The correct aswer is (d) (2 poits). Maual calculatio of the covolutio proves this. Ituitively, the first impulse from x[] occurs at time =, meaig the first impulse respose occurs at time = 2 (due to the -sample delay i h[]). This discouts optios (a) ad (c), which both show respose at time =. Referrig to the covolutio defiitio: x[] h[] = x[ k]h[k] = We see that whe x[] overlaps h[] (eg. for = 3), x[] h[] = 2. This discouts optio (b), leavig optio (d) as the correct aswer. 5) The correct aswer is (b) (2 poits). Notice that: The samplig frequecy is 5Hz, givig a Nyquist frequecy of 2.5Hz This implies that oly frequecies less tha 2.5 Hz ca be sampled without aliasig. Siusoid (a) has a frequecy of Hz Siusoid (b) has a frequecy of 2Hz Siusoid (c) has a frequecy of 5Hz Siusoid (d) has a frequecy of Hz It is therefore clear that (a) ad (b) are the oly siusoids with frequecy less tha 2.5Hz, ad therefore able to be sampled without aliasig.

12 Quiz Sigals & Systems Problem 3 Problem 3 poits A causal, liear time-ivariat (LTI) system is described by the trasfer fuctio z H (z) = (z +.75)(z.25). a) Determie the poles of the system. (2 poits) b) Is the system bouded-iput bouded-output (BIBO) stable? (2 poits) Justify your aswer. c) Draw the regio of covergece (ROC) of H (z). (2 poits) A LTI system described by a ukow trasfer fuctio H 2 (z) ad the ROC give i Figure is cascaded with H (z) as show i Figure 2. The impulse respose h[] of the sigle equivalet system is give by h[] = h [] h 2 [], where h [] ad h 2 [] are the impulse resposes associated with H (z) ad H 2 (z), respectively. Im(«).5.25 Re(«) [ ] [ ] [ ] 2 و [ ] و [ ] 2 و [ ] و = [ ]و [ ] [ ] Figure 2: Cascade combiatio. Figure : The shaded regio idicates the ROC of H 2 (z). d) Write the trasfer fuctio H(z) of the sigle equivalet ( poit) system as a fuctio of H (z) ad H 2 (z). e) Is the system H(z) BIBO stable? Justify your aswer. (3 poits)

13 Quiz Sigals & Systems Solutio 3 a) The poles of the system are defied as the roots of the deomiator, thus (z +.75)(z.25)! = z =.75 z 2 =.25. Gradig: poit for every correct pole. b) A causal LTI system is stable if ad oly if all its poles z ى are iside the uit circle, i.e. z ى <, for all poles i. Both poles lie iside the uit circle. The system is therefore BIBO stable. Gradig: poit for the stability coditio. poit for correct coclusio. c) The regio of covergece of a causal LTI system exteds outward from the largest magitude pole. The ROC of the system described by H (z) is show i Figure 3. The regio is R = {z z >.75}. Im(«).75 Re(«) Figure 3: The shaded regio idicates R. Gradig: poit for drawig the ROC outside a circle. poit for correct radius.

14 Quiz Sigals & Systems Problem 3 d) The impulse respose of two LTI systems i cascade is the covolutio of their respective impulse resposes. I the z-domai, this correspods to a multiplicatio. Thus, H(z) = H (z)h 2 (z). Gradig: poit for the correct result. e) The ROC resultig from the multiplicatio of two discrete-time trasfer fuctios cotais the itersectio of the two idividual ROCs (see Figure 4) ad is defied as R R R 2. By defiitio, a discrete-time LTI system is BIBO stable if ad oly if the ROC of H(z) cotais the uit circle. This is the case for R: the system is therefore BIBO stable. Remark: It ca also be argued that the system described by H 2 is stable because R 2 icludes the uit circle. The, it ca be cocluded that a system obtaied by cascadig two stable systems is also stable. Gradig: poit for correct reasoig. 2 poits for correct use of stability coditio (ROC cotais uit circle) ad correct coclusio. Im(«) Re(«) Figure 4: The dark regio idicates the itersectio of R ad R 2, which cotais the uit circle.

