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Unformatted text preview: THE STATE UNIVERSITY OF NEW JERSEY RUTGERS College of Engineering Department of Electrical and Computer Engineering 332:322 Principles of Communications Systems Spring 2004 Problem Set 3 1. Discovered Angle Modulation A signal s ( t ) is measured and found to be described by s ( t ) = A cos(2 πf a t + α sin 2 πf b t ) (a) We’re later told that s ( t ) is an angle modulated signal with sensitivity k p . Using the standard angle modulation signal format found in your text, what is the information signal m ( t ) ? SOLUTION: A phasemodulated signal s ( t ) is a form of angle modulation in where the angle θ i ( t ) is varied linearly with the message, thus this is described in time domain by s ( t ) = A cos(2 πf c t + k p m ( t )) where θ ( t ) = 2 πf c t + k p m ( t ) Using the above equations, we see that θ ( t ) = 2 πf a t + α sin 2 πf b t Hence, f c = f a k p = α m ( t ) = sin(2 πf b t ) with f m = f b (b) Now, imagine that you’re told “WHOOPS! I meant FREQUENCY modulation. with frequency sensitivity k f .” Again using the standard signal format described in your text, please provide the information signal m ( t ) and the instantaneous frequency f i ( t ) . SOLUTION: A frequencymodulated signal s ( t ) is a form of angle modulation in which the instantenous frequency f i ( t ) is varied linearly with the message signal m ( t ) , and it is given by s ( t ) = A cos 2 πf c t + 2 πk f Z t m ( τ ) dτ where the instantenous frequency is defined as f i ( t ) = fc + k f m ( t ) = 1 2 π dθ i ( t ) dt 1 Hence, given our signal s ( t ) we find that θ i ( t ) = 2 πf a t + α sin(2 πf b t ) and dθ i ( t ) dt = 2 πf a + α cos(2 πf b t )(2 πf b ) Therefore, f i ( t ) = 1 2 π dθ i ( t ) dt = f a + αf b cos(2 πf b t ) f c = f a k f = αf b m ( t ) = cos(2 πf b t ) 2. Two Tone Madness: Consider a message signal with two tones at frequencies f a and f b respectively, defined as m ( t ) = A m cos(2 πf a t ) cos(2 πf b t ) (a) Find the corresponding phase modulated and frequency modulated signals. SOLUTION: For a PM signal, we have s ( t ) = A cos(2 πf c t + k p m ( t )) = A cos(2 πf c t + k p A m cos(2 πf a t ) cos(2 πf b t )) = A cos 2 πf c t + k p A m 2 [cos(2 π ( f a f b ) t ) + cos(2 π ( f a + f b ) t )] For a FM signal, we have s ( t ) = A cos 2 πf c t + 2 πk f R t m ( τ ) dτ = A cos 2 πf c t + πk f A m R t [cos(2 π ( f a f b ) τ ) + cos(2 π ( f a + f b ) τ )] dτ = A cos 2 πf c t + k f A m 2 h sin(2 π ( f a f b ) t ) ( f a f b ) + sin(2 π ( f a + f b ) t ) ( f a + f b ) i (b) Find the narrowband FM (i.e NBFM) signal using the FM modulated signal obtained(b) Find the narrowband FM (i....
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This homework help was uploaded on 04/03/2008 for the course ECE ECE332 taught by Professor Rose during the Spring '08 term at Rutgers.
 Spring '08
 Rose

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