Signals and Systems
Signals and Systems ECE 350
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This 32 page Class Notes was uploaded by Judd Okuneva on Saturday October 3, 2015. The Class Notes belongs to ECE 350 at Boise State University taught by Elisa Smith in Fall. Since its upload, it has received 23 views. For similar materials see /class/217979/ece-350-boise-state-university in Engineering Electrical & Compu at Boise State University.
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Date Created: 10/03/15
EE350 7 Signals amp Transforms Math Review Solutions Spring 2006 EE350 Signals amp Transforms Math Review Solutions 1 a x141 b x1 c x0707 5 I 2 a 712 0j quot b1j 39 3 a 1j JEJA Jig l b lj1 e 1 4 Jig CEjw Zej d1 281 1 l n 4 agaew bm izenile J 1fW3 7 29 7 1 1 13 1 98 5 10 26 c 5 N51 J Mame4 551 d Zf O 111 1 N5 2 e 9 act0 154 t 5 a o 39 bi z cquot 0 1 1 1 2 4 l t bii L L biii a c 0 t 1 0 t 2 2 0 t g 7 x071 71 1ltt 2x27t 27 2gtt21x2t1 2t1 7ltth 1 t gt 2 1 I lt1 1 t gt 0 6 a b xx c 0 tgil 7 a 960 111 7016 b graphically triangles A 12 bh 2 0 tgt 2 A 1211 1212 12 1 32 EE350 7 Signals amp Transforms Math Review Solutions Spring 2006 A Tod 32 Mn 31 fad Todz 1 0 1 b Integrals A 6 1724 l Il t 0 A1 1 r i13 7 quot k D I ci 7 3 1 cii 391 39 1 I 9d IT 4 0111 1 39 c1V 39 2 3 je t ije t 2857 28757 jejt feijt 2eST 8757 J 8 a 72sint 4cos57 t 2 116 i196 J 3J 7k 2 b Ee s1n 16 1 9 a J39s dr 51e3 3 5e3 71 0 1 1 b JSe393 dt iii lo e 71 97 71 0 1 1 732 ii 7327 732 iii 73 L cJte d173te ge lei 98 9 0 1 1 732 7 1 732 1 732 7 1 737 1 Ire dti mte 3 e 0 79 3e 9 0 1 1 2 732 7712 73272 7327i 73 iii 73 i eJte dti 3te gte 278 lei 278 27 0 1 1 2 732 7 1 2 7327 2 7327 2 732 1 L 7372 D It 8 dti 3jt e 32 re 73 e 0 79 27 e 27 0 2 73 2 73 re e 32 733 g11712i3tdtir1 3t 1 t2 73 e 7318 3 e 0 1 7 i L39 3lig39 0 927Je 27 no no 2 732 1 732 2 732 2 732 2 73 JG 1 8 11 38 3jt e 612 t8 3me 0 0 100 1 27 Hint e39t decreases faster than tn grows EE350 7 Signals amp Transforms 10 11 12 13 14 15 16 17 Math Review Solutions Spring 2006 1 7m 1 2 rwz 1 rm 2 rm 2 rm 2 i 1 te dt Ne jwte Mzte Mze 0 w L 8 jw jW 0 L i 2 7w ZJW2 42w2 MK 1W 1 0 0 MK 1 e jwf jwf i Te39wdtie dt fequotdte iw 761 1io0771 2 OO OO 0 L mi 8732 3 no 0 no 32 0 k e wdt e3 dt e 3 dte 1 L l a 1 1 2 0 13 0110 1 3 2 7 732 Ism3le 3 dtI ee Walt a 2 l 5 1 762 lie l ILe 0 2j gtd 214 6 22 722 Icos21e 3 dl Mest dl b 2 7 it 752 7 72 f 5 7Je e jdlie e c a logl03X2log10x2log10x1 b logl03X2 510g10x2log10X 1 a 0 b oo0 9 use L Hopital s rule 1m x 1 0 Hint e39t decreases faster than tn grows xaw 63x eoo 2 36 9 0 W111 be when X96r and 7 when X90 SO 00 note limit without direction does not exist l39 d 7i e 39 use L Hopltal s rule 11 i0 2571 16 oo xaoo2x1 00 OR notice order of polynomial denominator is greater than order of polynomial numerator 1 0 00 139 Z Z I I 2 f m M O394but from coef clents 900 9 xgtoox2xil 002 002 lim 3x2 77007 lim x2 1 700 xgt700x2xil 002 xaw4x32x71 00 g just review on your own a 2 711x713 b x7 l7x35 c 2 12x16 x26x8 2 8 x26x8 L L a171 1 2 2 2x4x 2x 4x 2x24x 2x2 2x 4x c 2xZ 72 712x716 7 4 716 2x4x 2x4x 2x 4x 2 2 u d 2 x32 211x 13 2 2 2 x 6x8 x 6x8 x2 x4 L L A L 20 5quot S 10 57 S 0 57 S d s 5 1 71 7 1 71 T 1 71 T 1 31 7 a x2222 b x522J 2 c x 2222 d2X222 j2 18 19 20 21 EE350 9 Signals amp Transforms Math Review Solutions Spring 2006 Haj I 4 oddr 2 t C a I o I b l L 1 ij 510309 I39 V A i t c 39 d 1 z 1 L14 Mn 1 5 H t 39 3 H aampb m 1 0 d amp i Xf l 39Kh 1 eampf gamph quot a agt0 bgt0 positive slope positive intercept b alt0 bgt0 c 290 blt0 d agt0 bgt0 e 290 blt0 1 alt0 bgt0 Differential Equations 21 y Y 72y 0 y01y 0 1 s2 s 72 s2s1 9 s 21 yAe392 Bet y0AB1 y 2Ae392t Bet y 0 2A B 1 A B ya e b y 6y 9y 0 y0 0 y 0 2 s2 6s 9 s3s3 9 s 33 yAe3 Bte3t y0A00 y 3Ae3t 3Bte3 Be3t y 0 3A 0 B 2 A 0 B 2 yt 2te3t 0 y 8y e9y0 y11y 10 s 8s 9 9 s9s10 9 s19 yAe399 Bet y1Ae399Be1 y 9Ae399 Bet y 1 9Ae 9 Be 1 A 0 B951 ya e d y 4y 0 y0 0 y 0 1 s240 9 s i2j y Aezl Be392Jt y0 A B 0 y 2jAe2Jt 2jBe392Jt y 0 2jA 2jB 1 A 14139 B 14j yt 14je21 14je3921 1z sin2t 6 y y 2y 2t y0 0 y 0 1 Homogeneous Solution s2 s 2 s1s2 9 s 21 EE350 7 Signals amp Transforms 22 23 24 25 Math Review Solutions Spring 2006 yh Act Be39m Particular Solution ypa Ct D yp t C yp t 0 yp yp 2yp 0 C 72CtD 2t C 72D 72Ct 2t C 1D 12 yt Ae 13e39 7t712 y0 AB7120 y t Aet 213e392t 1 y 0 A 2B 1 1 A 1 B 42 yt e 12 e 2 7t 12 f y 4y t23et Homogeneous Solution s2409 si2j yh Aezjt Be39ZJt Particular Solution ypa Ctz Dt E Fet yp a 2Ct D Fet yp t 2C Fet yp 4yp 20 Fe 4Ct2 Dt E Fe tz3et C 1A D0 E 18 F35 yt AeZJt 3621171412 7 18 35et y t 2jAet 7 2j13e39ZJt 12 t35et Y0 0 Y 0 2 y0AB71835 0 y 0 2jA 7 2jB 35 2 A 19j2880139 B 19j2880139 yt 1980ezjt 2880je2 1980e3921 2880je39zjt M t2 7 18 35et yt 1940cos2t 2840sin2t 1A t2 7 18 35et yt 1940c0s2t 710sin2t 1A t2 7 18 35et Laplace Transforms answers will be the same as in 21 a 6x2 12x3 4x2 allch 36x2 l7xl 24x3 22x2 7xl atbccbdx2 bacdxa a V0 R2 R2Cs b V0 R2 R1R2R2Ls K R2Rl R1R2Csl K R2 R1R2LSR1R2 a V0 R2 d L7 K LCRZs2 LR1R2CsR1R2 K s2 ls16 C02 R130 