Lecture "Signals and Systems I"

 

Basic Information
Lecturers: Gerhard Schmidt (lecture) and Johannes Hoffmann (exercise)
Room: C-SR-I
E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
Language: German
Target group: Students in electrical engineering and computer engineering
Prerequisites: Mathematics for engineers I, II & III; Foundations in electrical engineering I & II
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:

  • Basic classes of signals and systems
    • Introduction and notation
    • Signals
    • Systems
  • Signals
    • Elementary signals
    • Reaction of linear systems on elementary signals
    • Signal decomposition into elementary signals
  • Spectral representations of deterministic signals
    • Fourier series, Discrete Fourier Transform (DFT)
    • Fourier transform
    • Laplace and z transform
  • Linear systems
    • Reaction on elementary signals
    • Reaction on arbitrary signals
    • Quantities for describing linear systems and their relations
    • Stability of linear systems
    • Rational transfer-functions
  • Modulation
    • Basics
    • Sampling theorem
References: H.W. Schüßler: Netzwerke, Signale und Systeme II: Theorie kontinuierlicher und diskreter Signale und Systeme, Springer, 1991
H.D. Lüke: Signalübertragung, Springer, 1995

 

News

Scroll down to find a list of the topics with a detailed schedule for the exercises in SS23.

13.04.2023: There will be an additional lecture instead of the exercise.

Lecture Slides

Link Content
Slides of the lecture "Introduction"
(details of the lecture, notation, signals, systems)
Slides of the lecture "Signals"
(basic signals, reaction on basic signals, signal decomposition)
Slides of the lecture "Spectral Representations - Part 1"
(Fourier series, Discrete Fourier Transform)
Slides of the lecture "Spectral Representations - Part 2"
(Fourier Transform)
Slides of the lecture "Spectral Representations - Part 3"
(Laplace and z-transform)
Slides of the lecture "Linear Systems"
(Basics and relations between different system descriptions)
Slides of the lecture "Modulation"
(Basics and linear modulation schemes)

 

Lecture Script

A script for the lecture is available here. It is currently under construction, but a first version is already available via the link mentioned before. If you find some error, please let me (Gerhard Schmidt) know.

 

Lecture Videos

Video Content

Introduction - part 1 of 3

Introduction - part 2 of 3

Introduction - part 3 of 3

Signals - part 1 of 3

Signals - part 2 of 3

Signals - part 3 of 3

Spectra, Fouier series and DFT - part 1 of 3

Spectra, Fouier series and DFT - part 2 of 3

Spectra, Fouier series and DFT - part 3 of 3

Spectra, Fouier transformations - part 1 of 3

Spectra, Fouier transformations - part 2 of 3

Spectra, Fouier transformations - part 3 of 3

Spectra, z and Laplace transform - part 1 of 3

Spectra, z and Laplace transform - part 2 of 3

Spectra, z and Laplace transform - part 3 of 3

Linear systems - part 1 of 4

Linear systems - part 2 of 4

Linear systems - part 3 of 4

Linear systems - part 4 of 4

Modulation - part 1 of 1

 

Exercises

Date Event
13.04.2023 No exercise (lecture instead)
20.04.2023 Theory: Matlab introduction and system properties
27.04.2023 Question time: Please prepare exercises 1 and 2!
04.05.2023 Theory: Periodicity and Fourier Series
11.05.2023 Question time: Please prepare exercises 5, 6, 7 and 9!
18.05.2023 No exercise (public holiday)
25.05.2023 Theory: DFT and Fourier transform
01.06.2023 Question time: Please prepare exercises 11, 12, 14, 15 and 16!
08.06.2023 No exercise (lecture instead)
15.06.2023 Theory: Convolution and Laplace transform
22.06.2023 Question time: Please prepare exercises 18, 19, 21, 23 and 25!
29.06.2023 Theory: Z Transform
06.07.2023 Question time: Please prepare exercises 27, 28, 30 and 31!
XX.XX.2023 - XX:XX h Exam preparation

 

Exercises and Solutions

Link Content
All exercises.
Solutions to all exercises.

 

Exercise Videos

Video Content  

Exercise 1

 

Exercise 2

 

Exercise 3

Optional exercise

Exercise 4

Optional exercise

Exercise 5

 

Exercise 6

 

Exercise 7

 

Exercise 8

Optional exercise

Exercise 9

 

Exercise 10

Optional exercise

Exercise 11

 

Exercise 12

 

Exercise 13

Optional exercise

Exercise 14

 

Exercise 15

 

Exercise 16

 

Exercise 17

Optional exercise

Exercise 18

 

Exercise 19

 

Exercise 20

Optional exercise

Exercise 21

 

Exercise 22

Optional exercise

Exercise 23

 

Exercise 24

Optional exercise

Exercise 25

 

Exercise 26

Optional exercise

Exercise 27

 

Exercise 28

 

Exercise 29

Optional exercise

Exercise 30

 

Exercise 31

 

Exam preparation (exam of winter term 2017/2018)

 

Exam preparation (exam of winter term 2020/2021)

 

 

Matlab Theory Examples

How to use Matlab as a student:

  1. Campus-wide license (recommended)
  2. Windows Remote Server
Matlab introduction (click to expand)

close all              % Close all plots
clear                   % Clear workspace
clc                      % Clear console

% Latex font
set(groot, 'DefaultTextInterpreter', 'LaTeX');
set(groot, 'DefaultAxesTickLabelInterpreter', 'LaTeX');
set(groot, 'DefaultAxesFontName', 'LaTeX');
set(groot, 'DefaultAxesFontSize', 14);
set(groot, 'DefaultLegendInterpreter', 'LaTeX'); 
cmap = lines;

% Get info from matlab docu e.g for command 'close' -------------
% help close % Short overview
% doc close  % Detailed documentation

% Matrix and element wise arithmetic operation -----------------------
% Use ; to suppress output of statement
% a = [1, 2, 3, 4];    % Row vector
% a = [1 2 3 4];       % Same row vector
% b = [2; 3; 4; 5];    % Column vector
% a = a';                 % Transpose vector
% c = b * a;             % Matrix multiplication
% c = a.* a;             % Element wise multiplication


