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  $H(f)$  is real and even so that according to the  Assignment Theorem  the associated impulse response  $h(t)$  is always real and even, too.
  • Thus, it is obvious that the systems considered here are non-causal and hence not realisable.  The description of causal systems is given in the chapter  Description of Causal Realisable Systems  of this book.
  • The advantage of these  "system theoretical filter functions"  is the simple description by at most two parameters such that the filter influence can be represented in a transparent way.
  • The most important frequency response parameter is the  equivalent bandwidth  according to the definition via the equal-area rectangle:
$$\Delta f = \frac{1}{H(f=0)}\cdot \int_{-\infty}^{+\infty}H(f) \hspace{0.15cm} {\rm d}f.$$
$$\Delta t = \frac{1}{h(t=0)}\cdot \int_{-\infty}^{+\infty}h(t) \hspace{0.15cm} {\rm d}t = \frac{1}{\Delta f}.$$
  • The direct signal (DC) transmission factor is 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  "Derivation of system theoretical high-pass functions".

Ideal low-pass filter – Rectangular-in-frequency

$\text{Definition:}$  An  $\text{ideal low-pass filter}$  is on hand if its frequency response has the following rectangular shape:

$$H(f) = H_{\rm RLP}(f) =\left\{ \begin{array}{l} \hspace{0.25cm}1 \\ 0.5 \\\hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad\begin{array}{*{10}c} \text {for} \\ \text {for} \\ \text {for} \\ \end{array}\begin{array}{*{20}c}{\vert \hspace{0.005cm}f\hspace{0.05cm} \vert< \Delta f/2,} \\{\vert \hspace{0.005cm}f\hspace{0.05cm} \vert = \Delta f/2,} \\{\vert \hspace{0.005cm}f\hspace{0.05cm} \vert > \Delta f/2.} \\\end{array}$$
  • Here  $Δf$  denotes the system theoretical bandwidth and  $f_{\rm G}=Δf/2$  the cut-off frequency  (German:  "Grenzfrequenz")   ⇒   $\rm G$.
  • We sometimes also use the term  "rectangular low-pass filter”  $\rm (RLP)$.

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

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 theoretic 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   ⇒   "stop band".
  • 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 transform the impulse response (right diagram) is:
$$h(t) = h_{\rm RLP}(t) =\Delta f \cdot {\rm si}(\pi \cdot \Delta f \cdot t)\hspace{0.55cm}{\rm{with}}\hspace{0.7cm}{\rm si}(x) ={\sin(x)}/{x},\hspace{0.5cm}{\rm or}$$
$$h_{\rm RLP}(t) =\Delta f \cdot {\rm sinc}(\Delta f \cdot t)\hspace{0.7cm}{\rm{with}}\hspace{0.7cm}{\rm sinc}(x) ={\sin(\pi x)}/{(\pi x)}.$$
  • $h(t)$  extended to infinity on both sides and exhibits equidistant zero-crossings at an interval of  $Δt = 1/ Δf$.
  • The asymptotic decay is inversely proportional to time  $|t|$:
$$|h(t)| = \frac{\Delta f}{\pi \cdot \Delta f \cdot |t|} \cdot \left |{\rm sin}(\pi \cdot \Delta f\cdot t )\right | \le \frac{1}{\pi \cdot |t|}.$$
  • 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:
$${\rm Si}(0) = 0, \hspace{0.3cm}{\rm Si}(\infty) = \frac{\pi}{2}, \hspace{0.3cm}{\rm Si}(-x) = -{\rm Si}(x).$$

Slit low-pass filter – Rectangular-in-time

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

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

$$\hspace{2.1cm}H_{\rm SLP}(f)= {\rm sinc}({f}/{ \Delta f})\hspace{0.7cm}{\rm{where} }\hspace{0.7cm}{\rm sinc}(x) ={\sin(\pi x)}/{(\pi x)}.$$

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

Slit low-pass filter:  Frequecy response and respective impulse response

According to the  Duality Theorem  the impulse response  $h_{\rm SLP}(t)$  of the slit low-pass filter must also have the same form as the frequency response $H_{\rm RLP}(f)$ of the ideal low-pass filter   ⇒   "rectangular-in-time".

Thus, with the  "equivalent duration of the impulse response"  $Δt = 1/ Δf$  the following holds:

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

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

  • The slit low-pass filter in this form is also non-causal.  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 theoretic investigations is the Gaussian low-pass filter, which can also be described by only one parameter, namely the  equivalent bandwidth  $Δf$.

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

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

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-called  "Gaussian formula"  which he discovered for probability theory.

