Wireless Communications - Exercises
1. Consider the bitstream 101101… . Draw a BPSK waveform that could be used to
transmit the bits. Draw an FSK waveform for this bitstream. Compare the
bandwidth of these two modulation schemes.
2. Assume in a cellular system that there are 10 channels (or slots) available and that
the average use of these slots is 50%. What is the blocking probability of this
system? Discuss whether this is a reasonable number.
3. Suppose you have been allocated 79 MHz of bandwidth by the FCC for a wireless
system using FDMA with 25 KHz channels not including guard band. For a
telephone call with full-duplexing, both an uplink and downlink channel is
required. This means that at least 50 KHz would be required per call. If your
carrier can drift by 80 Hz, which means that a guard band is required between
each channel, how many users can the system support at any one time. Explain
your reasoning.
4. Consider the bitstream 101001… . This bitstream can be converted into a
waveform by holding the voltage at +1 for T seconds when a 1 appears in the
bitstream and then holding the voltage at –1 when a 0 appears. Draw the resulting
waveform. Now suppose amplitude modulation is to be used to transmit the
bitstream. Draw the modulated signal by using the waveform from the first part in
Eq. (6). Show all steps.
5. Assume unit gain transmit and receive antennas, a transmission frequency of 900
MHz, and a transmission power of 1mW. Find the received power in Eq. (2) at a
transmitter-receiver separation of 10m, 100m and 1km. Now, assume that the path
loss exponent is 4 (you may assume the noise figure is 1). Find the received
power in Eq. (3) at a transmitter-receiver separation of 10m, 100m and 1km.
Compare the six results.
6. Discuss the advantages and disadvantages of different cell layouts and degrees of
frequency use. Consider the performance, the cost, and the societal impact.
MATLAB Exercises
Project 1 - AM, PM and Signal Bandwidth
The goal of this project is to discover the effect of amplitude modulation and phase
modulation on the time- and frequency-domain representations of the modulated carrier.
Steps:
1. Save the code given below for the first part of the project in a MATLAB directory
as proj1a.m, save the second part of the project as proj1b.m
2. Run the first part by typing proj1a
3. Print the resulting plots.
4. Discuss the shape of the time-domain AM and PM signals. Consider where the
message is contained, and how it might be extracted from the modulated carrier.
5. Discuss the differences and similarities in the frequency-domain signal, the
magnitude spectrum of the modulated carrier.
6. Change the frequency of the message slightly by changing the parameter f0
7. Repeat steps 3-5 comparing the results of the new signal with the previous results.
8. Save the PM signal by typing save PM xpm
9. Run the second project by typing proj1b
10. Print the resulting plot
11. Discuss the resulting bandwidth occupied by the signal.
12. Comment out the lines generating the cosine and uncomment the lines for
loading a signal.
13. Replace <filename> with PM or the name of another signal file
14. Run the program and print the resulting plot.
15. Discuss the resulting bandwidth occupied by the signal. Compare this to the
results from step 11.
