View source for Applets:Linear Distortions of Periodic Signals

{{LntAppletLink|physAnSignal_en}}  &nbsp; &nbsp; &nbsp; &nbsp; [https://en.lntwww.de/Applets:Physical_Signal_%26_Analytic_Signal '''English Applet with English WIKI description''']

==Applet Description==
<br>
This applet illustrates the effects of linear distortions (attenuation distortions and phase distortions) with
[[File:Modell_version2.png|right|frame|Meanings of the  used signals]]
*the input signal $x(t)$ &nbsp; &rArr; &nbsp; power $P_x$:
:$$x(t) = x_1(t) + x_2(t) = A_1\cdot \cos\left(2\pi f_1\cdot t- \varphi_1\right)+A_2\cdot \cos\left(2\pi f_2\cdot t- \varphi_2\right), $$
*the output signal $y(t)$ &nbsp; &rArr; &nbsp; power $P_y$:
:$$y(t) = \alpha_1 \cdot  x_1(t-\tau_1)  +  \alpha_2  \cdot  x_2(t-\tau_2),$$
*the matched output signal $z(t)$ &nbsp; &rArr; &nbsp; power $P_z$:
:$$z(t) = k_{\rm M} \cdot  y(t-\tau_{\rm M})  +  \alpha_2  \cdot  x_2(t-\tau_2),$$
*the difference signal &nbsp;  $\varepsilon(t) = z(t) - x(t)$ &nbsp; &rArr; &nbsp; power $P_\varepsilon$. 


The next block in the model above is ''Matching'': The output signal $y(t)$ is adjusted in amplitude and phase with equal variables $k_{\rm M}$ and $\tau_{\rm M}$ for all frequencies which means that this is not a frequency-dependent equalization. Using the signal $z(t)$, one can differentiate between:
*attenuation distortion and frequency&ndash;independent attenuation, as well as
*phase distortion and frequency&ndash;independent delay.


The ''Distortion Power'' $P_{\rm D}$ is used to measure the strength of the linear distortion and is defined as:
:$$P_{\rm D} = \min_{k_{\rm M},  \ \tau_{\rm M}} P_\varepsilon.$$


==Theoretical Background==
<br>
Distortions refer to generally unwanted alterations of a message signal through a transmission system. Together with the strong stochastic effects (noise, crosstalk, etc.), they are a crucial limitation for the quality and rate of transmission.

Just as the intensity of noise can be assessed through 
*the ''Noise Power'' $P_{\rm N}$ and
*the ''Signal&ndash;to&ndash;Noise Ratio'' (SNR) $\rho_{\rm N}$, 


distortions can be quantified through

*the ''Distortion Power'' $P_{\rm D}$ and
*the ''Signal&ndash;to&ndash;Distortion Ratio'' (SDR)
:$$\rho_{\rm D}=\frac{\rm Signal \ Power}{\rm Distortion \ Power} = \frac{P_x}{P_{\rm D} }.$$
 

=== Linear and Nonlinear Distortions ===
<br>
A distinction is made between linear and nonlinear distortions:
*'''Nonlinear distortions'''  occur, if at all times $t$ the nonlinear correlation $y = g(x) \ne {\rm const.}  \cdot x$ exists between the signal values $x = x(t)$ at the input and $y = y(t)$ at the output, whereby  $y = g(x)$ is defined as the system's nonlinear characteristic. By creating a cosine signal at the input with frequency $f_0$ the output signal includes  $f_0$, as well as  multiple harmonic waves. We conclude that new frequencies arise through nonlinear distortion.
  
[[File:LZI_T_2_2_S3_vers2.png|center|frame|For clarification of nonlinear distortions |class=fit]]

[[File:P_ID899__LZI_T_2_3_S1_neu.png|right |frame| Description of a linear system|class=fit]]
*'''Linear distortions''' occur,  if the transmission channel is characterized by a frequency response $H(f) \ne \rm const.$ Various frequencies are attenuated and delayed differently. Characteristic of this is that although frequencies can disappear (for example, through a low&ndash;pass, a high&ndash;pass, or a band&ndash;pass),  no new frequencies can arise. 


