Linear and Time Invariant Systems/Some Low-Pass Functions in Systems Theory

1. Linear and Time Invariant Systems
2. Basics of System Theory
3. Some Low-Pass Functions in Systems Theory

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General Remarks

All low-pass functions described on the next pages have the following properties:

• The frequency response $H(f)$ ist real and even, so that according to the Zuordnungssatz the associated impulse response$h(t)$ is always real and even, too.
• Thus, it is obvious that the systems considered here are noncausal and hence not realisable. The description of causal systems is given in the chapter Beschreibung kausaler realisierbarer Systeme of this book.
• The advantage of these systemtheoretic filter functions is the simple description by at most two parameters such that the filter influx can be represented in a transparent way.
• The most important function parameter is the equivalent bandwidth according to the definition via the rectangle of area equal to the square:

$$\Delta f = \frac{1}{H(f=0)}\cdot \int_{-\infty}^{+\infty}H(f) \hspace{0.15cm} {\rm d}f.$$

• According to the so-called Reziprozitätsgesetz the equivalent time period of the impulse response is thus also fixed, which is also defined via the rectangle of area equal to the square:

$$\Delta t = \frac{1}{h(t=0)}\cdot \int_{-\infty}^{+\infty}h(t) \hspace{0.15cm} {\rm d}t = \frac{1}{\Delta f}.$$

• The direct signal transmission factor wiris always assumed to be $H(f = 0) = 1$ unless explicitly stated otherwise.
• From every low-pass function corresponding high-pass functions can be derived as shown on the page Herleitung systemtheoretischer Hochpassfunktionen.

Ideal Low-Pass Filter – Küpfmüller Low-Pass Filter

$\text{Definition:}$  An ideal low-pass filter is on hand if its frequency response is as follows:

$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}1 \\ 0.5 \\\hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad\begin{array}{*{10}c} \text {for} \\ \text {for} \\ \text {for} \\ \end{array}\begin{array}{*{20}c}{\vert \hspace{0.005cm}f\hspace{0.05cm} \vert< \Delta f/2,} \\{\vert \hspace{0.005cm}f\hspace{0.05cm} \vert = \Delta f/2,} \\{\vert \hspace{0.005cm}f\hspace{0.05cm} \vert > \Delta f/2.} \\\end{array}$$

We sometimes also use the term "Küpfmüller-low-pass filter” (KLP) in in memory of the pioneer of systems theory, Karl Küpfmüller.

The graph shows such an ideal low-pass filter in the frequency and time domain. The following can be concluded from this curve shape:

Ideal Low-pass Filter: Frequency Response and Impulse Response

• Due to the abrupt, infinitely steep roll-off the 3dB cut-off frequency $f_{\rm G}$ is here exactly half the system-theoretical bandwidth $Δf$.
• All spectral components with $f \lt f_{\rm G}$ are transmitted undistorted (pass band).
• All components with $f \gt f_{\rm G}$ are completely suppressed (cut-off region).
• By definition, $H(f) = 0.5$ holds for $f = f_{\rm G}$.

Description of the ideal low-pass filter in the time domain:

• According to the inverse Fourier transformation the impulse response (see diagram on the right) is

$$h(t) = \Delta f \cdot {\rm si}(\pi \cdot \Delta f \cdot t)\hspace{0.7cm}{\rm{with}}\hspace{0.7cm}{\rm si}(x) ={\sin(x)}/{x}.$$

• The impulse response $h(t)$ extended to infinity on both sides exhibits equidistant zero-crossings at an interval of $Δt = 1/ Δf$.
• The asymptotic decay is inversely proportional to time:

$$|h(t)| = \frac{\Delta f}{\pi \cdot \Delta f \cdot |t|} \cdot \left |{\rm sin}(\pi \cdot \Delta f\cdot t )\right | \le \frac{1}{\pi \cdot |t|}.$$

• It follows that the impulse response is certainly less than $1‰$ of the impulse maximum only for times $t \gt t_{1‰} = 318 \cdot \Delta t$ .
• The step response $\sigma(t)$ is obtained from the impulse response by integration and is:

$${\sigma}(t) = \int_{ - \infty }^{ t } {h ( \tau )} \hspace{0.1cm}{\rm d}\tau = \frac{1}{2} + \frac{1}{\pi} \cdot {\rm Si}(\pi \cdot\Delta f \cdot t ).$$

• Here the so-called integral sine function is used:

$${\rm Si}(x) = \int_{ 0 }^{ x } {{\rm si} ( \xi )} \hspace{0.1cm}{\rm d}\xi = x - \frac{x^3}{3 \cdot 3!} + \frac{x^5}{5 \cdot 5!} - \frac{x^7}{7 \cdot 7!}+\text{ ...}$$

