Signals and Systems II

Lecture "Signals and Systems II"

Basic Information

	Lecture overview
	Lecturers:		Gerhard Schmidt (lecture), Marco Gimm/Christin Bald (exercise)
	Room:		KS2/C-SR I
	Language:		German
	Target group:		Students in electrical engineering and computer engineering
	Prerequisites:		Signal and Systems, Part 1
	Contents:		This course teaches basics in systems theory for electrical engineering and information technology. This basic course is restricted to continuous and deterministic signals and systems. Topic overview: Modulation State-space description Stochastic signals and corresponding spectra Linear systems and stochastic signals Ideal systems Add-ons (transforms)
	References:		E. Hänsler: Statistische Signale, 3.Auflage, Springer, 2001 H. D. Lüke: Signalübertragung, Springer, 1995 H. W. Schüßler: Netzwerke, Signale und Systeme II: Theorie kontinuierlicher und diskreter Signale und Systeme, Springer, 1991

News

The exam procedure for Signals and Systems II has been adapted slightly. The notes on the exam execution and rules will be shortened. Instead of reading out the whole exam problems there will be 10 minutes of additional reading time. Please note that during this reading time no writing will be allowed.

Exam Results and Exam Inspection:

The results of the winter semester's exam of 04.03.2020 are now online in the QIS-system. There is no date yet for an exam inspection.

Please note that any information regarding your exam, including achieved points and "how close" you are to passing, can only be given out during an official exam inspection. Your grades will always be available through the QIS-system as soon as possible.

Lecture Slides

	Link		Content
			Slides of the lecture "Modulation" (Linear and nonlinear modulation, digitalization)
			Slides of the lecture "State-space description" (State-space representation of linear, time-invariant systems)
			Slides of the lecture "Stochastic signals and corresponding spectra" (Random processes, power spectral densities)
			Slides of the lecture "Stochastic signals and linear time-invariant systems" (Random processes, power spectral densities)
			Slides of the lecture "Ideal systems" (Lowpass filter, magnitude and phase distortions)
			Slides of the lecture "Extensions on transformations" (Spectra of sampled continuous and discrete signals, convergence conditions and inverse transformations)

Exercises

Exercises can be found on an extra web page.

Evaluation

The evaluation can be found on an extra web page.

Completed evaluations can be found on an extra web page.

Matlab demos

Central limit theorem

%**************************************************************************
% Parameter fuer die Generierung von gleichverteilten Zufallsvariablen
%**************************************************************************
m_v   =      0.0;        % Mittelwert
var_v =      1.0;        % Varianz
N     = 400000;          % Anzahl der Variablen pro Experiment
M     =      4;          % Anzahl der summierten Zufallsvariablen

%**************************************************************************
% Erzeugen der Zufallszahlen
%**************************************************************************
V = ((rand(M,N) - 0.5) / sqrt(12) * sqrt(var_v)) + m_v;

%**************************************************************************
% Nicht-normierte Histogramm-Analyse der ersten Zufallsvariablen
%**************************************************************************
fig = figure(1);
set(fig,'Units','Normalized');
set(fig,'Position',[0.1 0.1 0.8 0.8]);

[h,x] = hist(V(1,:),20);
bar(x,h,0.7);
ylabel('Haeufigkeit');
xlabel('Variable v_1');
grid on

%return;

%**************************************************************************
% Histogramm-Analyse der einzelnen Zufallsvariablen
%**************************************************************************
A_min    =   -0.6;
A_max    =    0.6;
IntWidth =    (A_max - A_min) / 50;

IntVec   = A_min:IntWidth:A_max;
HistRes  = zeros(M,length(IntVec));

for k = 1:M
    HistRes(k,:) = hist(V(k,:),IntVec) / (N * IntWidth);
end;

%**************************************************************************
% Darstellung der einzelnen Histogrammanalysen
%**************************************************************************
fig = figure(2);
set(fig,'Units','Normalized');
set(fig,'Position',[0.1 0.1 0.8 0.8]);

for k = 1:min(4,M)
   subplot(2,2,k)
   bar(IntVec,HistRes(k,:),0.7)
   grid on;
   xlabel(['Variable v_',num2str(k)']);
   axis([A_min A_max -0.05*max(HistRes(k,:)) 1.05*max(HistRes(k,:))]);
   ylabel('Relative Haeufigkeit');   
end;

%return;