In this applet only linear distortions are considered.


=== Description Forms for the Frequency Response ===
<br>
The generally complex valued frequency response can be represented as follows: 
:$$H(f) = |H(f)| \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot
\hspace{0.05cm} b(f)} = {\rm e}^{-a(f)}\cdot {\rm e}^{-{\rm j}
\hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$

This results in the following description variables: 
*The absolute value $|H(f)|$ is called '''amplitude response''' and in logarithmic form '''attenuation function''': 
:$$a(f) = - \ln |H(f)|\hspace{0.2cm}{\rm in \hspace{0.1cm}Neper
\hspace{0.1cm}(Np) } = - 20 \cdot \lg |H(f)|\hspace{0.2cm}{\rm in
\hspace{0.1cm}Decibel \hspace{0.1cm}(dB) }.$$
*The '''phase function''' $b(f)$ indicates the negative frequency&ndash;dependent angle of $H(f)$ in the complex plane based on the real axis: 
:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in
\hspace{0.1cm}Radians \hspace{0.1cm}(rad)}.$$


=== Low&ndash;pass of Order <i>N</i>  ===
<br>
[[File:Tiefpass_version2.png|right|frame|Attenuation function $a(f)$ and phase function $b(f)$ of a low&ndash;Pass of order $N$]]
The frequency response of a realizable low&ndash;pass (LP) of order $N$ is:
:$$H(f) = \left [\frac{1}{1 + {\rm j}\cdot f/f_0 }\right ]^N\hspace{0.05cm}.$$
For example the RC low&ndash;pass is a first order low&ndash;pass. Consequently we can obtain 
*the attenuation function:
:$$a(f) =N/2 \cdot \ln  [1+( f/f_0)^2] \hspace{0.05cm},$$
*the phase function:
:$$b(f) =N \cdot \arctan( f/f_0) \hspace{0.05cm},$$
*the attenuation factor for the frequency $f=f_i$:
:$$\alpha_i =|H(f = f_i)| =  [1+( f_i/f_0)^2]^{-N/2}$$
:$$\Rightarrow \hspace{0.3cm} x(t)= A_i\cdot \cos(2\pi f_i t) \hspace{0.1cm}\rightarrow \hspace{0.1cm}  y(t)= \alpha_i  \cdot A_i\cdot \cos(2\pi f_i t)\hspace{0.05cm},$$
*the phase delay for the frequency $f=f_i$:
:$$\tau_i =\frac{b(f_i)}{2 \pi f_i} = \frac{N \cdot \arctan( f_i/f_0)}{2 \pi f_i}$$
:$$\Rightarrow \hspace{0.3cm} x(t)= A_i\cdot \cos(2\pi f_i t) \hspace{0.1cm}\rightarrow \hspace{0.1cm}  y(t)=A_i\cdot \cos(2\pi f_i (t- \tau_i))\hspace{0.05cm}.$$


