Problem 2. Describe the following signals in terms of elementary functions (δ, u,r, ) and compute. x(t+2) x(2-t) RT_1[x] -3-2 = 1 2 = 1

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1 EEE 03, HW NAME: SOLUTIONS Problm. Considr h signal whos graph is shown blow. Skch h following signals:, -, RT [], whr R dnos h rflcion opraion and T 0 dnos shif dlay opraion by 0. - RT_[] Problm. Dscrib h following signals in rms of lmnary fncions δ,,r, and comp δ 3 d and y δ d. y 3 3 r r 3 3 δ 3d [3 3 ] y r r 3 yδ d [y y ]

2 EEE 03, HW NAME: Problm. Comp h convolion h* whn --r-r-, hδ-. h is h ni sp, r is h ni ramp h r r 3, or Problm. Considr h filrs. y[]. y d. Find and graph hir impls rsponss.. h δ δ. y δd 0 δd δd. Find and graph hir sp rsponss.. h. y d d [ ] 0 3. Which filrs ar casal? Jsify. Non-casal bcas h 0, for som < 0.. Casal bcas h 0, for all < 0.. Which filrs ar sabl? Jsify. Sabl bcas h <. Unsabl bcas h divrgs. h h y y

3 EEE 03, HW 3 NAME: SOLUTIONS_ Problm : L b h priodic signal shown in h figr blow sawooh wav wih offs. Comp h cofficins a k of h Forir sris pansion of Th drivaiv of is d.5.5δ n d n. From h abls, h Forir sris FS cofficins of h impls rain ar a k.5, k. Thn, h FS cofficins of, say b k ar givn by b k a jkw k, k 0, w 0 π. Ths, b 0 k.5, k 0. jkπ For k 0, w comp h FS cofficin dircly from h dfiniion, b 0 d T T 3 b 6 0 Problm : L X b h Forir ransform of. Find Xj0 and From h dfiniion of h Forir ransform and is invrs, Xj0 j0 d d Xj0. On h ohr hand, d d 0 π X 0 dw wxdw πj d d 0 wxdw πj wx dw.

4 EEE 03, HW NAME: Problm 3: Considr h filr wih impls rspons h.. Find h ransfr fncion and skch h Bod Plo. Find h Forir ransform of h op whn and whn sin 3. Find h op whn a sing convolion, and b aking h invrs Forir ransform of yor answr o par.. H F{ H w In MATLAB, >> Hf,[ ] >> H.iodlay >> bodh }, H an w w cornr frqncy a 0 Bod Diagram -5-0 Magnid db Phas dg Frqncy rad/s

5 ], [ } {sin } { : Sinsoid inp, } { : inp Eponnial. X H Y w w j F F X H Y F δ δ π } { * 3. a h sin frqncy vala H 0 0 F F Y F d d h y w j w j j j j δ δ π

6 EEE 03, HW 5 NAME: SOLUTIONS Problm : Find h largs sampling inrval T s o allow prfc rconsrcion of h signals: NOTE: h* dnos convolion of h and sin sin 3. * sin sin. sin 3. *sin 7 Solions sing ihr h shorc simas of h ma frqncy or a fll compaion of h Forir ransform of h composi signals ar accpabl.. Using h shorcs, w m minw m, w m rad T s s < π s. Th sam answr is obaind by prforming h Forir ransform of h composi signal mliplicaion of wo plss in h frqncy domain.. Using h shorcs, w m maw m, w m rad T s s < π s. Th sam answr is obaind by prforming h Forir ransform of h composi signal smmaion of wo plss in h frqncy domain. 3. Using h shorcs, w m minw m, w m rad T s s < π s. This is only a consrvaiv sima in his cas i.., his sampl im is garand o work b largr ims ar possibl. By prforming h Forir ransform of h composi signal mliplicaion of a pls and wo implss in h frqncy domain w hav no ovrlap bwn h wo signals and a prodc which is idnically zro. Hnc, h acal ppr bond for T s is infiniy i.., any sampling im wold work. Problm : For a sampling procss wih sampling im ms, wha is h coff frqncy of h idal lowpass filr ndd o avoid any aliasing ffcs and wha is h idal low pass filr for rconsrcion? Sampling a ms, i.., a frqncy of 000Hz, allows rcovry of signals wih maimm frqncy 500Hz 500π 3 rad. This shold b h coff frqncy of h idal lowpass filr sd o pr-procss h analog signal o avoid aliasing ani-aliasing filr. I shold s also b h coff frqncy of h idal low-pass filr sd for rconsrcion of h maimm possibl rang of signals. Th rconsrcion filr can b dfind by: rad T 3, for w < 3 or 500Hz - Is frqncy rspons H s or, 0 ohrwis - Is impls rspons h F 0.00 sin3 {H} π Eihr on is accpabl.