%**************************************************************************
% Summation der Zufallsvariablen
%**************************************************************************
v = zeros(1,N);
for k = 1:M
    v = v + V(k,:);
end;

%**************************************************************************
% Histogrammanalyse der Summation
%**************************************************************************
histRes = hist(v,IntVec) / (N * IntWidth);

%**************************************************************************
% Darstellung der Histogrammanalyse der Summe
%**************************************************************************
fig = figure(3);
set(fig,'Units','Normalized');
set(fig,'Position',[0.1 0.1 0.8 0.8]);

bar(IntVec,histRes,0.7)
grid on;
xlabel('Variable y');
axis([A_min A_max -0.05*max(histRes) 1.05*max(histRes)]);
ylabel('Relative Haeufigkeit');   

%**************************************************************************
% Darstellung einer Gauss-Dichte
%**************************************************************************
f_g = 1/(std(v)*sqrt(2*pi)) * exp( -0.5 * ((IntVec-mean(v))/std(v)).^2);
hold on
plot(IntVec,f_g,'r','LineWidth',2);
hold off

Matlabdemo for exercise ''Amplitudenmodulation''

%% Example for single-sideband modulation and "double"-sideband modulation
%
% -> general concept
% -> advantage of singe-sibeband modulation (reduced error caused by frequency shift)
%
% Jens Reermann
%% Reset
clear all;
close all;
clc;
%% Parameter
f_s=44.1e3;
PLAY_SIGNAL=0;
AUDIO_ENABLED = 0;

%%
fc=16e3;
f_error=1500;
phi_error = 0; %pi*0.48;

%% Load signal
if AUDIO_ENABLED == 1
    [u, fs_sig] = audioread( 'signals/interview_heino.wav' );
    n=0:length(u)-1;
else
    fsig=2000;
    fs_sig=100e3;
    T=10;
    Nsig=T*fs_sig;
    n=0:Nsig-1;
    u = sin(2*pi*fsig/fs_sig.*n)'+randn(Nsig,1);
end


%% LP 3kHz | 4kHz
f_l=3e3;
f_h=4e3;
f = [ f_l f_h];
a = [1 0];
rp = 1;           
rs = 120;     
dev = [ (10^(rp/20)-1)/(10^(rp/20)+1) 10^(-rs/20)]; 
[n_ord,fo,ao,w] = firpmord(f,a,dev,f_s);
b = firpm(n_ord+1,fo,ao);
u_f = filter(b,1,u);

%% Carrier signals

c=cos(2*pi*n*fc/f_s)';
c_e=cos(2*pi*n*(fc+f_error)/f_s+phi_error)';

%% Modulation

% DSB
m=c.*u_f;

% BP: DSB->SSB
f_bp = [ fc fc+1000 fc+f_l fc+f_h];
a_bp = [0 1  0]; 
dev_bp = [10^(-rs/20) (10^(rp/20)-1)/(10^(rp/20)+1) 10^(-rs/20)]; 
[n_ord_bp,fo_bp,ao_bp,w] = firpmord(f_bp,a_bp,dev_bp,f_s);
b_bp = firpm(n_ord_bp+1,fo_bp,ao_bp);

m_bp = filtfilt(b_bp,1,m);    %SSB

%% Demodulation

% *cos
u_d=m.*c;
u_d_bp=m_bp.*c;
u_de=m.*c_e;
u_de_bp=m_bp.*c_e;
% LP
u_df = filter(b,1,u_d);
u_df_bp = filter(b,1,u_d_bp);
u_def = filter(b,1,u_de);
u_def_bp = filter(b,1,u_de_bp);

%% Play signalss
if PLAY_SIGNAL==1
    sound(u_f,f_s)
    pause
    sound(u_df,f_s)
    pause
    sound(u_def,f_s)
    pause
    sound(u_def_bp,f_s)
end

%% PSD estimation
N_FFT=2^12;
[Puu, F] = pwelch(u,hann(N_FFT),N_FFT-N_FFT/4,N_FFT,f_s);
[Pufuf, F] = pwelch(u_f,hann(N_FFT),N_FFT-N_FFT/4,N_FFT,f_s);
[Pmm, F] = pwelch(m,hann(N_FFT),N_FFT-N_FFT/4,N_FFT,f_s);
[Pmm_bp, F] = pwelch(m_bp,hann(N_FFT),N_FFT-N_FFT/4,N_FFT,f_s);
[Pudfudf, F] = pwelch(u_df,hann(N_FFT),N_FFT-N_FFT/4,N_FFT,f_s);
[Pudefudef, F] = pwelch(u_def,hann(N_FFT),N_FFT-N_FFT/4,N_FFT,f_s);
[Pudfudf_bp, F] = pwelch(u_df_bp,hann(N_FFT),N_FFT-N_FFT/4,N_FFT,f_s);
[Pudefudef_bp, F] = pwelch(u_def_bp,hann(N_FFT),N_FFT-N_FFT/4,N_FFT,f_s);

