Solutio of EECS 315 Fial Examiatio F9 1. Fid the umerical value of δ ( t + 4ramp( tdt. δ ( t + 4ramp( tdt. Fid the umerical sigal eergy of x E x = x[ ] = δ 3 = 11 = ( = ramp( ( 4 = ramp( 8 = 8 [ ] = ( δ 3 [ ] 3δ 6 [ ] ( u[ + 1] u[ 1]. ( [ ] 3δ 6 [ ] ( u[ + 1] u[ 1] E x = δ 3 [ ] 3δ 6 [ ] = ( 1 3 + 1 + ( 1 3 + 1 = 1 = 1 3. Fid the umerical stregth of the impulse 3δ ( 4t. 3δ ( 4t = 3 1 4 δ t = = 3 ( Stregth is 3 / 4 4. Fid the umerical fudametal period of x( t = 3cos( πt 8si( 5πt. =6 The fudametal frequecy is the greatest commo divisor of 1 ad 15 which is 5. Therefore the fudametal period is 1/5 or 4 ms. =9
5. Let x( t = 3rect ( t / 5 7 rect ( t /. If y( t = x( ( t 1 fid the umerical value of y(. ( = 3rect ( ( t 1 / 5 7rect ( ( t 1 / = 3rect (.4( t 1 7 rect ( t 1 y t ( = 3rect (.4( 1 7 rect ( 1 = 3rect (.4 7rect ( 1 = 3 y 6. A cotiuous-time system is described by 3 y ( t + Ay( t = x( t. If its impulse respose is h( t = Ke st u( t, fid the value of s i terms of A (all umbers except A. Fid the umerical value of K. (c For what umerical rage of values of A is the system stable? 3s + A = s = A / 3 h( t = Ke At /3 u( t 3 h( + h( + K = + A h( tdt = u( + u( 1 = 3K = K = / 3 = System is stable for A >. h( t = ( / 3e At /3 u( t 7. What umerical rages of values of A ad B mae this system stable? The output of the summig juctio is y[ ] / B. Therefore [ ] = B( x[ ] Ay[ 1] / B y x[] + B - y[] D A [ ] = B( x[ ] Ay[ 1] / B y[ ] + Ay[ 1] = Bx[ ] y The eigevalue is -A ad the impulse respose is h[ ] = B( A u[ ] System is stable for ay A < 1 ad for ay B.
( = tri( t / 3 δ ( t ad y( t = x t ( is ot zero? 8. If x t y t If x t ad a for which y t (, what is the umerical rage of values of t for which 1 / < t < 5 / ( = tri( t / w δ ( t + t ad y( t = x( at, what is the rage of values of t (i terms of w, t ( is ot zero? w t a < t < w t a 9. If x zero? [ ] = ( u[ + 4] u[ 3] δ [ + 3] ad y[ ] = x[ 4], what is the rage of values for which y[ ] is ot x is o-zero for 7 <. Therefore y is o-zero for 3 < 4 or 3 3. 1. Fid the umerical fudametal periods of 4 cos( 6π / 7 ( = 4 cos( π( 3 / 7 N = 7 4 cos 6π / 7 si( 15π /1 ( = si( π( 15 / 4 = si( π( 8 / 8 N = 8 si 15π /1 (c What is the smallest positive value of that maes the sigal 4 cos( 6π / 7 si( 15π /1 ( [ ] u [ ] u zero for all? This occurs whe is the legth of the commo period betwee the two siusoids because the the sum of the poits is the sum over a iteger umber of periods of each siusoid ad that must be zero. I this case that value is LCM 7,8 ( = 56.
