## Contents

## Ideal and Distortionless System

In all subsequent chapters, the following model will be assumed:

The task of any message transmission system is

- to provide a sink signal $v(t)$ at a spatially distant sink
- that differs as little as possible from the source signal $q(t)$ .

$\text{Definition:}$ An **ideal system** is achieved when the following conditions hold:

- $$v(t) = q(t) + n(t), \hspace{1cm}n(t) \to 0.$$

This takes into account that $n(t) \equiv 0$ is physically impossible due to Thermal Noise.

In practice, the signals $q(t)$ and $v(t)$ will not differ by more than the noise term $n(t)$ for the following reasons:

- Non-ideal realization of the modulator and the demodulator,
- linear attenuation distortions and phase distortions, as well as nonlinearities,
- external disturbances and additional stochastic noise processes,
- frequency-independent damping and delay.

$\text{Definition:}$ A **distortionless system** is achieved, if from the above list only the last restriction is effective:

- $$v(t) = \alpha \cdot q(t- \tau) + n(t), \hspace{1cm}n(t) \to 0.$$

- Due to the attenuation factor $α$, the sink signal $v(t)$ is only "quieter" compared to the source signal $q(t)$.
- Even a delay $τ$ is often tolerable, at least for a unidirectional transmission.
- In contrast, in bidirectional communications – such as a telephone call – a delay of $300$ milliseconds is already perceived as a significant disturbance.

## Signal–to–noise (power) ratio

In the general case, the sink signal $v(t)$ will still differ from $α · q(t - τ)$, and the error signal is characterized by:

- $$\varepsilon (t) = v(t) - \alpha \cdot q(t- \tau) = \varepsilon_{\rm V} (t) + \varepsilon_{\rm St} (t).$$

This error signal is composed of two components:

- linear and nonlinear distortions (German: "Verzerrungen" ⇒ subscript "V") $ε_{\rm V}(t)$, which are caused by the frequency responses of the modulator, the channel, and the demodulator and thus exhibit deterministic (time-invariant) behavior;
- a stochastic component $ε_{\rm St}(t)$, which originates from the high-frequency noise $n(t)$ at the demodulator input. However, unlike $n(t)$, $ε_{\rm St}(t)$ is usually due to a low-frequency noise disturbance in a demodulator with a low-pass characteristic curve.

$\text{Definition:}$ As a measure of the quality of the communication system, the **signal-to-noise (power) ratio** $\rm (SNR)$ $ρ_v$ at the sink is defined as the quotient of the signal power (variance) of the useful component $v(t) - ε(t)$ and the disturbance component $ε(t)$, respectively:

- $$\rho_{v} = \frac{ P_{v -\varepsilon} }{P_{\varepsilon} } \hspace{0.05cm},\hspace{0.7cm}\text{with}\hspace{0.7cm} P_{v -\varepsilon} = \overline{[v(t)-\varepsilon(t)]^2} = \lim_{T_{\rm M} \rightarrow \infty}\hspace{0.1cm}\frac{1}{T_{\rm M} } \cdot \int_{0}^{ T_{\rm M} } {\big[v(t)-\varepsilon(t)\big]^2 }\hspace{0.1cm}{\rm d}t,\hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon^2(t)} = \lim_{T_{\rm M} \rightarrow \infty}\hspace{0.1cm}\frac{1}{T_{\rm M} } \cdot \int_{0}^{ T_{\rm M} } {\varepsilon^2(t) }\hspace{0.1cm}{\rm d}t\hspace{0.05cm}.$$

For the power of the useful part, we obtain regardless of the delay time $τ$:

- $$P_{v -\varepsilon} = \overline{\big[v(t)-\varepsilon(t)\big]^2} = \overline{\alpha^2 \cdot q^2(t - \tau)}= \alpha^2 \cdot P_{q}.$$

Here, $P_q$ denotes the power of the source signal $q(t)$:

- $$P_{q} = \lim_{T_{\rm M} \rightarrow \infty}\hspace{0.1cm}\frac{1}{T_{\rm M}} \cdot \int_{0}^{ T_{\rm M}} {q^2(t) }\hspace{0.1cm}{\rm d}t .$$

This gives:

- $$\rho_{v} = \frac{\alpha^2 \cdot P_{q} }{P_{\varepsilon} } \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.15cm}\rho_{v} = 10 \cdot {\rm lg} \hspace{0.15cm} \frac{\alpha^2 \cdot P_{q} }{P_{\varepsilon} } \hspace{0.05cm}.$$

- In the following, we will refer to $ρ_v$ as the
**sink signal–to–noise ratio**(or short:**sink SNR**). - One often uses the logarithmic form ⇒ $10 · \lg \ ρ_v$ which is expressed in $\rm dB$ when using the logarithm of base ten $(\lg)$ .

$\text{Example 1:}$ On the right, you can see an exemplary section of the (blue) source signal $q(t)$ and the (red) sink signal $v(t)$, which are noticeably different.

However, the middle graph makes it clear that the main difference between $q(t)$ and $v(t)$ is due to the damping factor $α = 0.7$ and the transmission delay $τ = 0.1\text{ ms}$.

The bottom sketch shows the remaining error signal $ε(t) = v(t) - α · q(t - τ)$ after correcting for attenuation and delay. We refer to the mean square ⇒ "variance" of this signal as the noise power $P_ε$.

To calculate the sink SNR $ρ_v$ , $P_ε$ must be related to the useful signal power $α^2 · P_q$. This is obtained from the variance of the signal $α · q(t - τ)$, plotted in light blue in the middle graph.

From the assumed properties $\alpha = 0.7$ ⇒ $\alpha^2 \approx 0.5$ as well as $P_{q} = 8\,{\rm V^2}$ and ${P_{\varepsilon} } = 0.04\,{\rm V^2}$ , we obtain the sink SNR

- $$ ρ_v ≈ 100 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}10 · \lg ρ_v ≈ 20\ \rm dB.$$

- The error signal $ε(t)$ – and thus also the sink SNR $ρ_v$ – takes into account all imperfections of the transmission system under consideration (e.g. distortions, external interferences, noise, etc.).
- In the following, we will consider each of these different effects separately for the sake of explanation.

## Investigations with regard to signal distortions

All modulation methods described in the following chapters lead to distortions under non-ideal conditions, i.e. to a sink signal

- $$v(t) ≠ α · q(t - τ),$$

which differs from $q(t)$ by more than just damping and delay. For the study of these signal distortions, we always assume the following model and premises:

- The additive noise signal $n(t)$ at the channel output (demodulator input) is negligible and ignored.
- All components of modulator and demodulator are treated as linear,
- Similarly, the channel is assumed to be linear, and is thus completely characterized by its frequency response $H_{\rm K}(f)$ .

Depending on the type and realization of modulator and demodulator, the following signal distortions occur:

$\text{Linear distortions}$, as described in the "chapter of the same name" in the book "Linear and Time-Invariant Systems":

- Linear distortions can generally be compensated by an equalizer, but this will always result in higher $P_\epsilon$ and thus in a lower sink SNR in the presence of a stochastic disturbance $n(t)$.
- These linear distortions can be further divided into "attenuation distortions" and "phase distortions".

$\text{Nonlinear distortions}$, as described in the "chapter of the same name" in the book "Linear and Time-Invariant Systems":

- Nonlinear distortions are irreversible and thus a more severe problem than linear distortions.
- A suitable quantitative measure of such distortions is the distortion factor $K$, for example, which is related to the sink SNR in the following way: $\rho_{v} = {1}/{K^2} \hspace{0.05cm}.$
- However, specifying a distortion factor assumes a harmonic oscillation as the source signal.