It has the following properties:

$${\rm Si}(0) = 0, \hspace{0.3cm}{\rm Si}(\infty) = \frac{\pi}{2}, \hspace{0.3cm}{\rm Si}(-x) = -{\rm Si}(x).$$

Note:   In some books, instead of the function ${\rm si}(x)$ the similar function ${\rm sinc}(x)$ is used:

$${\rm si}(x) = \frac{\sin(x)}{x}\hspace{0.5cm}\Rightarrow\hspace{0.5cm}{\rm sinc}(x) = \frac{\sin(\pi x)}{\pi x} = {\rm si}(\pi x).$$

Thus, the impulse response of the ideal low-pass filter is:   $h(t)$ = $Δf · {\rm sinc}(Δf · t).$

Slit Low-Pass Filter

$\text{Definition:}$  An LTI–system is called a slit–low-pass filter if the frequency response has the following form:

$$H(f) = {\rm si}(\pi {f}/{ \Delta f})\hspace{0.7cm}{\rm{where} }\hspace{0.7cm}{\rm si}(x) ={\sin(x)}/{x}.$$

From the graph on the left it can be seen that the frequency response $H_{\rm SLP}(f)$ of the slit–low-pass is identical in shape to the impulse response $h_{\rm KLP}(t)$ of the Küpfmüller-low-pass.

slit–low-pass: frequecy response and respective impulse response

According to the Vertauschungssatz the impulse response $h_{\rm SLP}(t)$ of the slit low-pass filter must also have the same form as the frequency response $H_{\rm KLP}(f)$ of the ideal low-pass filter.

Thus, with $Δt = 1/ Δf$ the following holds:

$$h(t) = \left\{ \begin{array}{l} \hspace{0.25cm}\Delta f \\ \Delta f/2 \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{10}c} \text{for} \\ \text{for} \\ \text{for} \\ \end{array}\begin{array}{*{20}c} {\vert \hspace{0.005cm}t\hspace{0.05cm} \vert < \Delta t/2,} \\ {\vert \hspace{0.005cm}t\hspace{0.05cm} \vert = \Delta t/2,} \\ {\vert \hspace{0.005cm}t\hspace{0.05cm} \vert > \Delta t/2.} \\ \end{array}$$

Based on the graph on the right the following statements can be derived:

• The slit–low-pass filter is also noncausal in this form.
• However, adding a running time of $Δt/2$ renders the system causal and thus realisable.
• The slit–low-pass filter acts as an integrator over the time period $Δt$:

$$y(t) = x (t) * h (t) = \frac{1}{\Delta t} \cdot \int\limits_{ t - \Delta t/2 }^{ t + \Delta t/2 } {x ( \tau )} \hspace{0.1cm}{\rm d}\tau.$$

• If $x(t)$ is a harmonic oscillation with frequency $f_0 = k \cdot Δf$ (where $k$ is an integer), then it integrates exactly over $k$ periods and $y(t) = 0$ holds. This is also shown by the zeros of $H(f)$.

Gaussian Low-Pass Filter

A filter function frequently used for system-theoretical investigations is the Gaussian low-pass filter, which can also be described by only one parameter, namely the äquivalente Bandbreite $Δf$.

$\text{Definition:}$  For the frequency response and impulse response of the Gaussian low-pass filter the following holds:

$$H(f) = {\rm e}^{-\pi(f/\Delta f)^2}\hspace{0.15cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.15cm}h(t) = \Delta f \cdot {\rm e}^{-\pi(\Delta f \cdot \hspace{0.03cm} t)^2} .$$

The name goes back to the mathematician, physicist and astronomer Carl-Friedrich Gauß. Gauss did not deal with this subject matter himself, but the mathematical form of the frequency response and impulse response bear a resemblance to the so-calledGaußformel which he discovered for probability theory.

