Signals and Systems II

Lecture "Signals and Systems II"

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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 Stochastic signals and corresponding spectra Linear systems and stochastic signals Ideal systems State-space description 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

In KW 43/19 and 44/19, there will be a lecture instead of the exercise. Date, time and room remain the same.

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 summer semester's exam of 26.09.2019 are now online in the QIS-system. Please note the date of the first exam inspection:

Date: Wednesday, 30.10.2019
Time: 12:30 h - 13:30 h
Room: Library of the DSS-group (D-037 A)

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.

		Slides of the lecture "Modulation" (Linear and nonlinear modulation, digitalization)
		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 "State-space description" (State-space representation of linear, time-invariant systems)
		Slides of the lecture "Extensions on transformations" (Spectra of sampled continuous and discrete signals, convergence conditions and inverse transformations)

An online version of the slides (a script) can be found on an extra web page.

Exercises can be found on an extra web page.

The evaluation can be found on an extra web page.

Completed evaluations can be found on an extra web page.

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 ''Zweiseitenbandmodulation''

%--------------------------------------------------------------------------
% Signals and Systems II - Exercise 1
%
% Sound example for the "encoded" speech signal.
%
%
% Nov. 2012, Nov. 2013, Nov. 2014
% Digital Signal Processing and System Theory
% Faculty of Engineering Christian-Albrechts-Universit�t zu Kiel
% Jochen Withopf, 
%--------------------------------------------------------------------------

clear

%--------------------------------------------------------------------------
% Set parameters, initialize variables
%--------------------------------------------------------------------------
fs = 100000; % has to be high enough for the modulated signals

% --- Minimum and maximum signal frequency
fmin =  300;
fmax = 3400;

% --- Modulation frequencies
f1    = 20000;
f2    = 25000;

% --- Filter cutoff frequencies (3dB cutoffs, we don't use stop band and
%     pass band edge frequencies for simplicity here) and filter order N
fc_hp = 20000;
fc_tp = 10000;
N     = 12;

% --- Design butterworth/Chebychev filters
% [b_hp a_hp] = butter( N, fc_hp/fs*2, 'high');
% [b_tp a_tp] = butter( N, fc_tp/fs*2, 'low');
[b_hp, a_hp] = cheby2( N, 30, fc_hp/fs*2, 'high');
[b_tp, a_tp] = cheby2( N, 30, fc_tp/fs*2, 'low');

% --- Load input signal, bandlimit and upsample it to sample rate fs;
%[u, fs_sig] = wavread( 'signals/interview_heino.wav' );
[u, fs_sig] = wavread( 'signals/sprachnotiz_1.wav' );

[b, a] = butter( 4, [fmin fmax]/fs_sig*2 );
u = filter( b, a, u );
u = resample( u, fs, fs_sig );

%--------------------------------------------------------------------------
% Modulation system
%--------------------------------------------------------------------------
t = linspace(0, length(u)/fs, length(u));
t = t(:);    % make t a column vector

g1 = u.*sin(2*pi*f1*t);
g2 = filter(b_hp, a_hp, g1);
g3 = g2.*sin(2*pi*f2*t);
g4 = filter(b_tp, a_tp, g3);

%--------------------------------------------------------------------------
% Plots and sound
%--------------------------------------------------------------------------

N_win = 1024*4;

[pu, f] = pwelch(u, N_win,128,N_win,fs);
pg1     = pwelch(g1,N_win,128,N_win,fs);
pg2     = pwelch(g2,N_win,128,N_win,fs);
pg3     = pwelch(g3,N_win,128,N_win,fs);
pg4     = pwelch(g4,N_win,128,N_win,fs);

% [pu, f] = pwelch(u,N_win,128,N_win,fs,'twosided');
% pg1    = pwelch(g1,N_win,128,N_win,fs,'twosided');
% pg2    = pwelch(g2,N_win,128,N_win,fs,'twosided');
% pg3    = pwelch(g3,N_win,128,N_win,fs,'twosided');
% pg4    = pwelch(g4,N_win,128,N_win,fs,'twosided');

figure(1);
plot(f/1000, 10*log10([pu pg1 pg2 pg3 pg4]), 'LineWidth', 1);
legend('|U(f)|', '|G_1(f)|', '|G_2(f)|', '|G_3(f)|', '|G_4(f)|');
grid on;
xlabel('Frequency in kHz');
ylabel('Magnitude in dB');

%--------------------------------------------------------------------------
% Spectrograms
%--------------------------------------------------------------------------
u_dn  = resample(u, fs_sig, fs);
g4_dn = resample(g4, fs_sig, fs);
M     = round(40/1000*fs_sig);
ovl   = round(M*0.9);
t_dn  = linspace(0, length(u_dn)/fs_sig, length(u_dn));

figure(2);
sp(1) = subplot(4,2,1);
plot(t_dn, u_dn);
title('Eingangssignal u(t)');
grid on;
xlim([t_dn(1) t_dn(end)]);

sp(2) = subplot(4,2,2);
plot(t_dn, g4_dn);
title('Ausgangssignal g_4(t)');
grid on;

sp(3) = subplot(4,2,[3,5,7]);
[S,F,T] = spectrogram(u_dn, M, ovl, M, fs_sig);
imagesc(T,F,20*log10(abs(S)));
axis xy
ylim([0 fs_sig/4]);
xlabel('Zeit in Sekunden');
ylabel('Frequenz in Hz');

sp(4) = subplot(4,2,[4, 6, 8]);
[S,F,T] = spectrogram(g4_dn, M, ovl, M, fs_sig);
imagesc(T,F,20*log10(abs(S)));
axis xy
ylim([0 fs_sig/4]);
xlabel('Zeit in Sekunden');
ylabel('Frequenz in Hz');

linkaxes(sp, 'x');

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

Date: XXX, xx.xx.2020
Time: 09:00 h - 10:30 h
Room: Coming soon

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

Exam of the summer term 2017

Corresponding solution