#!/usr/bin/env python
# coding: utf-8

# # Continuous Signals
# 
# *This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*

# ## Elementary Operations
# 
# Operations like superposition, temporal shifting and scaling are used to construct signals with a more complex structure than the previously introduced [standard signals](standard_signals.ipynb). A set of elementary operations are introduced that are frequently used in signal processing.

# ### Superposition
# 
# The weighted superposition $x(t)$ of two signals $x_\text{A}(t)$ and $x_\text{B}(t)$ is given as
# 
# \begin{equation}
# x(t) = A \cdot x_\text{A}(t) + B \cdot x_\text{B}(t)
# \end{equation}
# 
# with the complex weights $A, B \in \mathbb{C}$.

# **Example**
# 
# The following example illustrates the superposition of two harmonic signals $x_\text{A}(t) = A \cdot \cos(\omega_\text{A} t)$ and  $x_\text{B}(t) = B \cdot \cos(\omega_\text{B} t)$ with weights $A$, $B$ and angular frequencies $\omega_\text{A}$ and $\omega_\text{B}$.

# In[1]:


import sympy as sym
sym.init_printing()

t = sym.symbols('t', real=True)

A = .3
omA = 3
B = .5
omB = 5

x = A*sym.cos(omA*t) + B*sym.cos(omB*t)

sym.plot(x, (t, -5, 5), ylim=[-1.2, 1.2], ylabel=r'$x(t)$');


# ### Temporal Shift
# 
# The temporal shift of a signal $x(t)$ by the time $\tau$ is a frequently applied operation in signal processing. For instance, to model the propagation of signals from an actuator to a sensor.
# 
# The temporally shifted signal $x(t)$ is defined as
# 
# \begin{equation}
# y(t) = x(t-\tau)
# \end{equation}
# with $\tau \in \mathbb{R}$. The signal $x(t)$ is
# 
# * shifted to the right (*delayed*) for $\tau > 0$
# * shifted to the left (*leading*) for $\tau < 0$

# **Example**
# 
# In order to illustrate the temporal shifting of signals, the construction of a staircase signal by a superposition of shifted [rectangular signals](standard_signals.ipynb#Rectangular-Signal) is considered
# 
# \begin{equation}
# x(t) = \text{rect}\left(t - \frac{1}{2} \right) + \frac{2}{3} \cdot \text{rect}\left(t-\frac{3}{2} \right) + \frac{1}{3} \cdot \text{rect} \left(t-\frac{5}{2} \right)
# \end{equation}

# In[2]:


rect = sym.Heaviside(t + 1/2) - sym.Heaviside(t - 1/2)
x = rect.subs(t, t-1/2) + 2/3*rect.subs(t, t-3/2) + 1/3*rect.subs(t, t-5/2)

sym.plot(x, (t, -1, 5), ylim=[-0.2, 1.2], ylabel='$x(t)$');


# **Exercise**
# 
# * Add another step to the beginning of the staircase signal by modifying above example.

# ### Temporal Scaling
# 
# The temporal scaling of a signal $x(t)$ is defined as
# 
# \begin{equation}
# y(t) = x(a \cdot t)
# \end{equation}
# 
# with $a \in \mathbb{R}$. The signal $x(t)$ is
# 
# * stretched for $0 < a < 1$
# * compressed for $a > 1$
# * time-reversed and scaled for $a < 0$
# 
# An application of temporal scaling in signal processing is the adaption of the time scale for standard signals and the modeling of the [Doppler effect](https://en.wikipedia.org/wiki/Doppler_effect).

# **Example**
# 
# The following example illustrates the temporal scaling of the staircase signal $y(t) = x(a \cdot t)$ introduced in the previous example. The original $x(t)$ is plotted in gray, the scaled signal $y(t)$ in blue. Here stretching is realized, such that $y(t)$ is twice as long as $x(t)$.

# In[3]:


a = 1/2
y = x.subs(t, a*t)


px = sym.plot(x, (t, -3, 7), ylim=[-0.2, 1.2],
              ylabel=r'$x(t)$', show=False, line_color='gray')
py = sym.plot(y, (t, -3, 7),
              ylim=[-0.2, 1.2], ylabel=r'$y(t)$', show=False)
py.extend(px)
py.show()


# **Exercise**
# 
# * Modify above example such that the signal is compressed.
# * Modify above example such that the signal is scaled and time reversed. What scaling factors `a` lead to  stretching/compression in this context?

# ### Temporal Flipping

# The temporal flipping of a signal $x(t)$ is defined as
# 
# \begin{equation}
# y(t) = x(\tau - t)
# \end{equation}
# 
# for $\tau \in \mathbb{R}$. As $x(\tau - t) = x(- (t - \tau))$ the flipping operation can also be represented as a time-reversal of the signal $x(t)$ followed by a shift of $\tau$ of the reversed signal. For $\tau = 0$ this results in only a time-reversal of the signal. 
# 
# The temporal flipping operation can be interpreted geometrically as a mirroring of the signal $x(t)$ at the vertical axis $t=\frac{\tau}{2}$.

# **Example**
# 
# The following example illustrates the temporal flipping $y(t) = x(\tau - t)$ of the staircase signal $x(t)$ introduced before.

# In[4]:


tau = -1
y = x.subs(t, tau - t)

px = sym.plot(x, (t, -5, 5), ylim=[-0.2, 1.2],
              ylabel=r'$x(t)$', show=False, line_color='gray')
py = sym.plot(y, (t, -5, 5), ylim=[-0.2, 1.2], ylabel=r'$y(t)$', show=False)
py.extend(px)
py.show()


# **Excercise**
# 
# * For what value $\tau$ does the flipped signal $y(t)$ start at $t=0$?
# * Realize the temporal flipping by splitting it into two consecutive operations: (i) time-reversal and (ii) temporal shift.

# **Copyright**
# 
# This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*.