<jupyter_start><jupyter_text>System Properties*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 [<EMAIL>](mailto:<EMAIL>).*  Idealized SystemsIdealized systems are systems with idealized properties that typically render their practical implementation infeasible. They play an important role in various fields of signal processing as they allow a convenient formulation of concepts and principles. In the following, the ideal low-pass is introduced as prototype for an idealized frequency selective system. Other frequency selective system can be deduced directly from this prototype.  Ideal Low-PassThe transfer function $H(j \omega)$ of a real-valued [ideal low-pass](https://en.wikipedia.org/wiki/Sinc_filter) reads\begin{equation}H(j \omega) = \text{rect} \left( \frac{\omega}{2 \omega_\text{c}} \right)\end{equation}where $\omega_\text{c} > 0$ denotes its cut-off frequency. The ideal low-pass removes all frequency components above the cut-off frequency $\omega_\text{c}$, without affecting lower frequencies. The impulse response $h(t) = \mathcal{F}^{-1} \{ H(j \omega) \}$ is computed by applying the [duality principle](../fourier_transform/properties.ipynbDuality) to the [Fourier transform of the rectangular signal](../fourier_transform/definition.ipynbTransformation-of-the-Rectangular-Signal)\begin{equation}h(t) = \frac{\omega_\text{c}}{\pi} \cdot \text{sinc} ( \omega_\text{c} t )\end{equation}Since the impulse response is an acausal signal, the ideal low-pass is an [acausal system](causality_stability.ipynbCondition-for-the-Impulse-Response). The sinc-function is furthermore not absolutely integrable. Hence the ideal-low pass is not a [stable system](causality_stability.ipynbCondition-for-the-Impulse-Response) in the sense of the bounded-input bounded-output (BIBO) criterion. In conclusion, the ideal low-pass is not realizable. It can only be approximated in practical systems. It plays nevertheless an important role in the foundations of sampling and interpolation.  Realizable low-passVarious techniques have been developed in order to approximate the ideal low-pass by a realizable system. One is the *windowed sinc filter*. In order to make the ideal-low pass filter causal and stable, its impulse response is windowed to a finite-length $T$ followed by a temporal shift of $\frac{T}{2}$. Using the [rectanglar signal](../continuous_signals/standard_signals.ipynbRectangular-Signal) to truncate (window) the impulse response, the impulse response of the realizable low-pass is given as\begin{equation}h(t) = \frac{\omega_\text{c}}{\pi} \cdot \text{sinc} \left( \omega_\text{c} \left(t - \frac{T}{2} \right) \right) \cdot \text{rect}\left( \frac{1}{T} \left( t - \frac{T}{2} \right) \right)\end{equation}Fourier transformation yields its transfer function\begin{equation}H(j \omega) = \frac{1}{2 \pi} e^{-j \omega \frac{T}{2}} \cdot \text{rect}\left( \frac{\omega}{2 \omega_c} \right) * T \cdot \text{sinc} \left( \frac{T}{2} \omega \right)\end{equation}The impulse response is plotted for $w_\text{c}=10$ and $T=5$<jupyter_code>%matplotlib inline
import sympy as sym
sym.init_printing()

t, w = sym.symbols('t omega', real=True)
wc = 10
T = 5

h = wc/sym.pi * sym.sinc(wc*(t-T/2))
sym.plot(h, (t, 0, T), xlabel='$t$', ylabel='$h(t)$');<jupyter_output><empty_output><jupyter_text>**Exercise*** Discuss the properties of the magnitude response $|H(j \omega)|$ of the realizable low-pass in dependence of its length $T$.   Ideal Band-PassThe transfer function $H(j \omega)$ of a real-valued ideal band-pass reads\begin{equation}H(j \omega) = \begin{cases}1 & \text{for }  \omega_\text{c} - \frac{\Delta \omega}{2} < |\omega| < \omega_\text{c} + \frac{\Delta \omega}{2} \\0 & \text{otherwise}\end{cases}\end{equation}The ideal band-pass does not affect the frequency components of a signal around a given center frequency $\omega_\text{c}$ where the total width of this transition band is $\Delta \omega$. Components outside the transition band are removed. The transfer function can be rewritten as\begin{equation}H(j \omega) =\text{rect} \left( \frac{\omega - \omega_\text{c}}{\Delta \omega} \right) + \text{rect} \left( \frac{\omega + \omega_\text{c}}{\Delta \omega} \right)= \text{rect} \left( \frac{\omega}{\Delta \omega} \right) * \left( \delta(\omega - \omega_\text{c}) + \delta(\omega + \omega_\text{c}) \right)\end{equation}Its impulse response is computed by inverse Fourier transformation\begin{equation}h(t) = \pi \Delta \omega \cdot \text{sinc} ( \frac{\Delta \omega}{2} t ) \cdot \cos(\omega_\text{c} t)\end{equation}The ideal band-pass can be interpreted as a modulated low-pass filter. Due to its relation to the ideal low-pass, it is neither causal nor stable. The ideal band-pass can only be approximated in practical realizations. Its plays an important role in the theoretical foundations of wireless communications. **Example**For illustration, the impulse response of the ideal band-pass for $\omega_\text{c} = 10$ and $\Delta \omega = 2$ is plotted<jupyter_code>wc = 10
dw = 2

h = sym.pi*dw * sym.sinc(dw/2*t) * sym.cos(wc*t)
sym.plot(h, (t, -10, 10), xlabel='$t$', ylabel='$h(t)$');<jupyter_output><empty_output>
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