R225 and L05The method is to calculate the magnitude and phase of HOW as in Circuits I or 11 Then the response would be a vow 21Hltljgt1cos t ang1eHIj b Vomt 3lH1jlcos t15 angleH1j c Vomt 2lH5jlcos 5t angleH5j d Vomt 3H0jcos0 For gs H11j 04508 00410j 04527e 0390907 H15j 04544 00083j 04545eJ0390182 H10j 0 so a Vanna 09054cost00907 b Vanna 13581cost 150 520 c Vanna 0909cos5t 00182 d Vanna 0 The others proceed like this EE350 7 Signals amp Transforms 26 Math Review Solutions Spring 2006 a b I IllI Ill llllllllll a II III 39El39ll39 I39I lllliul u 1 Fvequenm name Fvequenm name cHs 2 2 2 LCs RCs1 00004s 0008sl g 221 Fvequenm name Note you only need to be able to graph the straight line approximations The actual curves are shown for reference You should know how to generate these Via Matlab Advice to EE350 Spring 2005 Students From EE350 Fall 2004 Students Your best homework scores come when you put your heads together in study groups The assignments are intense use all available brain power Leave your ego at the door You will be stymied by this stuff that s ok get over it and learn Do your best in doing homework s try to go beyond and to understand the depth of the matter work in a group and ask questions Some students understand today some tomorrow but in the end they will all understand provided that they made effort to understand Don t walk into an exam raw make the time to work all sorts of expected problems for the preceding 3hours before the exam it makes a difference to be warmed up have your brain in gear The math review sheet will be very helpful to go through a couple of times since you will need them several times thru the semester Also take good notes and review them freq Will help the concepts sink in better Review your math Participate with other students Do your homework when you get it Do not wait until the last minute Go to see her as often as possible She is almost always there and will always help Start the homework early Read the book before doing the homework s Do practice exams before looking at solution to see where you would stand on the test Also take the equivalent exam from the year before as if it were the real test and see what grade you would have received Don t relay solely on groups to do homework Many students don t even attempt to do the problems by themselves for they meet up with others to do them You should try to do the problems on your own before you ask someone else That way you at least understand the general approach and maybe some specific issues Remember you can t do the test in groups 1 Get started on the homework early and make sure you understand it well Tests re ect homework so it s wise to know what your doing Additionally it s a large chuck of your grade 2 Do the math review and it helps 3 Don t skip class If you have to skip get the notes for that day from a reliable source 4 Have fun and leam this is a very interesting course that has many realworld applications When she says something s on the test It is on the test Don t be afraid to ask questions Review your differential equations Do the practice exams If you are willing to put the time into the homework and go to Dr Barney Smith s office you can get answers to all of your questions This will prove to be the most beneficial thing you can do to be successful in this course Study with a group if you can explain something to someone else you truly understand it Make friends with J like it or not you will be close Remember partial fractions graphical addition and polynomial log division Don t let the reputation of the class and instructor intimidate you it IS possible to gat a good grade Bring a highlighter to class to highlight the things that Dr Barney Smith says will be on the test Review those things before the test She doesn t lie Work the old te4sts if possible That s where you get an idea of the key concepts that will be covered and extra practice doesn t hurt In this class you will learn how convolution is convoluted Don t worry you will nd out what that means very soon Get together work together It s easier that way I ve spent at least 5hrswk to keep up wclass Now that I have a nal that wasn t enough study time Finish formal report ASAP And don t forget to go talk to writing center people to get their receipt Do not skip off HW questions write down something It s 50 of HW grade Make sure you understand all of the homework and take really good notes in class Read the textbook If you have questions ask her she is always wiling to help however I found that the amount of help you get it usually proportional to the amount of work you have put into trying to figure the problem out yourself so give the homework a concerted effort before asking questions Do the practice exams and look at the exams she has posted from previous semesters they are very helpful Don t spend more than 20minutes by yourself on a homework problem If you don t have clue by then it will probably take hours Really focus on the practice exams the test are vm similar Go