View source for Linear and Time Invariant Systems/System Description in Frequency Domain

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== # OVERVIEW OF THE FIRST MAIN CHAPTER # ==
<br>
*In the book&nbsp; &raquo;Signal Representation&laquo;&nbsp; you were familiarized with the mathematical description of deterministic signals in the time and frequency domain. 

*The second book&nbsp; &raquo;Linear Time-Invariant Systems&laquo;&nbsp; now describes&nbsp; what changes a signal or its spectrum undergoes through a transmission system&nbsp; and how these changes can be captured mathematically.

*In the first chapter,&nbsp; the basics of the so-called&nbsp; &raquo;Systems Theory&laquo;&nbsp; are mentioned,&nbsp; which allows a uniform and simple description of such systems.&nbsp; We start with the system description in the frequency domain with the partial aspects listed above.


{{BlaueBox|TEXT=  
$\text{Please note:}$&nbsp; 
*The&nbsp; &raquo;'''system'''&laquo;&nbsp; can be a simple circuit as well as a complete, highly complicated transmission system with a multitude of components.

*Here it is only assumed that the system has the two properties&nbsp; &raquo;'''linear'''&laquo;&nbsp; and&nbsp; &raquo;'''time-invariant'''&laquo;.}}


==The cause-and-effect principle==
<br>
[[File:P_ID775__LZI_T_1_1_S1_neu.png|right|frame|Simplest system model|class=fit]]
We always consider here the simple model outlined on the right.&nbsp; This arrangement is to be interpreted as follows:  

*The focus is on the so-called&nbsp; &raquo;system&laquo;,&nbsp; which is largely abstracted in its function &nbsp; &rArr; &nbsp; &raquo;black box&laquo;.&nbsp; Nothing is known in detail about the realization of the system.
 
*The time-dependent input&nbsp; $x(t)$&nbsp; acting on this system is also referred to as the&nbsp; &raquo;'''cause function'''&laquo;&nbsp; in the following.
 
*At the system output,&nbsp; the&nbsp; &raquo;'''effect function'''&laquo;&nbsp; $y(t)$&nbsp; appears  as the system response  to the input function&nbsp; $x(t)$. 


{{BlaueBox|TEXT= 
$\text{Please note:}$&nbsp; 

#The&nbsp; &raquo;system&laquo;&nbsp; can generally be of any kind and is not limited to communications technology alone.&nbsp; In fact,&nbsp; attempts are also made in other fields of science,&nbsp; such as the natural sciences,&nbsp; economics and business administration,&nbsp; sociology and political science,&nbsp; to capture and describe causal relationships between different variables by means of the cause-and-effect principle.  
#However,&nbsp; the methods used for these phenomenological systems theories differ significantly from the approach in Communications Engineering,&nbsp; which is outlined in this first main chapter of the present book&nbsp; &raquo;Linear Time-Invariant Systems&laquo;.}}

==Application in Communications Engineering==
<br>
The cause-and-effect principle can be applied in Communications Engineering,&nbsp; for example to describe one-port circuits which are also reffered to as one-ports.&nbsp; Here,&nbsp; one can consider the current curve&nbsp; $i(t)$&nbsp; as a cause function and the voltage&nbsp; $u(t)$&nbsp; as an effect function.&nbsp; By observing the I/U relationships,&nbsp; conclusions can be drawn about the properties of the actually unknown one-port. 

[[File:EN_LZI_T_1_1_S2.png|right|frame|General model of signal transmission|class=fit]]
In Germany,&nbsp; [https://en.wikipedia.org/wiki/Karl_K%C3%BCpfm%C3%BCller $\text{Karl Küpfmüller}$]&nbsp;  introduced the term&nbsp; &raquo;Systems Theory&laquo;&nbsp; for the first time in 1949.&nbsp;  
He considers it as a method for describing complex causal relationships in natural sciences and technology,&nbsp; based on a spectral transformation,&nbsp; e.g. the&nbsp;  
[[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;Fourier transform&laquo;]].

