Hw9Soln_1 - ECE 5670 : Digital Communications ECE...

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Unformatted text preview: ECE 5670 : Digital Communications ECE department, Cornell University, Spring 2011 Homework 9 Solutions Instructor: Salman Avestimehr Office 325 Rhodes Hall 1. Effects that make the tap gains vary with time: • Variation of the phase of each path: T c = f c v c . The coherence time depends on the carrier frequency and the speed of the nodes. It is of the order of a few milli-seconds. • Variation of { a i ( t ) } i with time. a i ( t ) changes slowly, with a time scale of variation much larger than the other effects discussed. However as W increases and it becomes comparable to f c assuming that a single gain affects the corresponding path equally across all frequencies may not be a good approximation. The reflection coefficient of the scatterers may be frequency dependent and for very large bandwidths we need to change the model. • Movement of paths from tap to tap. τ i ( t ) changes with t and the corresponding path moves from one tap to another. As W increases fewer paths contribute to each tap and the tap gains change significantly when a path moves from tap to tap. A path moves from tap to tap when Δ τ i ( t ) W = 1 or Δ τ i ( t ) / Δ t · W = 1 / Δ t . So this effect takes place in a time scale of Δ t ∼ 1 / ( Wτ i ( t )) . As W increases this effect starts taking place in a small time scale and it becomes the dominant cause of time variation in the channel tap gains. The third effect dominates when Δ t < T c or equivalently when W > f c . 2. (a) Let ρ = SNR . For Rayleigh fading | h [0] | 2 ∼ Exp (1) so we have: P e = E h Q p 2 | h [0] | 2 ρ i = Z ∞ Z ∞ √ 2 xρ 1 √ 2 π e- t 2 / 2 e- x dtdx = Z ∞ Z t 2 / (2 ρ ) 1 √ 2 π e- t 2 / 2 e- x dxdt = Z ∞ 1 √ 2 π e- t 2 / 2 h 1- e- t 2 / (2 ρ ) i dt = 1 2 1- r ρ 1 + ρ Z ∞...
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This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell University (Engineering School).

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Hw9Soln_1 - ECE 5670 : Digital Communications ECE...

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