%**************************************************************************
% Basic parameters
%**************************************************************************
f_s = 100;            % Sample rate in Hz
T = 10;                 % Signal duration in s
f_0 = 0.5;             % Test signal frequency in Hz
L = f_s * T;           % Signal length in samples

%**************************************************************************
% Generate signals
%**************************************************************************
% Generate equidistant time scale from 0 to 10 s
t = (0 : L - 1) / f_s;
% Generate sine test signal with frequency f_0
v = sin(2*pi*f_0*t); 

%**************************************************************************
% Create plots
%**************************************************************************

%%
% Minimal plot of test signal ----------------------------------------------
figure            % Open new figure
plot(t, v)        % Actual plotting operation

%%
% Plot with some beautification--------------------------------------------
figure                          
plot(t, v, 'LineWidth', 2, 'Color', 'red')           
title('Test signal with some beautification')
xlabel('Time / s')          % X axis label
ylabel('Magnitude')      % Y axis label
legend('Test signal')    % Add legend
grid on                        % Add legend
xlim([0 1/f_0])              % Limit x-axis to one period

%%
% Discrete stem plot ------------------------------------------------------
figure                          
stem(t, v)           
title('Test signal with stem plot')
xlabel('Time / s')          % X axis label
ylabel('Magnitude')      % Y axis label
legend('Test signal')    % Add legend
grid on                        % Add legend
xlim([0 0.5])                 % Limit x-axis


%%
% Multiple plots with hold on ----------------------------------------------
figure
plot(t, v)
title('Multiple plots with hold on')
hold on                 % Keep previous plot
plot(t, -v)
xlabel('Time / s')                                  
ylabel('Magnitude')                    
legend('Test signal', 'Inv. test signal') 
grid on                                              
xlim([0 1/f_0])                                   

%%
% Multiple plots with subplot ----------------------------------------------
% 2 vertical,1 horizontal plot
figure
subplot(2, 1, 1)    % First subplot
plot(t, v)      
title('Multiple plots with subplot')
ylabel('Magnitude')                    
legend('Test signal') 
grid on 
xlim([0 1/f_0])    

subplot(2, 1, 2)    % Second subplot
plot(t, -v)
xlabel('Time / s')                                  
ylabel('Magnitude')                    
legend('Inv. test signal') 
grid on                                              
xlim([0 1/f_0])    

%**************************************************************************
% 3D Plots (will be discussed next week)
%**************************************************************************

%%
% Plot v, v_2 and v_3 as 3D trajectory ----------------------------------

% Generate sine signal as first component
v_x = sin(2*pi*f_0*t).*t; 
% Generate cosine signal as second component
v_y = cos(2*pi*f_0*t).*t; 
% Generate linear function as third component
v_z = t;

figure                         
plot3(v_x, v_y, v_z)              % 3D plot
title('3D plot as trajectory')
xlabel('X / m')   
ylabel('Y / m')       
zlabel('Z / m')  
axis image                           % Proportional axis
grid on
%%
% Plot noise as surface plot ----------------------------------------------

% Generate gaussian noise as 2D matrix
L_2 = 50;                              % Use less samples
v_2d = randn(L_2, L_2);        % Generate noise
x =  1 : L_2;                           % X axis
y =  1 : L_2;                           % Y axis

% Surface plot
figure      
subplot(2, 1, 1)
surf(x, y, v_2d, 'EdgeColor', 'none', 'FaceColor', 'interp')
title('Surface plot 3D')
c = colorbar;                     % Use colorbar               
ylabel(c, 'Magnitude')
colormap jet
xlabel('X / m')   
ylabel('Y / m')       
zlabel('Z / m')  

subplot(2, 1, 2)
surf(x, y, v_2d, 'EdgeColor', 'none', 'FaceColor', 'interp')
title('Surface plot topview')
c = colorbar;
ylabel(c, 'Magnitude')
xlabel('X / m')   
ylabel('Y / m')       
zlabel('Z / m')  
view(0, 90)                        % Rotate view to XY
axis tight                           

System properties demo (click to expand)

close all hidden
clear
clc

set(groot, 'DefaultTextInterpreter', 'LaTeX');
set(groot, 'DefaultAxesTickLabelInterpreter', 'LaTeX');
set(groot, 'DefaultAxesFontName', 'LaTeX');
set(groot, 'DefaultAxesFontSize', 14);
set(groot, 'DefaultLegendInterpreter', 'LaTeX'); 
cmap = lines;

%**************************************************************************
% Basic paramters
%**************************************************************************
N = 10; % Number of samples

%**************************************************************************
% Calculate support variables and signals
%**************************************************************************
n = (0 : N - 1)';  % Time index

% v -----------------------------------------------------------
v = zeros(N, 1);   % Input signal
v(2) = 1;
v(5) = 0.5;
v(8) = 1;

% y1 -----------------------------------------------------------
y_1 = v;

% y2 -----------------------------------------------------------
% y3 = 2*v(n) + v(n - 1)
y_2 = 2*v;
y_2(2 : end) = y_2(2 : end) + v(1 : end - 1);

% y3 -----------------------------------------------------------
% y3 = 2*v(n + 1)*1/10*(n + 1) + 0.1
y_3 = zeros(N, 1);
y_3(1 : end - 1) = 2*v(2 : end);
y_3 = y_3.* (n + 1) / 10 + 0.1;

%**************************************************************************
% Plots
%**************************************************************************
% Evaluate section by section to get plots after each other

% Input signal -----------------------------------------------------
figure
% set(gcf, 'Position', [2000 0 1400 1400]) % Set window size

subplot(4, 1, 1)
stem(n, v, 'LineWidth', 2)
grid on
xlabel('$n$', 'FontSize', 20);
ylabel('$v(n)$', 'FontSize', 20);
grid on
title('Input signal  $v(n)$', 'FontSize', 20)

%%
% Output signal of system 1 -----------------------------------------------
subplot(4, 1, 2)
stem(n, y_1, 'LineWidth', 2, 'Color', cmap(2, :))
grid on
xlabel('$n$', 'FontSize', 20);
ylabel('$y_1(n)$', 'FontSize', 20);
title('First system: Real output signal $y_1(n)$', 'FontSize', 20)