7 EEE 03, HW 6 NAME: Problm : Considr h coninos im casal filr wih ransfr fncion s H s s s Comp h rspons of h filr o [] [] 3 y L s s s s L 6 s 6 s 6 s Problm : Considr h coninos im casal filr dscribd by h diffrnial qaion d y dy y d d Comp h sady-sa rspons of h filr o [] cos[ ]. Hs s s s 0.5 j0.866s 0.5 j0.866 Th sysm is sabl.f. pols in h LHP, hnc h sady-sa rspons is wll-dfind. y ss Hj cos [Hj] j j cos aan 0 cos 3.7 o

8 EEE 03, HW 7 NAME: SOLUTIONS Problm : Considr h casal filr dscribd by h diffrnc qaion y [ n] y[ n ] [ n ] 3. Drmin h ransfr fncion. Comp h rspons of h filr o [n] [n] 3. Comp h sady sa rspons o [n] [n]. Comp h sady sa rspons o [n] sin nπ / 6 [n-] No: For a sabl sysm if convrgs o a priodic signal _s, hn y convrgs o a priodic signal ha is h sysm rspons o h _s. This can b compd sing Forir hory. A sfl simplificaion is: jωo jωo y H jωo For Coninos Tim: cos ωo y H jωo cos ωo H jωo jωon jωo jωon n y n H For Discr Tim: jωo jωo n cos Ω on y n H cos Ω on H whr, H s, H z ar h coninos and discr sysm ransfr fncions, rspcivly.. Hz z 3 z z 3 z z 3 z z z 3 3 z z z 3 3 z 3 z z 3. yz Hz yn z Z z yn 3 n 3 n 3 n n y 0 n Th filr is sabl sinc h pol /3 has magnid lss han on. Hnc, h sady-sa rspons is wll dfind. For h consan sady-sa, yn Hz cos 0n 3. y ss n H jω sin Ωn H jω, Ω π 6 Thn, y ss n sin π n aan sin π sin π n 6 6 7o cos π 6 3 sin π 6 cos π 6 3 Problm : Considr h discr im casal filr wih ransfr fncion z H z z 0.8 z 0.9 Comp h rspons of h filr o [n] [n] z yz Hz z z z 0.8z yn Z z.7 z z 0.9 Z 0.7z z 0.8 yn n n n n n y n z z

9 EEE 03, Ts NAME: Solions Problm. Considr h signal whos graph is shown blow. Skch h following signals: -, T R[], whr R dnos h rflcion opraion and T 0 dnos shif dlay opraion by 0. - R[] T R[] Typ qaion hr. Problm. Dscrib h following signals in rms of lmnary fncions δ,,r, and comp δ 3 d and y δ 3 d. y 3 3 r r, δ 3d 3 0 y r r, yδ 3d y3 0

10 EEE 03, TEST. NAME: SOLUTIONS Problm. Comp h convolion h* whn, h. W hav shown ha r. Thn, by im invarianc, h r r Problm. Considr h filrs: y d A. B. y d. Find and graph hir impls rsponss. A. h δd B. h δ d δd δd. Which filrs ar casal? Jsify δd A is casal bcas h 0 for <0. B is no casal bcas h assms nonzro vals for som <0. 3. Which filrs ar sabl? Jsify A is sabl bcas h < B is no casal bcas h is no absolly ingrabl h r.

11 EEE03, TEST 3. NAME: SOLUTIONS 75, closd books and nos, calclaors and ransform abls allowd Problm : Considr h filr wih impls rspons h.. Find h Forir ransform of h op Y whn. Find h im-domain prssion for h op y whn H F F F{ } ; X Y HX Y PFE y F { } Problm : Considr h coninos im casal filr dscribd by h diffrnial qaion dy d 3 y a, whr a is an adjsabl paramr.. Find h magnid and phas of h frqncy rspons of h filr.. Comp h val of a so ha sady-sa rspons of h filr o [] cos[] has amplid. Hs a s 3, H a w 3, H a w an 3 Th sady sa rspons is y Hj cos Hj. For h amplid o b, w shold hav a 9 0 Problm 3: L b h priodic signal shown in h figr blow sqar wav wih offs. -. Comp h cofficins a k of h Forir sris pansion of.. Comp h cofficins b k of h op of an idal lowpass filr H wih coff frqncy.5π, and inp. Th sandard sqar wav in h abls has o a k sin kw 0T FS kπ. In rms of h sandard sqar wav, w 0 π, T 0.5 o 0.5 a k jkπ FS sin kπ kπ ; a 0 0. By h filring propry, b k Hjkw 0 a k Hjkπa k. Sinc h idal low pass is for w <.5π, or k <.5, and 0 ohrwis, w hav sin kπ b k jkπ kπ, for k ±; b k 0 ohrwis