11. Classify these systems accordig to stability, liearity ad time ivariace. y[ ] = x[ ] Stable If x is bouded the so is y. No-Liear If x is A, A >, the y is also A. The if we multiply x by -1, x is -A but y is still A. Not homogeeous, therefore ot liear. Time Ivariat This is a static system. y is the magitude of x, o matter whe x is applied (for ay. y( t = si( 1πtx( t Stable The siusoid is bouded betwee -1 ad 1. If x is bouded, so is y. Liear Multiplyig x by ay costat multiplies y by the same costat ad if x is the sum of two sigals, y is the sum of the resposes. Therefore the system is homogeeous ad additive ad therefore liear. Time Variat The respose y at time t depeds o x at time t but also o the value of the siusoid at time t. Therefore the respose if x is delayed is ot, i geeral, simply a delayed versio of the respose whe x is ot delayed. 1. Fid the regios of covergece of the Laplace trasforms of the followig fuctios. 3e t u( t ROC is σ > 1e t / u( t ROC is σ < 1 /
13. A cotiuous-time system is described by the differetial equatio 4 y ( t y t Its trasfer fuctio ca be writte i the stadard form ( + 3 y ( t y( t = 8 x ( t + x ( t 4 x( t. H( s = M = N = b s a s = b M s M + b M 1 s M 1 + + b s + b 1 s + b a N s N + a N 1 s N 1 + + a s + a 1 s + a Fid the umerical values of M, N ad all the a ad b coefficiets ( a N a ad b M b. M =, N = M =, N = 3 a coefficiets are a 3 = 4, a =, a 1 = 3, a = 1 b coefficiets are b = 8, b 1 = 1, b = 4 (Your idetificatio of the a ad b coefficiets must be i the form a = 18 or b 3 = 7 for example. That is, idetify them idividually, ot just as a set of umbers. 14. A cotiuous-time system has a trasfer fuctio H( s = s s +.1+ j1 ( ( s +.1 j1. At what umerical cyclic frequecy (i Hz will the magitude frequecy respose of this system be a maximum? The maximum will occur at the closest approach to a pole which occurs at ω = ±1 f = ±1.59 What will that umerical maximum magitude be? At ω = ±1, the magitude frequecy respose will be (substitutig j1 for s, 4.9999.
15. Fid the umerical values of the costats i 3 u 1 Alterate Solutio: x 3 u 1 Z ( [ ] u[ + 1] Z ( [ ] u[ + 1] = 3( δ [ + 1] + δ [ ] Z [ ] = 3( u[ 1] u[ + 1] 16. A digital filter has a trasfer fuctio H z 3 z 1 z z 1 z z 1 ( =.955 Az a + Bz b. 3z 1 3z = 31 z z 1 = 3 z 1 z 1 = 3z 3z 1 z z + 1 z.95z +.95. What are the umerical locatios of its poles ad zeros? Poles at z =.95e ± jπ /3 or.475 ± j.87 Zeros at z = e ± jπ /3 or.5 ± j.866 Fid the umerical frequecy respose magitude at these radia frequecies. Ω = H( e j = H 1 Ω = π / 3 ( =.955 jπ /3 H e 1 1 + 1 1.951 +.95 =.955 1 1.951 +.95 = 1.5 ( =.955 e j π /3 e jπ /3 + 1 e j π /3.951 e jπ /3 +.95 =
17. Below are some graphs of 8 discrete-time sigals ad below them some graphs of the magitudes of 1 DFT's, all based o 16 poits. Match the discrete-time sigals to the DFT magitudes by writig i the letter desigatio of the DFT magitude correspodig to each discrete-time sigal i the space provided at the top of its graph. x[] - 1-1 - 1-1 x[] X[] X[] X[] - 1 1 E 5 A 1 I 5 1-1 - 1 C J F E I A H B B 1 F 1.5 1 J 1 1 1 C 1 G 1 1 K 5 1-1 1 D 1 H 4 1 L 1 1
Solutio of EECS 315 Fial Examiatio F9 1. Fid the umerical value of δ ( t + 4ramp( 3t dt. δ ( t + 4ramp( 3t dt. Fid the umerical sigal eergy of x E x = x[ ] = δ 3 = 8 = ( = ramp( 3( 4 = ramp( 1 = 1 [ ] = ( δ 3 [ ] δ 6 [ ] ( u[ + 4] u[ 9]. ( [ ] δ 6 [ ] ( u[ + 4] u[ 9] E x = δ 3 [ ] δ 6 [ ] = 1 + ( 1 + 1 + ( 1 = 4 = 4 = 3 3. Fid the umerical stregth of the impulse 5δ ( 4t. 5δ ( 4t = 5 1 4 δ t = = 3 ( Stregth is 5 / 4 4. Fid the umerical fudametal period of x( t = 3cos( 3πt 8si( 4πt. The fudametal frequecy is the greatest commo divisor of 15 ad which is 5. Therefore the fudametal period is 1/5 or ms. =6
5. Let x( t = 6 rect ( t / 5 7 rect ( t /. If y( t = x( ( t 1 fid the umerical value of y(. ( = 6 rect ( ( t 1 / 5 7rect ( ( t 1 / = 6rect (.4( t 1 7rect ( t 1 y t ( = 6rect (.4( 1 7rect ( 1 = 6 rect (.4 7 rect ( 1 = 6 y 6. A cotiuous-time system is described by 5 y ( t + Ay( t = x( t. If its impulse respose is h( t = Ke st u( t, fid the value of s i terms of A (all umbers except A. Fid the umerical value of K. For what umerical rage of values of A is the system stable? 5s + A = s = A / 5 h( t = Ke At /5 u( t 5 h( + h( + K = + A h( tdt = u( + u( 1 = 5K = K = / 5 = System is stable for A >. h( t = ( / 5e At /5 u( t 7. What umerical rages of values of A ad B mae this system stable? The output of the summig juctio is y[ ] / B. Therefore [ ] = B( x[ ] Ay[ 1] / B y x[] + B - y[] D A [ ] = B( x[ ] Ay[ 1] / B y[ ] + Ay[ 1] = Bx[ ] y The eigevalue is -A ad the impulse respose is h[ ] = B( A u[ ] System is stable for ay A < 1 ad for ay B.
