Examen Sari Fioddrinaoct2012v2

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Majeure Tlcommunications MAJ1

and

Master Program SAR C3

Date: 30 0ctober 2012

Time: 9:00 a.m.

Duration: 3 Hours

This document includes 4pages

EXAM

Principles of Digital Communications

Important note:Use of calculators, lecture notes and personal notes of the students is allowed for

this exam, but the text books are not allowed. The answers to PART I and PART II must be given on

separate sheets.

PART I:

1) Consider a baseband signal of the form kTtatak , where t is the Dirac delta

pulse, Tis the symbol duration, and k

a designate uncorrelated symbols taking their values

from the alphabet A . This signal passes through a linear encoder whose z-domain

transfer function is 21 zzH followed by an ideal low-pass filter of bandwidth

TT 21

21 , . The filter output is denoted ts . Write the expression of the power spectrum

density of ts .

2) How would you decode this type of signal at the receiver? Sketch the receiver block

diagram and comment the operation of the receiver. If you can think of different alternatives

for the decoder, describe all of them and make comparisons.

3) Consider the following 16-state modulations: 16-ASK, 16-PSK and 16-QAM. For each of

them, express the minimum distance as a function of the average signal power. Knowing

that the minimum distance determines the bit error probability for a given noise power, how

would you rank these 3 modulations? Express the respective signal-to-noise ratio (SNR)

losses in dB. What general conclusions can you reach from this comparison concerning the

ASK, PSK and QAM modulations?

4)

Consider an additive white Gaussian noise channel over the frequency band of (960 980

MHz). On this channel, we wish to use a quadrature amplitude modulation (QAM) and

achieve a bit error rate (BER) lower than 10-6

. What is the maximum achievable bit rateassuming an Eb/N0 of 16 dB and Nyquist filtering with roll-off factor 0.25? Which


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modulation must be used to achieve this bit rate? What is the maximum achievable bit rate

for Eb/N0= 16 dB? With which modulation?

5) Consider an OFDM system with 64 carriers. The input symbol block to the Inverse DFT is

denoted ),.....,,,(1210 N

aaaa . The channel is a 2-path wireless communications channel

characterized by the impulse response Tttth . , where T denotes the symbol

period (the block duration is NT) and is a constant between 0 and 1. Plot the channel

amplitude response fH versus frequency. Next, assuming that the first symbol in the

block is transmitted at frequency 0f such that 10 fH , express the SNR loss at the

receiver (after propagation over the given 2-path channel) for each of the symbol in the

block ),.....,,,( 1210 Naaaa . Can you describe an OFDM system in which different powers

are allocated to different symbols in such a way that the SNR at the receiver is constant (the

same for all symbols in the block)? How would you allocate the power for the channel givenabove? What happens if the constant goes to 1?


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PART II:

Exercise 1

The received signal for antipodal signaling over a Rayleigh fading is

where

and , . It is known that under coherent detection, the error probabilitypedecreases like 1/SNR.

1. A precise way of saying thatpe decays like 1/SNR with increasing SNR is the following:

where cis a constant.

Identify the value of cfor the Rayleigh fading channel.

2.

Now we change the fading distribution. Suppose that ||has an arbitrary continuous pdf fsatisfying f(0)>0.

Give the value of c as a function of f(0).

Hint:you may need to interchange limit and integration in your calculations. You can

assume that this can be done without worrying about making a rigorous justification.

3. Suppose now we have L independent branches of diversitiy with gains , and ||having an arbitrary distribution as in the previous part.

Give the value of as a function of g1(0),,gL(0), where gl is the pdf of||

Hint 1:the pdf of the sum of two independent variable is the convolution of their pdf. For

, we can write:

whereis the pdf of || ||.Hint 2:

Exercise 2

We consider a DS spread spectrum system in the presence of CW jamming interference.

a) Compare the error rate performance when the signal pulse is:


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, (the energy of this pulse energy in low pass band is

2.Ec )

to the error rate performance when the signal pulse is rectangular.

b)

Compare it also to the case where the CW interference is replaced by a broadband

interference with the same total interfering power in the band W=1/Tc.

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