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Homework 1

This document contains an introduction and review problems for an introduction to wireless communications systems course. The homework will not be collected. The reading assignment is chapters 1-2 of Wireless Communications. Problem 1.1 asks students to calculate battery life of a cellular phone based on current draw in different usage scenarios and compare to specifications of a Nexus One smartphone. Problem 1.2 involves calculating output power in dBW and dBm for microwave transmitters and combining networks. Problem 1.3 reviews probability concepts like cumulative distribution functions, probability mass functions, and continuous Gaussian random variables. It asks students to plot examples and calculate probabilities for normal random variables.

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Atul Dubey
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81 views2 pages

Homework 1

This document contains an introduction and review problems for an introduction to wireless communications systems course. The homework will not be collected. The reading assignment is chapters 1-2 of Wireless Communications. Problem 1.1 asks students to calculate battery life of a cellular phone based on current draw in different usage scenarios and compare to specifications of a Nexus One smartphone. Problem 1.2 involves calculating output power in dBW and dBm for microwave transmitters and combining networks. Problem 1.3 reviews probability concepts like cumulative distribution functions, probability mass functions, and continuous Gaussian random variables. It asks students to plot examples and calculate probabilities for normal random variables.

Uploaded by

Atul Dubey
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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UNIVERSITY OF TEXAS AT DALLAS

Department of Electrical Engineering


EE 6390 - Introduction to Wireless Communications Systems
Problem Set #1: Introduction and Review

Homework will not be collected.

Reading: Wireless Communications, ch. 1-2

Please use MATLAB to help you solve these problems, check answers, etc.

Problem 1.1
Assume a 1 Amp-hour battery is used on a cellular telephone. Also, assume that the
cellular telephone draws 35 mA in idle mode and 250 mA during a call. How long would
the phone work (i.e., what is the battery life) if the user leaves the phone on continually
and has one 3-minute call every day? Every 6 hours? Every hour? What is the maximum
talk time available on the cellular phone in this example? Compare this example to the
technical specifications given for Nexus One smart phone. Find the average currents drawn
by Nexium in talk time and standby time.
Problem 1.2

(a) A microwave transmitter has an output of 500 mW. What is its output in dBW?

(b) A combining network has two inputs: +30 dBm and +19 dBm. It has an insertion
loss of 10 dB. What is the combined output in dBm?

Problem 1.3 Probability Review Questions


Discrete Random Variable Example: In the coin-tossing experiment, the proba-
bility of heads equals p and the probability of tails equals q. We define the random
variable X such that (
1 head
X=
0 tail
We find its cumulative and probability mass functions, CDF F (x) and PMF f (x),
respectively, for every x from −∞ to ∞. These functions are shown in the figure
below:

(a) In a die experiment, the random variable X is such that X = 2i where i is the outcome
of the experiment, i.e., i = 1, 2, 3, 4, 5, or 6. Plot the CDF and PMF of X. Find F (4)
and F (9.99).
F(x) f(x)
1

q q

0 1 x 0 1 x

Definition (Continuous Gaussian random variable): A widely used continuous


random variable in this class is Gaussian or normal random variable if its density is
given by
1 2 2
f (x) = √ e−(x−µ) /2σ
σ 2π
This is a bell-shaped curve, symmetrical about the line x = µ and its area equals 1 as
it should. We shall use the notation N (µ, σ) to indicate that an RV X is normal.

(b) An RV X is N (0, 2). Find the probability that X is between 1 and 2. In other words,
evaluate
P(1 ≤ X ≤ 2) =?

(c) (Conditional Probability) Find

P(1 ≤ X ≤ 2|X ≥ 1) =?

(d) Suppose that Y = aX + b, where a and b are real numbers and where RV X is N (0, 2).
Express CDF FY (y) in terms of FX (x), a, and b.

(e) Suppose that Z = X2 , where RV X is N (0, 2). Find PDF fZ (z).

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