8. If x t ( = tri( t / 5 δ ( t ad y( t = x t ( is ot zero? y t (, what is the umerical rage of values of t for which 3 / < t < 7 / If x t ( = tri( t / w δ ( t + t ad y( t = x at ( is ot zero? ad a for which y t (, what is the rage of values of t (i terms of w, t w t a < t < w t a 9. If x zero? [ ] = ( u[ + 6] u[ 3] δ [ + 3] ad y[ ] = x[ 4], what is the rage of values for which y[ ] is ot x is o-zero for 9 <. Therefore y is o-zero for 5 < 4 or 5 3. 1. Fid the umerical fudametal periods of 4 cos( 1π /11 ( = 4 cos( π( 6 /11 N = 11 4 cos 1π /11 si( 35π /14 ( = si( π( 35 / 8 = si( π( 5 / 4 N = 4 si 35π /14 (c What is the smallest positive value of that maes the sigal 4 cos( 6π / 7 si( 15π /1 ( [ ] u [ ] u zero for all? This occurs whe is the legth of the commo period betwee the two siusoids because the the sum of the poits is the sum over a iteger umber of periods of each siusoid ad that must be zero. I this case that value is LCM 7,8 ( = 56.
11. Classify these systems accordig to stability, liearity ad time ivariace. y[ ] = x [ ] Stable If x is bouded the so is y. No-Liear Time Ivariat If x is A, A >, the y is A. The if we multiply x by -1, x is -A but y is still A. Not homogeeous, therefore ot liear. This is a static system. y is x squared, o matter whe x is applied (for ay. y( t = 5x( t / 3 Stable If x is bouded, so is y. Liear Multiplyig x by ay costat multiplies y by the same costat ad if x is the sum of two sigals, y is the sum of the resposes. Therefore the system is homogeeous ad additive ad therefore liear. Time Variat If x 1 ( t = g t (, the y 1 ( t = g( t / 3. If x ( t = g( t t the y ( t = g( t / 3 t. ( = g ( t t / 3 y 1 t t (. Therefore y is ot (i geeral simply a delayed versio of y 1. 1. Fid the regios of covergece of the Laplace trasforms of the followig fuctios. 3e 5t u( t ROC is σ > 5 1e t /4 u( t ROC is σ < 1 / 4
13. A cotiuous-time system is described by the differetial equatio 3 y ( t 8 y t Its trasfer fuctio ca be writte i the stadard form ( + 7 y ( t 5y( t = 4 x ( t 6 x( t. H( s = M = N = b s a s = b M s M + b M 1 s M 1 + + b s + b 1 s + b a N s N + a N 1 s N 1 + + a s + a 1 s + a Fid the umerical values of M, N ad all the a ad b coefficiets ( a N a ad b M b. M =, N = 3 a coefficiets are a 3 = 3, a = 8, a 1 = 7, a = 5 b coefficiets are b = 4, b 1 =, b = 6 (Your idetificatio of the a ad b coefficiets must be i the form a = 18 or b 3 = 7 for example. That is, idetify them idividually, ot just as a set of umbers. 14. A cotiuous-time system has a trasfer fuctio H( s = s s +.1+ j1 ( ( s +.1 j1. At what umerical cyclic frequecy (i Hz will the magitude frequecy respose of this system be a maximum? Frequecy for maximum is Hz The maximum will occur at the closest approach to a pole which occurs at ω = ±1 f = ±1.91 What will that umerical maximum magitude be? Maximum frequecy respose magitude is At ω = ±1, the magitude frequecy respose will be (substitutig j1 for s, 4.9999.