12 EEE 03, TEST 5 CLOSED BOOK & NOTES. TRANSFORM TABLES ALLOWED. 30 NAME: SOLUTIONS Problm : Find h largs sampling inrval T s o allow prfc rconsrcion of h signals: NOTE: h* dnos convolion of h and. sin. sin cos. Using h shorcs: For convolion, f Nyq min f Nyq, f Nyq. Th firs signal is bandlimid o rad/s, w Nyq and h scond is no bandlimid w Nyq. Hnc, w Nyq, T s π π. For modlaion, f Nyq f Nyq f Nyq. Th firs signal is bandlimid o rad/s, w Nyq and h scond is bandlimid o rad/s w Nyq. Hnc, w Nyq, T s π π. Problm : Th frqncy spcrm of a vibraion signal is shown in h figr blow. Drmin h sampling im consrains o avoid aliasing. Th signal is pracically bandlimid o 7kHz approimaly. Hnc, T s k 0.07ms.

13 EEE 03, Ts 6 CLOSED BOOK & NOTES. TRANSFORM TABLES ALLOWED. 30 NAME: Problm : Considr h coninos im casal filr wih ransfr fncion s H s s s Comp h rspons of h filr o. y L s s s s L s s 0 s Problm : Considr h coninos im casal filr dscribd by h diffrnial qaion d y dy d y d d d Comp h sady-sa rspons of h filr o sin. s Hs s s s s 0.5 j0.5s 0.5 j0.5 Th sysm is sabl.f. pols in h LHP, hnc h sady-sa rspons is wll-dfind. y ss Hj sin [Hj] Hj0 j j j sin aan aan 7 80o 9 6 sin aan aan 7 80o 3 sin 86.8o 0.55sin 86.8 o

14 EEE 03, HW 6 NAME: Problm : Considr h casal filr dscribd by h diffrnc qaion y [ n] y[ n ] [ n ]. Drmin h ransfr fncion. Comp h rspons of h filr o [n] [n] 3. Comp h rspons o [n] δ[n]. Comp h sady sa rspons o [n] sin nπ / 6 [n-] No: For a sabl sysm if convrgs o a priodic signal _s, hn y convrgs o a priodic signal ha is h sysm rspons o h _s. This can b compd sing Forir hory. A sfl simplificaion is: jωo jωo y H jωo For Coninos Tim: cos ωo y H jωo cos ωo H jωo jωon jωo jωon n y n H For Discr Tim: jωo jωo n cos Ω on y n H cos Ω on H whr, H s, H z ar h coninos and discr sysm ransfr fncions, rspcivly.. Hz z z z. yz Hz z z z z z z z z yn Z z z z yn n n n n y n 0 3. yz Hz yn z Z z z n n z yn n n n y 0 n Th filr is sabl sinc h pol / has magnid lss han on. Hnc, h sady-sa rspons is wll dfind and y ss n H jω sin Ωn H jω, Ω π 6 Thn, y ss n cos π 6 sin π 6 sin π 6 n aan sin π 6 cos π sin π 6 n.o Problm : Considr h discr im casal filrs wih ransfr fncions z z H z, H z z 0.9 z 0.8 z 0.9 z.8. Drmin hir sabiliy propris.. Comp h rspons of h filrs o [n] [n]. Th pols of H ar 0.9, -0.8, boh ar lss han in magnid, so h sysm is sabl. Th pols of H ar -0.9, -.8, h scond has magnid grar han so h sysm is nsabl.. y z H z z z z z.7.7 z0.9z0.8 z0.9 z0.8 y n Z.7 0.9n n.7 0.8n n [ 0,, 0., 0.73, 0.5, 0.50, ] z. y z H z z z0.9z.8 z0.9 z z z.8 y n Z n n 0.9.8n n [0,,.7, 5.67, 0.935, ].7 z.7 z z0.9 z z 0.9 z z0.9 z.8