We refer you to three of our (German language) basic learning videos:

- "Lineare und nichtlineare Verzerrungen" ⇒ "Linear and nonlinear distortions",
- "Eigenschaften des Übertragungskanals" ⇒ "Properties of the transmission channel",
- "Einige Anmerkungen zur Übertragungsfunktion" ⇒ "Some remarks on the transmission function".

$\text{Two further points:}$

- The distortions with respect to $q(t)$ and $v(t)$ are nonlinear in nature whenever the channel contains nonlinear components and, as such,

nonlinear distortions are already present with respect to the signals $s(t)$ and $r(t)$. - Similarly, nonlinearities in the modulator or demodulator always lead to nonlinear distortions.

## Some remarks on the AWGN channel model

To investigate the noise behavior of each individual modulation and demodulation method, the starting point is usually the so-called $\rm AWGN$ channel, where the abbreviation stands for "$\rm A$dditive $\rm W$hite $\rm G$aussian $\rm N$oise". The name already sufficiently describes the properties of this channel model.

We would also like to refer you to the (German language) three-part learning video "Der AWGN-Kanal" ⇒ "The AWGN channel".

- The additive noise signal includes all frequency components equally ⇒ $n(t)$ has a constant power-spectral density $\rm (PSD)$ and a Dirac-shaped auto-correlation function $\rm (ACF)$:

- $${\it \Phi}_n(f) = \frac{N_0}{2}\hspace{0.15cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.15cm} \varphi_n(\tau) = \frac{N_0}{2} \cdot \delta (\tau)\hspace{0.05cm}.$$
- In each case, the factor $1/2$ in these equations accounts for the two-sided spectral representation.

- For example, in the case of thermal noise, for the physical noise power density (from a one-sided view) with a noise figure $F ≥ 1$ and an absolute temperature $θ$:

- $${N_0}= F \cdot k_{\rm B} \cdot \theta , \hspace{0.5cm}\text{Boltzmann constant:}\hspace{0.3cm}k_{\rm B} = 1.38 \cdot 10^{-23}{ {\rm Ws} }/{ {\rm K} }\hspace{0.05cm}.$$

- "True white noise" would result in infinitely large power. Therefore, a bandwidth limit of $B$ must always be taken into account, and the following applies to the effective noise power:

- $$N = \sigma_n^2 = {N_0} \cdot B \hspace{0.05cm}.$$

- The noise signal $n(t)$ has a Gaussian probability density function $\rm (PDF)$ ⇒ a normal amplitude distribution with standard deviation $σ_n$:

- $$f_n(n) = \frac{1}{\sqrt{2\pi}\cdot\sigma_n}\cdot {\rm e}^{-{\it n^{\rm 2}}/{(2\sigma_{\it n}^2)}}.$$

- For the AWGN channel, one should actually set $H_{\rm K}(f) = 1$. However, we modify this model for our purposes by allowing frequency-independent attenuation

(note: a frequency-independent attenuation factor does not lead to further distortions):

- $$H_{\rm K}(f) = \alpha_{\rm K}= {\rm const.}$$

## Investigations at the AWGN channel

In all investigations regarding noise behavior, we start from the block diagram sketched below. We will always calculate the sink SNR $ρ_v$ as a function of all system parameters and arrive at the following results:

- The more transmit power (German: "Sendeleistung" ⇒ subscript "S") $P_{\rm S}$ we apply, the greater is the sink SNR $ρ_v$. For some methods, this relationship can even be linear.
- $ρ_v$ decreases monotonically with increasing noise power density $N_0$ . An increase in $N_0$ can usually be compensated by a larger transmit power $P_{\rm S}$.
- The smaller the channel's $α_{\rm K}$ parameter, the smaller $ρ_v$ becomes. There is often a quadratic relationship, since the received power (German: "Empfangsleistung" ⇒ subscript "E") is $P_{\rm E} = {α_{\rm K}}^2 · P_{\rm S}$.
- A wider bandwidth of the source signal $($larger $B_{\rm NF})$ requires an increased high-frequency bandwidth $B_{\rm HF}$, too ⇒ this leads to smaller sink SNR $ρ_v$ ⇒ negative effect on the transmission system's quality.