15. Fid the umerical values of the costats i u Z ( [ ] u[ + ] Az a + Bz b. Alterate Solutio: Z ( [ ] u[ + ] = ( δ [ + ] + δ [ + 1] u z z 1 Z [ ] = ( u[ ] u[ + ] x 16. A digital filter has a trasfer fuctio H z z z 1 z3 z 1 ( = 1.95 z + 1 z +.95. = z 1 z z 1 = z z 1 z 1 = z1 z What are the umerical locatios of its poles ad zeros? Poles at z =.95e ± jπ / or ± j.95 ± jπ / Zeros at z = e or ± j Fid the umerical frequecy respose magitude at these radia frequecies. Ω = Ω = π / H( e j = H 1 ( = 1.95 jπ / H( e = H( j = 1.95 1 + 1 1+.95 = 1 + 1 1+.95 =
17. Below are some graphs of 8 discrete-time sigals ad below them some graphs of the magitudes of 1 DFT's, all based o 16 poits. Match the discrete-time sigals to the DFT magitudes by writig i the letter desigatio of the DFT magitude correspodig to each discrete-time sigal i the space provided at the top of its graph. x[] - 1-1 - 1-1 x[] - 1 A - 1-1 G C E A J I B K B C - 1 D X[] 5 1 E 1 1 F 1 1 G 5 1 H X[].5 1 I 4 1 1 J 1 K 1 1 L X[] 1 5 1 1 1 1 1
Solutio of EECS 315 Fial Examiatio F9 1. Fid the umerical value of δ ( t + 3ramp( tdt. δ ( t + 3ramp( tdt = ramp( ( 3 = ramp( 6 = 6. Fid the umerical sigal eergy of x E x = x[ ] = 4δ 3 = 11 = ( [ ] = ( 4δ 3 [ ] 3δ 6 [ ] u[ + 1] u[ 1] ( [ ] 3δ 6 [ ] ( u[ + 1] u[ 1] E x = 4δ 3 [ ] 3δ 6 [ ] = ( 4 3 + 4 + ( 4 3 + 4 = 34 = 1 3. Fid the umerical stregth of the impulse 3δ ( 7t. = = 3 =6 =9 (. ( = 3 1 7 δ ( t Stregth is 3 / 7 3δ 7t 4. Fid the umerical fudametal period of x( t = 3cos( 6πt 8si( 48πt. The fudametal frequecy is the greatest commo divisor of 3 ad 4 which is 6. Therefore the fudametal period is 1/6 or 16.667 ms.
5. Let x( t = 4 rect ( t / 5 9 rect ( t /. If y( t = x( ( t 1 fid the umerical value of y(. ( = 4 rect ( ( t 1 / 5 9rect ( ( t 1 / = 4 rect (.4( t 1 9rect ( t 1 y t ( = 4 rect (.4( 1 4 rect ( 1 = 4 rect (.4 9rect ( 1 = 4 y 6. A cotiuous-time system is described by 8 y ( t + Ay( t = 3x( t. If its impulse respose is h( t = Ke st u( t, fid the value of s i terms of A (all umbers except A. Fid the umerical value of K. For what umerical rage of values of A is the system stable? 8s + A = s = A / 8 h( t = Ke At /8 u( t 8 h( + h( + K = + A h( tdt = 3 u( + u( 1 = 8K = 3 K = 3 / 8 = System is stable for A >. h( t = ( 3 / 8e At /8 u( t 7. What umerical rages of values of A ad B mae this system stable? The output of the summig juctio is y[ ] / B. Therefore [ ] = B( x[ ] Ay[ 1] / B y x[] + B - y[] D A [ ] = B( x[ ] Ay[ 1] / B y[ ] + Ay[ 1] = Bx[ ] y The eigevalue is -A ad the impulse respose is h[ ] = B( A u[ ] System is stable for ay A < 1 ad for ay B.
8. If x t ( = tri( t / 4 δ ( t ad y( t = x t ( is ot zero? y t (, what is the umerical rage of values of t for which 1 < t < 3 If x t ( = tri( t / w δ ( t + t ad y( t = x( at, what is the rage of values of t (i terms of w, t ( is ot zero? ad a for which y t w t a < t < w t a 9. If x zero? [ ] = ( u[ + ] u[ 5] δ [ + 3] ad y[ ] = x[ 4], what is the rage of values for which y[ ] is ot x is o-zero for 5 <. Therefore y is o-zero for 1 < 6 or 1 5. 1. Fid the umerical fudametal periods of 4 cos( 6π /13 ( = 4 cos( π( 3 /13 N = 13 4 cos 6π /13 si( 1π / 9 ( = si( π( 1 /18 = si( π( / 3 N = 3 si 1π / 9 (c What is the smallest positive value of that maes the sigal 4 cos( 6π / 7 si( 15π /1 ( [ ] u [ ] u zero for all? This occurs whe is the legth of the commo period betwee the two siusoids because the the sum of the poits is the sum over a iteger umber of periods of each siusoid ad that must be zero. I this case that value is LCM 7,8 ( = 56.