to class everyday My only advise is please start homework as early as possible also try to do it by yourself even if parts of it is wrong Later go ask for clarification Do not go ask questions before you put some time and thought Believe me it works very well I learned a lot from my mistakes Don t miss class and don t get behind You ll need everything you learn at the beginning throughout the course Use the practice and previous exams to study This is a difficult course but you will do well if spend the time required to learn the material There are three main aspects of the class I would focus on were I to take the class again 1 Homework Of course do the homework as it will only help your grade However it is important the concepts of the homework are understood So even if you completely botch and assignment go back and look over the solutions Dr Barney Smith posts on the class website Lecture Notes Pay attention in class and take thorough notes Don t be caught napping because invariably Dr BarneySmith will call on you to answer the questions The main point to understand is the nature of the class does not allow a person to zone out in lecture and make up for it by reading the test The book is helpful but vague in many areas Practice Tests These are academic gems I did not feel as though there were any trick questions on the test yet if a student were to complete the study test backwards and forwards the chances of success are greatly increased N V L V Study in group this class is more difficult if you are preparing alone Dr Barney Smith will help you if you don t understand the material Do your homework on time and don t wait until the last moment Show up to class for every lecture because the tests will be similar what the professor explained in class 1 Work as many of the former tests as you can Work in groups Don t procrastinate the homework Use the office hours she is often available outside of office hours too ALAN VVV 4 6 EE350 Signals amp Transforms Math Review Solutions a X114 b X1 c X0707 l H a 712 0j quot b1j 139 a1j ejxEej97 m c1 j 2ej d1j 2ej J b1 j1 28j lej 1 INK5 2e 0 5 1430 j cabs75H 591 d e098gtltJ ef gt IE em b lj eij 59797 ma5 2e 2 e gtlt5ef gt 10 W3 K wt 1 a 0 39 bi z at q 13 IX 2 L x t bii l 1 biii quot 391 0 171 a xt H1 71ltt 0 b ra hicall 39trian les A 1 bh 7 171 0a 2 g p y39 g 2 0 t gt 2 A 1211 1212 12 1 32 A Tod 31 Mn 31 fad Todz foo 71 0 b Integrals A0 Tz 111 17 0 A1 i1 7 I Ak ci 391 3 1 cii 391 quot 39 0 j WM ciii L 39 CW 2 3 7 8 9 10 11 75 je t ijei t 257 26757 jjejl 7e t2e 4 e7 STM a 72sinI 4cossT I 7 3 3 7 1k 17e I 2 74 me gt20 4 jk 3 7 1k jlk 7 1k 2 e I e2 7e 2 4jk7r 2f2 3j b f 2 k me sm 2 1 1 a J39s dz 5 Tle 320 e 3 71 0 1 1 b seelair 5ge73 0 e 3 71 53 fe73 71 0 1 1 732 71 73271732 771 73 i CJIe dI7TIe 9e 3907 96 9 0 1 1 732 1 732 1 73 L L 737 1 IIe dI7 3 Ie 3N2 e 0 793e 9 0 1 732 7 7 17 73 2 l 7 27 e 27 1 e IIZe StdI7IZe 32 7Ie 32 722 7e 0 1 732 7 L 73 e 079Z7e 1 2 732 77 1 2 7327 2 327 2 72 39 f It 6 dz WI 6 32 re 737 27 0 1 3 0 Z 7Je 17 1 7 2 732 71 73 L 2 732 2 732 2 73 g J1 I e dI 3 e 3jI e 32 Ie 73j3 e 0 no 0 07 l00031 j 732 ZZIe MJr 2 3e e37 no 72 732 71 73 L2 73 h J1 I e dI73je 3jI 6 6 0 Hint e39t decreases faster than tn grows 1 1 39 7 2 732 71 71472 L 2 71472 2 71472 2 71472 7w i 2 1 IO I e dI W e WI 6 m2 Ie Hm e 6 W2 W3 0 32 732 e 7e 22 22 Ism3Ie 3f dI 3J dt J39coslt20e73zdtquote 9 73zdt 2 b 2 e a 1 1 2 jJ17e 6 dI2 JIe 6 c Je 1 e 5f dIe e 5 c a logl03X2 10g10x210g10x1 b logl03X2 510g10x210g10X 1 a 0 b oo0 9 use L Hopital s rule 11m x i 0 Hint e39t decreases faster than tn grows x n 00 636 eoo 12 13 14 15 16 17 18 3 9 0 Will be when X90Jr and 7 when X9 0 SO 00 note limit Without direction does not exist 2 l lim 3 3 d 7 e 39 use L Hopltal s rule 254 16 oo xgt002xl 00 OR notice order of polynomial denominator is greater than order of polynomial numerator 1 0 OO lim 3 2 2 2 2 Libut from coef ments 900 9 xgtoo x2xil 002 002 lim 3x2 7700 h lim x2l 700270 g xgt700x2xil 002 x gt004x32xil 003 just review on your own a 2 711x713 9674 l7x35 x2 6x8 x2 6x8 1 1 2 l 1 2 l a b 2 2x4x 2 4 2x24x 2x2 2 4 2 0 2x 7 712x716 7m 8 1 32 2x4x 2x4x 2 4 12x2x3m 711x713 m 9 1 31 2 2 x2 x4 x 6x8 x 6x8 1 1 2 1 a c 17X1tj 17X1tf 17X1tj 1 1j a x2222 b x52 2 c x2222 d 2x22 2J5 2 0le j oltltgt2tl 39 t a a I b 739 JON lam3H Iquot 1 A i t c k 39 d 1 z 1 11 i aampb M 1 c d amp i 19 20 21 22 23 24 25 6x2 12x3 4x2 allch a b 0 36x2 17x1 24x3 22x2 7x1 atbccbdx2 bacdxa a agt0 bgt0 positive slope positive intercept b alt0 bgt0 c agt0 blt0 d agt0 bgt0 e agt0 blt0 f alt0 bgt0 just review on your own just review on your own R3 R3 a V V b V V 7 quotR1R2R3 7 quot R1R2 R3 R1R2 R3R4 R3R4 R3 R4 R3 R4 e V V d V V 7 quot R3R4 7 quot R1R2 R3R4 R1R2 R3R4 R1R2 R3R4 a 7 R2 7 R2Cs b 7 R2 7 R1R2R2Ls K R2 R1 R1R2Cs1 V R2 R1R2LsR1R2 22 SR V 7 R2 d V0 K LCR2s2LR1R2CsR1R2 Q 7 s2 1s16 Fvequenm name 1 1 1 H 2 2 S 2 S LCs RCs1 00004s 0008s1 5 22 1 Fvequenm name ECE 350 Signals amp Systems Math Review The questions below represent types of mathematical problems