A transmission system can entirely be described in terms of systems theory&nbsp;. Here 
*the&nbsp; &raquo;cause function&laquo;&nbsp; is the input signal&nbsp; $x(t)$&nbsp; or its spectrum&nbsp; $X(f)$,
 
*the&nbsp; &raquo;effect function&laquo;&nbsp; is the output signal&nbsp; $y(t)$&nbsp; or its spectrum&nbsp; $Y(f)$. 


Also in the following graphs,&nbsp; input variables are mostly drawn in blue,&nbsp; output variables in red and system variables in green.
<br clear=all>
{{GraueBox|TEXT=  
$\text{Example 1:}$&nbsp; 

(1)&nbsp; For example,&nbsp; if the&nbsp; &raquo;system&laquo;&nbsp; describes a given linear electric circuit,&nbsp; then given a known input signal&nbsp; $x(t)$&nbsp;the output signal&nbsp;$y(t)$&nbsp; can be predicted with the help of&nbsp; "Systems Theory". 

(2)&nbsp; A second task of&nbsp; "Systems Theory"&nbsp; is to classify the transmission system by measuring&nbsp; $y(t)$&nbsp;knowing&nbsp; $x(t)$&nbsp; but without knowing the system in detail. 

(3)&nbsp; If&nbsp; $x(t)$&nbsp;describes, for instance, the voice of a caller in Hamburg and&nbsp; $y(t)$&nbsp;the recording of an answering machine in Munich, then the&nbsp; &raquo;transmission system&laquo;&nbsp; consists of the following components: 

:Microphone – telephone – electrical line – signal converter $($electrical-optical$)$ – fiber optic cable – optical amplifier – signal converter $($optical-electrical$)$ –  receiver filter $($for equalization and noise limitation$)$ – &nbsp; . . . . . . &nbsp; – electromagnetic transducer. }}

==Prerequisites for the application of Systems Theory==
<br>
The model of a transmission system given above holds generally and independently of the boundary conditions.&nbsp; However,&nbsp; the application of systems theory requires some additional limiting preconditions.  

Unless explicitly stated otherwise,&nbsp; the following shall always apply: 
*Both&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; are&nbsp; &raquo;'''deterministic'''&laquo;&nbsp; signals.&nbsp; Otherwise,&nbsp; one must proceed according to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Stochastische_Systemtheorie|&raquo;Stochastic System Theory&laquo;]].

*The system is&nbsp; &raquo;'''linear'''&laquo;.&nbsp; This can be seen,&nbsp; for example,&nbsp; from the fact that a harmonic oscillation &nbsp;$x(t)$&nbsp; at the input also results in a harmonic oscillation&nbsp; $y(t)$&nbsp; of the same frequency at the output: 
:$$x(t) = A_x \cdot \cos(\omega_0 \hspace{0.05cm}t - \varphi_x)\hspace{0.2cm}\Rightarrow \hspace{0.2cm} y(t) = A_y \cdot\cos(\omega_0 \hspace{0.05cm}t - \varphi_y).$$
*New frequencies do not arise.&nbsp; Only amplitude and phase of the harmonic oscillation can be changed.&nbsp; Nonlinear systems are treated in the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions|&raquo;Nonlinear Distortions&laquo;]].

*Because of linearity, the superposition principle is applicable.&nbsp; This states that due to&nbsp; $x_1(t) ⇒  y_1(t)$&nbsp;   and&nbsp; $x_2(t)  ⇒  y_2(t)$&nbsp; the following mapping also necessarily holds:
:$$x_1(t) + x_2(t) \hspace{0.1cm}\Rightarrow \hspace{0.1cm} y_1(t) + y_2(t).$$
*The system is&nbsp; &raquo;'''time-invariant'''&laquo;.&nbsp; This means that an input signal shifted by&nbsp; &nbsp;$\tau$&nbsp; results in the same output signal,&nbsp; but this is also delayed by&nbsp;$\tau$:
:$$x(t - \tau)  \hspace{0.1cm}\Rightarrow \hspace{0.1cm} y(t -\tau)\hspace{0.4cm}{\rm if} \hspace{0.4cm}x(t )\hspace{0.2cm}\Rightarrow \hspace{0.1cm} y(t).$$ 
:Time-varying systems are discussed in the book&nbsp; [[Mobile_Communications|&raquo;Mobile Communications&laquo;]]. 