Gaussian low-pass filter:  Frequency response and respective impulse response

Based on this graph the following statements can be made:

  • The  equivalent pulse duration  $Δt$  is also defined via the area-equal rectangle 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  Reciprocity Theorem  of time duration 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 Fourier transform.
  • The Gaussian low-pass filter is  –   like the ideal low-pass filter  –   strongly non-causal and (exactly) realisable only with infinitely large runtime 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 tabular form in formula collections:
$$\sigma(t) = \int_{ -\infty }^{ t } {h(\tau)} \hspace{0.1cm}{\rm d}\tau = {\rm \phi}\left( \sqrt{2 \pi }\cdot{t}/{\Delta t} \right) \hspace{0.7cm}{\rm{where}}\hspace{0.7cm}{\rm \phi}(x) = \frac{1}{\sqrt{2 \pi }} \cdot \int_{ -\infty }^{ x } {{\rm e}^{-u^2/2}} \hspace{0.1cm}{\rm d}u.$$

Trapezoidal low-pass filter

The low-pass functions described so far 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  $\text{trapezoidal low-pass filter}$  with cut-off frequencies  $f_1$  and  $f_2 \ge f_1$:

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

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

  • the  equivalent bandwidth  determined via the equal-area rectangle:
$$\Delta f = f_1 + f_2.$$
  • 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  $H(f)= H_{\rm TLP}(f)$  on the left and on the right the impulse response

$$h(t) = h_{\rm TLP}(t) =\Delta f \cdot {\rm sinc}(\Delta f \cdot t )\cdot {\rm sinc}(r_{\hspace{-0.05cm}f} \cdot \Delta f \cdot t )\hspace{0.7cm}{\rm{where}}\hspace{0.7cm}{\rm sinc}(x)= \frac{\sin(\pi x)}{\pi x}.$$

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

Trapezoidal low-pass filter:  Frequency response and 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  $\rm sinc$ functions with arguments  $Δf · t$  and  $r_{\hspace{-0.05cm}f} · Δf · t$.
  • The first  $\rm sinc$ function is part of the  $h(t)$  equation 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:
$$H(f) = \left\{ \begin{array}{c} \hspace{0.25cm} \frac{{\rm \Delta}f -|f|}{{\rm \Delta}f} \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{10}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} {\hspace{1cm} \left| \hspace{0.005cm}f\hspace{0.05cm} \right| \le {\rm \Delta}f ,} \\ {\hspace{1cm}\left|\hspace{0.005cm} f \hspace{0.05cm} \right| \ge {\rm \Delta}f } \\ \end{array}, \hspace{1cm} h(t) = \Delta f \cdot {\rm sinc}^2( \Delta f \cdot t ),\hspace{0.2cm}{\rm{where}}\hspace{0.4cm}{\rm sinc}(x)= \frac{\sin(\pi x)}{\pi x}.$$

Raised-cosine low-pass filter

Like the  trapezoidal low-pass filter  this low-pass filter is also described by two parameters, which are

  • the equivalent bandwidth  $Δf$  and
  • the roll-off factor $r_{\hspace{-0.05cm}f}$.

Its range of values lies between  $r_{\hspace{-0.05cm}f} = 0$  (rectangular low-pass filter)  and  $r_{\hspace{-0.05cm}f} = 1$  (cosine–squared low-pass filter).

$\text{Definition:}$  With cut-off frequencies  $f_1 = Δf · (1 – r_{\hspace{-0.05cm}f})$  and  $f_2 = Δf · (1 + r_{\hspace{-0.05cm}f})$  the frequency response of the  $\text{raised-cosine low-pass filter}$:

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

Raised-cosine low-pass filter and respective impulse response

The graph shows  $H(f)$  on the left and on the right the impulse response

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

For these graphs the roll-off factor  $r_{\hspace{-0.05cm}f} = 0.5$  was used.  In other words, $f_2 = 3 \cdot f_1$ applies.

The following is drawn in dashed lines for comparison:

  • in the frequency domain the trapezoidal low-pass filter and
  • in the time domain the  $\rm sinc$ function.

Please note:

  • The  $\rm sinc$ function is not the inverse Fourier transform of the trapezoidal low-pass filter drawn in blue on the left.
  • It rather describes the ideal rectangular low-pass filter in the time domain, which is not shown in the left graph.

Based on this graph and the above equations the following statements can be made:

  • The impulse response  $h(t)$  of the raised-cosine low-pass filter has zeros at all multiples of  $Δt = 1/Δf$, which are due to the  $\rm sinc$ function shown in dashed lines in the picture on the right-hand side.
  • The last term in the  $h(t)$  equation results in further zeros at multiples of  $Δt/r_f$.  If  $1/r_f$  is an integer as in the above graph  $(1/r_f = 2)$, these new zeros coincide with the other zeros and thus are not discernible.
  • The larger the roll-off factor  $r_f$  and thus the flatter the roll-off is, the more favourable is the transient behaviour of the raised-cosine low-pass filter.
  • The raised-cosine low-pass filter usually exhibits a better asymptotic transient behaviour than the trapezoidal low-pass filter with the same  $r_f$  although the latter has a flatter edge at least at  $Δf/2$.
  • This suggests that the transient behaviour is not only affected by points of discontinuity  (as in the case of the rectangle)  but also by kink pointss as in the case of the trapezoidal low-pass filter.