that you should have encountered in earlier math and engineering courses The problems are also representative of mathematical operations you will encounter frequently in this course This worksheet is designed to let you refresh your memory and seek outside help to ll any de ciencies you may have in these areas before they interfere with your ability to do the work in this course The text by Oppenheim amp Willsky has refresher sections on complex numbers p71 and partial fractions Appendix A that you may nd useful 1 a If 20log10x 3 what is x b If 20log10x 0 what is x c If 20log10x 3 what is x 2 Express the following complex number in Cartesian form xjy note j V l a e b Eai 3 Express the following complex numbers in polar form re a 1j b 1j C 15 115 d 1 145 4 Find the magnitude and phase of the following ratios and products of complex numbers hint express each part in polar form rem 1 J39 5 135 a b gt171 1mg 0 115an d 21 3X2a 1 N5 1 j373 4 j 5 If the function xt is piecewise de ned as follows 0 I S 0 xI I 0 lt I g l l I gt 1 a draw xt b from the drawing draw 0 Xtl ii x2t iii x2tl c from the equation write the equations for the functions i 7 iii O V If the function xt is piecewise de ned as follows iliI I 0 draw xt I71 Igt 0 a if xI Signals amp Systems page 1 Math Review ISO D 0 draw yt bif yak 0 xt 0 l xt gt 0 draw ya c using Xt as de ned in a if yt 7 A function Xt is drawn as follows a write the piecewise de ned function in a form similar to the previous problem s question b find Ixtdt both graphically and using formal integrals c write the equation and draw the graph of Xt2 Xtl Xt and X2t 8 Using Euler39s formula 6 cosx jsinx and the exponential representations of sin amp cos which are derived from Euler39s formula I J1C 1 J1 Jquot 5111x L 00505 L 2 j 2 write the following in terms of sin amp cos a few je jlt 265 2875 3 b l e39jk 2 4jk 9 Integrate the following 1 1 a 5de b ISe39mdt 0 0 1 1 c J39te 3 dt d J39te 3 dt 0 0 1 1 e J39zze 3 dt f eeta 0 0 1 no g 1 t2e 3f dt h 1 e 3f dt 0 0 1 i 1 t2 e39jwtdt where w a constant or another variable that isn39t the variable 0 of integration 039 j Nd k j e 3 dt Signals amp Systems page 2 Math Review 10 Integrate the following Use Euler s formula to help a J sin3te 3 dl b Icos2le 3 dl ll Reduce these logarithms to sum of logarithms with simple arguments 3x 3x 2 a 1 b 1 0g10x2x71 0g10x25x71 12 Find the following limits a x mw xe 31C b x1300 xe 31C lim 3x 2 lim 3x 2 d c H0 x23x HO x2x71 hm 3x 2 hm 3x 22 e xewx2x71 f Hquot0x2x1 hm 3x 2 hm x2 1 g x97wx2x 1 h x9w4x32x71 13 Find the roots of second order polynomials Recall use of quadratic formula r7 71241le 74110 211 I 14 Convert the following into a polynomial plus a proper polynomial fraction one where the order of the numerator is less than the order of the denominator Long Division is one method to accomplish this 2 3 2 2 2 a 2 x3 b x 22x x3 c x x 6x8 x 6x8 2x4x 15 Find the partial fractions expansion of 2 a 2x4x b 2x24x 2x2 2x2 x3 c 2x4x d x26x8 16 Put the following polynomial fractions into the form G 1X 1 2 1 b a 2s4s s28s7 2 1 d c s28s7 s27sl2 l7 Completing the square convert the following polynomials into the sum of two squares ie to the form Xa2b2 a x2 4x 8 b x2 5x 8 c x2 4x 8 Signals amp Systems page 3 Math Review d 4x2 16x 8 18 Given the three functions X y amp 2 determine the following by adding or multiplying graphs 1 xt A W 12 1 1 t 1 1 t zt 1 2 t a xtyt c mm b Kama d mm 19 Sketch the following a x30 tl f xfa equot b m0 1t g Kga e c Ken 5 h Xha equot d Xda mu 0 x a 5 1 1 e Xea e 20 What do you know about the parameters a amp b in the following graphs ie magnitude and sign a C e xtat xtatb xt be t t t b d f xtatb xtbeat xtbeat t t t 21 Solve the following differential equation using time domain methods a Y t Y 2Y 0 Y01 Y 01 b Y 6Y t 9Y 0 Y00 Y 02 C Y t 8Y 9Y 0 Y11 Y 10 d Y t 4Y 0 Y00 Y 01 6 Y t Y 2Y 2 Y00 Y 01 f y 4y t2 3e y00 y 02 22 Solve the differential equations in the preVious problem using Laplace Transforms Signals amp Systems page 4 Math Review 23 Simplify the following compound fractions to form a simple fraction containing the ratio of polynomials 2x 2x abx 12 12 abx a 2x Jl b 3x Jl c abx l 5 1 d l2x 3x l3x 2x a bx cx 24 Using impedances to nd the transfer functions Hs for the following circuits 25 From the transfer functions Hs in the preVious problem find the steady state responses to the following sinusoidal inputs if C02 Rl30 R225 and L05 a Vint 2cos t b Vint 3cos t15 c Vint 2cos 5t d Vina 3 26 Sketch the straight line approximation of the Bode plot for 5 110s b ms 10 H5 s2110s1000 s25100s500000 c Hs VISS where a H S 27 Using a table of Gaussian Normal probabilities calculate the following probabilities a u0 61 ProbXlt2 b u0 61 ProbXgtl c ul 61 ProbXlt5 d 1 61 ProbXgt2 e u0 62 ProbXlt5 f u0 65 ProbXgt2 g ul 62 ProbXlt5 h ul 65 ProbXgt0 Signals amp Transforms page 5 Math Review EE350 Signals amp Systems Math Review The questions below represent types of mathematical problems that you should have encountered in earlier math