15 EEE 03, HW 5 NAME: SOLUTIONS Problm : Considr h coninos im casal filr wih ransfr fncion s H s s s Comp h rspons of h filr o y L s s s s L /6 s /3 s s 6 3 Problm : Considr h coninos im casal filr dscribd by h diffrnial qaion d y dy y d d Comp h sady-sa rspons of h filr o cos[ ]. Hs s s Th sysm is sabl ransfr fncion pols or roos of h dnominaor ar ± 6/ ar in h LHP, i.., hav ngaiv ral pars, hnc h sady-sa rspons is wll-dfind. y ss Hj cos [Hj] j j cos aan 0 cos aan 0 cos 3.7o

16 EEE 03, HW NAME: SOLUTIONS Problm : Find h largs sampling inrval T s o allow prfc rconsrcion of h signals: NOTE: h* dnos convolion of h and sin sin 5. * 3 sin sin. sin 3. *sin 5 Using h shorc mhod, w comp h Nyqis ras ma signal frqncy of h individal signals and hn sima h Nyqis ra of h composi signal:. w N 8, w N 0 w N minw N, w N 8 T s π w N π. w N, w N w N maw N, w N T s 0 3. w N, w N 0 w N minw N, w N T s π π No: In fac, a dirc compaion of #3 shows ha h rsling signal is 0, hnc any sampling ra can b sd. This is consisn wih or ndrsanding of h shorc mhod prodcing consrvaiv simas of h sampling ra. Problm : For an idal sampling procss wih sampling im 5ms, drmin h coff frqncy of h idal low-pass filr ndd o avoid any aliasing ffcs. Dsign an idal low pass filr for rconsrcion. Wha is h maimm frqncy of signals ha can b sampld and rconsrcd wih his sysm? Wih a sampling im of 5ms, h sampling frqncy is 00Hz so h maimm signal frqncy ha can b rcovrd afr sampling and rconsrcion is 00Hz. Thn, h coff frqncy of h idal low-pass Ani-Aliasing Filr is 00Hz, or lss. Th rconsrcion filr shold hav amplid T 0.00 and coff frqncy 00Hz. Th maimm frqncy of signals ha can b rconsrcd is also 00Hz.

17 EEE 03, HW 3 NAME: SOLUTIONS Problm : L b h priodic signal shown in h figr blow sawooh wav wih offs and shif. Comp h cofficins a k of h Forir sris pansion of Th drivaiv of is d.5.5δ n d n. From h abls, h Forir sris FS 3 cofficins of h shifd impls rain ar a k.5 jkπ/3, k. Thn, h FS cofficins of, say b k ar givn by b k a jkw k, k 0, w 0 π. Ths, b 0 k.5 jkπ jkπ/3, k 0. For k 0, w comp h FS cofficin dircly from h dfiniion, b 0 d T T 3 3 b 3 0 Problm : L X b h Forir ransform of. Find Xj0 and From h dfiniion of h Forir ransform and is invrs, Xj0 j0 d d Xj0. On h ohr hand, d d 0 π X 0 dw wxdw π j π j wxdw πj d d 0 wx dw. Problm 3: Considr h filr wih impls rspons h.. Find h ransfr fncion and skch h Bod Plo. Find h Forir ransform of h op whn and whn sin 3. Find h op whn a sing convolion, and b aking h invrs Forir ransform of yor answr o par. h H

18 H w, H w an w In MATLAB, >> Hfp-,[ ] >> H.iodlay >> bodh ], [ } {sin } { : Sinsoid inp, } { : inp Eponnial. X H Y w w j F F X H Y F δ δ π ] [ ] [ } { ] [ * 3. a h sin frqncy vala H 0 F F Y F d d h y w j w j j j j δ δ π

19 EEE 03, HW NAME: SOLUTIONS Problm. Comp h convolion h* whn r-r---, hδ. h is h ni sp, r is h ni ramp - h r r, or y Problm. Considr h filrs h. y. y d. Find and graph hir impls rsponss.. h δ δ. y δ d δ d. Find and graph hir sp rsponss.. h. y d d [ ] [ ] 3. Which filrs ar casal? Jsify. Casal bcas h 0, for < 0.. Non-Casal bcas h 0, for som < 0.. Which filrs ar sabl? Jsify. Sabl bcas h <. Unsabl bcas h divrgs. h y y -

20 EEE 03, HW NAME: SOLUTIONS Problm. Considr h signal whos graph is shown blow. Skch h following signals: -, -, RT- [], whr R dnos h rflcion opraion and T 0 dnos shif dlay opraion by RT_-[] Problm. Dscrib h following signals in rms of lmnary fncions δ,,r, and comp δ d and y δ 3 d. y 3 3 r r 3 3 δ d [ ] y r r 3 yδ 3d [y3 y3 ] 0