%% 
% Complex output signal of system 1 ---------------------------------------
subplot(4, 1, 2)
stem(n, zeros(N, 1), 'LineWidth', 2, 'Color', cmap(2, :))
hold on
stem(n, y_1, 'LineWidth', 2, 'Color', cmap(3, :))
legend({'$\Re\big\{y_1(n)\big\}$', '$\Im\big\{y_1(n)\big\}$'}, 'FontSize', 20)
grid on
xlabel('$n$', 'FontSize', 20);
ylabel('$y_1(n)$', 'FontSize', 20);
title('First system: Complex output signal $y_1(n)$', 'FontSize', 20)

%%
% Output signal of system 2 -----------------------------------------------
subplot(4, 1, 3)
stem(n, y_2, 'LineWidth', 2, 'Color', cmap(4, :))
grid on
xlabel('$n$', 'FontSize', 20);
ylabel('$y_2(n)$', 'FontSize', 20);
title('Second system: Output signal $y_2(n)$', 'FontSize', 20)

%%
% Output signal of system 3 -----------------------------------------------
subplot(4, 1, 4)
stem(n, y_3, 'LineWidth', 2, 'Color', cmap(5, :))
grid on
xlabel('$n$', 'FontSize', 20);
ylabel('$y_3(n)$', 'FontSize', 20);
title('Third system: Output signal $y_3(n)$', 'FontSize', 20)

Periodicity demo (click to expand)

close all hidden
clear
clc

set(groot, 'DefaultTextInterpreter', 'LaTeX');
set(groot, 'DefaultAxesTickLabelInterpreter', 'LaTeX');
set(groot, 'DefaultAxesFontName', 'LaTeX');
set(groot, 'DefaultAxesFontSize', 14);
set(groot, 'DefaultLegendInterpreter', 'LaTeX'); 
cmap = lines;

%**************************************************************************
% Basic paramters
%**************************************************************************
f_s_h = 100;    % Sampling freq for "continuous" signal in Hz
T_0 = 4;          % Fundamental period for normalization in s
f_1 = 1;           % Fundamental freq for sin signal in Hz
T_max = 8;      % Signal duration in seconds

%**************************************************************************
% Calculate support variables
%**************************************************************************
% Continuous signal --------------------------------------
N_h = T_max * f_s_h + 1;    % Duration in samples
t_h = linspace(0, T_max / T_0, T_max * f_s_h + 1); % Time scale in seconds

%**************************************************************************
% Signal selection (uncomment a single signal to use)
%**************************************************************************
% Single cos signal ---------------------------------------
v = cos(2*pi*f_1*t_h);

% Multi cos signal -----------------------------------------
% v = cos(2*pi*f_1*t_h) + cos(2*pi*2*f_1*t_h);

% Rect signal -----------------------------------------------
% v = ones(N_h, 1);
% v(ceil(N_h/2) : end) = v(ceil(N_h/2) : end) * -1;

%**************************************************************************
% Plots
%**************************************************************************
% Evaluate section by section to get plots after each other

figure
% set(gcf, 'Position', [2000 0 1400 1350]) % Set window size

subplot(2, 1, 1)
plot(t_h, v, 'LineWidth', 1)
hold on
grid on
xlabel('$\frac{t}{T_0}$', 'FontSize', 30);
ylabel('$v(t)$', 'FontSize', 20);
title('Continuous signal  $v(t)$', 'FontSize', 20)
%%
% Periodicity check ---------------------------------------
T_p = 1 * T_0;  

plot([0 T_p / T_0], [0.01 0.01], 'LineWidth', 3, 'Color', cmap(3, :))
legend('Cont. signal', 'Periodicity')
%%
% Highest frequency check -----------------------------
T_n = 1 * T_0;

plot([0 T_n / T_0], [-0.01 -0.01], 'LineWidth', 3, 'Color', cmap(4, :))
legend('Cont. signal', 'Periodicity', 'Highest freq. component')

%%
% Sampling -------------------------------------------------
T_s = 0.125 * T_0;
t_s = downsample(t_h, f_s_h * T_s);
u = downsample(v, f_s_h * T_s);
N_1 = length(t_s);
n_1 = 0 : N_1 - 1;

stem(t_s, u, 'LineWidth', 2, 'Color', cmap(2, :), 'LineWidth', 2)
legend('Cont. signal', 'Periodicity', 'Highest freq. component', 'Sampling points')

%%
% Reconstruction using SI interpolation ------------
y_1 = zeros(N_h, 1);

r = T_0 / T_s;
t_n = t_h * r;

for i = 0 : N_1 - 1
    y_1 = y_1 + sinc((t_n - i))' * u(i + 1);
end

subplot(2, 1, 2)
stem(n_1, u, 'LineWidth', 2, 'Color', cmap(2, :))
hold on
plot(t_n, y_1)
grid on
xlabel('$n$', 'FontSize', 20);
ylabel('$u(n)$', 'FontSize', 20);
title('Discrete signal  $u(n)$', 'FontSize', 20)
legend('Discrete signal', 'Reconstruction')

Fourier series demo (click to expand)

close all hidden
clear
clc

set(groot, 'DefaultTextInterpreter', 'LaTeX');
set(groot, 'DefaultAxesTickLabelInterpreter', 'LaTeX');
set(groot, 'DefaultAxesFontName', 'LaTeX');
set(groot, 'DefaultAxesFontSize', 14);
set(groot, 'DefaultLegendInterpreter', 'LaTeX'); 
cmap = lines;

%**************************************************************************
% Basic paramters
%**************************************************************************
f_s_h = 200;    % Sampling freq for "continuous" signal in Hz
T_0 = 4;          % Fundamental period for normalization in s
T_max = 4;      % Signal duration in seconds

%**************************************************************************
% Calculate support variables
%**************************************************************************
% Continuous signal --------------------------------------
N_h = T_max * f_s_h + 1;    % Duration in samples
t_h = linspace(0, T_max / T_0, T_max * f_s_h + 1); % Time scale in seconds