Gaussian Low-Pass Filter: frequency response and associated impulse response

Based on this graph the following statements can be made:

• The äquivalente Impulsdauer $Δt$ is also defined via the rectangle of area equal to the square and is equal to the reciprocal of the equivalent bandwith $Δf$.
• A narrow-band filter function (small $Δf$) results in a wide (large $Δt$) and simultaneously low impulse response $h(t)$.
• The so-called Reziprozitätsgesetz of time period and bandwidth can be shown particularly clearly in the example of the Gaussian low-pass filter.
• The frequency and time domain representations are in principle of the same form. The Gaussian function is also said to be invariant to the Fourier transformation.
• The Gaussian low-pass filter is - like the ideal low-pass filter - strongly noncausal and (exactly) realisable only with infinitely large transit time due to the infinite propagation of its impulse response.
• However, it must be taken into account that $h(t)$ has already decayed to $1‰$ of its maximum value at $t = 1.5 \cdot Δt$ . For $t = 3 \cdot Δt$ we even get $h(t) ≈ 5 · 10^{–13} · h(0)$.
• These numerical values show that the Gaussian low-pass filter can be used feasibly for practical simulations as long as runtimes do not play a system-limiting role.
• The step response $σ(t)$ is given for the Gaussian error function $ϕ(x)$, which is usually given in formularies:

$$\sigma(t) = \int_{ -\infty }^{ t } {h(\tau)} \hspace{0.1cm}{\rm d}\tau = {\rm \phi}\left( \sqrt{2 \pi }\cdot{t}/{\Delta t} \right) \hspace{0.7cm}{\rm{where}}\hspace{0.7cm}{\rm \phi}(x) = \frac{1}{\sqrt{2 \pi }} \cdot \int_{ -\infty }^{ x } {{\rm e}^{-u^2/2}} \hspace{0.1cm}{\rm d}u.$$

Trapezoid Low-Pass Filter

The low-pass functions described so far in this chapter depend on only one parameter - the equivalent bandwidth $Δf$. Here, the edge steepness for a given filter type was fixed.

Now a low-pass filter with parameterisable edge steepness is described.

$\text{Definition:}$  The frequency response of the trapezoid low-pass filter with cut-off frequencies $f_1$ and $f_2 \ge f_1$:

$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}1 \\ \frac{f_2 - \vert f \vert }{f_2 -f_1} \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{10}c} \text{for} \\ \text{for} \\ \text{for} \\ \end{array}\begin{array}{*{20}c} {\hspace{0.94cm}\vert \hspace{0.005cm} f\hspace{0.05cm} \vert < f_1,} \\ {f_1 \le \vert \hspace{0.005cm} f\hspace{0.05cm} \vert \le f_2,} \\ {\hspace{0.94cm}\vert \hspace{0.005cm} f\hspace{0.05cm} \vert > f_2.} \\ \end{array}$$

Instead of $f_1$ and $f_2$ the following parameters can be used to describe $H(f)$:

• the equivalent bandwidth determined via the rectangle of area equal to the square:

$$\Delta f = f_1 + f_2.$$

• the roll-off factor (in the frequency domain) as a measure for the edge steepness:

$$r_{\hspace{-0.05cm}f} = \frac{f_2 - f_1}{f_2 + f_1}.$$

Special cases included in the general representation are:

• the ideal rectangular low-pass filter $(r_{\hspace{-0.05cm}f} = 0)$,
• the triangular low-pass filter $(r_{\hspace{-0.05cm}f} = 1)$.

For a roll-off factor of $r_f = 0.5 \ \Rightarrow \ f_2 = 3f_1$ the following graph shows the frequency response $H(f)$ on the left and the impulse response

$$h(t) = \Delta f \cdot {\rm si}(\pi \cdot \Delta f \cdot t )\cdot {\rm si}(\pi \cdot r_{\hspace{-0.05cm}f} \cdot \Delta f \cdot t )\hspace{0.7cm}{\rm{where}}\hspace{0.7cm}{\rm si}(x) = {\sin(x)}/{x}$$ on the right.

The time-dependent $\rm si$–curve of the rectangular low-pass filter with the same equivalent bandwidth is shown dashed for comparison.

Trapezoid Low-Pass Filter: frequency response and associated impulse response

With the help of the graph and the above equations the following statements can be made:

• The trapezoidal shape is obtained, for example, by convolution of two rectangles of widths $Δf$ and $r_f \cdot Δf$.
• According to the convolution theorem the impulse response is thus the product of two $\rm si$–functions with arguments $π · Δf · t$ and $π · r_{\hspace{-0.05cm}f} · Δf · t$.
• The first $\rm si$–function is part of the equation for $h(t)$ for all values of $r_{\hspace{-0.05cm}f}$ and always results in equivalent zero-crossings at an interval of $1/Δf$.
• For $0 \lt r_{\hspace{-0.05cm}f} \lt 1$ there are further zero-crossings at multiples of $Δt/r_{\hspace{-0.05cm}f}$.
• The larger $r_{\hspace{-0.05cm}f}$ is (i.e. for a given $Δf$ with a flatter edge), the faster is the asymptotic decay of the impulse response $h(t)$.
• The fastest possible decay is obtained for the triangular low-pass filter   ⇒   $r_{\hspace{-0.05cm}f} = 1$, $f_1 = 0$, $f_2 = Δf$. For this, the following holds in the frequency and time domains:

$$H(f) = \left\{ \begin{array}{c} \hspace{0.25cm} \frac{{\rm \Delta}f -|f|}{{\rm \Delta}f} \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{10}c} {\rm{f\ddot{u}r}} \\ {\rm{for\ddot{u}r}} \\ \end{array}\begin{array}{*{20}c} {\hspace{1cm} \left| \hspace{0.005cm}f\hspace{0.05cm} \right| \le {\rm \Delta}f ,} \\ {\hspace{1cm}\left|\hspace{0.005cm} f \hspace{0.05cm} \right| \ge {\rm \Delta}f } \\ \end{array}, \hspace{1cm} h(t) = \Delta f \cdot {\rm si}^2(\pi \cdot \Delta f \cdot t )\hspace{0.4cm}{\rm{where}}\hspace{0.4cm}{\rm si}(x) = \frac{\sin(x)}{x}.$$

Raised-Cosine Low-Pass Filter

Ebenso wie der Trapez–Tiefpass wird dieser Tiefpass durch zwei Parameter beschrieben, nämlich durch

• die äquivalente Bandbreite $Δf$ und
• den Rolloff–Faktor $r_{\hspace{-0.05cm}f}$.

Dessen Wertebereich liegt zwischen $r_{\hspace{-0.05cm}f} = 0$ (Rechtecktiefpass) und $r_{\hspace{-0.05cm}f} = 1$ (Cosinus–Quadrat–Tiefpass).

$\text{Definition:}$  Mit den Eckfrequenzen $f_1 = Δf · (1 – r_{\hspace{-0.05cm}f})$  und  $f_2 = Δf · (1 + r_{\hspace{-0.05cm}f})$ lautet der Frequenzgang des Cosinus–Rolloff–Tiefpasses:

$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}1 \\ \cos \left( \frac{ \vert f \vert - f_1}{f_2 -f_1}\frac{\pi}{2}\right) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{10}c} \text{for} \\ \text{for} \\ \text{for} \\ \end{array}\begin{array}{*{20}c} {\hspace{0.94cm}\vert \hspace{0.005cm} f\hspace{0.05cm} \vert < f_1,} \\ {f_1 \le \vert \hspace{0.005cm} f\hspace{0.05cm} \vert \le f_2,} \\ {\hspace{0.94cm}\vert \hspace{0.005cm} f\hspace{0.05cm} \vert> f_2.} \\ \end{array}$$

Cosinus–Rolloff–Tiefpass und zugehörige Impulsantwort

Die Grafik zeigt links $H(f)$ sowie rechts die Impulsantwort

$$h(t) = \Delta f \hspace{-0.05cm}\cdot\hspace{-0.05cm} {\rm si}(\pi \hspace{-0.05cm}\cdot\hspace{-0.05cm} \Delta f \hspace{-0.05cm}\cdot\hspace{-0.05cm} t )\hspace{-0.05cm}\cdot\hspace{-0.05cm} \frac {\cos(\pi \cdot r_{\hspace{-0.05cm}f} \cdot \Delta f \cdot t )}{1 - (2 \cdot r_f \cdot \Delta f \cdot t)^2}.$$

Für diese Grafiken wurde der Rolloff–Faktor $r_{\hspace{-0.05cm}f} = 0.5$ verwendet, das heißt, es gilt $f_2 = 3 \cdot f_1$.

Gestrichelt sind zum Vergleich eingezeichnet:

• im Frequenzbereich der Trapeztiefpass und
• im Zeitbereich die $\rm si$–Funktion.

Es ist zu beachten:

• Die $\rm si$–Funktion ist nicht die Fourierrücktransformierte des links blau eingezeichneten Trapeztiefpasses.
• Sie beschreibt vielmehr den (nicht dargestellten) idealen, rechteckförmigen Tiefpass im Zeitbereich.

