Quality Criteria

Ideal and Distortionless System

Block diagram describing modulation and demodulation

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 due to the demodulator with low-pass characteristic usually a low-frequency noise disturbance.

$\text{Definition:}$ 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).
Often one uses the logarithmic form ⇒ $10 · \lg \ ρ_v$ which is expressed in $\rm dB$ when using the logarithm of base ten $(\lg)$ .

Illustrating the remaining error signal $ε(t) = v(t) - α · q(t - τ)$

$\text{Example 1:}$ 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 "root 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 useable 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

Simplified model of a communication system

$$v(t) ≠ α · q(t - τ),$$

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 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:

Linear Distortions, as described in the chapter of the same name in the book "Linear and Time-Invariant Systems":

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

Non-linear 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 basic tutorial videos:

$\text{Two further points:}$

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 $s(t)$ and $r(t)$ signals.
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 AWGN channel, where the abbreviation stands for "$\rm A$dditive $\rm W$hite $\rm G$aussian $\rm N$oise" and already sufficiently describes the properties of this channel model. We would also like to refer you to the three-part tutorial video The AWGN Channel.

The additive noise signal includes all frequency components equally; $n(t)$ has a constant power density spectrum $\rm (LDS)$ and a Dirac-shaped autocorrelation 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 value $F ≥ 1$ and an absolute temperature $θ$:

$${N_0}= F \cdot k_{\rm B} \cdot \theta , \hspace{0.3cm}k_{\rm B} = 1.38 \cdot 10^{-23}{ {\rm Ws} }/{ {\rm K} }\hspace{0.2cm}{\rm (Boltzmann-constants)}\hspace{0.05cm}.$$

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 interference signal $n(t)$ has a Gaussian probability density function $\rm (PDF)$ ⇒ a normal amplitude distribution with interference root-mean-square value $σ_n$:

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

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 distortion):

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

Investigating 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 transmission power $P_{\rm S}$ we apply, the better the quality of 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 transmission power $P_{\rm S}$ .
The smaller the channel's $α_{\rm K}$ parameter, the smaller $ρ_v$ becomes. There is often a quadratic dependency, since the received power is given by $P_{\rm E} = {α_{\rm K}}^2 · P_{\rm S}$ .
A wider bandwidth source signal $($larger $B_{\rm NF})$ leads to smaller $ρ_v$ ⇒ this requires increased RF bandwidth ⇒ interference has a larger effect.

Block diagram for investigating noise behavior

$\text{Conclusion:}$ 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 what follows, 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..

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

The AWGN Channel

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

For $\text{System A}$ , $ρ_ν = ξ$ holds. For example, 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 straight 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 amplitude modulation without a carrier ⇒ with modulation depth $m → ∞$ , while $\text{System B}$ describes Double-sideband amplitude modulation with a carrier with modulation depth $m ≈ 0.5$ .

$\text{System C}$ 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 $ξ = 10000$ .

Durch eine Erhöhung der Leistungskenngröße $ξ$ wird das $\text{System C}$ im Gegensatz zum $\text{System A}$ nicht signifikant verbessert. Ein solches Verhalten ist zum Beispiel bei Digitalsystemen feststellbar, bei denen das Sinken–SNR durch das Quantisierungsrauschen begrenzt wird. Befindet man sich bereits auf dem horizontalen Abschnitt der Kurve, so ist durch eine größere Sendeleistung – und damit verbunden eine kleinere Bitfehlerwahrscheinlichkeit – kein besseres Sinken–SNR zu erzielen.

Meist werden die Größen $ρ_v$ und $ξ$ in logarithmierter Form dargestellt, wie in der rechten Grafik zu sehen ist:

Durch die doppelt–logarithmische Darstellung ergibt sich für das $\text{System A}$ weiterhin die Winkelhalbierende. Die geringere Steigung $($Faktor $3)$ von $\text{System B}$ führt nun zu einer Verschiebung um $10 · \lg 3 ≈ 5\text{ dB}$ nach unten.
Der Schnittpunkt der Systeme $\text{A}$ und $\text{C}$ verschiebt sich durch die doppelt–logarithmische Darstellung von $ξ = ρ_v = 10000$ auf $10 · \lg ξ = 10 · \lg ρ_v = 40\text{ dB}$.

Aufgaben zum Kapitel

Aufgabe 1.2: Verzerrungen? Oder keine Verzerrung?

Aufgabe 1.2Z: Linear verzerrendes System

Aufgabe 1.3: Systemvergleich beim AWGN–Kanal

Aufgabe 1.3Z: Thermisches Rauschen