{{BlaueBox|TEXT=
$\text{Please note:}$&nbsp;
If all the conditions listed here are fulfilled, one deals with a&nbsp; &raquo;'''linear time-invariant system'''&laquo;,&nbsp; abbreviated&nbsp; $\rm LTI$&nbsp; system.}} 

==Frequency response  &ndash; Transfer function==
<br>
We assume an LTI system whose input and output spectra&nbsp; $X(f)$&nbsp; and&nbsp; $Y(f)$&nbsp; are known or can be derived from the time signals&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; via&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;Fourier transform&laquo;]].

[[File:EN_LZI_T_1_1_S4.png|right|frame|Definition of the frequency response|class=fit]]

{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; 
The behaviour of a system is described in the frequency domain by the&nbsp; &raquo;'''frequency response'''&laquo;:
:$$H(f) = \frac{Y(f)}{X(f)}= \frac{ {\rm response\:function} }{ {\rm cause\:function} }.$$
Other terms for&nbsp; $H(f)$&nbsp; are&nbsp; &raquo;system function&laquo;&nbsp; and&nbsp; &raquo;transfer function&laquo;. }}


{{GraueBox|TEXT=  
$\text{Example 2:}$&nbsp; 
The signal &nbsp; $x(t)$&nbsp; with real spectrum&nbsp; $X(f)$&nbsp; $($blue curve$)$&nbsp; is applied to the input of an LTI system.&nbsp; 
[[File:P_ID778__LZI_T_1_1_S4b_neu.png |right|frame|Input spectrum,&nbsp; output spectrum&nbsp; and&nbsp; frequency response|class=fit]]
The measured output spectrum&nbsp; $Y(f)$&nbsp; &ndash; marked red in the graph &ndash; 
*is larger than&nbsp; $X(f)$&nbsp; at frequencies lower than&nbsp; $2 \ \rm kHz$,&nbsp; 

*has a steeper slope in the region around&nbsp; $2 \ \rm kHz$,&nbsp; and

*above&nbsp; $2.8 \ \rm kHz$&nbsp; the signal&nbsp; $y(t)$&nbsp; has no spectral components.


The green circles mark some measuring points of the frequency response&nbsp; $H(f) = Y(f)/X(f)$&nbsp; which is real,&nbsp; too. 
#At low frequencies it holds&nbsp; $H(f)>1$:&nbsp; In this range the LTI system has an amplifying effect. 
#The frequency roll-off of&nbsp; $H(f)$&nbsp; is similar to that of&nbsp; $Y(f)$ but not identical.}}

==Properties of the frequency response==
<br>
The frequency response&nbsp; $H(f)$&nbsp; is a central variable in the description of communication systems. 

