Exercise 2.5: Distortion and Equalization

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Trapezoidal spectrum (top),
associated impulse response)

A communication system with input  $x(t)$  and output  $y(t)$,  which is fully described by the trapezoidal frequency response  $H(f)$  according to the top graph, is considered.  Using the roll-off factor  $r = 0.5$  and the equivalent bandwidth  $\Delta f = 16 \ \rm kHz$  the corresponding impulse response,  which is computable by applying the inverse Fourier transform, is:

$$h(t) = \Delta f \cdot {\rm si}(\pi \cdot \Delta f \cdot t )\cdot {\rm si}(\pi \cdot r \cdot \Delta f \cdot t ) = \Delta f \cdot {\rm sinc}(\Delta f \cdot t )\cdot {\rm sinc}(r \cdot \Delta f \cdot t ).$$

Here the following functions which can be converted into each other are used:

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

The available input signals are:

  • The sum of two harmonic oscillations:
$$x_1(t) = {1\, \rm V} \cdot \cos(\omega_1 \cdot t) + {1\, \rm V} \cdot \sin(\omega_2 \cdot t).$$
Here, the following holds:   $\omega_1 = 2\pi \cdot 2000 \ {\rm 1/s}$  and  $\omega_2 \gt \omega_1$.
  • A periodic triangular signal:
$$x_2(t) = \frac{8\, \rm V}{\pi^2} \cdot \big[\cos(\omega_0 t) + {1}/{9} \cdot \cos(3\omega_0 t) + {1}/{25} \cdot \cos(5\omega_0 t) + \hspace{0.05cm}\text{...}\big].$$
It should be noted that the basic frequency is  $f_0 = 2 \ \rm kHz$  or  $3\ \rm kHz$.  At time  $t = 0$  the signal value in both cases is  $1 \ \rm V$.
  • A rectangular pulse  $x_3(t)$  with amplitude  $A = 1 \ \rm V$  and duration  $T = 1 \ \rm ms$.
    Since its spectrum  $X_3(f)$  extends to infinity,   $H(f)$  always results in linear distortions here.

From subtask  (6)  onwards, it shall be attempted to eliminate the distortions possibly generated by  $H(f)$  by means of a downstream equalizer with

  • frequency response  $H_{\rm E}(f)$,
  • input signal  $y(t)$,  and
  • output signal  $z(t)$.

Please note:

  • The task belongs to the chapter  Linear Distortions.
  • In particular, reference is made to the page  Equalization methods.
  • The term  "overall distortion"  used in the formulation of the questions refers to the input signal  $x(t)$  and the output signal  $z(t)$.



What types of distortion can be ruled out for this system??

Nonlinear distortions.
Attenuation distortions.
Phase distortions.


What characteristics does the system exhibit for the test signal  $x_1(t)$  with  $\underline{f_2 = 4 \ \rm kHz}$?

It acts like an ideal system.
It acts like a distortion-free system.
It can be seen that the system at hand is a distorting system.


What characteristics does the system exhibit for the test signal  $x_1(t)$  with  $\underline{f_2 = 10 \ \rm kHz}$?

It acts like an ideal system.
It acts like a distortion-free system.
It can be seen that the system at hand is a distorting system.


For the test signal  $x_2(t)$  with  $\underline{f_0 = 3 \ \rm kHz}$,  what is the maximum deviation  $\varepsilon_{\rm max} = |y_2(t_0) - x_2(t_0)|$.
At what time  $t_0$  does  $\varepsilon_{\rm max}$  occur for the first time?

$\varepsilon_\text{max} \ = \ $

$\ \rm V$
$t_0 \ = \ $

$\ \rm ms$


What is the maximum deviation  $\varepsilon_{\rm max}$  with  $\underline{f_0 = 2 \ \rm kHz}$?

$\varepsilon_\text{max} \ = \ $

$\ \rm V$


What curve shape should the equalizer  $H_{\rm E}(f)$  have to compensate all distortions of  $H(f)$  in the best possible way?
What magnitude value arises as a result for  $\underline{f = 10 \ \rm kHz}$?

$|H_E(f = 10 \ \rm kHz)| \ = \ $


For which of the listed signals is complete equalization possible?
$z(t) = x(t)$  should be understood by "complete equalization".

For signal  $x_1(t)$  with  $f_2 = 10 \ \rm kHz$,
for signal  $x_2(t)$,
for signal  $x_3(t)$.