%**************************************************************************
% Signal generation
%**************************************************************************

% Rect signal -----------------------------------------------
v = ones(N_h, 1);
v(ceil(N_h/2) : end) = v(ceil(N_h/2) : end) * -1;

% Fourier series approximation
y = zeros(N_h, 1);

% Number of trigonometric coefficients
M = 10;

% Offset
c_0 = 0;

% Even coefficients
a = zeros(M, 1);

% Odd coefficients
b = zeros(M, 1);

% Calculate odd coefficients according to equation
% b_mu = (4 * h) / (pi * mu)
% Also calculate limited Fourier series approx
for mu = 1 : M 
    if mod(mu, 2)
        b(mu) = (4*1) / (pi * mu);
        y = y + b(mu) * sin(2*pi*mu*t_h)';
    else
        b(mu) = 0;
    end
end

% Calculate complex coefficients from trigonometric ones
% c_mu = 1/2 * (a_mu + j * b_mu) with c(-mu) = c(mu)*
c = 1/2 * ([flip(a); c_0; a] + 1i*[-flip(b); 0; b]);

%**************************************************************************
% Plots
%**************************************************************************
% Evaluate section by section to get plots after each other

figure
% set(gcf, 'Position', [2000 0 1400 1350]) % Set window size


subplot(3, 1, 1)
plot(t_h, v, 'LineWidth', 2)
hold on
grid on
xlabel('$\frac{t}{T_0}$', 'FontSize', 30);
ylabel('$v(t)$', 'FontSize', 20);
title('Continuous rect signal', 'FontSize', 20)

%%
plot(t_h, y, 'LineWidth', 2);
legend('Original signal', 'Finite Fourier series approx.', 'FontSize', 20)

subplot(3, 1, 2)
stem(0, c_0, 'LineWidth', 2)
hold on
stem(a, 'LineWidth', 2)
stem(b, 'LineWidth', 2)
grid on
xlim([0 max(10, M)])
xlabel('$\mu$', 'FontSize', 20);
ylabel('$a_\mu$, $b_\mu$', 'FontSize', 25);
title(['Trigonometric Fourier coefficients ($\mu \leq$ ', num2str(M), ')'], 'FontSize', 20)
legend({'$c_0 = \frac{a_0}{2}$', '$ a_\mu$', '$ b_\mu$'}, 'FontSize', 20)

%%
subplot(3, 1, 3)
stem(-M : M, real(c), 'LineWidth', 2)
hold on
stem(-M : M, imag(c), 'LineWidth', 2)
grid on
xlim([-max(10, M) max(10, M)])
xlabel('$\mu$', 'FontSize', 20);
ylabel('$c_\mu$', 'FontSize', 25);
title(['Complex Fourier coefficients ($\mu \leq$ ', num2str(M), ')'], 'FontSize', 20)
legend({'$\Re\big\{c_\mu\big\}$', '$\Im\big\{c_\mu\big\}$'}, 'FontSize', 20)

Discrete Fourier transform demo (click to expand)

close all hidden
clear
clc

set(groot, 'DefaultTextInterpreter', 'LaTeX');
set(groot, 'DefaultAxesTickLabelInterpreter', 'LaTeX');
set(groot, 'DefaultAxesFontName', 'LaTeX');
set(groot, 'DefaultAxesFontSize', 14);
set(groot, 'DefaultLegendInterpreter', 'LaTeX'); 
cmap = lines;

%**************************************************************************
% Basic paramters
%**************************************************************************
f_s = 10;
f_s_h = 200;    % Sampling freq for "continuous" signal in Hz
T_0 = 4;          % Fundamental period for normalization in s
f_1 = 1;           % Fundamental freq for sin signal in Hz
T_max = 4;      % Signal duration in seconds

%**************************************************************************
% Calculate support variables
%**************************************************************************
% Discrete signal --------------------------------------
N = T_max * f_s + 1;                             % Duration in samples
n = 0 : N - 1;                                         % Time indices
mu = -floor((N - 1) / 2) : floor((N - 1) / 2);  % Spectral indices
mu_abs = 0 : floor((N - 1) / 2);                % One-sided spectral indices
%**************************************************************************
% Signal generation
%**************************************************************************

% Unity impulse -----------------------------------------------
% v = zeros(N, 1);
% v(1) = 1;

% Offset ---------------------------------------------------------
v = ones(N, 1);

% Sine signal ------------------------------------------------
% m = 1;
% h = 0;
% v = sin(2*pi*m/N * n) + h;

% Rect signal ------------------------------------------------
% v = ones(N, 1);
% v(n > N / 2) = -v(n > N/2);
% v(1) = 0;


% Compute DFT / FFT
V = fftshift(fft(v) / N);

% Compute one-sided spectrum
V_abs = abs(V(mu >= 0));
V_abs(mu_abs > 0) = 2 * V_abs(mu_abs > 0);

%**************************************************************************
% Plots
%**************************************************************************
% Evaluate section by section to get plots after each other

figure
% set(gcf, 'Position', [2000 0 1400 1350]) % Set window size

subplot(3, 1, 1)
stem(n, v, 'LineWidth', 2)
hold on
grid on
xlabel('$n$', 'FontSize', 20);
ylabel('$v(n)$', 'FontSize', 20);
title('Discrete signal', 'FontSize', 20)

%%

subplot(3, 1, 2)
stem(mu, real(V), 'LineWidth', 2)
hold on
stem(mu, imag(V), 'LineWidth', 2)
grid on
xlabel('$\mu$', 'FontSize', 20);
ylabel('$V(\mu)$', 'FontSize', 20);
title('DFT spectrum', 'FontSize', 20)
legend({'$\Re\big\{V(\mu)\big\}$', '$\Im\big\{V(\mu)\big\}$'}, 'FontSize', 20)

%%
subplot(3, 1, 3)
stem(mu_abs, V_abs, 'LineWidth', 2)
grid on
xlabel('$\mu$', 'FontSize', 20);
ylabel('$|V(\mu)|$', 'FontSize', 20);
title('One-sided DFT spectrum', 'FontSize', 20)