21 EEE 03, Ts 6 NAME: SOLUTIONS Problm : Considr h casal filr dscribd by h diffrnc qaion y [ n ] y[ n] [ n ]. Drmin h ransfr fncion Taking z-ransforms of boh sids, zyz Yz z Xz Hz Yz Xz. Comp h rspons of h filr o [n] [n] z z Yz z z z z Xz z z z z z z yn n n n Problm : Comp h sady-sa rspons of h following discr-im, casal filrs o [n] [n-]: z H z, Sabl, y ss n H j0 j0n H z z 0.5 H z z, Unsabl, sady-sa is no wll-dfind z z 0.8 z 0. H 3 z, Sabl, y ss n H j0 j0n H 9 z 0.5 z 0.8 H z 0 z, Unsabl, sady-sa is no wll-dfind z z

22 EEE 03, Ts 5 CLOSED BOOK & NOTES. TRANSFORM TABLES ALLOWED. 30 Problm : Considr h coninos im casal filr wih ransfr fncion s H s s 3 s Comp h rspons of h filr o -. NAME: SOLUTIONS y y s y s, whr, y s L Hs, h sp rspons. s y s L s3s L s3 s 3 y 3 3 Problm : Considr h coninos im casal filr dscribd by h diffrnial qaion d y dy d 8y d d d Comp h sady-sa rspons of h filr o sin Hs s ± 8. Is pols ar and hav ngaiv ral pars, so h sysm is sabl and h s s8 sady-sa rspons is wll-dfind. Th sady-sa inp is sin 3. Th sady-sa op is Hj sin 3 Hj Hj0. sin 3 o. sin

23 EEE 03, TEST CLOSED BOOK & NOTES. TRANSFORM TABLES ALLOWED. 30 NAME: SOLUTIONS Problm : Find h largs sampling inrval T s o allow prfc rconsrcion of h signals: NOTE: h* dnos convolion of h and. sin 3 : w N min, T s π π.. sin cos w N min, T s π π 3. cos sin sin cos: w N sm, T s π π Problm : Th frqncy spcrm of a vibraion signal is shown in h figr blow. W wold lik o sampl and analyz h spcral paks arond 3kHz b or compr can only sppor sampling ras p o 8kHz. Commn on h fasibiliy of his objciv and dscrib h idal componns ha shold b sd in sch a sampling sysm. To analyz h spcral paks arond 3kHz w nd o sampl fasr han 6kHz pls any margin dsirabl so h compr has sfficin capabiliis, a las in principl. From h givn spcrm, hr ar componns in h signal wih frqncis highr han Nyqis khz so w nd o s a good Ani-Aliasing Filr wih coff frqncy blow khz, b abov 3kHz o incld h inrsing porion of h signal. Thn, if ndd, rconsrc wih a filr wih coff highr han 3kHz o covr h signal, b blow h khz Nyqis frqncy of h sysm. Th analysis may or may no rqir rconsrcion AAF, 3-kHz SAMPLING 6-8kHz Analysis -- if ndd, Rconsrcion Lowpass 3-KHz, amplid T /3000-/000

24 EEE03, TEST 3. NAME: SOLUTIONS 75, closd books and nos, calclaors and ransform abls allowd Problm : Considr h filr wih impls rspons h. 3. Find h Forir ransform of h op Y whn 3. Find h im-domain prssion for h op y whn. F H. y F {Y} F 3 F 3 Y HX 3 3 F / F / 3 dy Problm : Considr h RC filr dscribd by h diffrnial qaion a y, whr a is an adjsabl d paramr.. Find h magnid and phas of h frqncy rspons of h filr.. Comp h val of a so ha sady-sa rspons of h filr o [] cos[0] has amplid 0.0. yilds a. H H. y H cos w H yilds a w, H an aw a w w0 0.0 a 0 Problm 3: L b h priodic signal shown in h figr blow sqar wav wih offs. - Comp h cofficins a k of h Forir sris pansion of. Svral possibl approachs,.g., s d δ n wo dla rains, from which, F{} F d, c. d jkw 0 d Hr, w s 0 sandard sqar wav, T, w 0 π,. Thn 0. Hnc, b k a k jkπ/ sin kπ kπ jkπ/ j kπ sin kπ, b 0 d 0. T

