Prob1 - ECE 1520 Data Communications Fall 2011 Problem Set...

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ECE 1520 Data Communications Fall 2011 Problem Set 1 Due in class, Wednesday, Sept. 28th, 2011. In this problem set, ˆ x ( t ) denotes the Hilbert transform of x ( t ) and x l ( t ) denotes the low-pass equivalent of a real bandpass signal x ( t ). [ · ] denotes the real part of a complex number. 1. Show that if x 1 ( t ) = x ( t ), ˆ x 1 ( t ) = ˆ x ( t ). 2. Show that if x ( t ) = cos( ω 0 t ), ˆ x ( t ) = sin( ω 0 t ). 3. Show that i -∞ x ( t x ( t ) dt = 0 . Note that a x ( t ) ,y ( t ) A = I -∞ x ( t ) y ( t ) dt is a common inner product of two real signals x ( t ) and y ( t ). In this deFnition of inner product, therefore, x ( t ) and ˆ x ( t ) are orthogonal. 4. Show that if x ( t ) and y ( t ) are bandpass, i -∞ x ( t ) y ( t ) dt = 1 2 bi -∞ x l ( t ) y * l ( t ) dt B . Use this result to show that
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Prob1 - ECE 1520 Data Communications Fall 2011 Problem Set...

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