15 Quiz Sigals & Systems

16 Quiz Sigals & Systems Problem 4 Problem 4 poits The cotiuous-time sigal x(t) = si(2πt) is uiformly sampled with samplig time T, resultig i the discrete-time sigal x[]. a) Choosig the samplig time T = /5 secods results i a periodic sigal x[]. What is its fudametal period? b) For which other values of T is x[] periodic? Justify your aswer. Now, cosider the followig MATLAB script: Ts = /5; = [ 2 3 4]; x = + si(2*pi**ts); X = fft(x); N = legth(); m = abs(x)/n; C =... f =... stem(f, m(:c)); xlabel( Frequecy (Hz) ) ylabel( Magitude X(f) /N ) Aswer the followig questios that refer to the above script: ( poit) (2 poits) c) What is the Nyquist frequecy i Hz? ( poit) d) Complete the lies f=... ad C=... i order to plot m for (2 poits) frequecies up to the Nyquist frequecy. e) Draw the output of the stem commad. (4 poits) Hit: The followig excerpt from the MATLAB help for the fft commad might be useful for this problem: >> help fft fft Discrete Fourier trasform. fft(x) is the discrete Fourier trasform (DFT) of vector X. For legth N iput vector x, the DFT is a legth N vector X, with elemets N X(k) = sum x()*exp(-j*2*pi*(k-)*(-)/n), <= k <= N. =

17 Quiz Sigals & Systems Solutio 4 a) The sampled siusoid has a period of N = 5. Ituitively, we use the defiitio of periodicity ad calculate this as: x[] = x[ + N] si(2πt ) = si(2π( + N)T ) = si(2πt + 2πNT ) Which implies that NT Z. Give that N N ad T >, we ca costrai this further to NT N. The fudametal period N is the smallest positive iteger N such that this coditio holds. We have that T = /5, thus N/5 N ad it follows that N = 5, which is the fudametal period of the discrete-time sequece. A alterative (more rigorous) approach is preseted i the book: Equatio.54 ad.55, ad Problem.. Gradig: poit for correct aswer (N = 5) b) As per the above, the coditio for periodicity of the sampled sequece x[] is that NT N. Let NT = k where k N. It follows that T = k/n. Give that both k ad N are positive itegers, T is the divisio of two positive itegers ad is therefore a positive, ratioal umber. A alterative (more rigorous) approach is preseted i the book: Equatio.54 ad.55, ad Problem.. Gradig: poit for justificatio, poit for the correct coclusio c) Per defiitio, the Nyquist frequecy is half the samplig frequecy. Thus f خ = f /2 = 5/2 = 2.5Hz. Gradig: poit for correct aswer

18 Quiz Sigals & Systems Problem 4 d) As per the Matlab script ad correspodig istructio, we wish to plot the magitude of the DFT coefficiets up to the Nyquist frequecy. We calculate the cotiuous-time frequecies correspodig to the DFT coefficiets as: f = k f N Hz, for k Z, k N/2 = k 5 Hz, for k {,, 2} 5 = k Hz, for k {,, 2}, Where N is the umber of items i the DFT. Note that i this case, N = N, as we are samplig for oe period of the iput. We therefore have: f = [ 2] The frequecy axis of the stem plot. C = 3 The umber of items we are plottig (m(:c)) It is also possible to calculate discrete-time frequecies as: Ω = k 2π N rad, for k Z, k N/2 Gradig: poit for correct f, poit for correct C. Follow through poits were awarded if your aswer to part c was icorrect. e) The siusoid + si(2*pi**ts) is sampled with fs=5, Ts=.2 ad for =:4. This correspods to samplig oe period of a offset, Hz siusoid at a samplig frequecy of 5 Hz. As per calculatios i part d, we wish to plot the magitude of the first three DFT coefficiets, which correspod to frequecies Hz, Hz, ad 2Hz respectively. m correspods to the absolute DC value, ie. the absolute value of the mea of the sampled sigal, which is i this case equal to the offset. Thus m =.

19 Quiz Sigals & Systems m correspods to half (DFT symmetry!) the siusoid magitude at Hz, thus m =.5 m 2 correspods to half (DFT symmetry!) the siusoid magitude at 2Hz, thus m 2 = X[f].5 2 Frequecy (Hz) Gradig: 2 poits for correct m, 2 poits for correct m, -2 poits for other o-zero elemets. Follow through poits were awarded if your aswer to part c was icorrect. Additioal commets:. As we have a periodic iput ad are samplig for the period legth, the DFT elemets are equal to the DFS coefficiets scaled by N. That is, X[i] = Nc ى for a periodic iput. We scale by /N whe calculatig the DFT magitudes, thus m ى = c ى. Therefore, oe possible solutio ivolves direct calculatio of the DFS coefficiets ad plottig the DFT by usig this property of DFT/DFS equivalece. 2. Due to the complex-cojugate symmetry of the DFT, X[3] = X[2] ˉ ad X[4] = X[], ˉ implyig that m 3 = m 2 ad m 4 = m. You were ot required to plot these compoets.