25 EEE 03, Ts NAME: SOLUTIONS Problm. Comp h convolion h* whn r---r---, hδ- δ. h is h ni sp, r is h ni ramp - h r r {r r } Problm. Considr h filrs. y. y d. Find and graph hir impls rsponss.. Which filrs ar casal? Jsify 3. Which filrs ar sabl? Jsify h. h δ δ δ. This filr is no casal bcas h is no zro for som <0. 3. This filr is sabl bcas h <. h δ d. This filr is casal bcas h 0 for all <0. 3. This filr is sabl bcas h is absolly ingrabl h <. - h

26 EEE 03, Ts NAME: SOLUTIONS Problm. Considr h signal whos graph is shown blow. Skch h following signals:, T R[], whr R dnos h rflcion opraion and T 0 dnos shif dlay opraion by 0. 3 R[] T R[] 0 Problm. Dscrib h following signals in rms of lmnary fncions δ,,r, and comp δ d and y δ d. y 3 3 r r, δ d y r 3 r r 3 3, yδ d y 0

27 EEE 03, HW NAME: SOLUTIONS Problm. Considr h signal whos graph is shown blow. Skch h following signals:, ---, RT [], whr R dnos h rflcion opraion and T 0 dnos shif dlay opraion by RT_[] - - Problm. Dscrib h following signals in rms of lmnary fncions δ,,r, and comp δ d and y δ 3 d. y 3 3 r r r 3 δ d [ ] y r r 3 yδ 3d [y3 y3 ]

28 EEE 03, HW NAME: SOLUTIONS Problm. Comp h convolion h* whn, h ar as shown blow. Eprss boh signals in rms of lmnary fncions and s boh an analyical and a graphical approach h \ h δ [r r ] r r y h Problm. Considr h filrs. y. y d. Find and graph hir impls rsponss.. h δ δ. h δ d δ d. Find and graph hir sp rsponss.. y. y d d 3. Which filrs ar casal? Jsify is no casal bcas h 0 for som < 0 is casal bcas h 0 for all < 0. Which filrs ar sabl? Jsify is sabl bcas h d < is no sabl bcas h d divrgs h - y y

29 EEE 03, HW 3 NAME: SOLUTIONS Problm : L b h priodic signal shown in h figr blow sawooh wav wih offs and shif. Comp h cofficins a k of h Forir sris pansion of. - - Th drivaiv of is d 3 3δ n d n. From h abls, h Forir sris FS cofficins of h shifd impls rain ar a k 3/ jkπ/, k. Thn, h FS cofficins of, say b k ar givn by b k a jkw k, k 0, w 0 π. Ths, b 0 k 3 jkπ jkπ, k 0. For k 0, w comp h FS cofficin dircly from h dfiniion, b 0 d T T 3 d T Problm : L X b h Forir ransform of. Find Xj0 and From h dfiniion of h Forir ransform and is invrs, Xj0 j0 d d Xj0. On h ohr hand, d d 0 π X 0 dw wxdw π j π j / avrag of lf and righ limis wxdw 0 d d 0 X dw. Problm 3: Considr h filr wih impls rspons h.. Find h ransfr fncion and skch h Bod Plo. Find h Forir ransform of h op whn and whn sin 3. Find h op whn a sing convolion, and b aking h invrs Forir ransform of yor answr o par.

30 h H H w, H w an w In MATLAB, Hf,[ ] %Malab dos no accp % -v dlays [m,p,w]bodh; mm:;pp:;ww:; figr;sbplo,loglogw,m,il'magnid' sbplo,loglogw,m,smilogw,pw*80/pi,il'phas' ], [ } {sin } { : Sinsoid inp, } { : inp Eponnial. X H Y w w j F F X H Y F δ δ π ] [ ] [ } { ] [ * 3. a h sin frqncy vala H 0 F F Y F d d h y w j w j j j j δ δ π

31 EEE 03, HW NAME: SOLUTIONS Problm : Find h largs sampling inrval T s o allow prfc rconsrcion of h signals: NOTE: h* dnos convolion of h and sin sin sin sin. sin 3. *sin Using h shorc mhod, w comp h Nyqis ras ma signal frqncy of h individal signals and hn sima h Nyqis ra of h composi signal:. w N 8, w N 0 w N w N w N 8 T s π w N π 9. w N, w N w N maw N, w N T s π 3. w N, w N w N minw N, w N T s π π Problm : A sampling sysm is o b dsignd o handl signals wih maimm frqncy of inrs khz. Using idal componns, dsign h sampling im, drmin h coff frqncy of h ani-aliasing filr and dsign h rconsrcion filr. Th Nyqis ra for khz signals is khz. Hnc h sampling im shold b lss han 0.5ms, Thn, h coff frqncy of h idal low-pass Ani-Aliasing Filr is khz. Th rconsrcion filr shold hav amplid T and coff frqncy khz.

