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DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING

EC 4011Fundamentals of Wireless Communications

Problem sheet #1 02.08.2013

1. If the received power at a reference distance d0 = 1 km is 1 microwatt, find the received power atdistanced = 10 km for (i) free space model and (ii) model with path loss exponent 3.

2. Consider the two ray ground reflected model for path loss calculation. It was shown that thereceived power at a distance, d from the transmitter has lognormal distribution under large scale

fading. Find the probability that the received power at a distance, d will exceed a certain minimum

value. Also find the outage probability.

3. Find the critical distance under the two path model for a large macro cell in a suburban area withbase station mounted on a tower or building with hT=20 meters; hR=3 meters; f=2GHz.

4. Consider the combined path loss-shadowing model. A transmitter provides 15 Watts to an antennahaving gain 12dB. The receiver antenna has a gain of 3dB and the receiver bandwidth is 30 kHz.

Determine the maximum separation between transmitter and receiver that will ensure a minimum

received power of -101 dBm for 95% of the time (Assume: Path loss exponent = 4; Carrier

frequency = 1800 MHz; Standard deviation of log normal shadowing = 8dB; and 0d = 1 km).

5. Consider a cellular system where the received power is distributed according to a log-normalprobability distribution with mean dB and standard deviation dB. Assume that the received

power must be above 10dBm for acceptable performance. For 4dBm = , what value of is

required such that the outage probability is less than 1%.

6. Prove, for and independent zero-mean Gaussian random variables with variance, that thedistribution of is Rayleigh and that the probability distribution of is

exponential. If has non zero mean, find the probability distribution of.

7. Find the probability distribution of the received signal envelope under the narrow band fadingmodel when there is a dominant LOS component in addition to the faded multipath components.

State the assumptions clearly.

8. Assume normalized signal envelope follows a Rayleigh fading distribution with the average signalpower 20 dB, what is the power outage probability relative to threshold P0 = 5 dB.If the application

requires a power outage probability of 0.01 for the threshold P0 = 10 dB, what value of average

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signal power 22 is required. Assume that the received signal has a LOS component, how will

you find the outage probability given that the LOS component has average power 2S .

9. An application requires a power outage probability of 0.01 for the threshold 0 80P dBm= .Assuming Rayleigh fading, determine the average signal power required to meet this criterion(neglect large scale fading effects).

10.The power delay profile of a wireless channel is given by 0,)( 0 f = ePP . Determine theaverage delay and delay spread for this channel.

11.The power delay profile for a wideband channel is given as 0;1)( / = mWeT

A mT

m

cwhere,

mSecTm

1= . What are the values for meanm

T and rms

mT

delay spread. What is the coherence

bandwidth?12.A multi-path fading channel has a delay spread of 1 seconds and a Doppler spread of 0.01Hz. The

total channel bandwidth at band pass available for transmission is 5 Hz. Input signal pulse duration

is 10seconds. Find the coherence time and coherence bandwidth. Is the channel frequency

selective? Is the channel fading slowly or rapidly?

13.Consider a channel with impulse response:( , ) ( ) 0.5 ( 10 sec) 0.5 ( 40 sec)c t = + + .

i. Is the channel time varying or time invariant? Explain.ii. For what data rates does a BPSK signal sent over this channel experience ISI?

14.Let be normal with mean and variance (all quantities in dB). Define 10 .Find the PDF, mean and variance of.