$\text{Conclusion:}$ Considering these four assumptions, we conclude that it makes sense to express the sink SNR in normalized form as

- $$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} }$$

In the following, we refer to $ξ$ as the **performance parameter**.

The input variables summarized in $ξ$ are marked with blue arrows in the above block diagram, while the quality criterion $ρ_v$ is highlighted by the red arrow.

- The larger $ξ$ is, the larger is $\rho_{v }$ in general.
- But the relationship is not always linear, as the following example shows.

$\text{Example 2:}$ The left graph shows the sink SNR $ρ_v$ of three different systems, each as a function of the normalized performance parameter

- $$\xi = { {\alpha_{\rm K} }^2 \cdot P_{\rm S} }/({N_0 \cdot B_{\rm NF} }).$$

- For $\text{System A}$, $ρ_ν = ξ$ holds. The system parameters

- $$P_{\rm S}= 10 \;{\rm kW}\hspace{0.05cm}, \hspace{0.2cm} \alpha_{\rm K} = 10^{-4}\hspace{0.05cm},$$
- $$ {N_0} = 10^{-12}\hspace{0.05cm}{ {\rm W} }/{ {\rm Hz} }\hspace{0.05cm}, \hspace{0.2cm} B_{\rm NF}= 10\; {\rm kHz}$$

- give $ξ = ρ_v = 10000$ (see the circular mark on the graph).

- Exactly the same sink SNR would result from the parameters

- $$P_{\rm S}= 5 \;{\rm kW}\hspace{0.05cm}, \hspace{0.2cm} \alpha_{\rm K} = 10^{-6}\hspace{0.05cm},$$
- $${N_0} = 10^{-16}\hspace{0.05cm}{ {\rm W} }/{ {\rm Hz} }\hspace{0.05cm}, \hspace{0.2cm} B_{\rm NF}= 5\; {\rm kHz}\hspace{0.05cm}.$$

- In $\text{System B}$, there is also a linear relationship of $ρ_v = ξ/3$. The line also passes through the origin. However, the slope is only $1/3$.
- It should be noted that the noise behavior corresponding to $\text{System A}$ is observed for Double-sideband suppressed-carrier amplitude modulation $($modulation depth $m → ∞)$, while $\text{System B}$ describes Double-sideband amplitude modulation with carrier $(m ≈ 0.5)$.
- $\text{System C}$ shows a completely different noise behavior. For small $ξ$–values, this system is superior to $\text{System A}$, though the quality of both systems is the same at $ξ = 10000$.

Increasing the performance parameter $ξ$ does not significantly improve $\text{System C}$, unlike in $\text{System A}$. Such behavior can be observed, for example, in digital systems where the sink SNR is limited by the quantization noise. Along the horizontal section of the curve, a higher transmit power will not result in a better sink SNR – and thus a smaller bit error probability.

Usually, the quantities $ρ_v$ and $ξ$ are represented in logarithmic form, as shown in the graph on the right:

- The double logarithmic representation still results in the angle bisector for $\text{System A}$ .
- The lower slope $($factor $3)$ of $\text{System B}$ now results in a downward shift of $10 · \lg 3 ≈ 5\text{ dB}$.
- The intersection of $\text{A}$ and $\text{C}$ shifts from $ξ = ρ_v = 10000$ to $10 · \lg ξ = 10 · \lg ρ_v = 40\text{ dB}$ due to the double-logarithmic representation.

## Exercises for the chapter

Exercise 1.2: Distortion? Or no distortion?

Exercise 1.2Z: Linear distorting system

Exercise 1.3: System comparison at the AWGN channel