# Hw9Soln_1 - ECE 5670 : Digital Communications ECE...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ∞...
View Full Document

## 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).

### Page1 / 4

Hw9Soln_1 - ECE 5670 : Digital Communications ECE...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online