=== High&ndash;pass of Order <i>N</i>  ===
<br>
[[File:Hochpass_version2.png|right|frame|Attenuation function $a(f)$ and phase function $b(f)$ of a high&ndash;pass of order $N$]]
The frequency response of a realizable high&ndash;pass  (HP) of order $N$ is:
:$$H(f) = \left [\frac{ {\rm j}\cdot f/f_0 }{1 + {\rm j}\cdot f/f_0 }\right ]^N\hspace{0.05cm}.$$
For example the LC high-pass is a first order high-pass. Consequently we can obtain  
*the attenuation function:
:$$a(f) =N/2 \cdot \ln  [1+( f_0/f)^2] \hspace{0.05cm},$$
*the phase function:
:$$b(f) =-N \cdot \arctan( f_0/f) \hspace{0.05cm},$$
*the attenuation factor for the frequency $f=f_i$:
:$$\alpha_i =|H(f = f_i)| =  [1+( f_0/f_i)^2]^{-N/2}$$
:$$\Rightarrow \hspace{0.3cm} x(t)= A_i\cdot \cos(2\pi f_i t) \hspace{0.1cm}\rightarrow \hspace{0.1cm}  y(t)= \alpha_i  \cdot A_i\cdot \cos(2\pi f_i t)\hspace{0.05cm},$$
*the phase delay for the frequency $f=f_i$:
:$$\tau_i =\frac{b(f_i)}{2\pi f_i} = \frac{-N \cdot \arctan( f_0/f_i)}{2\pi f_i}$$
:$$\Rightarrow \hspace{0.3cm} x(t)= A_i\cdot \cos(2\pi f_i t) \hspace{0.1cm}\rightarrow \hspace{0.1cm}  y(t)=A_i\cdot \cos(2\pi f_i (t- \tau_i))\hspace{0.05cm}.$$


[[File:Verzerrungen_HP_TP_1_englisch.png|right|frame|Phase function $b(f)$ of high&ndash;pass and low&ndash;pass]]
{{GraueBox|TEXT=  
$\text{Example:}$&nbsp;
This graphic shows the phase function $b(f)$ with the cutoff frequency $f_0 = 1\ \rm kHz$ and order $N=1$
* of a low&ndash;pass (green curve),
* of a high&ndash;pass (violet curve).


The input signal is sinusoidal with frequency $f_{\rm S} = 1.25\ {\rm kHz}$ whereby this signal is only turned on at $t=0$: 

:$$x(t) = \left\{ \begin{array}{l} \hspace{0.75cm}0  \\ \sin(2\pi \cdot f_{\rm S}  \cdot t ) \\  \end{array} \right.\quad\begin{array}{l} (t < 0), \\   (t>0). \\ \end{array}$$

The left graphic shows the signal $x(t)$. The dashed line marks the first zero at  $t = T_0 = 0.8\ {\rm ms}$. The other two graphics show the output signals $y_{\rm LP}(t)$ und $y_{\rm HP}(t)$ of  low&ndash;pass  and high&ndash;pass, whereby the change in amplitude was balanced in both cases.

[[File:Verzerrungen_HP_TP_2_version2.png|center|frame|Input signal $x(t)$  (enframed in blue) as well as output signals  $y_{\rm LP}(t)$ &rArr; &nbsp; green and $y_{\rm HP}(t)$ &rArr; &nbsp; magenta]]

*The first zero of the signal $y_{\rm LP}(t)$ after the low&ndash;pass is delayed by $\tau_{\rm LP} = 0.9/(2\pi) \cdot T_0 \approx 0.115 \ {\rm ms}$ compared to the first zero of $x(t)$ &nbsp; &rArr; &nbsp; marked with green arrow, whereby $b_{\rm LP}(f/f_{\rm S} = 0.9 \ {\rm rad})$ was considered.
* In contrast, the phase delay of the high&ndash;pass is negative:  $\tau_{\rm HP} = -0.67/(2\pi) \cdot T_0 \approx -0.085 \ {\rm ms}$  and therefore the first zero of $y_{\rm HP}(t)$ occurs before the dashed line.
*Following this transient response, in both cases the zero crossings again come in the raster of the period duration $T_0 = 0.8 \ {\rm ms}.$


''Remark:'' The shown signals were created using the interactive applet [[Applets:Kausale_Systeme_-_Laplacetransformation|"Causal systems &ndash; Laplace transform"]]. }}