courses The problems are also representative of mathematical operations you will encounter frequently in this course This worksheet is designed to let you refresh your memory and seek outside help to ll any de ciencies you may have in these areas before they interfere with your ability to do the work in this course The text by Oppenheim amp Willsky has refresher sections on complex numbers p71 and partial fractions Appendix A that you may nd useful 1 a If 20log10x 3 what is x b If 20log10x 0 what is x c If 20log10x 3 what is x 2 Express the following complex number in Cartesian form xjy note j V l a b 456 3 Express the following complex numbers in polar form re a 1j b 1j C 15 115 d 1 145 4 Find the magnitude and phase of the following ratios and products of complex numbers hint express each part in polar form re H b 5 1 5 171 1mg 0 115an d 21 3X2a 1 N5 1 j373 4 j 5 If the function xt is piecewise defined as follows 0 I S 0 xI I 0 lt I g l l I gt 1 a draw xt b from the drawing draw 0 Xtl ii x2t iii x2tl 6 A function xt is drawn as follows Signals amp Transforms page 1 Math Review a write the piecewise de ned function in a form similar to the previous problem s question b nd Ixtdt both graphically and using formal integrals c write the equation and draw the graph of Xt2 X tl Xt and X2t 7 Using Euler39s formula 6 cosx jsinx and the exponential representations of sin amp cos which are derived from Euler39s formula 6 e39 sinx 2f 6 6 cosx 2 write the following in terms of sin amp cos a few 13967 28 3 b l 6 2 4k 8 Integrate the following 1 a ISe 3 dt 0 1 c J39te 3 dt 0 1 e J39zze 3 dt 0 g 1 t2e 3 dt 1 b ISe Sf dt 0 1 d J39te 3 dt 0 1 f tze 3 dt l h T1 t2e 3 dt 1 i 1 t2 e39jwtdt where w a constant or another variable that isn39t the variable 0 of integration 9 Integrate the following using Euler s formula to help a J39sin3ze3f dr 10 Reduce these logarithms 3x 1 a 0g10x ZXxi 1 11 Find the following limits a ximw lim 3x 2 x a 0 x2 3x c hm 3x2 e xaw x2xil Signals amp Transforms b Icos21e 3 dt 3x2 b l 0g10x25x71 b xlinooxe 31C lim 3x2 d xe0x2x71 lim 3x22 f x wx2xil page 2 Math Review hm 3x2 hm x2 1 g xv wx2xel h xaw4x32xil 12 Find the roots of second order polynomials Recall use of quadratic formula V 71241le 74110 2a 13 Convert the following into a polynomial plus a proper polynomial fraction one where the order of the numerator is less than the order of the denominator Long Division is one method to accomplish this 2x2x3 b x32x2x3 x26x8 x26x8 14 Find the partial fractions expansion of 2 a b 0 2x4x 2x24x 2 2 0 2x d 2x x3 2x4x x2 6x8 15 Put the following polynomial fractions into the form 11 2 1 b a 2s4s s28s7 2 1 d c s28s7 s27s12 l6 Completing the square convert the following polynomials into the sum of two squares ie to the form xa2b a x2 4x 8 b x2 5x 8 c x2 4x 8 d 4x2 16x 8 17 Given the three functions X y amp 2 determine the following by adding or multiplying graphs ya 12 zt t l 2 a X0W0 c X0W b Xtzt d Yt39zt Signals amp Transforms page 3 Math Review 18 Sketch the following a x30 tl f xfa equot b Xba 1t g Kga e c Ken 5 h Xha equot d ma M i x1ltrgt5 k 2 l I lgt 2 e Xea e 19 Simplify the following compound fractions to form a simple fraction containing the ratio of polynomials 2x 2x abx a 12x b 12x c abx 2x Jri 3x L1 abx l d l2x 3x l3x 2x abxg 20 What do you know about the parameters a amp b in the following graphs ie magnitude and sign xtat xtbeat t t a d xtatb xtbeat t t b e xtatb xtbeat t t 0 f 21 Look over your differential equations on how to solve linear constant coefficient differential equations with time domain methods These are the problems in the form of ay by Xt or ayquot by cy Xt 22 Look over using Laplace Transforms to solve linear constant coefficient differential equations like above Signals amp Transforms page 4 Math Review 23 Using equivalent resistances and voltage dividers relate Vin and Vout c d 25 Sketch the straight line approximation of the Bode plot for 1 10s 105 s 5 s2110s1000 s25100s500000 aHs b Hs 50mH IO V0uts Vins c H s where Other comments from past semester students 0 If you own an HP calculator make sure you can still do partial fractions by hand Note Matlab will be introduced in the lab EE350L Signals amp Transforms page 5 Math Review Oppenheim amp Willsky Chapter 1Overview Signals amp Systems Overview Text Chapter 1 11 12 xxxx C7me Examples of Continuous and Discrete time signals amp systems give motivation and example Introduction skip mathematics energy and power on their own Transformations of the Independent Variable 121 Mathematics cover with convolution or math review Classes of Signals cover with Ch 3 122 Periodic Signals 123 Even amp Odd Signals Exponential and Sinusoidal Signals skip Unit Impulse amp Unit Step Special Signals Continuous amp Discrete Time Systems skip Basic System Properties Learning Objectives for Chapter 1 Prerequisite knowledge nctions complex numbers After completion of Chapter 1 students should be able to give examples of signals and systems determine the even and odd components of