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Advanced Digital Communications EPFL Winter Semester 2003/2004

FinalWednesday February 11, 2003 14:15-17:15

This exam has 4 problems and 100 points in total.

Instructions

You are allowed to use 2 sheets of paper for reference. You can attempt the problems in any order, as long as it is clear which problem is being

attempted and which solution to the problem you want us to grade.

If you are stuck in any part of a problem do not dwell on it, try to move on and attemptit later.

There are also 5 bonus points to be had making the maximum number of points 105/100.

Good Luck!

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Problem 1

[ OFDM/Multicarrier Transmission(25pts)]Consider a finite impulse response channel

yk =

n=0

pnxkn+ zk

where yk, zk C 2 and pn C2, i.e they are 2-dimensional vectors. This could arise, forexample, through Nyquist sampling like the model considered in class (L= 2).

(a) Suppose one observes a block ofN samples of{yk},Yk =

yk...ykN+1

. Write down the[4pts]

relationship betweenYk andXk =

xk...

xkN+1

...xkN+1

in the form

Yk=PXk+ ZkwhereYk,Zk C2N,Xk CN+,P C2N(N+) by specifying the form ofP.

(b) Suppose we use a cyclic prefix, i.e[4pts]

xkNl =xkl, l= 0, , 1

kN+1 kN+1 k

cyclic prefix

Figure 1: Cyclic Prefix

Develop the equivalent model:

Yk=PXk+ Zk (1)whereXk =

xk...

xkN+1

CN andP C2NN. FindP.

(c) Let Y() = 1N

N1k=0 yke

j 2Nk,Z() = 1

N

N1k=0 zke

j 2Nk,[6pts]

P() =

n=0 pnej 2

Nn , X() = 1

N

N1k=0 xke

j 2Nk.

Develop the vector OFDM form for (1), i.e, show that

Y() =P()X() + Z(), = 0, , N 1 (2)

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This can be done by either arguing about equivalent periodic sequences or any otherproof technique. Here we would like to see a derivation, just stating the result is notenough.

(d) In the form given in (2) , we get Nparallel vector channels. If we want to detect each[6pts]component{X()} separately what would be the best linear estimator of X() fromY(), i.e the appropriate frequency-domain MMSE linear equalizer (FEQ).

(e) For the form in (2), if we concatenate with powerful codes, what would be the total[Bonus5pts] achievable rate and the corresponding power optimization problem formulation. You

need only formulate the power optimization problem and not neccessarily solve it.

Problem 2

[Target Channels(20pts)]Suppose we have a linear time-invariant channel as in the previous problem i.e

yk =

n=0

pnxkn+ zk

where yk,pn,zk CL and xk C .In class we developed the following block model for Nfsamples of{yk}:

Yk=PXk+ ZkwhereYk,ZkCNfL,Xk CNf, PCNfL(N+). Suppose we want the following receiverstructure

W{yk} {rk}

and we want rk

n=0hnxkn, where . That is we want the output of the equalizerto be close to a given target channel{hn}. Therefore we find WC 1NfL :

Wopt= arg minW

E||WYk HXk||2 (3)

where H = [ 0 0 times

h0 h1 h 0 0 Nf+1

]i.eH C1(Nf+)

You may assume that Exkxk = Ex, E|zk|2 =2 and{zk} is AWGN.

(a) (Complete equalization) In class we derived the finite length equalizer when[5pts]

hn=

1 n= 00 otherwise

(4)

Re-derive the optimal MMSE finite length equalizerW, for this case, i.e. for{

hn}

givenin (4) and the criterion in (3) (i.e. same as done in class).

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(b) Now if we want a particular target channel{hn}n=0, and the criterion is given in (3),[8pts]derive the optimal filter Wopt for the given target channel.

(c) For a given target channel{hn}n=0, compute the 2FIR-MMSE-LE, i.e.[7pts]

E||WoptYk HXk||2

Problem 3

[Fluttering and fading (25pts)]

Suppose there is a transmitter which is sending signals to be received by two receive antennas.However due to a strange and unfortunate coincidence there is a flag fluttering in the windquite close to one of the receive antennas and sometimes completely blocks the received signal.In the absence of the flag, the received signal is given by a flat fading model, (discrete timemodel as done in class).

Yk= y1(k)y2(k)= h1(k)h2(k) x(k) + z1(k)z2(k) (5)where y1(k), y2(k) are the received signals on first and second receive antennas respec-tively, x(k) is the transmitted signal and h1(k), h2(k) are respectively the fading attenuationfrom the transmitter to the first and second receive antennas. Assume that x(k) is binary,i.e. x(k) {Ex,

Ex}. The additive noise z1(k), z2(k) are assumed to be independentcirculary symmetric complex Gaussian with variance (each) of2. Assume that h1(k), h2(k)are i.i.d complex Gaussian C (0, 1).

(a) Over several transmission blocks, compute the upperbound to the error probability and[5pts]

comment about the behavior of the error probability with respect to SNR for high SNR.Hint: Use the fact that Q(x) ex2/2.

(b) Now let us consider the presence of fluttering flag which could potentially block only[10pts]the second receive antenna. The model given in (5) now changes to:

Yk =

y1(k)y2(k)

=

h1(k)Fkh2(k)

x(k) +

z1(k)z2(k)

where:

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Suppose due to the random fluttering, the flag blocks a fraction qof the transmissions,i.e for a fractionqof the transmission, one receives only the signal from the first antenna.

Conditioned on F, write down the error probabilities, i.e find and expression for Pe(x x|F) and compute its upper bound (see hint in (a)).

(c) Find the overall error probability in the presence of the fluttering. How does the error[10pts]probability behave at high SNR, i.e what diversity order does one obtain.

Hint: If the error probability behaves as 1SNRD

at high SNR, the diversity order is D.

Problem 4

[Multiple Access and Multiuser Detection (35pts)]Suppose we have two users u= 1, 2, transmitting information{x1(k)}and{x2(k)}. Considera simple multiple access channel where the received signal (discrete time) is

y(k) =x1(k) +x2(k) +z(k)

where{z(k)} is additive white complex Gaussian noise with variance 2, is i.i.d. and indepen-dent of{x1(k)}and{x2(k)}. You may assume that x1 and x2 are independent with identicalvarianceEx.

(a) Suppose we use blocks of length two for transmission and the users use the following[8pts]transmission strategy,

x1(k) = s1 , x1(k+ 1) = s1x2(k) = s2 , x2(k+ 1) = s2

(6)

Express the received signal, [y(k), y(k +1)] in terms of the transmitted symbols, i.e. spe-cialize

y(k) = x1(k) +x2(k) +z(k)

y(k+ 1) = x1(k+ 1) +x2(k+ 1) +z(k+ 1)

to the transmission strategy in (6), prove that

y= Rs + z.

and find the form ofR.

(b) Find[4pts]

y= Ry

and comment about its implications to detecting s1 and s2. Is this equivalent to thedecorrelating detector?

(c) Find the MMSE multiuser detector Msuch that[5pts]

E||My s||2 is minimized.

Explicitly calculate M and comment about its relationship to the receiver in (b).

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