Eam 1 Information 332: 345 Fall 2003 Eam 1 results are posted on WebCT. The average is 17.5/35. Eam 1 solutions are on reserve reading in SERC Library. Eam 1 books are in Professor Gajic s office, ELE 222. Students may look at and pick up eams during Professor Gajic s office hours, M3, Th3. Once the eam book is taken out from Professor Gajic s office, no complains about grading will be considered. GRADING POLICY: The total number of points for an A will be somewhere in between 80 and 90, to be decided when all three eams are completed. Hence, 90 points and above guarantees an A. The total number of point for a D is somewhere in between 40 and 50, to be decided when all three eams are completed. Hence, 50 points guarantees a D. Other grades will be evenly distributed beetwen D and A. 1
Eam I will be based on the material from Chapters 1, 2, and 3 covered in class and outlined in the course syllabus. Closed book and notes. No calculators allowed. Tables 3.3 and 3.4 will be distributed to students before the eam. Study Guide: Do all homework problems and read the study guides given in chapter summaries. No proofs of the Fourier transform properties. Eam Time: Monday Oct 27, 2003; 8:10 9:30am (during regular class hours) Place: SEC 111, A-L (115 students) SEC 210 M-R (52 students) SEC 117, S-Z (57 students) Attachments: Table 3.3 Table 3.4 Sample Eam I, Fall 1999 (This year Eam 1 will also include some minor and simple problmes from Chapter 1 similar to Homework Problems assigned from Chapter 1, dealing with linearity, time invariance, and system response. Point distribution for Eam 1: (35pointe = 35% of the course grade) Chapter 1: 6 points (6%) Chapter 2: 12 points (12%) Chapter 3: 17 points (17%) SERC Library: Copies of all solutions to homework problems from Chapters 1, 2, and 3 and Eam 1 Information are also on reserve reading in the SERC Library. 2
o / Time Frequency 1!" 2 $#% '&( )!*,+.-,/10 2 34 3 657 8 9:8 <; +=- 9?> 2&3 57 #@ & 8 9:8 ) *7+ BACD /E0 ; +=- 9 > 4 GF!, FIHJ H - J 2 E4K 5. )+=- 0L/ E MN#O$&( 6 7 H J H / J L.7P(Q R M &. SR.TVUW M$&.. YX Z E M\[]$&^[]2 _ MO#O$& =G` +YX Z E M\[] & a#o2 _ MO#O & =B` 3! F @ 3! F H H - J 7a # = J bdc +=-fe 8 hgi 34B 34 9. _j L. k 34gK!" 10 =#l 2.#!" 11 2 3. m!nh =#" 11a 2.# S m!nh ' 12 6qrs,q +=- 2 E4Kt[un2 34vw7 Parseval s Theorem p o. s =k p o!" s! Table 3.3: Properties of the Fourier transform 3
Á À Time Frequency 1 y,z6{ } 2 ~! = ƒh z{. ˆ Š Œ rž 3 4 z{ z{,. V Y š?œ. _ 5 }! y,zl ž Ÿ š 6 (ªj _ B«(ªj _.«l! y7zm 7 (ª! z { ä y7zm ²±] ±y7z %³O 8. zm { µ ä y,z ±] $³ y,zm O³ B 9 j z{ 10 ƒ_ z{. Œ Ž= Ž= ± y7z ~ ¹ ~ 11 ¹fº º %»2ˆ¼Š ¹!½G Y½ 12 Ž= ¾! y,zm O³O 13 À Á ²Ã Á ~ÄŽ ¾ $ œ =š Å! ²Ã Á y,zm ³%Æh 14 À À y7zl{$³%æhçt ( $ y,zm O³ Æh $ $ œ š Å ¾ Table 3.4: Common Fourier transform pairs 4
ý Í û ë ò ï ï î Ö ñ Í ý Î 330:345 Linear Systems and Signals Eam 1 Fall 1999 #1a) Using the properties of the impulse delta function simplify the following epressions LÉ' 'Ô7 ÉÉG Û.Ô, G Ë! ÍÎ!ÏÐVÑ Òh Õ ÖY ÙØ Ë!hÍÎ:Ï.ÐÑ Òh Õ%Ör ÞØ Ô7 LÉÉÉGàÛ LÉ6áY ãåä.ô,. Õ%Ör ÞØ Ë!hÍÎjÏ.ÐÑ Òh Í Ë!hÍ.ÎjÏ.ÐÑ Òh Õ%Ör ÞØ èç Gë #1b) Find the generalized derivative of the signal Ë hí.îjé ê Ï Òh Õ ì #1c) Plot the graph èçíë of the signal represented ï in ïë terms of unit step and unit ramp signals as Õ]î $Õuì $Õ%ð $Õ@Ø èç ïuñ! ç #2a) Find the Fourier series of a periodic signal represented by ì ÕóìõôuöôuÖ Õ ì Öøô öôùì Lühý ç òoþ ü\ç çùñ #2b) Consider a periodic signal whose ü\ç Fourier series coefficients are given by úû ÿ ì ý" ç Gñ ï ý" Ö This signal is the input to a linear time invariant system whose transfer function is given by. Find the output signal and sketch its line spectra (magnitude and phase). #3a) Using the tables of common pairs and properties of the Fourier transform find Fourier ÉG transforms of the ÉÉG following signals: MÉÉ6ÉG GË4 Í Î Ë!hÍÎ MÉG ý LÉÉG ý #3b) Find the inverse Fourier transform of the signals é(ê Ï ì 5