Fourier transform demo (click to expand)

close all hidden
clear
clc

set(groot, 'DefaultTextInterpreter', 'LaTeX');
set(groot, 'DefaultAxesTickLabelInterpreter', 'LaTeX');
set(groot, 'DefaultAxesFontName', 'LaTeX');
set(groot, 'DefaultAxesFontSize', 14);
set(groot, 'DefaultLegendInterpreter', 'LaTeX'); 
cmap = lines;

%**************************************************************************
% Basic parameters
%**************************************************************************
f_s = 200;        % Sampling freq for "continuous" signal in Hz
T_0 = 2;          % Fundamental period for signal description in s
T_max = 100;      % Signal duration in seconds

%**************************************************************************
% Calculate support variables
%**************************************************************************
N = T_max * f_s + 1;                     % Duration in samples
t = linspace(-T_max/2, T_max/2, N);      % Time scale in s
w = 2 * pi * linspace(-f_s/2, f_s/2, N); % Frequency scale in s^-1

%**************************************************************************
% Signal generation
%**************************************************************************

signalSel = 5;

switch(signalSel)
    case 1
    % Rect signal -----------------------------------------------
    % Set time signal using ones and zeros
    v = ones(N, 1);                 % Fill signal with ones
    v(t < -T_0) = 0 * v(t < -T_0);  % Set signal before rect pulse to 1
    v(t > T_0) = 0 * v(t > T_0);    % Set signal after rect pulse to 0

    % Set discretized Fourier spectrum to sin(x)/x function
    % NaN value at zero is okay here
    V = abs(2 * sin(w * T_0) ./ w); 
    
   % Use sinc (si) function as better alternative 
   % Only with dsp toolbox
   %    V = abs(2 * T_0 * sinc(w / pi * T_0));
    
    % Calculate approx abs Fourier spectrum based on DFT
    V_dft = abs(fftshift(fft(v) / f_s));
    
    case 2
    % Sinc signal ----------------------------------------------
    % Define auxiliary parameters
    f_0 = 1;
    w_0 = 2 * pi * f_0;
    
    % Set time signal to sin(x)/x function
    % NaN value at zero is not okay here as
    % it would make the FFT result NaNs only
    v = (sin(w_0 * t)  + eps) ./ (w_0 * t + eps);
    
    % Use sinc (si) function as better alternative 
    % Only with dsp toolbox
    %  v = sinc(w_0 / pi * t);
    
    % Set discretized Fourier spectrum to rect pulse
    V = pi / w_0 * ones(N, 1);
    V(w < -w_0) = 0 * V(w < -w_0);
    V(w > w_0) = 0 * V(w > w_0);
   
    % Calculate approx abs Fourier spectrum based on DFT
    V_dft = abs(fftshift(fft(v) / f_s));
    
    case 3
    % Exp signal ----------------------------------------------
    % Define auxiliary parameters
    alpha = 0.1;
    
    % Set time signal to exp function for t > 0
    v = zeros(N, 1);
    v(t >= 0) = exp(-alpha * t(t >= 0));
    
    % Set discretized Fourier spectrum to 1/(a + jw)
    V = abs(1 ./ (alpha + 1i * w)); 
   
    % Calculate approx abs Fourier spectrum based on DFT
    V_dft = abs(fftshift(fft(v) / f_s));
    
    case 4
    % Cos signal ----------------------------------------------
    % Define auxiliary parameters
    f_0 = 1;
    w_0 = 2 * pi * f_0;
    
    % Set time signal to exp function for t > 0
    v = cos(w_0 * t);
    
    % Set discretized Fourier spectrum to zero 
    V = zeros(N, 1);
    % Find freq index closest to w_0
    [~, w_0_c] = min(abs(w - w_0));
    % Set value of found index to pi
    V(w_0_c) = pi;
    % Find freq index closest to -w_0
    [~, w_0_c] = min(abs(w + w_0));
    % Set value of found index to pi
    V(w_0_c) = pi;
   
    % Calculate approx abs Fourier spectrum based on DFT
    % Scaling is different here, why?
    V_dft = abs(fftshift(fft(v) /  N * 2 * pi));
    
    case 5
    % Combined rect signal -----------------------------
    % Define auxiliary parameters
    a = 1;
    
    % Set signal composed of multiple rect functions
    v = ones(N, 1);                        % Fill signal with ones
    v(t < -T_0) = 0 * v(t < -T_0);     % Set signal to 0 before first rect pulse
    v(t > -T_0) = a * v(t > -T_0);     % Set signal to a from first rect pulse
    v(t > 0) = 2 * v(t > 0);               % Set signal to 2 * a from second rect pulse
    v(t > T_0) = -1/2 * v(t > T_0);    % Set signal to - a from third rect pulse
    v(t > 4*T_0) = 0 * v(t > 4*T_0);  % Set signal to 0 after third rect pulse
    
    
    % Fourier transform according to sample solution
    % Using eps to avoid NaN value in the center
    V = a ./ (1i * w + eps) .* (exp(1i*T_0.*w) + 1 - 3 .* exp(-1i*T_0.*w) + exp(-4i*T_0.*w));

    % Alternative approach to get Fourier transform using rect pulses
    % DSP toolbox only
    % V = 1 * a * T_0 * sinc(w / pi * 0.5 * T_0) .* exp(0.5i * w * T_0);
    % V = V +  2 * a * T_0 * sinc(w / pi * 0.5 * T_0) .* exp(-0.5i * w * T_0);
    % V = V -  3 * a * T_0 * sinc(w / pi * 1.5 * T_0) .* exp(-2.5i * w * T_0);

    % Take absolute value
    V = abs(V);

