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problem_set7

# problem_set7 - EE 511 Problem Set 7 Due on 16 Nov 2007 1...

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EE 511 Problem Set 7 Due on 16 Nov 2007 1. Let ˆ X t be the Hilbert transform of the W.S.S. random process X t . Show that (i) S ˆ X ( f ) = S X ( f ), (ii) S X ˆ X ( f ) = - S ˆ XX ( f ), (iii) E [ X t ˆ X t ] = 0, and (iv) for Z t = X t + j ˆ X t , determine S Z ( f ) in terms of S X ( f ). 2. The power spectrum of a W.S.S. band-pass random process X t is as shown in Figure 1. Sketch S X I ( f ), S X Q ( f ) and S X I X Q ( f ) assuming 5 MHz to be the carrier frequency. S (f) X f(MHz) 4 5 7 -7 -5 -4 1 Figure 1: 3. Repeat problem 2 assuming 4 MHz to be the carrier frequency. 4. A narrow-band noise process N t has zero-mean and auto-correlation function R N ( τ ). Its power spectral density S N ( f ) is centered about ± f c . The in-phase and quadrature compo- nents N I t and N Q t are defined by N I t = N t cos 2 πf c t + ˆ N t sin 2 πf c t and N Q t = ˆ N t cos 2 πf c t - N t sin 2 πf c t . Show that R N I ( τ ) = R N Q ( τ ) = R N ( τ ) cos 2 πf c τ + ˆ R N ( τ ) sin 2 πf c τ and R N I N Q ( τ ) = - R N Q N I ( τ ) = R N ( τ ) sin 2 πf c τ - ˆ R N ( τ ) cos 2 πf c τ , where ˆ R N ( τ

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