16. Submit all plots and written discussions.
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Project 1
Part I - AM and PM Signal Generation
Wireless Communications
clear
% Generate a message which is a sinusoid
% First initialize variables
%
N = number of samples
%
f0 = cyclical frequency of message signal
%
T0 = period of the message signal
N = 1000;
f0 = 10;
w0 = 2*pi*f0;
T0 = 1/f0;
% Generate the message
n = 0:1:N-1;
t = n*T0/N;
x = 0.75*cos(w0*t);
figure
subplot(711);
%
% AM SIGNAL GENERATION
%
% Plot message signal
plot(t, x);
title('Message signal v(t)');
xlabel('t');
ylabel('v(t)');
% Generate carrier signal
fc = 100;
wc = 2*pi*fc;
Tc = 1/fc;
xc = cos(wc*t); % Ac=1
subplot(713);
plot(t, xc);
title('Carrier signal');
xlabel('t');
ylabel('c(t)');
% Generate AM signal
mu = 0.5;
xam = (1+mu*x) .* xc;
% Plot AM signal
subplot(715);
plot(t, xam);
title('AM signal')
xlabel('t');
ylabel('vc(t)');
% Find the spectrum of the AM signal
XAM = fft(xam);
XAMshift = 1/N*fftshift(XAM);
n1 = -500:499;
freq_n1 = n1/T0;
% Plot the magnitude spectrum of the AM signal at frequencies of
interest
subplot(717);
plot(freq_n1, abs(XAMshift), 'o');
axis([-200 200 0 1]);
title('Magnitude spectrum of the AM signal');
xlabel('f');
ylabel('Vc(f)');
%
% PM SIGNAL GENERATION
%
% Plot message signal this is the signal generated above
figure
subplot(511);
plot(t, x);
title('Message signal v(t)');
xlabel('t');
ylabel('v(t)');
% Generate PM signal
k_p = 3*pi/4;
xpm = cos(wc*t + k_p*x);
% Plot PM signal
subplot(513);
plot(t, xpm);
title('PM signal')
xlabel('t');
ylabel('vc(t)');
% Find spectrum of the PM signal
XPM = fft(xpm);
XPMshift = 1/N*fftshift(XPM);
n2 = -500:499;
freq_n2 = n2/T0;
% Plot spectrum of the PM signal
subplot(515);
plot(freq_n2, abs(XPMshift), 'o');
axis([-200 200 0 1]);
title('Magnitude spectrum of the PM signal');
xlabel('f');
ylabel('Vc(f)');
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Project 1
Part II - Signal Bandwidth
Wireless Communications
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Either load in a previously stored signal or create a new
uncomment the appropriate following section.
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LOAD a signal - uncomment five command lines
replace <filename> the name of the stored file
it must be in filename.mat (Matlab) form
and replace <dataname> with the name of the data.
You can use the command “whos” to see the current arrays
of data.
%
For instance, if you use the data from the previous
%
program you would type
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load PM;
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w = xpm;
%
%load <filename>;
%w = <dataname>;
%N = length(w);
%n=0:1:N-1;
%To=0.01;
%
CREATE a new signal - uncomment seven command lines
%
the signal will be a cosine with variables
%
N = number of samples
%
fo = cyclical frequency of sinusoid
%
wo = angular frequency of sinusoid
%
To = period of sinusoid
N = 50;
fo = 10000;
wo = 2*pi*fo;
To = 1/fo;
n = 0:1:N-1;
t = n*To/N;
w = 4*cos(wo*t);
% Find Fast Fourier Transform of the signal
W = fft(w);
% Convert samples 0,1,...,N-1 into positive and negative
Wshift = 1/N*fftshift(W);
% Plot signal vs. sample
subplot(3,1,1);
plot(n,w,'o');
xlabel('Sample Number');
ylabel('Sample Amplitude');
% Plot FFT vs sample
subplot(3,1,2);
plot(n,abs(Wshift),'o');
xlabel('Sample Number');
ylabel('Magnitude Spectrum');
% Shift the samples so that they are centered at 0
% Scale the axis
n1 = -1*round(N/2):1:round(N/2-1);
freq_n1= n1/To;
if length(freq_n1) > length(Wshift)
freq_n1=freq_n1(1:length(Wshift));
end
% Plot the transform vs. cyclical frequency
subplot(3,1,3);
plot(freq_n1,abs(Wshift),'o');
xlabel('Cyclical Frequency (Hz)');
ylabel('Magnitude Spectrum');
Project 2 - Noise in Baseband Signals
The goal of this project is to investigate the effect of varying degrees of noise on digital
baseband signals.
Steps:
1. Save the code given below for the first part of the project in a MATLAB directory
as proj2a.m
2. Run the first part by typing proj2a
3. Print the resulting plot.
4. Record the noise standard deviation and the bit error count.
5. Discuss the appearance of the signal with noise as compared to the noise-free
signal.
6. Change the noise standard deviation to a value within the permitted range given in
the code and re-run the program
7. Record the noise standard deviation and the bit error count.
8. Repeat steps 6 and 7 for at least 10 values of the standard deviation.
9. As the noise standard deviation increases and decreases note the appearance of the
noisy signal. Discuss the changes in the signal.
