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Topics: Fundamental physics concepts, Volt, Fourier transform Pages: 2 (397 words) Published: February 4, 2014
ECE 4784/6784, Wireless Communications, Assignment 1

1. (15) A half-wave dipole is situated at the top of a cellular tower, oriented vertically on the tower. The current into the dipole is a sinusoidal current at frequency 870 MHz, with peak value I = 0.1 ampere. Consider a mobile receiver at a distance of d kilometers along the horizon. Remember the dipole pattern gives ﬁeld strength E(d, θ, φ) =

60I cos( π cos(θ))
2
aθ v/m
d
sin(θ)

(1)

a. Assume free-space propagation laws. Find the electric ﬁeld strength at the mobile location. Also ﬁnd the power ﬂux density Φ in w/m2 for users along the horizon. b. Repeat for the two-ray model, assuming a tower height of 40 meters, a mobile height of 2 meters, and a plane earth approximation. Assume the transmitted signal is verticallypolarized, and that the earth reﬂection coeﬃcient magnitude is 1. c. Plot both power ﬂux densities versus d on log-log scales, over a range from 100 meters to 10 kilometers. You should see scalloping in the two-ray result, with an envelope that decays twice as fast (on log-log scale) as the free-space result.

2. (10) Given an RF pulse
s(t) = 10−3 rect(

t
) cos(2π870 · 106 t + π/6) volts
.001

(2)

• ﬁnd S(f ), the Fourier transform, and sketch the energy density spectrum in the frequency domain • Find the complex envelope s(t) with respect to the reference cos(2π870 · 106 t). ˜
• ﬁnd the energy delivered to a 50 Ω load

3. (Additional assignment for ECE 6784 group, building on ﬁrst problem, 10 points) The power ﬂux density expression can be integrated over the surface of a sphere enclosing the dipole to obtain the total power radiated by the antenna. Call this number P watts. An isotropic antenna radiating P watts would produce a ﬂux density at the same radius of P = P/4πr2

(3)

The gain of any antenna is the ratio of the maximum ﬂux density, which occurs at θ = π/2, to the ﬂux density of an isotropic antenna, with the power density measured at the same...

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