    V_dft = abs(fftshift(fft(v) /  f_s));
        
end

%**************************************************************************
% Plots
%**************************************************************************
% Evaluate section by section to get plots after each other

w_lim = 20;

figure
set(gcf, 'Position', [2000 0 1400 1350]) % Set window size

subplot(3, 1, 1)
plot(t, v, 'LineWidth', 2, 'Color', cmap(1, :))
hold on
grid on
xlabel('$t$ / $\mathrm{s}$', 'FontSize', 20);
ylabel('$v(t)$', 'FontSize', 20);
title('Continuous time signal (discretized)', 'FontSize', 20)

subplot(3, 1, 2)
plot(w, V, 'LineWidth', 2, 'Color', cmap(2, :))
grid on
xlabel('$\omega$ / $\mathrm{s^{-1}}$', 'FontSize', 20);
ylabel('$|V(j\omega)|$', 'FontSize', 20);
title('Fourier spectrum (discretized)', 'FontSize', 20)
xlim([-w_lim, w_lim])

subplot(3, 1, 3)
plot(w, V_dft, 'LineWidth', 2, 'Color', cmap(3, :))
grid on
xlabel('$\omega$ / $\mathrm{s^{-1}}$', 'FontSize', 20);
ylabel('$|V_{\mathrm{dft}}(j\omega)|$', 'FontSize', 20);
title('Fourier spectrum approx. using DFT', 'FontSize', 20)
xlim([-w_lim, w_lim])

Convolution demo (click to expand)

close all hidden
clear
clc

set(groot, 'DefaultTextInterpreter', 'LaTeX');
set(groot, 'DefaultAxesTickLabelInterpreter', 'LaTeX');
set(groot, 'DefaultAxesFontName', 'LaTeX');
set(groot, 'DefaultAxesFontSize', 14);
set(groot, 'DefaultLegendInterpreter', 'LaTeX'); 
cmap = lines;

%**************************************************************************
% Basic paramters
%**************************************************************************
f_s = 10;
T_max = 4;      % Signal duration in seconds

%**************************************************************************
% Calculate support variables
%**************************************************************************
% Discrete signal --------------------------------------
N = T_max * f_s + 1;                             % Duration in samples
n = 0 : N - 1;                                         % Time indices
mu = -floor((N - 1) / 2) : floor((N - 1) / 2);  % Spectral indices
mu_abs = 0 : floor((N - 1) / 2);                % One-sided spectral indices

N_c = 2 * N - 1;                                     % Length of conv. singal in samples
n_c = 0 : N_c - 1;                                  % Time indices of conv signal

%**************************************************************************
% Signal generation
%**************************************************************************

% Set v to rect pulse --------------------------------------------
v = zeros(N, 1);
v(1 : 10) = 1; 

% Set signal u ----------------------------------------------------
u = zeros(N, 1);
u(1) = 1;               % Unity impulse  
% u(10) = 1;             % Shifted unity impulse
% u(1 : 10) = 1;         % Short rect pulse
% u(1 : 15) = 1;         % Long rect pulse

% Compute convolution of v and u -------------------------
y = conv(v, u);

% Compute DFT / FFT
V = fftshift(fft(v) / N);

U = fftshift(fft(u) / N);

Y = fftshift(fft(y(n_c < N) / N));

%**************************************************************************
% Plots
%**************************************************************************
% Evaluate section by section to get plots after each other

figure
% set(gcf, 'Position', [2000 0 1400 1350]) % Set window size

subplot(3, 2, 1)
stem(n, v, 'LineWidth', 2)
hold on
grid on
xlabel('$n$', 'FontSize', 20);
ylabel('$v(n)$', 'FontSize', 20);
title('Discrete time signal $v(n)$', 'FontSize', 20)

%%
subplot(3, 2, 2)
stem(mu, abs(V), 'LineWidth', 2)
grid on
xlabel('$\mu$', 'FontSize', 20);
ylabel('$|V(\mu)|$', 'FontSize', 20);
title('Abs. DFT spectrum $|V(\mu)|$', 'FontSize', 20)

%%
subplot(3, 2, 3)
stem(n, u, 'LineWidth', 2, 'Color', cmap(2, :))
grid on
xlabel('$n$', 'FontSize', 20);
ylabel('$u(n)$', 'FontSize', 20);
title('Discrete time signal $u(n)$', 'FontSize', 20)

%%
subplot(3, 2, 4)
stem(mu, abs(U), 'LineWidth', 2, 'Color', cmap(2, :))
grid on
xlabel('$\mu$', 'FontSize', 20);
ylabel('$|U(\mu)|$', 'FontSize', 20);
title('Abs. DFT spectrum $|U(\mu)|$', 'FontSize', 20)

%%
subplot(3, 2, 5)
stem(n_c, y, 'LineWidth', 2, 'Color', cmap(3, :))
grid on
xlim([0, N - 1])
xlabel('$n$', 'FontSize', 20);
ylabel('$y(n)$', 'FontSize', 20);
title('Discrete time signal $y(n)$', 'FontSize', 20)

%%
subplot(3, 2, 6)
stem(mu, abs(Y), 'LineWidth', 2, 'Color', cmap(3, :))
grid on
xlabel('$\mu$', 'FontSize', 20);
ylabel('$|Y(\mu)|$', 'FontSize', 20);
title('Abs. DFT spectrum $|Y(\mu)|$', 'FontSize', 20)

Laplace transform demo (click to expand)

close all hidden
clear
clc

set(groot, 'DefaultTextInterpreter', 'LaTeX');
set(groot, 'DefaultAxesTickLabelInterpreter', 'LaTeX');
set(groot, 'DefaultAxesFontName', 'LaTeX');
set(groot, 'DefaultAxesFontSize', 14);
set(groot, 'DefaultLegendInterpreter', 'LaTeX'); 
cmap = lines;

%**************************************************************************
% Basic parameters
%**************************************************************************
f_s = 200;        % Sampling freq for "continuous" signal in Hz
T_0 = 1;          % Fundamental period for signal description in s
T_max = 100;      % Signal duration in seconds

%**************************************************************************
% Calculate support variables
%**************************************************************************
N = T_max * f_s + 1;                     % Duration in samples
t = linspace(-T_max/2, T_max/2, N);      % Time scale in s
w = 2 * pi * linspace(-f_s/2, f_s/2, N); % Frequency scale in s^-1

