# Exercise 1.2: Distortions? Or no Distortion?

The communication systems $S_1$, $S_2$ and $S_3$ are analyzed in terms of the distortions they cause. For this purpose, the cosine-shaped test signal with signal frequency $f_{\rm N} = 1\text{ kHz}$ is applied to the input of each system:

- $$q(t) = 2 \;{\rm V} \cdot \cos(2 \pi f_{\rm N} t )$$

The three signals at the system output are measured, as shown in the graph:

- $$v_1(t) = 2 \;{\rm V} \cdot \cos(2 \pi f_{\rm N} t )\hspace{0.05cm},$$
- $$v_2(t) = 1 \;{\rm V} \cdot \cos(2 \pi f_{\rm N} t + 1 \;{\rm V} \cdot \sin(2 \pi f_{\rm N} t) \hspace{0.05cm},$$
- $$v_3(t)= 1.5 \;{\rm V} \cdot \cos(2 \pi f_{\rm N} t) - 0.3 \;{\rm V} \cdot \cos(6 \pi f_{\rm N} t)\hspace{0.05cm}.$$

The noise components that are always present in practice will be assumed to be negligible here.

Hints:

- This exercise belongs to the chapter Quality criteria. Particular reference is made to the page Signal-to-noise power ratio and to the chapter Non-linear distortions in the book "Linear and Time-Invariant Systems".
- For nonlinear distortion, the sink SNR is $ρ_v = 1/K^2$, where the distortion factor $K$ is the ratio of the rms values of all harmonics to the rms value of the fundamental frequency.

### Questions

### Solution

**(1)**

__Answers 1, 2 and 3__are correct:

- System $S_1$ could well be an ideal system, namely if for all frequencies $f_{\rm N}$ the condition $v(t) = q(t)$ were satisfied.
- The second alternative is also possible, since the ideal system is a special case of distortion-free systems.
- However, if at a different message frequency $f_{\rm N} \ne 1$ kHz the condition $v(t) = q(t)$ were not satisfied, then a linearly distorting system would exist whose frequency response would happen to be equal to $1$ at frequency $f_{\rm N}$ .
- In contrast, a nonlinearly distorting system (Answer 4) can be excluded due to the lack of harmonics.

**(2)** Following the explanations in the chapter "Harmonic Oscillation" in the book "Signal Representation" the following equations apply:

- $$A \cdot \cos(\omega_{\rm N} t ) + B \cdot \sin(\omega_{\rm N} t ) = C \cdot \cos(\omega_{\rm N} t - \varphi)\hspace{0.3cm} \Rightarrow \hspace{0.3cm} C = \sqrt{A^2 + B^2},\hspace{0.5cm}\varphi ={\rm arctan}\hspace{0.1cm} ({A}/{B})\hspace{0.05cm}$$

- Applied to the present example, one obtains

- $$C = \sqrt{(1 \,{\rm V})^2 + (1 \,{\rm V})^2}= 1.414\,{\rm V}\hspace{0.05cm}.$$

- The damping ratio of the system thus takes the value $α = 1.414/2 \hspace{0.15cm}\underline{= 0.707}$, and the following applies to the phase:

- $$ \varphi ={\rm arctan}\hspace{0.1cm}\frac {1 \,{\rm V}}{1 \,{\rm V}} = 45^{\circ} = {\pi}/{4}\hspace{0.05cm}.$$

- The transformation $\cos(\omega_{\rm N} t - \varphi)= \cos[\omega_{\rm N} (t - \tau)]$ enables claims about the running time:

- $$\tau =\frac {\varphi}{2\pi f_{\rm N}} = \frac {\pi /4}{2\pi f_{\rm N}} = \frac {1}{8 \cdot 1 \,{\rm kHz}} \hspace{0.15cm}\underline {= 125\,{\rm µ s}}\hspace{0.05cm}.$$

**(3)** __Answers 2 and 3__ are correct:

- Applying the logic from subtask
**(1)**, the system $S_2$ is neither ideal nor nonlinearly distorting. - In contrast, options 2 and 3 are possible, depending on whether the calculated values of $α$ and $τ$ are preserved for all frequencies or not.
- However, with just a single measurement at only one frequency, this cannot be clarified.

**(4)** The signal $v_3(t)$ contains a third order harmonic. Therefore, the distortion is nonlinear ⇒ __Answer 2__.

**(5)** The amplitudes $A_1 = 1.5 \ \rm V$ and $A_3 = -0.3\ \rm V$ give the distortion factor:

- $$ K_3 =\frac {|A_3|}{|A_1|} = 0.2\hspace{0.05cm}.$$

- Therefore, according to the given equation, the sink SNR is $ρ_{v3} = 1/K_3^{ 2 } = 25$.

The same result is obtained from the more general calculation.

- From the amplitudes of the source signal and the fundamental wave of the sink signal, we get a frequency-independent damping factor of:

- $$ \alpha =\frac {1.5 \,{\rm V}}{2 \,{\rm V}} = 0.75\hspace{0.05cm}.$$

- Therefore, the error signal coming from the nonlinear distortions is:

- $$\varepsilon_3(t) = v_3(t) - \alpha \cdot q(t) = - 0.3 \,{\rm V} \cdot \cos(6 \pi f_{\rm N} t)\hspace{0.05cm}.$$

- This gives a distortion power of:

- $$P_{\varepsilon 3}= {1}/{2} \cdot (0.3 \,{\rm V})^2 = 0.045 \,{\rm V}^2\hspace{0.05cm}.$$

- Together with the power of the source signal,

- $$P_{q}= {1}/{2} \cdot (2\,{\rm V})^2 = 2 \,{\rm V}^2\hspace{0.05cm},$$
- and taking into account the damping factor $ \alpha = 0.75 $ just calculated, we obtain:
- $$\rho_{v3} = \frac{\alpha^2 \cdot P_{q}}{P_{\varepsilon 3}} = \frac{0.75^2 \cdot 2 {\rm V}^2}{0.045 } \hspace{0.15cm}\underline {= 25}\hspace{0.05cm}.$$