07hw04 - EE544 Homework Assignment 4 1(Proakis problem 2.18...

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Unformatted text preview: EE544 Homework Assignment # 4 1. (Proakis problem 2.18) Use the Chernoff bound to show that Q ( x ) ≤ e- x 2 / 2 where Q ( x ) is defined by Q ( x ) = 1 √ 2 π Z ∞ x e- t 2 / 2 dt, x ≥ 2. A real bandpass (BP) signal σ ( t ) is filtered by a real linear time-invariant (LTI) BP filter g ( t ). Let the output signal, which is real , be r ( t ). Define σ l ( t ), g l ( t ), and r l ( t ) be the complex envelopes of σ ( t ), g ( t ), and r ( t ), respectively, with respect to the angular frequency ω c 1. Also define the real and imaginary components of σ l ( t ), g l ( t ), and r l ( t ) in the following way: σ l ( t ) = σ I ( t ) + j σ Q ( t ) , g l ( t ) = g I ( t ) + j g Q ( t ) , r l ( t ) = r I ( t ) + j r Q ( t ) . (a) Draw the system block diagram with the labels σ ( t ) ,g ( t ) and r ( t ). (b) Write the real signals σ ( t ), g ( t ), and r ( t ) in terms of σ l ( t ), g l ( t ), r l ( t ), exp( jω c t ), and the operator Re {·} . (c) Write the convolution operation r ( t ) = σ ( t ) * g ( t ) in terms of real signals σ I ( t ) ,σ Q ( t ) ,g I ( t ), g Q ( t ) , cos( ω c t ) and sin( ω c t ). (d) Write the convolution operation r l ( t ) = σ l ( t ) * 1 2 g l ( t ) of two complex signals in terms of their real and imaginary components. (e) Based on (c) and (d), show that r ( t ) = Re ‰ σ l ( t ) * 1 2 g l ( t ) ¶ exp( jω c t ) , Re { r l ( t )exp( jω c t ) } . (f) Draw the complex equivalent system block diagram with the labels σ l ( t ) , 1 2 g l ( t ) and r l ( t ). Note that, for the impulse response of the complex equivalent system, we should use one-half of the complex envelope of the real impulse response. Now, suppose σ ( t ) = v ( t )cos( ω c t + θ ) and g ( t ) = 2 h ( t )cos( ω c t + φ ) , where v ( t ) and h ( t ) are real lowpass signals. 1 (g) Find the complex envelope of σ ( t ) and g ( t ), i.e., find σ l ( t ) and g l ( t ) such that σ ( t ) = Re { σ l ( t )exp( jω c t ) } and g ( t ) = Re { g l ( t )exp( jω c t ) } ....
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This note was uploaded on 02/13/2012 for the course ECE 544 taught by Professor Jameslehnert during the Spring '12 term at Purdue.

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07hw04 - EE544 Homework Assignment 4 1(Proakis problem 2.18...

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