10. Follow the directions in the code for the second part of the project.
11. Print the resulting plot.
12. Discuss the changes in the bit error rate as a function of the noise.
13. Submit all plots and discussions.
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Project 2
Part I - Noise in Baseband Signals
Wireless Communications
% Change this following line to change the standard deviation of the
noise,
% which will change the noise power. The valid range is [0.01,1]. A
large number
% indicates more noise
stdev=1;
%
% Create an arbitrary bitstream
%
bitstream = [1 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1
1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 0
1 1 0 1 1 0 1 01 0 0 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0
1];
%
% Generate the baseband signal,v[n], from the above bitstream with
pulses
% of width 5 and height 1/sqrt(5).
%
for i=0:99
if bitstream(i+1) == 1
v(i*5+1) = 1;
v(i*5+2) = 1;
v(i*5+3) = 1;
v(i*5+4) = 1;
v(i*5+5) = 1;
else
v(i*5+1) = 0;
v(i*5+2) = 0;
v(i*5+3) = 0;
v(i*5+4) = 0;
v(i*5+5) = 0;
end
end
%
% plot the NRZ baseband waveform
%
subplot(3,1,1), plot(v,'o');
axis([0 40 0 1.0]);
%
% add zero-mean Gaussian noise to the waveform
%
for i=1:500
y(i) = v(i) + stdev*randn;
end;
%
% plot the noisy waveform
%
subplot(3,1,2), plot(y,'o');
axis([0 40 -4 4]);
%
% create an integrate and dump filter with output z(i)
%
for i=0:99
temp = y(i*5+1) + y(i*5+2)+ y(i*5+3) + y(i*5+4) + y(i*5+5);
z(i*5+1) = temp;
z(i*5+2) = temp;
z(i*5+3) = temp;
z(i*5+4) = temp;
z(i*5+5) = temp;
end
%
% plot the matched filter output
%
subplot(3,1,3), plot(z,'o');
axis([0 40 -10 10]);
%
% sample and determine the number of bit errors
%
bit_error_count = 0;
for i=0:99
if z(i*5+3) > 0.5
temp = 1;
else
temp = 0;
end
if bitstream(i+1) ~= temp bit_error_count = bit_error_count +1;
end
end
bit_error_count
% Run this program for many choices of the standard deviation and
record
% the standard deviation and the number of bit errors for each trial
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Project 2
Part II - Bit Error Rate Curves
Wireless Communications
% For each standard deviation compute the signal-to-noise ratio
% using the following. Replace stdev with your value and record
% the result
SNR = 10*log10(1/(stdev^2));
% Now plot the first point with the following three commands
% Replace SNR and bit_error_count with the appropriate values
semilogy(SNR,bit_error_count,'o');
axis([0 40 0.000001 0.1]);
hold;
% Now add the other points by repeating the following command
% Replace SNR and bit_error_count with the appropriate values
semilogy(SNR,bit_error_count,'o');
Project 3 - Number of Users and Blocking Probability
The goal of this project is to determine the average amount of traffic, effectively the
number of users, which can be served in a cell given a desired grade of service and a
number of available channels.
Steps:
1. Save the code given below for the auxiliary function in a MATLAB directory as
ErlangB.m, save the code for the project as proj3.m
2. Run the program by typing proj3
3. Record the capacity
4. Change the number of channels, the parameter channels, to a number between 2
and 100.
5. Repeat steps 2 through 4 for 10 channel values.
6. Change the desired grade of service, the parameter desired_GOS, to a value
between 0.1 and 0.0001
7. Repeat steps 2 through 6 four times.
8. Discuss the results. Consider the tradeoffs in desired grade of service and the
number of users in the system.
9. Submit the discussion.
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Project 3 - Auxiliary function
Number of Users and Blocking Probability
Wireless Communications
function [diff] = ErlangB(A,C,GOS)
sumA = 0;
for i=0:C
sumA=sumA+(A^i/factorial(i));
end
GOS1 = (A^C/factorial(C))/sumA;
diff=GOS-GOS1;
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Project 3 Main Routine
Number of Users and Blocking Probability
Wireless Communications
% Change the following two parameters and plot the number of
% users, the capacity, as a function of desired_GOS for a
% selected number of channels
channels = 10;
desired_GOS = 0.01;
init_capacity=channels*4/5;
options=optimset;
capacity=fsolve('ErlangB',init_capacity,options,channels,desired_GOS)