Some properties of this important system characteristic are listed below:
*The frequency response describes the LTI system on its own.&nbsp; It can be calculated, for example, from the linear components of an electrical network.&nbsp; With a different input signal&nbsp; $x(t)$&nbsp; and a correspondingly different output signal&nbsp; $y(t)$&nbsp; the result is exactly the same frequency response&nbsp; $H(f)$. 
*The frequency response can have a&nbsp; "unit".&nbsp; For example, if one considers the voltage curve&nbsp; $u(t)$&nbsp; as cause and the current&nbsp; $i(t)$&nbsp; as effect for a one-port, the frequency response&nbsp; $H(f) = I(f)/U(f)$&nbsp; has the unit&nbsp; $\rm A/V$.&nbsp; $I(f)$&nbsp; and&nbsp; $U(f)$&nbsp; are the Fourier transforms of&nbsp; $i(t)$&nbsp; and&nbsp; $u(t)$, respectively. 
*In the following we only consider &raquo;'''two-port networks'''&laquo; or so-called &raquo;'''quadripoles'''&laquo;.&nbsp; Moreover, without loss of generality, we usually assume that&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; are voltages, respectively.&nbsp; In this case&nbsp; $H(f)$&nbsp; is always dimensionless. 
*Since the spectra&nbsp; $X(f)$&nbsp; and&nbsp; $Y(f)$&nbsp; are generally complex, the frequency response&nbsp; $H(f)$&nbsp; is also a complex function.&nbsp; The magnitude&nbsp; $|H(f)|$&nbsp; is called the&nbsp; &raquo;'''amplitude response'''&laquo;&nbsp; or the&nbsp; "magnitude frequency response".&nbsp; 
*This is also often represented in logarithmic form and called the&nbsp; "attenuation curve"&nbsp; or&nbsp; "gain curve":
:$$a(f) = - \ln |H(f)| = - 20 \cdot \lg |H(f)|.$$
*Depending on whether the first form with the natural logarithm or the second with the decadic logarithm is used, the pseudo-unit&nbsp; "neper"&nbsp; $\rm (Np)$&nbsp; or&nbsp; "decibel"&nbsp; $\rm  (dB)$&nbsp; must be added. 
*The&nbsp; &raquo;'''phase response'''&laquo;&nbsp; can be calculated from&nbsp; $H(f)$&nbsp; in the following way:
:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in\hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$

{{BlaueBox|TEXT=  
&nbsp; $\text{Thus, the total frequency response can also be represented as follows}$:
:$$H(f) = \vert H(f)\vert \cdot {\rm e}^{ - {\rm j} \hspace{0.05cm} \cdot\hspace{0.05cm} b(f)} = {\rm e}^{-a(f)}\cdot {\rm e}^{ - {\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$}}

==Low-pass, high-pass, band-pass and band-stop filters==
<br>
[[File:EN_LZI_T_1_1_S6.png|right|frame|Left:&nbsp; Low-pass and high-pass;&nbsp; cut-off frequency&nbsp; $f_{\rm G}$&nbsp; $($German:&nbsp; Grenzfrequenz &nbsp; &rArr; &nbsp; $\rm G)$&nbsp;,<br>Right:&nbsp; Band-pass;&nbsp; lower cut-off frequency&nbsp; $f_{\rm U}$,&nbsp; upper cut-off frequency&nbsp; $f_{\rm O}$|class=fit]]
According to the amplitude response&nbsp; $|H(f)|$&nbsp; one distinguishes between

*&raquo;'''Low-pass filters'''&laquo;: &nbsp;Signal components tend to be more attenuated with increasing frequency. 
*&raquo;'''High-pass filters'''&laquo;: &nbsp;Here, high-frequency signal components are attenuated less than low-frequency ones.&nbsp; A direct signal&nbsp; $($that is, a signal component with the frequency&nbsp; $f = 0)$&nbsp; cannot be transmitted via a high-pass filter. 
*&raquo;'''Band-pass filters'''&laquo;: &nbsp;There is a preferred frequency called the center frequency&nbsp; $f_{\rm M}$.&nbsp; The further away the frequency of a signal component is from&nbsp; $f_{\rm M}$,&nbsp; the more it will be  attenuated. 
*&raquo;'''Band-stop filters'''&laquo;: &nbsp;This is the counterpart to the band-pass filter and it is&nbsp; $|H(f_{\rm M})| ≈ 0$.&nbsp; However, very low-frequency and very high-frequency signal components are let pass.


The graph shows on the left the amplitude responses of the filter types&nbsp; "low-pass"&nbsp; $\rm (LP)$&nbsp; and&nbsp; "high-pass"&nbsp; $\rm (HP)$, and an the right&nbsp; "band-pass"&nbsp; $\rm (BP)$. 

In the sketch also shown are the&nbsp; &raquo;'''cut-off frequencies'''&laquo;.&nbsp;  These denote 3dB cut-off frequencies here, for example, according to the following definition.

