tut3_solutions_07_02_09

tut3_solutions_07_02_09 - Department of Electrical...

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Department of Electrical Engineering EE 308: Communication Systems (Spring 2009) Course Instructor: Prof. Abhay Karandikar Solutions to Problem Set 1 1. Let x c ( t ) = A c cos [2 πf c t + θ ( t )] and x i ( t ) = A i cos [2 πf c t + θ i ( t )] be any two bandpass signals with ρ = A i A c . (a) Derive expressions for the amplitude and phase of r ( t ) = x c ( t ) + x i ( t ). (b) What is the output when r ( t ) is applied to an ideal i) amplitude ii) phase and iii) frequency demodulator? (c) Repeat the above for ρ << 1 , θ ( t ) = 0. (d) Comment on your results when θ i ( t ) = 2 πf i t + φ ( t ), where φ ( t ) is the message component of a PM or FM signal. Solution: (a) Representing r ( t ) = A r ( t ) cos [2 πf c t + θ r ( t )], we obtain, after some algebra A r ( t ) = A c radicalbig 1 + ρ 2 + 2 ρ cos[ θ i ( t ) - θ ( t )] θ r ( t ) = tan 1 parenleftbigg sin[ θ ( t )] + ρ sin[ θ i ( t )] cos[ θ ( t )] + ρ cos[ θ i ( t )] parenrightbigg (1) (b) The output is A r ( t ) for the ideal envelope detector. For the PM demodulator, the output is θ r ( t ) and for the ideal FM demodulator, the output is θ r ( t ) 2 π .
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