=== Attenuation and Phase Distortions  ===
<br>
[[File:P_ID900__LZI_T_2_3_S2_neu.png|frame| Requirements for a non&ndash;distorting channel|right|class=fit]]
The adjacent figure shows
*the even attenuation function $a(f)$ &nbsp; &rArr; &nbsp; $a(-f) = a(f)$, and 
*the uneven  function curve $b(f)$ &nbsp; &rArr; &nbsp; $b(-f) = -b(- f)$


of a non&ndash;distorting channel. One can see: 
*In a distortion&ndash;free system the attenuation function $a(f)$ must be constant between$f_{\rm U}$ and $f_{\rm O}$ around the carrier frequency $f_{\rm T}$, where the input signal  exists &nbsp; &rArr; &nbsp;  $X(f) \ne 0$. 
*From the specified constant attenuation value $6 \ \rm dB$ follows for the amplitude response $|H(f)| = 0.5$ &nbsp; &rArr; &nbsp; the signal values of all frequencies are thus halved by the system &nbsp; &rArr; &nbsp; '''no attenuation distortions'''.
*In addition, in such a system,  the phase function $b(f)$ between $f_{\rm U}$ and $f_{\rm O}$ must increase linearly with the frequency. As a result, all frequency components are delayed by the same phase delay $τ$ &nbsp; &rArr; &nbsp;  '''no phase distortion'''.
*The delay $τ$ is fixed by the slope of $b(f)$. The phase function $b(f) \equiv 0$ would result in a delay&ndash;less system  &nbsp; &rArr; &nbsp; $τ = 0$. 


The following summary considers that &ndash; in this applet &ndash; the input signal is always the sum of two harmonic oscillations,
:$$x(t) = x_1(t) + x_2(t) = A_1\cdot \cos\left(2\pi f_1\cdot t- \varphi_1\right)+A_2\cdot \cos\left(2\pi f_2\cdot t- \varphi_2\right), $$
and therefore the channel influence is fully described by the attenuation factors $\alpha_1$ and $\alpha_2$ as well as the phase delays  $\tau_1$ and $\tau_2$:
:$$y(t) = \alpha_1 \cdot  x_1(t-\tau_1)  +  \alpha_2  \cdot  x_2(t-\tau_2).$$

{{BlaueBox|TEXT=  
$\text{Summary:}$&nbsp;
*A signal $y(t)$ is only '''distortion&ndash;free''' compared to $x(t)$ if $\alpha_1 = \alpha_2= \alpha$ &nbsp;<u> and </u>&nbsp; $\tau_1 = \tau_2= \tau$ &nbsp; &rArr; &nbsp; $y(t) = \alpha \cdot  x(t-\tau)$.
* '''Attenuation distortions''' occur when  $\alpha_1 \ne \alpha_2$. If $\alpha_1 \ne \alpha_2$ and $\tau_1 = \tau_2$, then there are exclusively attenuation distortions. 
* '''Phase distortions''' occur when $\tau_1 \ne \tau_2$. If $\tau_1 \ne \tau_2$ and $\alpha_1 = \alpha_2$, then there are exclusively phase distortions.  }}




==Exercises==
[[File:Exercises_verzerrungen.png|right]]
*First choose an exercise number.
*An exercise description is displayed. 
*Parameter values are adjusted to the respective exercises.
*Click "Hide solution" to display the solution. 


Number "0" is a "Reset" button:
*Sets parameters to initial values (when loading the page).
*Displays a "Reset text" to describe the applet further.

 
{{BlaueBox|TEXT=  
'''(1)''' &nbsp; We consider the parameters $A_1 = 0.8\ {\rm V}, \ A_2 = 0.6\ {\rm V}, \ f_1 = 0.5\ {\rm kHz}, \ f_2 = 1.5\ {\rm kHz}, \ \varphi_1 = 90^\circ, \ \varphi_2 = 30^\circ$ for the input signal $x(t)$. 
:Calculate the signal's period duration $T_0$ and power $P_x$. Can you read the value for $P_x$ off the applet? }}


$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}T_0 = \big [\hspace{-0.1cm}\text{ greatest common divisor }(0.5  \ {\rm kHz}, \ 1.5  \ {\rm kHz})\big ]^{-1}\hspace{0.15cm}\underline{ =  2.0 \ {\rm ms}};$ 

$\hspace{1.85cm} P_x = A_1^2/2 + A_2^2/2 \hspace{0.15cm}\underline{= 0.5 \ {\rm V^2}} = P_\varepsilon\text{, if }\hspace{0.15cm}\underline{k_{\rm M} = 0} \ \Rightarrow \ z(t) \equiv 0$.

