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

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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 0 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( 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( c t ) , Re { r l ( t ) exp( 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
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(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( c t ) } and g ( t ) = Re { g l ( t ) exp( c t ) } . (h) Using the result of (e), show that r ( t ) = ( v ( t ) * h ( t )) cos( ω c t + θ + φ ). Note the following: The input bandpass signal, the bandpass filter, and the filter output can be represented by their complex envelopes. The complex envelope of the filter output is the convolution of the complex envelope of the input signal and the complex envelope of the filter scaled by 1 / 2.
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