%% Plotting (Modulation)
[val1 ind1] =max(10*log10(Puu));
[val2 ind2] =max(10*log10(Pmm));
[val3 ind3] =max(10*log10(Pmm_bp));

if AUDIO_ENABLED
    y_lims=[-160 -50];
else
    y_lims=[-100 10];
end

figure

plot(F, 10*log10(Puu));
text(F(ind1),val1,[num2str(val1) 'dB']);
legend('S_{uu}');
xlabel('f / Hz');ylabel('LDS / dB');xlim([0 f_s/2]);ylim(y_lims);
pause
hold on;
plot(F, 10*log10(Pufuf),'r');
legend('S_{uu}','S_{ufuf}');
xlabel('f / Hz');ylabel('LDS / dB');xlim([0 f_s/2]);ylim(y_lims);
pause
plot(F, 10*log10(Pmm),'g');
text(F(ind2),val2,[num2str(val2) 'dB']);
legend('S_{uu}','S_{ufuf}','S_{mm}');
xlabel('f / Hz');ylabel('LDS / dB');xlim([0 f_s/2]);ylim(y_lims);
pause
plot(F, 10*log10(Pmm_bp),'k');
text(F(ind3),val3,[num2str(val3) 'dB']);
legend('S_{uu}','S_{ufuf}','S_{mm}','S_{mm_{BP}}');
xlabel('f / Hz');ylabel('LDS / dB');xlim([0 f_s/2]);ylim(y_lims);
pause
hold off;

%% Plotting (Demodulation)

[val3 ind3] =max(10*log10(Pufuf));
[val4 ind4] =max(10*log10(Pudfudf));
[val5 ind5] =max(10*log10(Pudefudef));
[val6 ind6] =max(10*log10(Pudefudef_bp));

figure
xlim([0 f_h+1000]);
ylabel('LDS / dB');
xlabel('f / Hz');
if AUDIO_ENABLED
    ylim=[-160 -50];
else
    ylim=[-100 10];
end

hold on;
plot(F, 10*log10(Pufuf));
text(F(ind3),val3,[num2str(val3) 'dB']);
legend('S_{ufuf}');
pause
plot(F, 10*log10(Pudfudf),'g');
text(F(ind4),val4,[num2str(val4) 'dB']);
legend('S_{ufuf}','S_{ZSB}');
pause
plot(F, 10*log10(Pudefudef),'r');
text(F(ind5),val5,[num2str(val5) 'dB']);
legend('S_{ufuf}','S_{ZSB}','S_{eeZSB}');
pause
plot(F, 10*log10(Pudefudef_bp),'k');
text(F(ind6),val6,[num2str(val6) 'dB']);
legend('S_{ufuf}','S_{ZSB}','S_{eeZSB}','S_{eeESB}');
pause

hold off;

disp( [ num2str(20*log10(cos(phi_error))) ] );

Matlab examples from the exercises as a zip file

Current Exam

Date: 04.03.2020
Time: 09:00 h - 10:30 h
Room: OHP5 - Chemie I + II

Previous Exams

	Link		Content		Link		Content
			Exam of the winter term 2019/2020				Corresponding solution
			Exam of the summer term 2019				Corresponding solution
			Exam of the winter term 2018/2019				Corresponding solution
			Exam of the summer term 2018				Corresponding solution
			Exam of the winter term 2017/2018				Corresponding solution

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Due to the work of Owe Wisch and Alexej Namenas (and of the rest of the SONAR team, of course) our SONAR simulator supports now a real-time mode for testing underwater speech communication. A multitude of "subscribers" can connect to our virtual ocean and send and receive signals. The simulator consists of large (time-variant) convolution engine as well as a realistic noise simulation that takes parameters such as wind speed, etc. into account. In addition, linear and non-linear behaviour of the sending and receiving hardware can be simulated. However, no simuator is of course as beautiful as the real sea.

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