a function review on your own write piecewise defined functions in terms of unit step functions determine stability causality memory timeinvariance and linearity given an inputoutput definition of a system Oppenheim amp Willsky Chapter 2 Overview LTI Systems Overview Text 21 25 Chapter 2 DiscreteTime LTI Systems The Convolution Sum give motivation Why we have the convolution sum amp integral significance of impulse response ContinuousTime LTI Systems The Convolution Integral mechanics of convolution integral Properties of LTI Systems 2313 7 Commutative Distributive Associative 23467 7 Memory Causality Stability 238 7 Unit Step Response amp unit impulse response Causal LTI Systems Difference amp Differential Equations 24l Linear ConstantCoefficient Differential Equations 242 Linear ConstantCoefficient Difference Equations 243 Block diagram Representations of FirstOrder Systems Described by Differential and Difference Equations Singularity Functions Unit Impulse revisited Learning Objectives for Chapter 2 Prerequisite knowled e 9 solvrng linear constant coefficient differential equations After completion of Chapter 2 students should be able to calculate the convolution of two continuous signals calculate the convolution of two discrete signals determine the system response by applying the sifting theorem and LTI properties determine the system response using LTI properties determine stability causality memory given an impulse response definition of a system determine the impulse response of a discrete system given the inputoutput difference equation determine the impulse response of a continuous system given the inputoutput differential equation After completion of laboratory assignments related to Chapter 2 students should be able to describe conceptually how the convolution integral works lab 2 or 3 explain how linearity causality and commutitivity apply to convolution lab 3 Oppenheim amp Willsky Chapter 3 Overview Fourier Series Overview Text Chapter 3 31 Historical Perspective 32 The response of LTI systems to Complex Exponentials 33 Fourier Series representation of continuoustime periodic signals 34 Convergence of the Fourier Series 35 Properties of ContinuousTime Fourier Series 3516 Linearity Time shifting Time reversal Time Scaling Multiplication Conjugation amp Conjugate Symmetry 357 7 Parseval 36 Fourier Series representation of discretetime Periodic Signals 37 Properties of DiscreteTime Fourier Series 38 Fourier Series and LTI Systems 39 Filtering 310 Examples of Continuoustime filters described by differential equations 311 Examples of Discretetime filters described by difference equations Ch 12 Classes of Signals cover with Ch 3 122 7 Periodic Signals 123 7 Even amp Odd Signals Learning Objectives for Chapter 3 After completion of Chapter 3 students should be able to describe what is the response of an LTI system to a complex exponential input describe how the response of an LTI system to a complex exponential input relates to the Fourier series determine the fundamental frequency of a sum of pure trigonometric functions calculate exponential Fourier series coefficients of a pure trigonometric signal using Euler s formula given a finite list of Fourier series coefficients determine the associated time function in terms of sine andor cosine functions calculate exponential Fourier series coefficients of a general periodic function using Fourier series integral calculate exponential Fourier series coefficients using properties of Fourier series graphically portray the magnitude and phase of the Fourier series coefficients versus and versus m define and describe the basic filter types LPF BPF HPF SBF Notch Filter calculate Hjoa by applying an input of xte quot to a linear constant coefficient differential equation Determine the response of a periodic signal to a system described by a linear constant coefficient differential equation After completion of laboratory assignments related to Chapter 3 students should be able to calculate exponential Fourier series coefficients using properties of Fourier series lab 5 describe how the Fourier series converges as k 9 lab 5 describe how yt is calculated based on Fourier series coefficients and Hjw lab 6 Oppenheim amp Willsky Chapter 4 Overview Fourier Transform Overview Text Chapter 4 41 Representation of Aperiodic Signals The ContinuousTime Fourier Transform 42 The Fourier Transform for Periodic Signals 43 Properties of the ContinuousTime Fourier Transform 4315 Linearity Time shifting Conjugation amp Conjugate Symmetry Differentiation and Integration Time and Frequency Scaling 436 Duality 437 7 Parseval 44 The Convolution Property 45 The Multiplication Property 46 Tables of Fourier Properties and of Basic Fourier Transform Pairs 47 Systems Characterized by Linear ConstantCoefficient Differential Equations Learning Objectives for Chapter 4 After completion of Chapter 4 students should be able to describe how and why Fourier Transforms and Fourier Series are related calculate Fourier