Anhand dieser Grafik und den obigen Gleichungen sind folgende Aussagen möglich:

• Die Impulsantwort $h(t)$ des Cosinus–Rolloff–Tiefpasses hat bei allen Vielfachen von $Δt = 1/Δf$ Nullstellen, die auf die im rechten Bild gestrichelt eingezeichnete si–Funktion zurückzuführen sind.
• Der letzte Term in der $h(t)$–Gleichung führt zu weiteren Nullstellen bei Vielfachen von $Δt/r_f$. Ist $1/r_f$ ganzzahlig wie in obiger Grafik $(1/r_f = 2)$, so fallen diese neuen Nullstellen mit den anderen Nullstellen zusammen, sind also nicht erkennbar.
• Je größer der Rolloff-Faktor $r_f$ ist und je flacher damit der Flankenabfall erfolgt, desto günstiger ist im Allgemeinen das Einschwingverhalten des Cosinus-Rolloff-Tiefpasses.
• Der Cosinus–Rolloff–Tiefpass zeigt meist ein besseres asymptotisches Einschwingverhalten als der Trapez–Tiefpass mit gleichem $r_f$, obwohl dieser zumindest bei $Δf/2$ eine flachere Flanke aufweist.
• Dies lässt darauf schließen, dass das Einschwingverhalten nicht nur durch Unstetigkeitsstellen (wie beim Rechteck), sondern auch durch Knickpunkte wie beim Trapez–Tiefpass beeinträchtigt wird.

$\text{Definition:}$  Als Sonderfall ergibt sich mit $f_1 = 0$, $f_2 = Δf$   ⇒   $r_f = 1$ der Cosinus–Quadrat–Tiefpass, dessen Impulsantwort auch wie folgt dargestellt werden kann:

$$h(t) = \frac{1}{ \Delta t}\cdot{\rm si}(\pi \frac{t}{ \Delta t}) \cdot \left[ {\rm si}(\pi \frac{t}{ \Delta t} + 0.5) - {\rm si}(\pi \frac{t}{ \Delta t} - 0.5) \right].$$

• Diese Funktion hat Nullstellen bei $t/Δt = ±1, ±1.5, ±2, ±2.5$ usw., nicht jedoch bei $t/Δt = ±0.5$.
• Der Cosinus–Quadrat–Tiefpass erfüllt als einziger Tiefpass beide Nyquistkriterien   ⇒   siehe Buch  Digital Signal Transmission.

Herleitung systemtheoretischer Hochpassfunktionen

Bisher wurden in diesem Kapitel fünf häufig verwendete systemtheoretische Tiefpassfunktionen betrachtet. Für jede einzelne Tiefpassfunktion lässt sich auch eine äquivalente Hochpassfunktion angeben.

$\text{Definition:}$  Ist $H_{\rm TP}(f)$ eine systemtheoretische TP–Funktion mit $H_{\rm TP}(f = 0) = 1$, so ist die äquivalente Hochpassfunktion:

$$H_{\rm HP}(f) = 1 - H_{\rm TP}(f).$$

Damit lauten die Beschreibungsgrößen im Zeitbereich:

$$h_{\rm HP}(t) = \delta (t) - h_{\rm TP}(t),\hspace{1cm} \sigma_{\rm HP}(t) = \gamma (t) - \sigma_{\rm TP}(t). $$

Hierbei bezeichnen:

• $h_{\rm HP}(t)$  und  $h_{\rm TP}(t)$ die Impulsantworten von Hoch– und Tiefpass,
• $σ_{\rm HP}(t)$  und  $σ_{\rm TP}(t)$ die dazugehörigen Sprungantworten,
• $γ(t)$ die Sprungfunktion als Ergebnis der Integration über die Diracfunktion $δ(t)$.

$\text{Beispiel 1:}$  Wir betrachten den Spalttiefpass, der sich durch einen $\rm si$–förmigen Frequenzgang, eine rechteckförmige Impulsantwort und eine linear ansteigende Sprungantwort auszeichnet. Diese sind in der nachfolgenden Grafik dargestellt.

Die untere Skizze zeigt die entsprechenden Hochpassfunktionen.

Konstruktion von Hochpassfunktionen aus den entsprechenden Tiefpässen

Man erkennt, dass

• $H_{\rm HP}(f = 0)$ immer den Wert $0$ besitzt, wenn $H_{\rm TP}(f = 0) = 1$ ist,
• demzufolge das Integral über $h_{\rm HP}(t)$ ebenfalls Null ergeben muss, und
• auch die Sprungantwort $σ_{\rm HP}(t)$ gegen den Endwert Null tendiert.

Exercises for the Chapter

Aufgabe 1.5: Idealer rechteckförmiger Tiefpass

Aufgabe 1.5Z: si-förmige Impulsantwort

Aufgabe 1.6: Rechtförmeckige Impulsantwort

Aufgabe 1.6Z: Interpretation der Übertragungsfunktion

Aufgabe 1.7: Nahezu kausaler Gaußtiefpass

Aufgabe 1.7Z: Systemanalyse

Aufgabe 1.8: Variable Flankensteilheit

Aufgabe 1.8Z: Cosinus-Quadrat-Tiefpass