{{BlaueBox|TEXT=  
'''(2)''' &nbsp;Vary $\varphi_2$ between $\pm 180^\circ$ while keeping all other parameters from Exercise (1). How does the value of $T_0$ and $P_x$ change?}}


$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}\text{No changes:}\hspace{0.2cm}\hspace{0.15cm}\underline{ T_0 = 2.0 \ {\rm ms}; \hspace{0.2cm} P_x =  0.5 \ {\rm V^2}}$.

{{BlaueBox|TEXT=  
'''(3)''' &nbsp; Vary $f_2$ between $0 \le f_2 \le 10\ {\rm kHz}$ while keeping all  other parameters from Exercise (1). How does the value of $P_x$ change?}}


$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}\text{No changes if }f_2 \ne 0\text{ and } f_2 \ne f_1\text{:}\hspace{0.3cm} \hspace{0.15cm}\underline{P_x =  0.5 \ {\rm V^2}}\text{.} \hspace{0.2cm} T_0 \text{ changes if }f_2\text{is not a multiple of }f_1$.

$\hspace{1.85cm}\text{If }f_2 = 0\text{:}\hspace{0.2cm} P_x = A_1^2/2 + A_2^2\hspace{0.15cm}\underline{ = 0.68 \ {\rm V^2}}$. $\hspace{3cm}$

$\hspace{1.85cm}\text{If }f_2 = f_1\text{:}\hspace{0.2cm} P_x = [A_1\cos(\varphi_1) + A_2\cos(\varphi_2)]^2/2 + [A_1\sin(\varphi_1) + A_2\sin(\varphi_2)]^2/2 \text{.} $
 
$\hspace{1.85cm}\text{With } \varphi_1 = 90^\circ, \ \varphi_2 = 30^\circ\text{:}\hspace{0.3cm}\hspace{0.15cm}\underline{ P_x =  0.74 \ {\rm V^2}}\text{.} $

{{BlaueBox|TEXT=  
'''(4)''' &nbsp; Keeping the previous input signal $x(t)$, set following parameters
: $\alpha_1 = \alpha_2 = 0.5, \ \tau_1 = \tau_2  = 0.5\ {\rm ms}$, $k_{\rm M} = 1 \text{ and } \tau_{\rm M} = 0$ . 
:Are there linear distortions? Calculate the received power $P_y$ and the power $P_\varepsilon$ of the differential signal $\varepsilon(t) = z(t) - x(t)$. }}


$\hspace{1.0cm}\Rightarrow \hspace{0.3cm}\hspace{0.15cm}\underline{ y(t) = 0.5 \cdot x(t- 1\ {\rm ms})}\text{ is only attenuated and delayed, but not distorted.}$  

$\hspace{1.85cm}\text{Received power:}\hspace{0.2cm} P_y = (A_1/2)^2/2 + (A_2/2)^2/2\hspace{0.15cm}\underline{ = 0.125 \ {\rm V^2}}\text{.   } P_\varepsilon \text{ is significantly larger:} \hspace{0.1cm} \hspace{0.15cm}\underline{P_\varepsilon = 0.625 \ {\rm V^2}}.$

{{BlaueBox|TEXT=  
'''(5)''' &nbsp; With the same settings as in Exercise (4), vary the matching parameters $k_{\rm M} \text{ and } \tau_{\rm M}$. How big is the distortion power $P_{\rm D}$?}}


$\hspace{1.0cm}\Rightarrow \hspace{0.3cm} P_{\rm D}\text{ is equal to }P_\varepsilon  \text{ when using the ideal matching parameters:} \hspace{0.2cm}k_{\rm M} = 2 \text{ and } \tau_{\rm M}=T_0 - 0.5\ {\rm ms} = 1.5\ {\rm ms}$