{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; 
The&nbsp; &raquo;'''3 dB cut-off frequency'''&laquo;&nbsp; of a low-pass filter specifies the frequency&nbsp; $f_{\rm G}$,&nbsp; for which holds:
:$$\vert H(f = f_{\rm G})\vert = {1}/{\sqrt{2} } \cdot \vert H(f = 0)\vert \hspace{0.5cm}\Rightarrow\hspace{0.5cm} \vert H(f = f_{\rm G})\vert^2 = {1}/{2} \cdot  \vert H(f = 0)  \vert^2.$$}}


#Note that there are also a number of other definitions for the cut-off frequency.&nbsp; 
#These can be found in the section&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#General_remarks|&raquo;General Remarks&laquo;]]&nbsp; in the chapter&nbsp; &raquo;Some Low-Pass Functions in Systems Theory&laquo;. 


 
==Test signals for measuring the frequency response==
<br>
For measuring the frequency response&nbsp; $H(f)$&nbsp; any input signal&nbsp; $x(t)$&nbsp; with spectrum&nbsp; $X(f)$&nbsp is suitable,&nbsp as long as&nbsp; $X(f)$&nbsp; has no zeros (in the range of interest).&nbsp; By measuring the output spectrum&nbsp; $Y(f)$&nbsp; the frequency response can thus be determined in a simple way:
:$$H(f) = \frac{Y(f)}{X(f)}.$$
In particular,&nbsp the following input signals are suitable:  
*'''&raquo;Dirac delta function'''&laquo; &nbsp; $x(t) = K · δ(t)$  &nbsp; ⇒ &nbsp;  spectrum $X(f) = K$:
:Thus, the frequency response by magnitude and phase is of the same shape as the output spectrum&nbsp; $Y(f)$&nbsp; and it holds&nbsp; $H(f) = 1/K · Y(f)$. <br>If one approximates the Dirac delta function by a narrow rectangle of equal area&nbsp; $K$, then&nbsp; $H(f)$&nbsp; must be corrected by means of a&nbsp; ${\rm sin}(x)/x$–function. 
*'''&raquo;Dirac comb'''&laquo; &nbsp;–&nbsp; the infinite sum of equally weighted Dirac delta functions at the time interval&nbsp; $T_{\rm A}$: 
:This leads according to the chapter&nbsp;  [[Signal_Representation/Discrete-Time_Signal_Representation#Dirac_comb_in_time_and_frequency_domain|&raquo;Discrete-Time Signal Representation&laquo;]]&nbsp;  to a Dirac comb in the frequency domain with distance&nbsp; $f_{\rm A} =1/T_{\rm A}$.&nbsp; This allows a discrete frequency measurement of&nbsp; $H(f)$&nbsp; with the spectral samples spaced&nbsp; $f_{\rm A}$. 
*'''&raquo;Harmonic oscillation'''&laquo; &nbsp; $x(t) = A_x · \cos (2πf_0t - φ_x)$  &nbsp; ⇒ &nbsp; Dirac-shaped spectrum at&nbsp; $\pm f_0$:
:The output signal&nbsp; $y(t) = A_y · \cos(2πf_0t - φ_y)$&nbsp; is an oscillation of the same frequency $f_0$.&nbsp; The frequency response for&nbsp; $f_0 \gt 0$&nbsp; is: 
:$$H(f_0) = \frac{Y(f_0)}{X(f_0)} = \frac{A_y}{A_x}\cdot{\rm e}^{\hspace{0.05cm} {\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} (\varphi_x - \varphi_y)}.$$ 
:To determine the total frequency response&nbsp; $H(f)$&nbsp; an infinite number of measurements at different frequencies $f_0$&nbsp; are required. 



==Exercises for the chapter==
<br>
[[Aufgaben:Exercise_1.1:_Simple_Filter_Functions| Exercise 1.1: Simple Filter Functions]]

[[Aufgaben:Exercise_1.1Z:_Low-Pass_Filter_of_1st_and_2nd_Order|Exercise 1.1Z: Low-Pass Filter of 1st and 2nd Order]] 

[[Aufgaben:Exercise_1.2:_Coaxial_Cable|Exercise 1.2: Coaxial Cable]]

[[Aufgaben:Exercise_1.2Z:_Measurement_of_the_Frequency_Response|Exercise 1.2Z: Measurement of the Frequency Response]]



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