Philadelphia Uiversity Studet Name: Faculty of Egieerig Studet Number: Dept. of Computer Egieerig Fial Exam, First Semester: 2014/2015 Course Title: Digital Sigal Aalysis ad Processig Date: 01/02/2015 Course No: 630340 Time Allowed: 2 Hours Lecturer: Dr. Qadri Hamarsheh No. Of Pages: 7 Iformatio for cadidates 1. This examiatio paper cotais 7 questios totalig (40) marks 2. The marks for parts of questios are show i roud brackets. Advices to cadidates 1. You should attempt all questios. 2. You should write your aswers clearly. Basic Notios: The aim of the questios i this part is to evaluate the required miimal studet kowledge ad skills. Aswers i the pass category represet the miimum uderstadig of basic cocepts: Mathematical Models of LTI cotiuous ad discrete systems, Covolutio, Liear Costat-Coefficiet Differece Equatio, trasform ad their computatio methods ad represetatios. Questio1 Multiple choices (circle the most appropriate oe): (10 marks) 1. The fudametal period of the siusoidal sigal x[ 5cos[0.2 is a) 10 b) 5 c) 0.2 d) 2 2 2. The system y(t ) 4x (t ) 2x(t ) is a) Liear b) Noliear c) Not eough iformatio has bee provided to determie the liearity of the system 3. Which of the followig is correct? 3t2 a) e u(t 2 ) ( t 3 )dt 1 b) 3t2 e u(t 2 ) ( t 3 )dt e 3t2 c) e u(t 2 ) ( t 3 )dt 0 d) Noe of the above 4. The step respose of a system ca be writte as 7 a) s ( t ) u( t )* h( t ) u( )h( t ) d b) s ( t ) u( t )* h( t ) ( t )h( t ) d c) s(t ) u(t )h(t ) d) Noe of above 1
2t 5. The liear time-ivariat system with h(t ) 4e u(t ) a) Not stable b) Stable, causal, but has memory c) Stable, causal, ad memeoryless d) Stable, but ot causal 6. The first oero value of a fiite-legth sequece x [ occurs at idex 5 ad has a value x[ 5] 4. What is the idex of the first oero value i the covolutio y[ x[ * x[ ad what is its value? a) The idex is 0, ad the value is 8 b) The idex is 10, ad the value is 16 c) The idex is 0, ad the value is 16 d) The idex is 5, ad the value is 8 7. Covolutio i the time domai correspods to a) Multiplicatio i the frequecy domai b) Time-shifted covolutio i the frequecy domai c) Scaled covolutio i the frequecy domai d) Multiplicatio i the frequecy domai, with the secod sigal shifted i frequecy 8. The Z -trasform of a discrete time sigal is give by a) X [ ] x[ b) X [ ] x[ c) X [ ] x[ d) X [ ] x[ 9. The Z -trasform of is a) 1 b) c) d) 2 1 1 2 1 1 3 10. The Z -trasform ad the ROC of the fiite-duratio sigal x[ 3, 2, 5,7,0,1 a) b) c) d) X [ ] X [ ] X [ ] X [ ] 1 3 3 2 5 7, ROC: etire 1 3 3 2 5 7, ROC: etire 1 3 3 2 5 7, ROC: etire 1 3 3 2 5 7, ROC: etire plae except plae except 0 plae except 0 ad plae 2
Questio 2 (6 marks) a) State the methods to fid iverse Z trasform. (2 marks) b) Explai the followig topics about the Recursive DT system (IIR filter) with first order differece equatio described by the followig iput-output relatioship: (4 marks) y[ ay[ 1] x[. Block diagram realiatios. Zero-state Respose ad Zero-iput Respose (equatios ad how to derive them). 3
3 Questio 3 The system ( t) t ( dx / dt) memoryless, stable, causal ad liear (explai). y have iput x (t) ad output y (t). Determie whether it is (4 marks) Familiar ad Ufamiliar Problems Solvig: The aim of the questios i this part is to evaluate that the studet has some basic kowledge of the key aspects of the lecture material ad ca attempt to solve familiar ad ufamiliar problems of Mathematical Models of LTI discrete systems, Covolutio, Liear Costat-Coefficiet Differece Equatio, trasform, their computatio methods, represetatios ad programmig usig Matlab. Questio 4 Derived a closed-form expressio (aalytically) for the covolutio of x [ ad h [ where 1 x[ 6 ad 6 u[ 1 h[ u[ 3] 3 (4 marks) 4
Questio 5 For the Recursive Filter (LTI system) described by the followig LCCDE equatio: y [ 5x[ 3.5x[ 1] 0.5x[ 4] 0.5 y[ 1] 0.3 y[ 3] 2y[ 5] (6 marks) do the followig: a) Draw the block diagram represetatio of the Direct Form I ad Direct Form II realiatios (determie the eeded equatios for Direct Form I ad Direct Form II realiatios). (4 marks) b) Use the MATLAB M-files imp ad step to write a Matlab program that calculates the first 15 samples of the impulse ad step respose of this filter ad plots the impulse ad step respose. (2 marks) 5
Questio 6 Suppose that a sequece x [ has a trasform (4 marks) with a regio of covergece (ROC) 7 1 1 2 4 X [ ] 4 4 3 1 1 2 1 4 8 1, Fid the sequece [ 2 x. 6
Questio 7 (6 marks) Evaluate the covolutio, y[ x[ * h[, of the two sequeces h[ 0.5 u[ ad x[ 3 u[ usig trasform. Wheever coveiet, use the properties of the trasform to make the solutio easier (see table below). GOOD LUCK 7