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Ilmenau University of Technology

CRLCommunications Research Laboratory

Prof. Dr.-Ing. Martin Haardt

Mobile Communications

Homework 03

due on May 21, 2013

(Please hand in your homework in the office H3513, H3504 or H3502 before 4 pm.)

Note: Homework points are only valid in the same semester, they do not carry over to thenext. If you are taking the exam one semester later you cannot get homework bonus points.

1. (10p) Perform a baseband Rayleigh fading simulation (MATLAB).

The AWGN samples (nI(t), nQ(t), t = 1..10000) for the I (inphase) and Q (quadrature) branchare lowpass filtered by the Doppler filter. This introduces some intersample correlation by

restricting the maximum rate of change at the filter outputs (aI(t), aQ(t)).

(a) (2p) Generate the AWGN samples nI(t) and nQ(t).

(b) (2p) The Doppler filter can be modeled as an IIR filter. Use the following coefficients:

% Poles, Zeros, Numerator and Denominator Polynomials of Jakes Filter

Poles = [

0.990155 + j * 4.50092E-02

0.990155 - j * 4.50092E-02

0.980484 + j * 1.87576E-02

0.980484 - j * 1.87576E-02

0.996529 + j * 5.49384E-02

0.996529 - j * 5.49384E-02

0.998280 + j * 5.66694E-02

0.998280 - j * 5.66694E-02

];

Zeros = [

0.998358 + j * 5.72764E-02

0.998358 - j * 5.72764E-02

0.997444 + j * 7.14561E-02

0.997444 - j * 7.14561E-02

0.994404 + j * 0.105644

0.994404 - j * 0.105644

0.965308 + j * 0.261113

0.965308 - j * 0.261113

];

B = real(poly(Zeros));

A = real(poly(Poles));

and filter nI(t) and nQ(t) separately using MATLABs filter function.

(c) (3p) Plot the phase (linear) and the magnitude (logarithmic) of the fading simulation.

a(t) =

(aI(t))2 + (aQ(t))2 and (t) = arctanaQ(t)

aI(t),

where aI(t) and aQ(t) are the output samples of the Doppler filter.

(d) (3p) Plot a histogram of the amplitudes a(t) and phases (t) using 100 bins (quantizationvalues). What do the distributions look like?

2. (12p) A very important step in the development of future mobile communication systems is the

test of various algorithms under realistic conditions. Since building prototypes and testing them

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in the real world is too expensive, researchers usually prefer to test their methods in computer

simulations.

In order to do so we require realistic channels that can represent all types of propagation condi-

tions that are relevant. One option is to use measured channels but this is not always feasible.

Therefore the automatic generation of mobile radio channels is a very important task in com-

munications research.

This task requires a suitable channel model. Channel models can coarsely be divided into the

following categories:

Deterministic channel models: Here, the goal is to compute the actual propagation condi-tions by solving Maxwells equations. This is a computationally very demanding task and

quickly becomes infeasible for larger scenarios. It also requires very precise knowledge

about all interacting objects which is often unavailable.

Stochastic channel models: For this type of channel model, the actual electromagneticpropagation phenomena are ignored. Instead, the channel is treated as a random pro-

cess. It is described by appropriate characterizations such as amplitude distributions (e.g.,

Rayleigh), correlation functions (e.g. Jakes model) and power distributions (e.g., path

loss, delay power spectrum).

Hybrid channel models: These represent a mixture of both. For example, IlmProp is a hy-brid channel model: It includes geometrical information through a ray-based propagation

model but the scattering coefficients are modeled as random numbers.

In this task, the focus is on stochastic channel models. Our goal will be to implement a simple

channel model that can generate random channels following a certain power delay profile as

well as a certain Doppler power spectrum. The generation of a random channel h(, t) can bedivided into the following steps:

Generate random samples w(, t) such that all samples have the desired distribution (e.g.,Rayleigh, Rice) and are independent identically distributed.

Introduce correlation in time by filtering the sequence w(, t) in time direction with an

appropriate filter g(t), i.e., h1(, t) = w(, t)t g(t) =

w(, t t1) g(t1)dt1 (use

filter).

Finally, fit the variance of the channel for a certain delay time to the desired variance,such that the delay power spectrum follows the specifications. The variance after the

second step is 2h() = limT1T

T|h1(, t)|

2dt, i.e., the average over the time variable

t. Therefore given a desired delay power spectrum h() the normalized channel is given

by h(, t) =h()

2h() h1(, t).

Of course, in a computer the time (t) and delay time () are not continuous but sampled.

(a) (4p) Write a Matlab script that implements a stochastic channel model following the steps

described above. Assume that there is no line of sight, so that the amplitude distribution

is Rayleigh and the phase distribution is uniform (as in the previous homework). You

can find an outline for the script in the homework3.zip which you can download at our

website. In there you will find:

channelmodel.m the main file which provides you with an outline

dopplerfilter.mat a file to load to get the filter coefficients

makeaxis.m a helper function to generate the correct axes in t, , f, fD.

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(b) (4p) Use your script to generate a random channel h(, t) for NT = 1000. Compute theremaining three Bello functions H(f, t), h(, fD), and H(f, fD). Plot all four Bello func-tions along with the correct axis for t, , f and fD (Hints: Matlab provides an implemen-

tation of the Fast Fourier transform by the command fft. Note that the normalization

is different from the one used in our lecture. To get a correct DFT use fft(...)/N.

Also note the ordering of the elements, which can be corrected using fftshift.). Do

the Bello functions look realistic? Why, why not?

(c) (2p) Compute a histogram of the amplitude and of the phase ofh( = 0, t). Does it followyour expectation (to get a smooth histogram you can increase NT)?

(d) (2p) Estimates of the Doppler power spectrum and the delay power spectrum are given

by:

h() =

|h(, fD)|

2dfD

H(fD) =

|h(, fD)|

2d

Compute the estimated delay power spectrum and compare it with the desired one for

NT = 1000. Do they agree? Also compute the estimated Doppler power spectrum.What can you see from its shape? Is this a realistic model for the Doppler power spectrum?

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Ilmenau University of Technology

CRLCommunications Research Laboratory

Prof. Dr.-Ing. Martin Haardt

Mobile Communications

Hints 03

due on May 21, 2013

(Please hand in your homework in the office H3513, H3504 or H3502 before 4 pm.)

1. Useful MATLAB functions: randn, filter, hist, plot, semilogy, xlabel, ylabel.

The MATLAB command rand returns uniformly distributed (pseudo-) random variables.To get complex Gaussians you need normal distributed (pseudo-) random variables, which

are returned by randn.

In complex baseband terminology, the real part is referred to as the inphase component,the imaginary part is called quadrature component. The magnitude is therefore equal to

the square root of the sum of the squared inphase and the squared magnitude.

Label your plots. A good labeling (as an example) would have been Magnitude on thehorizontal and Frequency of occurrence on the vertical axis.

Hand in printouts of your programs and the resulting plots. Label your axis correctly.

Electronic submissions are accepted in PDF and Word format (PDF preferred), separate graph-

ics in JPEG and PNG format (including them in one document is preferred). Do not send .FIG

files.

Always indicate your name and your Mat.-number for easier association.

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