32 EEE 03, HW 5 NAME: SOLUTIONS Problm : Considr h coninos im casal filr wih ransfr fncion Hs s ss Comp h rspons of h filr o y L s ss s L s s s Problm : Considr h coninos im casal filr dscribd by h diffrnial qaion d y dy d 8y d d d Comp h sady-sa rspons of h filr o cos[ ]. s Hs s s 8 Th sysm is sabl ransfr fncion pols or roos of h dnominaor ar ± j.66 ar in h LHP, i.., hav ngaiv ral pars, hnc h sady-sa rspons is wll-dfind. y ss Hj cos [Hj] 0.35cos 5 o 0.35 cos 0.3 o

33 EEE 03, HW 6 NAME: SOLUTIONS Problm : Considr h casal filr dscribd by h diffrnc qaion y [ n] y[ n ] [ n ] 3. Drmin h ransfr fncion. Comp h rspons of h filr o [n] [n] 3. Comp h rspons o [n] δ[n]. Comp h sady sa rspons o [n] sin nπ / 6 [n-] No: For a sabl sysm if convrgs o a priodic signal _s, hn y convrgs o a priodic signal ha is h sysm rspons o h _s. This can b compd sing Forir hory. A sfl simplificaion is: jωo jωo y H jωo For Coninos Tim: cos ωo y H jωo cos ωo H jωo jωon jωo jωon n y n H For Discr Tim: jωo jωo n cos Ω on y n H cos Ω on H whr, H s, H z ar h coninos and discr sysm ransfr fncions, rspcivly.. Hz z 3 z z 3 z z z 8 z. yz Hz z z 3 z z z 3 3 z yn Z z 3 3 yn n n 3 8 n n y n 0 3. yz Hz yn z 3 Z z z 3 n n z yn n 3 n n y 0 n Th filr is sabl sinc h pol /3 has magnid lss han on. Hnc, h sady-sa rspons is wll-dfind and y ss n H jω sin Ωn H jω, Ω π 6 Thn, y ss n cos π 6 3 sin π 6 sin π 6 n aan sin π 6 cos π 6 3 Vrify in MATLAB: >> Hf/,[ -/3],, [m,p]bodh,pi/ sin π 6 n 6.8o Problm : Considr h discr im casal filrs wih ransfr fncions z H z, z 0.3 z 0. z H z z 0.5 z. Drmin hir sabiliy propris.. Comp h rspons of h filrs o [n] [n]. Th pols of H ar 0.3, -0., boh ar lss han in magnid insid h ni circl, so h sysm is sabl. Th pols of H ar -0.5,, h scond has magnid grar han so h sysm is nsabl. z.3. y z H z z.3 y z z0.3 z0. n Z.3z.3z z0.3 z n n.30. n n [ 0,, 0.067, 0.38, 0.08, ] z. y z H z z 0. z 0. y z z0.5z z0.5 z n Z 0.z 0.z z0.5 z n n 0. n n [0,,.5, 3.5, 6.375, ]

34 EEE 03, Ts NAME: SOLUTIONS Problm. Considr h signal whos graph is shown blow. Skch h following signals: --, RT- R[], whr R dnos h rflcion opraion and T 0 dnos shif dlay opraion by RT - R[] 3 Problm. Dscrib h following signals in rms of lmnary fncions δ,,r, and comp δ d and y δ 3 d. y 3 3 r r 3, δ d y r r r r 3 3, yδ 3d y3 0

35 EEE 03, TEST. NAME: SOLUTIONS Problm. Comp h convolion h* whn h, δ δ y h δ δ Problm. Considr h filrs: A. y B. d y d. Find and graph hir impls rsponss. A. h δd B. h δ d δ d. Which filrs ar casal? Jsify A is no casal, h - is no zro for all < 0. A is casal, h h- is zro for all < Which filrs ar sabl? Jsify A is no sabl bcas h is no absolly ingrabl B is sabl, bcas h is absolly ingrabl d

