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finals03 - Sample Final Exam Covering Chapters...

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Sample Final Exam Covering Chapters 10-17 (finals03) 1 Sample Final Exam ( finals03) Covering Chapters 10-17 of Fundamentals of Signals & Systems Problem 1 (25 marks) Consider the discrete-time system shown below, where N represents decimation by N . This system transmits a signal [ ] x n coming in at 1000 samples/s over two low-bit-rate channels. Numerical values: lowpass filters' cutoff frequencies 1 2 2 clp clp π ω ω = = , highpass filter's cutoff frequency 2 chp π ω = , signal's spectrum over [ , ] π π : 4 3 1 , 3 4 ( ) 3 0, 4 j X e ω π ω ω π π ω π = < < . (a) [7 marks] Sketch the spectra ( ) j X e ω , 1 ( ) j X e ω , 2 ( ) j X e ω , 2 ( ) j W e ω , indicating the important frequencies and magnitudes. ( ) j hp H e ω ( ) j lp H e ω 1 N 2 N Receiver [ ] x n 1 [ ] x n 2 [ ] x n 1 [ ] v n 2 [ ] v n [ ] y n channels ω −π 1 0.75π π −0.75π ( ) j X e ω ω −π 1 0.5π π −0.5π 1/3 1 ( ) j X e ω cos( ) 2 n π × 2 [ ] w n
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Sample Final Exam Covering Chapters 10-17 (finals03) 2 (b) [10 marks] Let the decimation factors be 1 2 N = and 2 4 N = . Sketch the corresponding spectra 1 ( ) j V e ω , 2 ( ) j X e ω , 2 ( ) j V e ω , indicating the important frequencies and magnitudes. Assuming for the moment that 1 2 [ ], [ ] v n v n are quantized using 16-bit quantizers, find the bit rate of each channel, and the total bit rate. How would this compare to the bit rate for of a direct transmission of [ ] x n using a 16-bit quantizer? Answer: After sampling in decimation operation: With 1 2 N = , the first channel transmits at a bit rate of: 1000 samples/s 16bits/sample = 8000 bits/s 2 × And with 2 4 N = , the second channel transmits at a bit rate of: ω −π 1 0.5π π 0.75π −0.5π −0.75π 1/3 2 ( ) j X e ω ω −π 1/2 0.5π π −0.5π 1/6 1 ( ) j V e ω ω −π π 1/12 0.5π −0.5π 2 ( ) j V e ω 2 ( ) j p W e ω ω −π 0.25π π 0.75π −0.25π −0.75π 1/6 2 ( ) j W e ω ω −π 0.25π π 0.75π −0.25π −0.75π 1/12
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Sample Final Exam Covering Chapters 10-17 (finals03) 3 1000 samples/s 16bits/sample = 4000 bits/s 4 × Thus, the total bit rate is 12000 bits/s . A direct transmission of the signal would require a bit rate of 1000samples/s 16bits/sample = 16000 bits/s × (c) [8 marks] Design the receiver system (draw its block diagram) such that [ ] [ ] y n x n = (assume that there is no quantization of the signals.) You can use upsamplers (symbol { } lp N , with embedded ideal lowpass filters of cutoff frequency N π and gain N ), synchronous demodulators, ideal filters and summing junctions. Sketch the spectra of all signals in your receiver. Answer: + { } 1 lp N { } 2 lp N 2 [ ] k n [ ] y n 2 [ ] x n + 1 [ ] v n 2 [ ] v n ω −π 2 0.5π π −0.5π 0.75π −0.75π 1 [ ] x n cos( ) 2 n π × 2 ( ) j K e ω ω −π 0.25π π 0.75π −0.25π −0.75π 1/3 ω −π 1 0.5π π −0.5π 1/3 1 ( ) j X e ω 2 ( ) j Q e ω ω −π 0.25π π 0.75π −0.25π −0.75π 1/6 2 [ ] q n
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Sample Final Exam Covering Chapters 10-17 (finals03) 4 Problem 2 (20 marks) You are the engineer in charge of the design of a rocket's guidance control system so that the rocket can track a desired pitch angle trajectory α des t ( ) in a vertical plane during the take-off phase. The transfer function from the rocket's thrust vector angle command with respect to its longitudinal axis, call it θ ( ) t
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