Difference between revisions of "Theory of Stochastic Signals/Moments of a Discrete Random Variable"
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| − | == | + | ==Calculation as ensemble average or time average== |
<br> | <br> | ||
| − | + | The probabilities and the relative frequencies provide extensive information about a discrete random variable. Reduced information is obtained by the so-called moments $m_k$, where $k$ represents a natural number. | |
| − | {{ | + | {{BlueBox|TEXT= |
| − | $\text{ | + | $\text{Two alternative ways of calculation:}$ |
| − | + | Under the [[Theory_of_Stochastic_Signals/Autocorrelation_Function_(ACF)#Ergodic_random_processes|Ergodicity]] implied here, there are two different calculation possibilities for the moment $k$-th order: | |
| − | * | + | *the '''ensemble averaging'''' or ''expected value formation'' ⇒ averaging over all possible values $\{ z_\mu\}$ with the index $\mu = 1 , \hspace{0.1cm}\text{ ...} \hspace{0.1cm} , M$: |
| − | :$$m_k = {\rm E} \big[z^k \big] = \sum_{\mu = 1}^{M}p_\mu \cdot z_\mu^k \hspace{2cm} \rm | + | :$$m_k = {\rm E} \big[z^k \big] = \sum_{\mu = 1}^{M}p_\mu \cdot z_\mu^k \hspace{2cm} \rm with \hspace{0.1cm} {\rm E\big[\text{ ...} \big]\hspace{-0.1cm}:} \hspace{0.3cm} \rm expected\hspace{0.1cm}value ;$$ |
| − | * | + | *the '''time averaging''' over the random sequence $\langle z_ν\rangle$ with the index $ν = 1 , \hspace{0.1cm}\text{ ...} \hspace{0.1cm} , N$: |
| − | :$$m_k=\overline{z_\nu^k}=\hspace{0.01cm}\lim_{N\to\infty}\frac{1}{N}\sum_{\nu=\rm 1}^{\it N}z_\nu^k\hspace{1.7cm}\rm | + | :$$m_k=\overline{z_\nu^k}=\hspace{0.01cm}\lim_{N\to\infty}\frac{1}{N}\sum_{\nu=\rm 1}^{\it N}z_\nu^k\hspace{1.7cm}\rm with\hspace{0.1cm}horizontal\hspace{0.1cm}line\hspace{-0.1cm}:\hspace{0.1cm}time\hspace{0.1cm}average.$$}} |
| − | + | Note: | |
| − | * | + | *Both types of calculations lead to the same asymptotic result for sufficiently large values of $N$ . |
| − | * | + | *For finite $N$ , a comparable error results as when the probability is approximated by the relative frequency. |
| − | == | + | ==Linear mean - DC component== |
<br> | <br> | ||
| − | {{ | + | {{BlueBox|TEXT= |
| − | $\text{Definition:}$ | + | $\text{Definition:}$ With $k = 1$ we obtain from the general equation for moments the '''linear mean''': |
:$$m_1 =\sum_{\mu=1}^{M}p_\mu\cdot z_\mu =\lim_{N\to\infty}\frac{1}{N}\sum_{\nu=1}^{N}z_\nu.$$ | :$$m_1 =\sum_{\mu=1}^{M}p_\mu\cdot z_\mu =\lim_{N\to\infty}\frac{1}{N}\sum_{\nu=1}^{N}z_\nu.$$ | ||
| − | * | + | *The left part of this equation describes the ensemble averaging (over all possible values), |
| − | * | + | *while the right equation gives the determination as time average. |
| − | *In | + | *In the context of signals, this quantity is also referred to as the [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal|DC component]] }} |
| − | [[File:P_ID49__Sto_T_2_2_S2_neu.png|right|frame| | + | [[File:P_ID49__Sto_T_2_2_S2_neu.png|right|frame|DC component $m_1$ of a binary signal]] |
| − | {{GraueBox|TEXT= | + | {{GraueBox|TEXT= |
| − | $\text{ | + | $\text{Example 1:}$ A binary signal $x(t)$ with the two possible amplitude values. |