36 EEE03, TEST 3. NAME: SOLUTIONS 30, closd books and nos, calclaors and ransform abls allowd Problm : Considr h filr wih impls rspons h.. Find h Forir ransform of h op Y whn. Find h im-domain prssion for h op y whn Y HX y F {Y} F F F Problm : Considr h RC filr dscribd by h diffrnial qaion dy 0 d y Wri an prssion for h magnid and phas of h frqncy rspons of h filr Y/X. H 0 > H, H aan0w 00w Problm 3: L b h priodic signal shown in h figr blow sqar wav wih offs. Comp h cofficins a k of h Forir sris pansion of. W obsrv ha o 0.5, whr o is h sandard sqar wav in h abls wih priod T, T. From h abls, h FS pansion for o is - Hnc, sin kw 0 T kπ sin kπ kπ a k FS{} sin kπ kπ jkπ, k 0, a 0 T d <T> For h firs rm w may also obsrv ha jkπ cos kπ j sin kπ zro. So, and h prodc of cos kπ sin kπ kπ a k j sin, k 0, kπ a 0 is always

37 EEE 03, TEST CLOSED BOOK & NOTES. TRANSFORM TABLES ALLOWED. 30 NAME: SOLUTIONS Problm : Find h largs sampling inrval T s o allow prfc rconsrcion of h signals: NOTE: h* dnos convolion of h and. sin5 0 w NYQ 5 0 rad s, w NYQ > w NYQ ma w NYQ, w NYQ > T s 0.. sin5 sin w NYQ 5 0, w NYQ rad s > w NYQ min w NYQ, w NYQ > T s π π 3.cos 3 sin w NYQ 3 3 rad, w s NYQ 8 > w NYQ sm w NYQ, w NYQ 0 > T s π 0 π 0 Problm : Th frqncy spcrm of a vibraion signal is shown in h figr blow. W wold lik o sampl and analyz h spcral paks arond 3kHz b or sampling sysm dos no incld an ani-aliasing filr. Drmin h sampling ra rqird for his ask. Sinc an ani-aliasing filr is no availabl, w nd o sampl fasr han wic h largs frqncy in h signal o mak sr ha hr is no aliasing ha wold compromis h rsls of or analysis, rgardlss of h frqncy of inrs. From h plo, w sima h highs frqncy as 7kHz 6kHz or 8kHz ar also valid simas. Ths, h sampling ra rqird for h ask is f NYQ 7kHz khz, T s 7μs.

38 EEE 03, Ts 5 CLOSED BOOK & NOTES. TRANSFORM TABLES ALLOWED. 30 Problm : Considr h coninos im casal filr wih ransfr fncion H s s s Comp h rspons of h filr o -. NAME: SOLUTIONS y y s y s, whr, y s L Hs, h sp rspons. s y s L ss L / / s s y Problm : Considr h coninos im casal filr dscribd by h diffrnial qaion d y dy d 8 y d Comp h sady-sa rspons of h filr o sin -. d d s Hs. Is pols ar 0.3, 3.8 and hav ngaiv ral pars, so h sysm is sabl and h s 8s sady-sa rspons is wll-dfind. Th sady-sa inp is sin. Th sady-sa op is Hj sin Hj Hj0 0.7 sin 90 o No: Hs s s 8s s s 0.3s 3.8 w w H aanw aan w 0.3 w aan w 3.8

39 EEE 03, Ts 6 NAME: SOLUTIONS Problm : Considr h casal filr dscribd by h diffrnc qaion y [ n ] 0.3y[ n] 0.3[ n ]. Drmin h ransfr fncion Taking z-ransforms of boh sids, zyz 0.3Yz 0.3z Xz Hz Yz Xz 0.3 zz 0.3. Comp h rspons of h filr o [n] [n] Xz z 0.3, Yz z z 0.3 zz0.3 z zz z z0.3 yn n n n Problm : Comp h sady-sa rspons of h following discr-im, casal filrs o [n] [n-]: z z z z 0.5 H, Sabl, y ss n H j0 j0n H 0 H z z z z 0.5, Unsabl, sady-sa is no wll-dfind Problm 3: Comp h sady-sa rspons of h following discr-im, casal filrs o [n] cos π n [n 5]: 5 H 3 z 0. z z 0. z 0.5 Th filr is sabl sinc h pols hav magnid lss han on. Hnc, h sady-sa rspons is wlldfind and y ss n H jω cos Ωn H jω, Ω π 5 Thn, y ss n. sin π 5 n 39.8o H jω cos π 5 0. sin π 5 cos π 5 0. sin π 5 cos π sin π 5. H jω aan sin π 5 cos π 5 π 5 π 5 sin sin aan aan 0. cos π 0. cos π o all ral pars posiiv