%**************************************************************************
% Signal generation
%**************************************************************************

signalSel = 1;

switch(signalSel)
    
    case 1
        % Sin signal with heaviside function ------------------------------
        % Define auxiliary parameters
        sigma = 0.01;
        w_0 = 2 * pi * 1 / T_0;

        % Time domain signal and labels 
        v = sin(w_0 * t);
        v(t < 0) = 0;

        t_lim = 2*[-1 1];
        t_ticks = [-2 -1 0 1 2];
        t_labels = {'', '', '0', '$T_0$', ''};

        v_ticks = [-1 0 1];
        v_labels = {'-1', '0', '1'};

        % Area of conversion 
        s_re = [0 1 1 0]; 
        s_im = [1 1 -1 -1];

        s_re_labels = {'$-\infty$', '', '0', '', '$\infty$'};

        % Laplace domain signal and labels
        s_conv = sprintf('$\mathrm{Re}\{s\} = %.2f $', sigma);

        w_lim = 2*w_0*[-1 1];
        w_ticks = w_0*[-2 -1 0 1 2];
        w_labels = {'', '$-\omega_0$', '0', '$\omega_0$', ''};

        V = w_0 ./ (sigma^2 + 1i * 2 * w + w_0^2 - w.^2);
        V = abs(V);
        V_lim = max(V) * [0 1];
        V_ticks = max(V) * [0 1];
        V_labels = {'0', '0.5'};
    
    case 2
        % Sin signal without heaviside function -------------------------------
        % Define auxiliary parameters
        w_0 = 2 * pi * 1 / T_0;

        % Time domain signal and labels 
        v = sin(w_0 * t);

        t_lim = 2*[-1 1];
        t_ticks = [-2 -1 0 1 2];
        t_labels = {'', '', '0', '$T_0$', ''};

        v_ticks = [-1 0 1];
        v_labels = {'-1', '0', '1'};

        % Area of conversion 
        s_re = [-0.01, -0.01 0.01 0.01]; 
        s_im = [-1 1 1 -1];

        s_re_labels = {'$-\infty$', '', '0', '', '$\infty$'};

        % Laplace domain signal and labels
        s_conv = sprintf('$\mathrm{Re}\{s\} = 0 $');

        w_lim = 2*w_0*[-1 1];
        w_ticks = w_0*[-2 -1 0 1 2];
        w_labels = {'', '$-\omega_0$', '0', '$\omega_0$', ''};

        V = zeros(N, 1);
        % Find freq index closest to w_0
        [~, w_0_c] = min(abs(w - w_0));
        % Set value of found index to pi
        V(w_0_c) = 0.5;
        % Find freq index closest to -w_0
        [~, w_0_c] = min(abs(w + w_0));
        % Set value of found index to pi
        V(w_0_c) = 0.5;

        V = abs(V);

        V_lim = max(V) * [0 1];
        V_ticks = max(V) * [0 1];
        V_labels = {'0', '$\infty$'};
    
    case 3   
        % Decreasing exp with heaviside function --------------------------
        % Define auxiliary parameters
        alpha = -0.7;
        sigma = 0.6;
        T_h = log(0.5) / alpha;
        w_0 = 2 * pi * 1 / T_0;

        % Time domain signal and labels 
        v = exp(alpha*t); % sin(w_0 * t);
        v(t < 0) = 0;

        t_lim = T_h*2*[-1 1];
        t_ticks = T_h* [-2 -1 0 1 2];
        t_labels = {'', '', '0', '$T_h$', ''};

        v_ticks = [-1 0 1];
        v_labels = {'-1', '0', '1'};

        % Area of conversion 
        s_re = [-0.5, -0.5 1 1]; 
        s_im = [-1 1 1 -1];

        s_re_labels = {'$-\infty$', '$s_{\infty}$', '0', '', '$\infty$'};

        % Laplace domain signal and labels
        s_conv = sprintf('$\mathrm{Re}\{s\} = %.1f $', sigma);

        w_lim = 4*pi/T_h * [-1 1];
        w_ticks = 2*pi/T_h * [-2 -1 0 1 2];
        w_labels = {'', '$-\frac{2 \pi}{T_h}$', '0', '$\frac{2 \pi}{T_h}$', ''};

        V = 1 ./ (alpha - sigma - 1i*w);
        V = abs(V);

        V_lim = 1.1 * max(V) * [0 1];
        V_ticks = 1.1 * max(V) * [0 1];
        V_labels = {'0', '$b$'};
  
    case 4
        % Increasing exp with heaviside function --------------------------
        % Define auxiliary parameters
        alpha = 0.7;
        sigma = 0.9;
        T_h = log(2) / alpha;
        w_0 = 2 * pi * 1 / T_0;

        % Time domain signal and labels 
        v = exp(alpha*t); % sin(w_0 * t);
        v(t < 0) = 0;

        t_lim = T_h*2*[-1 1];
        t_ticks = T_h* [-2 -1 0 1 2];
        t_labels = {'', '', '0', '$T_d$', ''};

        v_ticks = [-1 0 1];
        v_labels = {'-1', '0', '1'};

        % Area of conversion 
        s_re = [0.5, 0.5 1 1]; 
        s_im = [-1 1 1 -1];

        s_re_labels = {'$-\infty$', '', '0', '$s_{\infty}$', '$\infty$'};

        % Laplace domain signal and labels
        s_conv = sprintf('$\mathrm{Re}\{s\} = %.1f $', sigma);

        w_lim = 4*pi/T_h * [-1 1];
        w_ticks = 2*pi/T_h * [-2 -1 0 1 2];
        w_labels = {'', '$-\frac{2 \pi}{T_d}$', '0', '$\frac{2 \pi}{T_d}$', ''};

        V = 1 ./ (alpha - sigma - 1i*w);
        V = abs(V);