| − | *$1\hspace{0.03cm}\rm V$ $($ | + | *$1\hspace{0.03cm}\rm V$ $($for the symbol $\rm L)$, |
| − | *$3\hspace{0.03cm}\rm V$ $($ | + | *$3\hspace{0.03cm}\rm V$ $($for the symbol $\rm H)$ |
| − | + | as well as the occurrence probabilities $p_{\rm L} = 0.2$ respectively $p_{\rm H} = 0.8$ has the linear mean (DC) | |
:$$m_1 = 0.2 \cdot 1\,{\rm V}+ 0.8 \cdot 3\,{\rm V}= 2.6 \,{\rm V}. $$ | :$$m_1 = 0.2 \cdot 1\,{\rm V}+ 0.8 \cdot 3\,{\rm V}= 2.6 \,{\rm V}. $$ | ||
| − | + | This is drawn as a red line in the graph. | |
<br clear=all> | <br clear=all> | ||
| − | + | If we determine this parameter by time averaging over the displayed $N = 12$ signal values, we obtain a slightly smaller value: | |
:$$m_1\hspace{0.01cm}' = 4/12 \cdot 1\,{\rm V}+ 8/12 \cdot 3\,{\rm V}= 2.33 \,{\rm V}. $$ | :$$m_1\hspace{0.01cm}' = 4/12 \cdot 1\,{\rm V}+ 8/12 \cdot 3\,{\rm V}= 2.33 \,{\rm V}. $$ | ||
| − | + | Here, the probabilities of occurrence $p_{\rm L} = 0.2$ and $p_{\rm H} = 0.8$ were replaced by the corresponding frequencies $h_{\rm L} = 4/12$ and $h_{\rm H} = 8/12$ respectively. The relative error due to insufficient sequence length $N$ is greater than $10\%$ in the example. | |
| − | <u> | + | <u>Note about our (admittedly somewhat unusual) nomenclature:</u>. |
| − | + | We denote binary symbols here as in circuit theory with $\rm L$ (Low) and $\rm H$ (High) to avoid confusion. | |
| − | *In | + | *In coding theory, it is useful to map $\{ \text{L, H}\}$ to $\{0, 1\}$ to take advantage of the possibilities of modulo algebra. |
| − | * | + | *In contrast, to describe modulation with bipolar (antipodal) signals, one better chooses the mapping $\{ \text{L, H}\}$ ⇔ $ \{-1, +1\}$. |
}} | }} | ||
| + | |||
==Quadratischer Mittelwert – Varianz – Streuung== | ==Quadratischer Mittelwert – Varianz – Streuung== | ||
Revision as of 22:41, 2 December 2021
Contents
Calculation as ensemble average or time average
The probabilities and the relative frequencies provide extensive information about a discrete random variable. Reduced information is obtained by the so-called moments $m_k$, where $k$ represents a natural number.
$\text{Two alternative ways of calculation:}$
Under the Ergodicity implied here, there are two different calculation possibilities for the moment $k$-th order:
- the ensemble averaging' or expected value formation ⇒ averaging over all possible values $\{ z_\mu\}$ with the index $\mu = 1 , \hspace{0.1cm}\text{ ...} \hspace{0.1cm} , M$:
- $$m_k = {\rm E} \big[z^k \big] = \sum_{\mu = 1}^{M}p_\mu \cdot z_\mu^k \hspace{2cm} \rm with \hspace{0.1cm} {\rm E\big[\text{ ...} \big]\hspace{-0.1cm}:} \hspace{0.3cm} \rm expected\hspace{0.1cm}value ;$$
- the time averaging over the random sequence $\langle z_ν\rangle$ with the index $ν = 1 , \hspace{0.1cm}\text{ ...} \hspace{0.1cm} , N$:
- $$m_k=\overline{z_\nu^k}=\hspace{0.01cm}\lim_{N\to\infty}\frac{1}{N}\sum_{\nu=\rm 1}^{\it N}z_\nu^k\hspace{1.7cm}\rm with\hspace{0.1cm}horizontal\hspace{0.1cm}line\hspace{-0.1cm}:\hspace{0.1cm}time\hspace{0.1cm}average.$$
Note:
- Both types of calculations lead to the same asymptotic result for sufficiently large values of $N$ .