$\hspace{1.0cm}\Rightarrow \hspace{0.3cm}z(t) = x(t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\varepsilon(t) = 0\hspace{0.3cm}\Rightarrow \hspace{0.3cm}P_{\rm D}\hspace{0.15cm}\underline{ = P_\varepsilon = 0} \hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{Neither attenuation nor phase distortion.}$  

{{BlaueBox|TEXT=  
'''(6)''' &nbsp; The channel parameters are now set to: $\alpha_1 = 0.5, \hspace{0.15cm}\underline{\alpha_2 = 0.2}, \ \tau_1 = \tau_2  = 0.5\ {\rm ms}$. Calculate the distortion power $P_{\rm D}$ and the Signal-to-Distortion ratio $(\rm SDR) \ \rho_{\rm D}$.}}


$\hspace{1.0cm}\Rightarrow \hspace{0.3cm} P_{\rm D} = P_\varepsilon  \text{ when using the best matching parameters:} \hspace{0.2cm}\hspace{0.15cm}\underline{k_{\rm M} = 2.24} \text{ and } \hspace{0.15cm}\underline{\tau_{\rm M} = 1.5\ {\rm ms} }\text{:} \hspace{0.2cm}\hspace{0.15cm}\underline{P_{\rm D} =  0.059 \ {\rm V^2}}$.

$\hspace{1.85cm}\text{Attenuation distortions only.} \hspace{0.3cm}\text{Signal-to-Distortion-Ratio}\ \hspace{0.15cm}\underline{\rho_{\rm D} = P_x/P_\varepsilon \approx 8.5}$.  

{{BlaueBox|TEXT=  
'''(7)''' &nbsp; The channel parameters are now set to: $\alpha_1 = \alpha_2 = 0.5, \ \tau_1 \hspace{0.15cm}\underline{= 2\ {\rm ms} }, \  \tau_2  = 0.5\ {\rm ms}$. Calculate the distortion power $P_{\rm D}$ and the the Signal-to-Distortion ratio $\rho_{\rm D}$.}}


$\hspace{1.0cm}\Rightarrow \hspace{0.3cm} P_{\rm D} = P_\varepsilon  \text{when using the best matching parameters:} \hspace{0.2cm}\hspace{0.15cm}\underline{k_{\rm M} = 1.84} \text{ and } \tau_{\rm M}\hspace{0.15cm}\underline{  = 0.15\ {\rm ms} }\text{:} \hspace{0.2cm}\hspace{0.15cm}\underline{P_{\rm D} =  0.071 \ {\rm V^2}}$.

$\hspace{1.85cm}\text{Phase distortions only.} \hspace{0.3cm}\text{Signal-to-Distortion-Ratio}\ \hspace{0.15cm}\underline{\rho_{\rm D} = P_x/P_\varepsilon \approx 7}$.  

{{BlaueBox|TEXT=  
'''(8)''' &nbsp; The channel parameters are now set to: $\hspace{0.15cm}\underline{\alpha_1 = 0.5} , \hspace{0.15cm}\underline{\alpha_2 = 0.2} , \ \hspace{0.15cm}\underline{\tau_1= 0.5\ {\rm ms} }, \  \hspace{0.15cm}\underline{\tau_2  = 0.3\ {\rm ms} }$. Are there attenuation distortions? Are there phase distortions? How can $y(t)$ be approximated?  ''Hint:'' $\cos(3x) = 4 \cdot \cos^3(x) - 3\cdot \cos(x).$}}


$\hspace{1.0cm}\Rightarrow\hspace{0.3cm} \text{Both attenuation and phase distortions, because }\alpha_1 \ne \alpha_2\text{ and }\tau_1 \ne \tau_2$. 