Transforms and Inverse Fourier Transforms of signals using Fourier Integrals calculate Fourier Transforms and Inverse Fourier Transforms of signals using properties of Fourier Transforms calculate Fourier Transforms of periodic signals describe how Fourier Transforms of periodic signals and Fourier Series are related calculate Fourier Transforms and Inverse Fourier Transforms pairs using duality property calculate Frequency response and impulse response given linear constant coefficient differential equations given Xt and ht calculate yt and Yjoi using Fourier Transforms given Xt and yt calculate ht and H003 using Fourier Transforms given ht and yt calculate Xt and Xjoi using Fourier Transforms calculate the response of applying a given input signal to a system described by a linear constant coefficient differential equation apply partial fractions to finding Inverse Fourier Transforms After completion of laboratory assignments related to Chapter 4 students should be able to name some of the common filter forms lab 7 design physical circuits to implement filters of varying order lab 7 41 Oppenheim amp Willsky Chapter 5 Overview DiscreteTime Fourier Transform Overview Text Chapter 5 51 Representation of Aperiodic Signals The DiscreteTime Fourier Transform 52 The Fourier Transform for Periodic Signals 53 Properties of the DiscreteTime Fourier Transform 531 Periodicity of the DiscreteTime Fourier Transform 5328 7 Linearity Time shifting Conjugation amp Conjugate Symmetry Differentiation and Integration Time and Frequency Scaling 539 7 Parseval 54 The Convolution Property 55 The Multiplication Property 56 Tables of Fourier Properties and of Basic Fourier Transform Pairs 57 Duality 58 Systems Characterized by Linear ConstantCoefficient Difference Equations Learning Objectives for Chapter 5 After completion of Chapter 5 students should be able to apply all methods from Chapter 4 continuoustime FourierTransforms to discretetime Fourier Transforms 0 describe how and why continuoustime and discretetime Fourier Transforms are related 0 calculate Fourier Transforms and lnverse Fourier Transforms of signals using properties of Fourier Transforms tables 0 calculate Fourier Transforms and lnverse Fourier Transforms pairs using duality propert o calculate Frequency response and impulse response given linear constant coefficient difference equations given Xn and hn calculate yn and Ye using Fourier Transforms given Xn and yn calculate hn and Hel using Fourier Transforms given hn and yn calculate Xn and Xe using Fourier Transforms calculate the response of applying a given input signal to a system described by a linear constant coefficient difference equation 0000 After completion of laboratory assignments related to Chapter 5 students should be able to describe how the FFT and DFT are related 51 Oppenheim amp Willsky Chapter 6 Overview Time amp Freq Char of Sigs amp Sys Overview Text Chapter 6 61 MagnitudePhase Representation of the Fourier Transform 62 MagnitudePhase Representation of the Frequency Response of LTI Systems 63 Timedomain properties of Ideal Frequencyselective lters 64 Timedomain and FrequencyDomain aspects of nonideal filters 65 Firstorder and SecondOrder ContinuousTime Systems 66 Firstorder and SecondOrder DiscreteTime Systems 67 Examples of Time and FrequencyDomain Analysis of Systems Learning Objectives for Chapter 6 Prerequisite knowledge sketching straight line approximations to bode plots calculate steady state sinusoidal system response if Xt A 008w0t then yt HjoJA 0080J0t 4 4Hjoa understanding of ti meconstants in firstorder systems understanding of dampening ratios and undamped natural frequencies for secondorder systems After completion of Chapter 6 students should be able to determine whether a system represents a LPF HPF BPF or stop band filter Identify the passband transition band and stopband of a nonideal filter After completion of laboratory assignments related to Chapter 5 students should be able to describe some ofthe differences between Butterworth Chebschev l amp II and Elliptical filters design a filter to given specifications Draw circuits for general n h order transfer functions including secondorder sections 61 Oppenheim amp Willsky Chapter 7 Overview Sampling Overview Text Chapter 7 71 Representation of a ContinuousTime Signal by its Samples The sampling Theorem 711 Impulse train sampling 712 Sampling with a zeroOrder Hold 72 Reconstruction of a Signal from Its Samples Using Interpolation 73 The Effect of Undersampling Aliasing 74 DiscreteTime Processing of ContinuousTime Signals 75 Sampling of DiscreteTime Signals Learning Objectives for Chapter 7 Prerequisite knowledge application of sifting property ch 2 After completion of Chapter 7 students should be able to determine for a given signal the Nyquist Frequency determine the spectrum of an impulse sampled signal determine the reconstructed signal and spectrum Define the terms Nyquist Rate Nyquist Frequency and aliasing sketch the impulse response for zeroorder hold