        V_lim = 1.1 * max(V) * [0 1];
        V_ticks = 1.1 * max(V) * [0 1];
        V_labels = {'0', '$b$'};
   
end

%**************************************************************************
% Plots
%**************************************************************************
% Evaluate section by section to get plots after each other


figure
% set(gcf, 'Position', [2000 0 1400 1350]) % Set window size

f = subplot(3, 1, 1);
plot(t, v, 'LineWidth', 2, 'Color', cmap(1, :))
hold on
grid on
xlabel('$t$', 'FontSize', 20);
ylabel('$v(t)$', 'FontSize', 20);
title('Continuous time signal', 'FontSize', 20)
xlim(t_lim)
f.XTickLabelMode = 'manual';
f.XTick = t_ticks; 
f.XTickLabel = t_labels; 

f.YTickLabelMode = 'manual';
f.YTick = v_ticks;
f.YTickLabel = v_labels; 


%%
g = subplot(3, 1, 2);
fill(s_re, s_im, cmap(2, :), 'EdgeColor', 'none')
grid on
xlabel('$\mathrm{Re}\{s\}$', 'FontSize', 20);
ylabel('$\mathrm{Im}\{s\}$', 'FontSize', 20);
title('Area of conversion', 'FontSize', 20)
xlim([-1 1])
ylim([-1 1])

g.XTickLabelMode = 'manual';
g.XTick = [-1 -0.5 0 0.5 1];
g.XTickLabel = s_re_labels;

g.YTickLabelMode = 'manual';
g.YTick = [-1 0 1];
g.YTickLabel = {'$-\infty$', '0', '$\infty$'};


%%
h = subplot(3, 1, 3);
plot(w, V, 'LineWidth', 2, 'Color', cmap(3, :))
grid on
xlabel('$\mathrm{Im}\{s\}$', 'FontSize', 20);
ylabel('$|V(s)|$', 'FontSize', 20);
title(['Laplace spectrum (' s_conv ')'], 'FontSize', 20)
xlim(w_lim)
h.XTickLabelMode = 'manual';
h.XTick = w_ticks; 
h.XTickLabel = w_labels; 

ylim(V_lim)
h.YTickLabelMode = 'manual';
h.YTick = V_ticks;
h.YTickLabel = V_labels;

z-Domain system demo (click to expand)

close all hidden
clear
clc

set(groot, 'DefaultTextInterpreter', 'LaTeX');
set(groot, 'DefaultAxesTickLabelInterpreter', 'LaTeX');
set(groot, 'DefaultAxesFontName', 'LaTeX');
set(groot, 'DefaultAxesFontSize', 14);
set(groot, 'DefaultLegendInterpreter', 'LaTeX'); 
cmap = lines;

%**************************************************************************
% Basic parameters
%**************************************************************************
f_s = 48e3;          % Sampling freq in Hz

%**************************************************************************
% System characterisation
%**************************************************************************
systemSel = 4;

% All systems are real-valued (partly complex-conj)
switch(systemSel)
    case 1
        % Stable FIR highpass filter
        % Minimal phase
        z = [0.5 + 0.5j, 0.5 - 0.5j]';
        p = [0 0]';
        [b, a] = zp2tf(z, p, 1);
     case 2
        % Instable IIR lowpass/bandpass filter
        % Maximal phase
        z = [-1 + 1i, -1 - 1i]';
        p = [1/sqrt(2) * (1.2 + 1i), 1/sqrt(2) * (1.2 - 1i)]';
        [b, a] = zp2tf(z, p, 1);
    case 3
        % Stable IIR lowpass (Butterworth) filter
        % Mixed phase (implementation-dependent)
        N = 4; 
        [b, a] = butter(N, 0.5);
        [z, p, ~] = tf2zpk(b, a);
    case 4
        % Stable FIR highpass filter
        % Linear phase
        z = [1/(2 * sqrt(2)) *[1 + 1i, 1 - 1i], sqrt(2) *[1 + 1i, 1 - 1i]]';
        p = [0 0 0 0]';
        [b, a] = zp2tf(z, p, 1);       
end

% Get system from coefficients
H = tf(b, a, f_s);            

% Get frequency response from coefficients
[H_f, w] = freqz(b, a, 2^10);
% Norm frequency vector
w = w / pi;

% Get impulse response from system
[h, t_h] = impulse(H);

%**************************************************************************
% Plots
%**************************************************************************
% Evaluate section by section to get plots after each other

figure
% set(gcf, 'Position', [2000 0 1400 1350]) % Set window size

phi = linspace(0, 2*pi, 1000);
x = cos(phi);
y = sin(phi);

% Pole zero plot
subplot(4, 1, 1);
plot(real(z), imag(z), 'LineStyle', 'none', 'Marker', 'o', 'Color', cmap(1, :))
hold on
plot(real(p), imag(p), 'LineStyle', 'none', 'Marker', 'x', 'Color', cmap(1, :))
plot(x, y, '--', 'Color', 'Black')
plot([-4 4], [0 0], '--', 'Color', 'Black')
plot([0 0], [-4 4], '--', 'Color', 'Black')
xlim([-2 2])
ylim([-2 2])
grid on
xlabel('$\mathrm{Re}\big(z\big)$')
ylabel('$\mathrm{Im}\big(z\big)$')
title('Pole-zero plot')

%%
% Magnitude of frequency response
subplot(4, 1, 2)
plot(w, 20*log10(abs(H_f)))
grid on
xlabel('$\frac{\Omega}{\pi}$')
ylabel('$|H\big(e^{j \Omega}\big)|$ / dB')
title('Frequency response - magnitude')

%%
% Impulse response
subplot(4, 1, 3)
stem(t_h, h)
grid on
xlabel('$n \, T_s$ / s')
title('Impulse response')

%%
% Phase of frequency response
subplot(4, 1, 4)
plot(w, rad2deg(unwrap(angle(H_f)))) 
grid on
xlabel('$\frac{\mathrm{\Omega}}{\pi}$')
ylabel('$\phi$ / deg')
title('Frequency response - phase')

 

Evaluation

Current evaluation Completed evaluations

 

Formulary

A formulary can be found below:

Link Content
Formulary

 

Previous Exams

Link Content Link Content
Exam of the winter term 2022/2023 Corresponding solution
Exam of the summer term 2022 Corresponding solution
Exam of the winter term 2021/2022 Corresponding solution
Exam of the summer term 2021 Corresponding solution
Exam of the winter term 2020/2021 Corresponding solution