$\hspace{1.85cm}y(t) = y_1(t) + y_2(t)\ \Rightarrow \ y_1(t) = A_1 \cdot \alpha_1 \cdot \sin[2\pi f_1\  (t- 0.5\ \rm ms)] = -0.4 \ {\rm V} \cdot \cos(2\pi f_1 t)$

$\hspace{1.85cm}  y_2(t) = \alpha_2 \cdot x_2(t- \tau_2) \text{ mit }x_2(t) = A_2 \cdot \cos[2\pi f_2\  (t- 30^\circ)] \approx  A_2 \cdot \cos[2\pi f_2\  (t- 1/36 \ \rm ms)]$

$\hspace{1.85cm}  \Rightarrow \ y_2(t) = 0.12 \ {\rm V} \cdot \cos[2\pi f_2\  (t- 0.328 \ {\rm ms})] \approx -0.12 \ { \rm V} \cdot \cos[2\pi f_2t] $.

$\hspace{1.85cm}  \Rightarrow \ y(t) = y_1(t) + y_2(t) \approx -0.4 \ {\rm V} \cdot [\cos(2\pi \cdot f_1\cdot  t) + 1/3 \cdot \cos(2\pi \cdot 3 f_1 \cdot t) =  -0.533 \ {\rm V} \cdot \cos^3(2\pi f_1  t)$.

{{BlaueBox|TEXT=  
'''(9)''' &nbsp; Using the parameters from  Exercise  (8),  calculate the distortion power $P_{\rm D}$ and the the Signal-to-Distortion ratio  $\rho_{\rm D}$.}}


$\hspace{1.0cm}\text{Best possible adaptation:} \hspace{0.2cm}\hspace{0.15cm}\underline{k_{\rm M} = 1.96} \text{, } \hspace{0.15cm}\underline{\tau_{\rm M} = 1.65\ {\rm ms} }\text{:} \hspace{0.2cm}\hspace{0.15cm}\underline{P_{\rm D} =  0.15 \ {\rm V^2} },\hspace{0.1cm}\hspace{0.15cm}\underline{\rho_{\rm D}  = 0.500/0.15 \approx 3.3}$.

{{BlaueBox|TEXT=  
'''(10)''' &nbsp;Now we set $A_2 = 0$ and $A_1 =  1\ {\rm V}, \ f_1 = 1\ {\rm kHz}, \ \varphi_1 = 0^\circ$. The channel is a <u>Low-pass of  order  1</u>  $\underline{(f_0 = 1\ {\rm kHz})}$. <br>Are there any attenuation  and/or phase distortions? Calculate the channel coefficients $\alpha_1$ and $\tau_1$.}}


$\hspace{1.0cm}\text{At only one frequency there are neither attenuation nor phase distortions.}$
$\hspace{1.0cm}\text{Attenuation factor for }f_1=f_0\text{ and }N=1\text{:   }\alpha_1 =|H(f = f_1)| =  [1+( f_1/f_0)^2]^{-N/2} = 2^{-1/2}= 1/\sqrt{2}\hspace{0.15cm}\underline{=0.707},$
$\hspace{1.0cm}\text{Phase factor for }f_1=f_0\text{ and }N=1\text{:   }\tau_1 = N \cdot \arctan( f_1/f_0)/(2 \pi f_1)=\arctan( 1)/(2 \pi f_1) =1/(8f_1) \hspace{0.15cm}\underline{=0.125 \ \rm ms}.$

{{BlaueBox|TEXT=  
'''(11)''' &nbsp; How do the channel parameters change when using a <u>Low-pass of order 2</u> compared to a Low-pass of order 1  $(f_0 = 1\ {\rm kHz})$?}} 


$\hspace{1.0cm}\alpha_1 = 0.707^2 = 0.5$ and $\tau_1 = 2 \cdot 0.125 = 0.25 \ {\rm ms}$.  

$\hspace{1.0cm}\text{The signal }y(t)\text{  is only half as big as }x(t)\text{ and is retarded: The cosine turns into a sine function}$. 

