Last Modified: 10/03/2022 ELECTENG 733: Signal Processing Assignment-1 Cover Sheet Available from 11 March to 21 March. All answers are to be submitted before 11:59 p.m. on 21 March INSTRUCTIONS Assignment total marks: 70 Assessment Grading: This assignment contributes 10% towards your final course mark. To get full credit, you must comprehensively answer all questions correctly. There are SIX questions in total. Attempt all questions. You must show all your working to receive full credit for each solution. If you believe you need further information than that provided, make the appropriate engineering assumption(s), state them clearly, and continue with your answer. Prepare your hand-written solutions in PDF format, and submit your completed work, with this cover sheet, electronically via Canvas before the deadline. Late submissions will not be accepted. One day delay allowed, but 30% of the total mark will be deducted. The submitted pages must be of A4 size. Begin each question on a new page. It is the responsibility of students to ensure that their names and ID numbers are correctly depicted, and the cover sheet is signed. ASSESSMENT LOGISTICS • The assessment must be taken completely alone. Showing or sharing it with anyone is considered copying and lead to zero mark in the assignment for all contributed parties. • You may not consult, collaborate, or discuss, with any other person regarding the materials relating to, or directly from, the assessment. Anything you submit must be your own unaided work. • Any questions and/or clarifications about the assessment can only be sent to Waleed via the e-mail (
[email protected]). DO NOT use Piazza for any query regarding the assessment. Failure to comply with any of the rules above may result in voiding your submission. I hereby accept all of the rules and instructions in this cover page, and declare that I will NOT discuss or otherwise communicate in any way, or via any channel, with any individual within or outside the university regarding the content of this assessment before the assignment submission due date. Family Name: Office Use Only Given Name: Q1: Q4: Username: Q2: AUID: Q3: Total Mark: Signature Lecturer: Waleed Abdulla Q5: Q6: Student Name: Student ID: 2 1. Let x(t) be the sum of sinusoidal signals formulated by: ( ) 4 3cos( ) 2cos(2 ) cos(3 )x t t t tπ π π= + + + Where t is the time in milliseconds. (a) Determine the minimum sampling rate that will not cause any aliasing effects, that is, the Nyquist rate. (2 marks) (b) Find the digital representation, x[n], of x(t) if it is sampled at 2 KHz. (3 marks) (c) Reconstruct the analog representation xa(t) from the signal x[n] in (b). (5 marks) (d) Using MATLAB, plot x(t) and x[n] superimposed on each other, locating the sampling points on the graph. Depict the MATLAB program and the resulting graph. On the graph, show its title and the labels of the axes. You may suggest a proper title name for the figure. (10 marks) Note: Upload the MATLAB program as a separate *.m file with your assignment so we can run it. Student Name: Student ID: 3 2. Find the difference equation that formulates the input-output relation of the following system in its simplest form. Determine the output [ ]g n if its input is [ ] [ ]v n u n= , where [ ]u n is a step function. The system is relaxed at n = 0, which implies that [ 1] 0g − = . Plot the output signal [ ]g n . Solution: (10 marks) Student Name: Student ID: 4 3. Determine mathematically the fundamental period of the following periodic sequence 2 3[ ] 4 cos cos 6 4 2 x n n nπ π π = + + + (10 marks) Solution: Student Name: Student ID: 5 4. Determine the average power and the energy of the following sequence [ ] cos [ ] 4 x n n u nπ = , Where u[n] is a unit step function. Indicate whether the sequence is a power or an energy signal. (10 marks) Solution: Student Name: Student ID: 6 5. For the following discrete-time system: [ ] ( 1) [2 ]y n n x n= + where [ ] and [ ]x n y n are the input and output sequences respectively, determine with proofs whether or not the system is: (1) Linear (2) Causal (3) Shift-invariant (4) BIBO stable. (10 marks) Solution: Student Name: Student ID: 7 −12 -10 −8 −6 −4 −2 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 1.4 k 6. Plot the shifted and scaled function w[k] of the following [ ]x k signal. [ ] [ 15 3 ] [ 4]w k x k u k= − − + , where u[k] is a unit step function. Show in your solution steps 1 [ 15 ]w x k= − + , 2 [ 15 3 ]w x k= − + , 3 [ 15 3 ]w x k= − − , and finally [ ] [ 15 3 ] [ 4]w k x k u k= − − + . (10 marks) Solution: [ ]x k
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