- For finite $N$ , a comparable error results as when the probability is approximated by the relative frequency.
Linear mean - DC component
$\text{Definition:}$ With $k = 1$ we obtain from the general equation for moments the linear mean:
- $$m_1 =\sum_{\mu=1}^{M}p_\mu\cdot z_\mu =\lim_{N\to\infty}\frac{1}{N}\sum_{\nu=1}^{N}z_\nu.$$
- The left part of this equation describes the ensemble averaging (over all possible values),
- while the right equation gives the determination as time average.
- In the context of signals, this quantity is also referred to as the DC component
DC component $m_1$ of a binary signal
$\text{Example 1:}$ A binary signal $x(t)$ with the two possible amplitude values.
- $1\hspace{0.03cm}\rm V$ $($for the symbol $\rm L)$,
- $3\hspace{0.03cm}\rm V$ $($for the symbol $\rm H)$
as well as the occurrence probabilities $p_{\rm L} = 0.2$ respectively $p_{\rm H} = 0.8$ has the linear mean (DC)
- $$m_1 = 0.2 \cdot 1\,{\rm V}+ 0.8 \cdot 3\,{\rm V}= 2.6 \,{\rm V}. $$
This is drawn as a red line in the graph.
If we determine this parameter by time averaging over the displayed $N = 12$ signal values, we obtain a slightly smaller value:
- $$m_1\hspace{0.01cm}' = 4/12 \cdot 1\,{\rm V}+ 8/12 \cdot 3\,{\rm V}= 2.33 \,{\rm V}. $$
Here, the probabilities of occurrence $p_{\rm L} = 0.2$ and $p_{\rm H} = 0.8$ were replaced by the corresponding frequencies $h_{\rm L} = 4/12$ and $h_{\rm H} = 8/12$ respectively. The relative error due to insufficient sequence length $N$ is greater than $10\%$ in the example.
Note about our (admittedly somewhat unusual) nomenclature:.
We denote binary symbols here as in circuit theory with $\rm L$ (Low) and $\rm H$ (High) to avoid confusion.
- In coding theory, it is useful to map $\{ \text{L, H}\}$ to $\{0, 1\}$ to take advantage of the possibilities of modulo algebra.
- In contrast, to describe modulation with bipolar (antipodal) signals, one better chooses the mapping $\{ \text{L, H}\}$ ⇔ $ \{-1, +1\}$.
Quadratischer Mittelwert – Varianz – Streuung
$\text{Definitionen:}$
- Analog zum linearen Mittelwert erhält man mit $k = 2$ für den quadratischen Mittelwert:
- $$m_2 =\sum_{\mu=\rm 1}^{\it M}p_\mu\cdot z_\mu^2 =\lim_{N\to\infty}\frac{\rm 1}{\it N}\sum_{\nu=\rm 1}^{\it N}z_\nu^2.$$
- Zusammen mit dem Gleichanteil $m_1$ kann daraus als weitere Kenngröße die Varianz $σ^2$ bestimmt werden (Satz von Steiner):
- $$\sigma^2=m_2-m_1^2.$$
- Als Streuung $σ$ bezeichnet man in der Statistik die Quadratwurzel der Varianz; manchmal wird diese Größe auch Standardabweichung genannt:
- $$\sigma=\sqrt{m_2-m_1^2}.$$
Hinweise zu den Einheiten:
- Bei Nachrichtensignalen gibt $m_2$ die (mittlere) Leistung eines Zufallssignals an, bezogen auf den Widerstand $1 \hspace{0.03cm} Ω$.
- Beschreibt $z$ eine Spannung, so besitzt dementsprechend $m_2$ die Einheit ${\rm V}^2$.
- Die Varianz $σ^2$ eines Zufallssignals entspricht physikalisch der Wechselleistung und die Streuung $σ$ dem Effektivwert.
- Diesen Definitionen liegt wiederum der Bezugswiderstand $1 \hspace{0.03cm} Ω$ zugrunde.
Das Lernvideo Momentenberechnung bei diskreten Zufallsgrößen verdeutlicht die definierten Größen am Beispiel eines Digitalsignals.