11. Classify these systems accordig to stability, liearity ad time ivariace. y[ ] = x[ ]u[ ] Stable If x is bouded the so is y. Liear If x is multiplied by a costat y is multiplied by the same costat. If two excitatios are added, the respose is the sum of the idividual resposes. Therefore it is liear. [ ] the Time Variat If x 1 [ ] = g[ ], the y 1 [ ] = g[ ]u[ ]. If x [ ] = g y [ ] = g[ ]u[ ]. y 1 [ ] = g[ ]u[ ]. Therefore y is ot (i geeral simply a delayed versio of y 1. y( t = 3t x( t Ustable Liear x is multiplied by 3t to get y. There is o upper boud o t. Therefore a bouded x ca produce a ubouded y. Multiplyig x by ay costat multiplies y by the same costat ad if x is the sum of two sigals, y is the sum of the resposes. Therefore the system is homogeeous ad additive ad therefore liear. Time Variat If x 1 ( t = g( t, the y 1 t y 1 ( t t = 3( t t g t t y 1. ( = 3t g( t. If x ( t = g( t t the y ( t = 3t g( t t. (. Therefore y is ot (i geeral simply a delayed versio of 1. Fid the regios of covergece of the Laplace trasforms of the followig fuctios. 3e 1t u( t ROC is σ > 1 1e t /7 u( t ROC is σ < 1 / 7
13. A cotiuous-time system is described by the differetial equatio y ( t 5 y t Its trasfer fuctio ca be writte i the stadard form ( 4 y( t = 9 x ( t + x ( t 7 x t (. H( s = M = N = b s a s = b M s M + b M 1 s M 1 + + b s + b 1 s + b a N s N + a N 1 s N 1 + + a s + a 1 s + a Fid the umerical values of M, N ad all the a ad b coefficiets. M =, N = 3 a 3 =, a = 5, a 1 =, a = 4 b = 9, b 1 =, b = 7 (Your idetificatio of the a ad b coefficiets must be i the form a = 18 or b 3 = 7 for example. That is, idetify them idividually, ot just as a set of umbers. 14. A cotiuous-time system has a trasfer fuctio H( s = s s +.1+ j15 ( ( s +.1 j15. At what umerical cyclic frequecy (i Hz will the magitude frequecy respose of this system be a maximum? The maximum will occur at the closest approach to a pole which occurs at ω = ±15 f = ±.39 What will that umerical maximum magitude be? At ω = ±15, the magitude frequecy respose will be (substitutig j15 for s, 4.9999.
15. Fid the umerical values of the costats i 11 u Alterate Solutio: x 1 u Z [ ] = 11( u[ ] u[ ] 16. A digital filter has a trasfer fuctio H z Z ( [ ] u[ ] Z ( [ ] u[ ] = 11( δ [ ] + δ [ 1] 11 z z z 1 z z 1 = 1 z 11z 1 ( =.955 Az a + Bz b. 11z 11z 1 z + z + 1 z +.95z +.95. z 1 = 11z 1 z 1 z 1 = 11z 11z 1 What are the umerical locatios of its poles ad zeros? Poles at z =.95e ± j π /3 =.475 ± j.87 Zeros at z = e ± j π /3 =.5 ± j.866 Fid the umerical frequecy respose magitude at these radia frequecies. Ω = H( e j = H 1 j π /3 Ω = π / 3 H e ( =.955 ( =.955 1 + 1 + 1 1 +.95 +.95 =.955 3.855 = 1.18 e j 4π /3 + e j π /3 + 1 e j 4π /3 +.95e j π /3 +.95 =
17. Below are some graphs of 8 discrete-time sigals ad below them some graphs of the magitudes of 1 DFT's, all based o 16 poits. Match the discrete-time sigals to the DFT magitudes by writig i the letter desigatio of the DFT magitude correspodig to each discrete-time sigal i the space provided at the top of its graph. x[] - 1-1 - 1-1 x[] - 1 A - 1-1 H E C K G D I B B C - 1 D X[] 5 1 E 1 1 F 5 1 G 1 H X[] 5 1 I 1 1 1 J 1 K 4 1 1 L X[].5 1 1 1 1 1 1