辅导B31SI2、讲解MSc Course、辅导Matlab程序设计、Matlab语言讲解

Lab Exercises for the MSc Course B31SI2: Principles of Mobile Communications 7

Exercise 3

A flat fading channel simulator can be constructed by using the sum-ofsinusoids

method. With this method, a baseband complex Gaussian random

signal can in general be modelled by

g(t) = g1(t) + jg2(t), (3.1)

where

gi(t) = X

Ni

n=1

ci,n cos(2πfi,nt + θi,n) , i = 1, 2 . (3.2)

The gains ci,n, discrete Doppler frequencies fi,n, and phases θi,n can be

determined by using the so-called method of exact Doppler spread (MEDS)

and are given as follows

In (3.3), b stands for the variance of gi(t) and fm denotes the maximum

Doppler frequency. Note that this channel simulator is of deterministic

nature since all the model parameters are kept constant instead of random

during simulations. Fig. 3.1 shows the structure of the corresponding

deterministic simulation model for Rayleigh fading channels, where ?α(t)

indicates the received signal envelope in the complex baseband.

Fig. 3.1: A deterministic Rayleigh fading channel simulator.

8 Lab Exercises for the MSc Course B31SI2: Principles of Mobile Communications

3.1 For the given number of sinusoids N1 = 9 and N2 = 10, compute the

simulation model parameters ci,n, fi,n, and θi,n according to (3.3a)–

(3.3c) by using b = 1 and fm = 91 Hz. Write your results in Table 3.1

given at the end of Exercise.

3.2 Develop a Matlab function m-file to simulate the deterministic processes

gi(t) for i = 1, 2. Use the quantities ci,n, fi,n, θi,n, and t as input

arguments.

3.3 Write a Matlab script m-file to carry out the simulation of the channel

amplitude α(t) = |g(t)| = |g1(t) + jg2(t)| and plotαdB(t) = 20logα(t).

Solve this problem by making use of your function m-file developed in

Exercise 3.2. The simulation model parameters ci,n, fi,n, θi,n, and Ni

are given as listed in Table 3.1. Use the simulation time Tsim = 0.4 s

and the sampling frequency fs = 270.8 kHz, which corresponds to the

symbol rate used in GSM. If a vehicle drives with a speed of v = 109.2

km/h, what is the distance it covers in the specified simulation time?

Hint: The discretization of the time t can be expressed as t = 0 :

1/fs : Tsim.

3.4 Simulate the deterministic functions g1(t), g(t), and α(t) by using the

sampling frequency fs = 50 kHz and the simulation time Tsim = 20 s.

Determine the mean value and variance of g1(t), g(t), and α(t). Determine

the probability density function (PDF) of g1(t) and α(t) by

using histograms. Which distributions do they approximate? Compare

the PDF of g1(t) with the theoretical result given in (1.1), where

mμ = 0 and σμ = 1. Compare the PDF of α(t) with the theoretical

result pα(x) given in (2.4), where p = 2, and K = 0.

Hint: Use the Matlab functions mean, std, and hist.

3.5 Determine a part of the signal g1(t)|t∈I from the simulation results of

g1(t) in the interval I = [10 000 /fs , 20 000 /fs], where fs = 1000 Hz.

Find the autocorrelation function (ACF) φg1g1

(τ ) of g1(t)|t∈I and plot

it. What is the value of φg1g1

(0)

Hint: Use the Matlab function xcorr with the option biased.

Lab Exercises for the MSc Course B31SI2: Principles of Mobile Communications 9

3.6 The ACF φg1g1

(τ ) of g1(t) can be expressed by

cos (2π f1,n τ ) . (3.4)

Compute φg1g1

(τ ) over the interval 0 ≤ τ ≤ 0.08 s and sketch your

result in the same figure as in Exercise 3.5. For the sake of comparison,

the ideal ACF φg1g1

(τ ) in (2.1) is required to be shown in this figure

as well. Use the value 2 for p and 91 Hz for fm.

10 Lab Exercises for the MSc Course B31SI2: Principles of Mobile Communications

i n fi,n (Hz) ci,n θi,n (rad)

Table 3.1: Parameters of the simulation model (fm = 91 Hz, b = 1, N1 = 9, N2 = 10).