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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 f a t + sin 2 f b t ) (a) Were 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 i ( t ) is varied linearly with the message, thus this is described in time domain by s ( t ) = A cos(2 f c t + k p m ( t )) where ( t ) = 2 f c t + k p m ( t ) Using the above equations, we see that ( t ) = 2 f a t + sin 2 f b t Hence, f c = f a k p = m ( t ) = sin(2 f b t ) with f m = f b (b) Now, imagine that youre 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 f c t + 2 k f Z t m ( ) d where the instantenous frequency is defined as f i ( t ) = fc + k f m ( t ) = 1 2 d i ( t ) dt 1 Hence, given our signal s ( t ) we find that i ( t ) = 2 f a t + sin(2 f b t ) and d i ( t ) dt = 2 f a + cos(2 f b t )(2 f b ) Therefore, f i ( t ) = 1 2 d i ( t ) dt = f a + f b cos(2 f b t ) f c = f a k f = f b m ( t ) = cos(2 f 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 f a t ) cos(2 f b t ) (a) Find the corresponding phase modulated and frequency modulated signals. SOLUTION: For a PM signal, we have s ( t ) = A cos(2 f c t + k p m ( t )) = A cos(2 f c t + k p A m cos(2 f a t ) cos(2 f b t )) = A cos 2 f c t + k p A m 2 [cos(2 ( f a f b ) t ) + cos(2 ( f a + f b ) t )] For a FM signal, we have s ( t ) = A cos 2 f c t + 2 k f R t m ( ) d = A cos 2 f c t + k f A m R t [cos(2 ( f a f b ) ) + cos(2 ( f a + f b ) )] d = A cos 2 f c t + k f A m 2 h sin(2 ( f a f b ) t ) ( f a f b ) + sin(2 ( 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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 Spring '08
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