Cosine–square low-pass filter

$\text{Definition:}$  For  $f_1 = 0$,  $f_2 = Δf$   ⇒   $r_f = 1$ the  $\text{cosine–square low-pass filter}$  is obtained as a special case.  Its impulse response can also be represented as follows:

$$H(f) = \cos^2\Big(\frac{\vert f \vert \hspace{0.05cm}\cdot\hspace{0.05cm} \pi}{2\hspace{0.05cm}\cdot\hspace{0.05cm} \Delta f}\Big). $$

Outside this inner frequency range,  $H(f)=0$.

For the impulse response one obtains according to the inverse Fourier transform after some transformations:

$$h(t)=\Delta f \cdot {\rm sinc}(\Delta f \cdot t)\cdot \big [{\rm sinc}(\Delta f\cdot t +0.5)+{\rm sinc}(\Delta f\cdot t -0.5)\big ],$$
$$T=1/\Delta f \hspace{0.5cm}\Rightarrow \hspace{0.5cm} h(t)=1/T \cdot {\rm sinc}(t/T)\cdot \big [{\rm sinc}(t/T +0.5)+{\rm sinc}(t/T -0.5)\big ].$$
  • Because of the first  ${\rm sinc}$  function,  $h(t)=0$  for multiples of  $T=1/\Delta f$   ⇒   the equidistant zero-crossings of the cosine–rolloff lowpass are preserved.
  • Because of the bracket expression,  $h(t)$  now exhibits further zero-crossings at  $t=\pm1.5 T$,  $\pm2.5 T$,  $\pm3.5 T$, ...  but not at  $t=\pm0.5 T$.
  • For  $t=\pm0.5 T$  the impulse response has the value  $\Delta f/2$.

The asymptotic decay of  $h(t)$  in this special case runs with  $1/t^3$.  Furthermore, it should be mentioned that the  "cosine-square low-pass filter"  is the only low-pass filter that fulfils both  Nyquist Criteria.    Here the  "eye"  is maximally open both vertically and horizontally   ⇒   see  Definition and statements of the eye diagram.

Derivation of system theoretical high-pass functions

So far in this chapter, six commonly used system theoretic low-pass functions have been considered.  For each individual low-pass function there is also an equivalent high-pass function.

$\text{Definition:}$  If  $H_{\rm TP}(f)$  is a system theoretic low-pass function  (German "Tiefpass"   ⇒   $\text{TP}$)  with  $H_{\rm TP}(f = 0) = 1$, then the    $\text{equivalent high-pass function}$  is:

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

Thus, the descriptive quantities in the time domain are:

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

Here, the following denotations are used:

  • $h_{\rm HP}(t)$  and  $h_{\rm TP}(t)$ denote the impulse responses of the high- and low-pass filter,
  • $σ_{\rm HP}(t)$  and  $σ_{\rm TP}(t)$ denote the respective step responses,
  • $γ(t)$  denotes the step function as a result of integration over the Dirac function  $δ(t)$.

Construction of high-pass functions  $H_{\rm HP}(f)$,  $h_{\rm HP}(t)$,  $\sigma_{\rm HP}(t)$  from the
corresponding low-pass functions  $H_{\rm TP}(f)$,  $h_{\rm TP}(t)$,  $\sigma_{\rm TP}(t)$

$\text{Example 1:}$  We consider the  "slit low-pass filter"  ⇒   "rectangular-in-time"  which is characterised by

  • a  $\rm sinc$–shaped frequency response,
  • a rectangular impulse response and
  • a linearly increasing step response. 

These are shown in the upper diagram.  The sketch below shows the corresponding high-pass functions.

It can be seen that

  • $H_{\rm HP}(f = 0)$  always is  $0$  if  $H_{\rm TP}(f = 0) = 1$,
  • consequently the integral over  $h_{\rm HP}(t)$  must also be zero, and
  • the step response  $σ_{\rm HP}(t)$, too, tends towards the final value of zero.

Exercises for the chapter

Exercise 1.5: Rectangular-in-Frequency Low-Pass Filter

Exercise 1.5Z: Sinc-shaped Impulse Response

Exercise 1.6: Rectangular-in-Time Low-Pass Filter

Exercise 1.6Z: Interpretation of the Frequency Response

Exercise 1.7: Nearly Causal Gaussian Low-Pass Filter

Exercise 1.7Z: Overall Systems Analysis

Exercise 1.8: Variable Edge Steepness

Exercise 1.8Z: Cosine-Square Low-Pass Filter