Standardabweichung eines Binärsignals
$\text{Beispiel 2:}$ Ein Binärsignal $x(t)$ mit den Amplitudenwerten
- $1\hspace{0.03cm}\rm V$ $($für das Symbol $\rm L)$,
- $3\hspace{0.03cm}\rm V$ $($für das Symbol $\rm H)$
sowie den Auftrittswahrscheinlichkeiten $p_{\rm L} = 0.2$ bzw. $p_{\rm H} = 0.8$ besitzt die gesamte Signalleistung
- $$P_{\rm Gesamt} = 0.2 \cdot (1\,{\rm V})^2+ 0.8 \cdot (3\,{\rm V})^2 = 7.4 \hspace{0.05cm}{\rm V}^2,$$
wenn man vom Bezugswiderstand $R = 1 \hspace{0.05cm} Ω$ ausgeht.
Mit dem Gleichanteil $m_1 = 2.6 \hspace{0.05cm}\rm V$ $($siehe $\text{Beispiel 1})$ folgt daraus für
- die Wechselleistung (Varianz) $P_{\rm W} = σ^2 = 7.4 \hspace{0.05cm}{\rm V}^2 - \big [2.6 \hspace{0.05cm}\rm V\big ]^2 = 0.64\hspace{0.05cm} {\rm V}^2$,
- den Effektivwert $s_{\rm eff} = σ = 0.8 \hspace{0.05cm} \rm V$.
- Einschub: Bei anderem Bezugswiderstand ⇒ $R \ne 1 \hspace{0.1cm} Ω$ gelten nicht alle diese Berechnungen. Beispielsweise haben mit $R = 50 \hspace{0.1cm} Ω$ die Leistung $P_{\rm Gesamt} $, die Wechselleistung $P_{\rm W}$ und der Effektivwert $s_{\rm eff}$ folgende Werte:
- $$P_{\rm Gesamt} \hspace{-0.05cm}= \hspace{-0.05cm} \frac{m_2}{R} \hspace{-0.05cm}= \hspace{-0.05cm} \frac{7.4\,{\rm V}^2}{50\,{\rm \Omega} } \hspace{-0.05cm}= \hspace{-0.05cm}0.148\,{\rm W},\hspace{0.5cm} P_{\rm W} \hspace{-0.05cm} = \hspace{-0.05cm} \frac{\sigma^2}{R} \hspace{-0.05cm}= \hspace{-0.05cm}12.8\,{\rm mW} \hspace{0.05cm},\hspace{0.5cm} s_{\rm eff} \hspace{-0.05cm} = \hspace{-0.05cm}\sqrt{R \cdot P_{\rm W} } \hspace{-0.05cm}= \hspace{-0.05cm} \sigma \hspace{-0.05cm}= \hspace{-0.05cm} 0.8\,{\rm V}.$$
- Einschub: Bei anderem Bezugswiderstand ⇒ $R \ne 1 \hspace{0.1cm} Ω$ gelten nicht alle diese Berechnungen. Beispielsweise haben mit $R = 50 \hspace{0.1cm} Ω$ die Leistung $P_{\rm Gesamt} $, die Wechselleistung $P_{\rm W}$ und der Effektivwert $s_{\rm eff}$ folgende Werte:
Die gleiche Varianz und der gleiche Effektivwert $s_{\rm eff}$ ergeben sich für die Amplituden $0\hspace{0.05cm}\rm V$ $($für das Symbol $\rm L)$ und $2\hspace{0.05cm}\rm V$ $($für das Symbol $\rm H)$ , vorausgesetzt, die Auftrittswahrscheinlichkeiten $p_{\rm L} = 0.2$ und $p_{\rm H} = 0.8$ bleiben gleich. Nur der Gleichanteil und die Gesamtleistung ändern sich:
- $$m_1 = 1.6 \hspace{0.05cm}{\rm V}, \hspace{0.5cm}P_{\rm Gesamt} = P_{\rm W} + {m_1}^2 = 3.2 \hspace{0.05cm}{\rm V}^2.$$
Aufgaben zum Kapitel
Aufgabe 2.2: Mehrstufensignale
Aufgabe 2.2Z: Diskrete Zufallsgrößen