and first order hold reconstruction filters Discuss the practical implementation issues of reconstruction filters After completion of laboratory assignments related to Chapter 7 students should be able to describe how undersamping and aliasing affect a signal in the time and frequency domain lab 8 71 Oppenheim amp Willsky Chapter 8 Overview Communication Systems Overview Text Chapter 8 81 Complex Exponential and Sinusoidal Amplitude Modulation 82 Demodulation for Sinusoidal AM 83 FrequencyDivision Multiplexing 84 SingleSideband Sinusoidal Amplitude Modulation 85 Amplitude Modulation with a pulseTrain Carrier 852 Timedivision Multiplexing 86 PulseAmplitude Modulation 87 Sinusoidal Frequency Modulation 88 DiscreteTime Modulation Learning Objectives for Chapter 8 Prerequisite knowledge application of sifting property ch 2 After completion of Chapter 8 students should be able to Convert equations describing modulation and filtering steps into flow diagrams determine spectrum for modulated signals and all signals on a flow diagram Describe and draw the flow diagram for modulation amp synchronous demodulation Describe and draw the flow diagram for modulation amp asynchronous demodulation Calculate constraints on signal spectra to be able to reconstruct a modulated signal calculate the modulation index Draw a flowdiagram for time and frequency division multiplexing After completion of laboratory assignments related to Chapter 7 students should be able to describe how modulation affects a signal in the time and frequency domain lab 9 describe how to do asynchronous demodulation lab 9 81 Oppenheim amp Willsky Chapter 9 Overview Laplace Transforms Overview Text Chapter 9 91 98 99 The Laplace Transform The Region of Convergence for Laplace Transforms The Inverse Laplace Transform Geometric Evaluation of the Fourier Transform from the PoleZero Plot Properties of the Laplace Transform 9510 7 Initial and final value theorem EEZZS Some Laplace Transform 39 Analysis and Characterization of LTI Systems Using the Laplace Transform 975 7 Butterworth Filters System Function Algebra and Block Diagram EE360 The Unilateral Laplace Transform EE225 Learning Objectives for Chapter 9 Prerequisite knowledge calculate Inverse Laplace Transforms using properties and partial fractions EE225 causality and stability from impulse response Ch 3 After completion of Chapter 9 students should be able to describe how and why Fourier Transforms and Laplace Transforms are related describe how Unilateral and Bilateral Laplace Transforms are related Find ROCs for timedomain functions tables amp integral Given Laplace transform and R00 find the proper inverse Laplace Transform Determine the stability and causality of a system based on R00 and relate this to the corresponding impulse response Sketch the approximate Magnitude and phase ofthe FourierTransform based on pole amp zero locations given the Laplace Transform of a function calculate initial and final values by both inversion and IVT amp FVT After completion of laboratory assignments related to Chapter 9 students should be able to Sketch the approximate magnitude and phase ofthe FourierTransform based on pole amp zero locations lab 10 91 Oppenheim amp Willsky Chapter 10 Overview zTransforms Overview Text Chapter 10 101 102 103 104 105 106 107 108 109 The z Transform The Region of Convergence for the zTransform The Inverse 1 Trans orm Geometric Evaluation of the Fourier Transform from the PoleZero Plot Properties of the z Transform 1059 7 Initial value theorem Some Common 1 Transform pairs Analysis and Characterization of LTI Systems Using the z Transform System Function Algebra and Block Diagram The Unilateral z Transform Learning Objectives for Chapter 10 Prerequisite knowledge partial fractions long division calculate impulse response from inputoutput difference equation After completion of Chapter 10 students should be able to describe how and why discrete Fourier Transforms and zTransforms are related describe how and why zTransforms and Laplace Transforms are related Find ROCs for timedomain functions tables Given ztransform and R00 find the proper inverse z Transform by partial fractions and tables Given ztransform and R00 find the proper inverse zTransform by long division and tables Determine the stability and causality of a system based on R00 and relate this to the corresponding impulse response Sketch the approximate Magnitude and phase ofthe discrete Fourier Transform based on pole amp zero locations given the zTransform of a function calculate initial and final values by both inversion and NT amp FVT Draw block diagrams for discretesystems delay units After completion of laboratory assignments related to Chapter 10 students should be able to Sketch the approximate magnitude and phase ofthe discrete Fourier Transform based on pole amp zero locations lab 11 convert a continuous time system into discrete time by Euler s method and the bilinear transform 101
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