{{BlaueBox|TEXT=  
'''(12)''' &nbsp; What differences arise when using a  <u>High-pass of order 2</u> compared to a Low-pass of order 2  $(f_0 = 1\ {\rm kHz})$? }}


$\hspace{1.0cm}\text{Since }f_1 = f_0\text{ the attenuation factor }\alpha_1 = 0.5\text{ stays the same and }\tau_1 = -0.25 \ {\rm ms}\text{ which means:}$

$\hspace{1.0cm}\text{The signal }y(t)\text{  is also only half as big as }x(t)\text{ and precedes it: The cosine turns into the Minus&ndash;sine function}$. 

{{BlaueBox|TEXT=  
'''(13)''' &nbsp; What differences at the signal $y(t)$ can be observed between the Low-pass and the High-pass of order 2  $(f_0 = 1\ {\rm kHz})$ when you start with the initial input signal according to Exercise (1) and continuously raise $f_2$ up to $10 \ \rm kHz$ ? }}


$\hspace{1.0cm}\text{With the Low-pass the second term is increasingly suppressed. For }f_2 =  10 \ {\rm kHz}\text{: }y_{\rm LP}(t) \approx 0.8 \cdot x_1(t-0.3 \ \rm ms).$   

$\hspace{1.0cm}\text{With the High-pass however the second  term dominates. For }f_2 =  10 \ {\rm kHz}\text{: }y_{\rm HP}(t) \approx 0.2 \cdot x_1(t+0.7 \ {\rm ms}) + x_2(t).$

==Applet Manual==
[[File:Handhabung_verzerrungen.png|center]]
<br>
&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Parameter selection for input signal $x(t)$ per slider: Amplitude, frequency, phase values

&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Preselection for channel parameters per slider: Low-pass or High-pass

&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Selction of channel parameters per slider: Dämpfungsfaktoren und Phasenlaufzeiten

&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Selection of channel parameters for High and Low pass: Order$n$, cutoff frequency $f_0$

&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Selection of matching parameters $k_{\rm M}$ and $\varphi_{\rm M}$

&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Selection of the signals to be displayed: $x(t)$,  $y(t)$, $z(t)$, $\varepsilon(t)$, $\varepsilon^2(t)$

&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Graphic display of the signals

&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Enter the time $t_*$ for the numeric output

&nbsp; &nbsp; '''( I )''' &nbsp; &nbsp; numeric output of the signal values $x(t_*)$,  $y(t_*)$, $z(t_*)$  and $\varepsilon(t_*)$

&nbsp; &nbsp; '''(J)''' &nbsp; &nbsp; Numeric output of the main result $P_\varepsilon$

&nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Save and reall parameters

&nbsp; &nbsp; '''(L)''' &nbsp; &nbsp; Exercises: Exercise selection, description and solution

&nbsp; &nbsp; '''(M)''' &nbsp; &nbsp; Variation possibilities for the graphic display
 
$\hspace{1.5cm}$Zoom&ndash;functions "$+$" (scale up), "$-$" (scale down) und $\rm o$ (reset)

$\hspace{1.5cm}$Move with "$\leftarrow$" (section to the left, ordinate to the right),  "$\uparrow$" "$\downarrow$" und "$\rightarrow$"

$\hspace{1.5cm}$'''Other options'':

$\hspace{1.5cm}$Hold shift and scroll:  Zoom in on/out of coordinate system,

$\hspace{1.5cm}$Hold shift and left click: Move the coordinate system.

==About the Authors==
This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München]. 
*The original version was created in 2005 by [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29|Bettina Hirner]] as part of her Diploma thesis using "FlashMX&ndash;Actionscript" (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]] ). 
*In 2018 this Applet was redesigned and updated to "HTML5" by [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Jimmy_He_.28Bachelorarbeit_2018.29|Jimmy He]] as part of his Bachelor's thesis (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]) .

==Once again: Open Applet in new Tab==

{{LntAppletLinkEn|linDistortions_en}} &nbsp; &nbsp; &nbsp; &nbsp; [https://www.lntwww.de/Applets:Lineare_Verzerrungen_periodischer_Signale '''English Applet with German WIKI description''']