(1)  Proposed solutions 1 and 3  are correct:

  • A linear system is already implicitly assumed by specifying a frequency response so that nonlinear distortions cannot occur.
  • Since  $H(f)$  is purely real, phase distortions can also be ruled out.

(2)  Proposed solutions 1 and 2 are correct:

  • The output signal is  $y_1(t) = x_1(t)$.
  • Thus, the system is not only distortion-free but can also be termed ideal for this application.

(3)  Proposed solution 3  is correct:

  • In this case, the following is obtained for the output signal:
$$y_1(t)= 1\,{\rm V}\cdot \cos(2 \pi \cdot f_1 \cdot t) + {1}/{4}\cdot 1\,{\rm V}\cdot \sin(2 \pi \cdot f_2 \cdot t).$$
  • While the component at  $f_1$  is transmitted unchanged, the sinusoidal component at  $f_2$  is attenuated and one-quarter of the original sinusoidal component.
  • So, there are attenuation distortions.

(4)  The output signal  $y_2(t)$  has the following form taking into account the basic frequency  $f_0 = 3 \ \rm kHz$:

$$y_2(t)= \frac{8\,{\rm V}}{\pi^2} \left( \cos(\omega_0 t) + \frac{3}{8}\cdot \frac{1}{9} \cdot \cos(3\omega_0 t)\right) .$$
  • The factor  $3/8$  describes  $H(f = 9 \ \rm kHz)$.  All other spectral components at  $15 \ \rm kHz$,  $21 \ \rm kHz$,  etc. are suppressed by the system.
  • The strongest deviations between  $x_2(t)$  and  $y_2(t)$  will occur at the triangle peaks since the missing high frequencies have the strongest effect here.
  • For example, for the time  $\underline{t= 0}$ one obtains:
$$y_2(t=0)= \frac{8\,{\rm V}}{\pi^2} \left( 1 + {3}/{72}\right)= 0.844\,{\rm V} \hspace{0.3cm}\Rightarrow\hspace{0.3cm} \varepsilon_{\rm max} = |y_2(t=0)- x_2(t=0)| \hspace{0.15cm}\underline{= 0.156\,{\rm V}}.$$

(5)  With the basic frequency  $f_0 = 2 \ \rm kHz$  and the values  $H(3f_0) = 0.75$,  $H(5f_0) = 0.25$,  $H(7f_0) = 0$  the following is obtained:

$$y_2(t=0)= \frac{8\,{\rm V}}{\pi^2} \left( 1 + \frac{3}{4}\cdot \frac{1}{9} + \frac{1}{4} \cdot\frac{1}{25}\right)= 0.886\,{\rm V}\hspace{0.5cm} \Rightarrow \hspace{0.5cm}\varepsilon_{\rm max} \hspace{0.15cm}\underline{= 0.114\,{\rm V}}.$$

(6)  In the range up to  $4 \ \rm kHz$,  $H_{\rm E}(f) = H(f) = 1$  is to be set.  In contrast, in the range from  $4 \ \rm kHz$  to  $12 \ \rm kHz$ the following holds:

$$H_{\rm E}(f)= \frac{1}{H(f)} = \frac{1}{1.5 \cdot \big[1 - f/(12\,{\rm kHz})\big]} \hspace{0.5cm} \Rightarrow \hspace{0.5cm} H_{\rm E}(f = 10\,{\rm kHz})\hspace{0.15cm}\underline{= 4} .$$

Here, the denominator expression describes the equation of the straight line of the frequency roll-off.

(7)  Proposed solution 1 is correct:

  • Both  $x_2(t)$  and  $x_3(t)$  also contain spectral components at frequencies greater than  $12 \ \rm kHz$.
  • If these have been truncated by  $H(f)$  ⇒   band limitation, they can no longer be reconstructed by the equalizer.
  • This means that only the signal  $x_1(t)$  can be recovered by  $H_{\rm E}(f)$  but only if  $f_2 < 12 \ \rm kHz$ holds:
$$z_1(t)= \underline{1} \cdot 1\,{\rm V}\cdot \cos(2 \pi \cdot f_1 \cdot t) + \underline{4} \cdot \frac{1}{4}\cdot 1\,{\rm V}\cdot \sin(2 \pi \cdot f_2 \cdot t).$$
  • The first (underlined) factors indicate the gain values of  $H_{\rm E}(f)$  respectively.