Difference between revisions of "Aufgaben:Exercise 5.4: Walsh Functions (PCCF, PACF)"

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{{quiz-Header|Buchseite=Modulationsverfahren/Spreizfolgen für CDMA
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{{quiz-Header|Buchseite=Modulation_Methods/Spreading_Sequences_for_CDMA
 
}}
 
}}
  
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[[File:P_ID1889__Mod_A_5_4.png|right|frame|Hadamard–Matrix <b>H</b><sub>8</sub>]]
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[[File:P_ID1889__Mod_A_5_4.png|right|frame|Hadamard matrix &nbsp;${\mathbf{H}_{8}}$]]
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Häufig verwendet man zur Bandspreizung und Bandstauchung so genannte ''Walsh–Funktionen'', die mittels der Hadamard–Matrix konstruiert werden können. Ausgehend von der Matrix
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The so-called &nbsp;"Walsh functions",&nbsp; which can be constructed by means of the Hadamard matrix,&nbsp; are often used for band spreading and band compression.&nbsp; Starting from the matrix
 
:$${\mathbf{H}_{2}} = \left[ \begin{array}{ccc} +1 & +1 \\ +1 & -1 \end{array} \right] $$
 
:$${\mathbf{H}_{2}} = \left[ \begin{array}{ccc} +1 & +1 \\ +1 & -1 \end{array} \right] $$
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lassen sich durch folgende Rekursiont die weiteren Hadamard–Matrizen $ {\mathbf{H}_{4}}$, $ {\mathbf{H}_{8}}$, usw. herleiten:
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the further Hadamard matrices &nbsp;$ {\mathbf{H}_{4}}$, &nbsp;$ {\mathbf{H}_{8}}$,&nbsp; etc. can be derived by the following recursion:
 
:$$ {\mathbf{H}_{2J}} = \left[ \begin{array}{ccc} \mathbf{H}_J & \mathbf{H}_J \\ \mathbf{H}_J & -\mathbf{H}_J \end{array} \right] \hspace{0.05cm}.$$
 
:$$ {\mathbf{H}_{2J}} = \left[ \begin{array}{ccc} \mathbf{H}_J & \mathbf{H}_J \\ \mathbf{H}_J & -\mathbf{H}_J \end{array} \right] \hspace{0.05cm}.$$
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Die Grafik zeigt die Matrix $ {\mathbf{H}_{8}}$ für den Spreizfaktor $J = 8$. Daraus lassen sich die Spreizfolgen
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The diagram shows the matrix &nbsp;$ {\mathbf{H}_{8}}$&nbsp; for the spreading factor &nbsp;$J = 8$.&nbsp; From this we can derive the spreading sequences
 
:$$ \langle w_\nu^{(1)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
 
:$$ \langle w_\nu^{(1)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
 
:$$ \langle w_\nu^{(2)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm}{-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
 
:$$ \langle w_\nu^{(2)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm}{-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
 
:$$...$$
 
:$$...$$
−
:$$\langle w_\nu^{(7)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm}.$$
+
:$$\langle w_\nu^{(7)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm}{-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm}$$
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für sieben CDMA–Teilnehmer ablesen. Die Spreizfolge $ \langle w_\nu^{(0)}\rangle$ entsprechend der ersten Zeile in der Hadamard–Matrix wird meistens nicht vergeben, da sie nicht spreizt.
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for seven CDMA subscribers.&nbsp; The spreading sequence &nbsp;$ \langle w_\nu^{(0)}\rangle$&nbsp; corresponding to the first row in the Hadamard matrix is usually not assigned because it does not spread.
  
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Die Fragen beziehen sich meist auf den Spreizfaktor $J = 4$. Damit können entsprechend mit den Spreizfolgen $ \langle w_\nu^{(1)}\rangle$, $ \langle w_\nu^{(2)}\rangle$ und $ \langle w_\nu^{(3)}\rangle$ maximal drei CDMA–Teilnehmer versorgt werden, die sich aus der zweiten, dritten und vierten Zeile der Matrix $ {\mathbf{H}_{4}}$ ergeben.
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The questions mostly refer to the spreading factor &nbsp;$J = 4$.&nbsp; Thus,&nbsp; correspondingly,&nbsp; a maximum of three CDMA subscribers can be supplied with the spreading sequences &nbsp;$ \langle w_\nu^{(1)}\rangle$, &nbsp;$ \langle w_\nu^{(2)}\rangle$&nbsp; and &nbsp;$ \langle w_\nu^{(3)}\rangle$,&nbsp; which result from the second, third and fourth rows of the matrix $ {\mathbf{H}_{4}}$.
  
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Hinsichtlich der Korrelationsfunktionen soll in dieser Aufgabe folgende Nomenklatur gelten:
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Regarding the correlation functions, the following nomenclature shall apply in this exercise:
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* Die [[Modulationsverfahren/Spreizfolgen_für_CDMA#Periodische_AKF_und_KKF|periodische Kreuzkorrelationsfunktion]] (PKKF) zwischen den Folgen $ \langle w_\nu^{(i)}\rangle$〉 und $ \langle w_\nu^{(j)}\rangle$ wird mit $φ_{ij}(λ)$ bezeichnet. Hierbei gilt:
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* The &nbsp;[[Modulation_Methods/Spreading_Sequences_for_CDMA#Periodic_ACF_and_CCF|periodic cross-correlation function]]&nbsp; $\rm (PCCF)$&nbsp; between the sequences &nbsp;$ \langle w_\nu^{(i)}\rangle$&nbsp; and &nbsp;$ \langle w_\nu^{(j)}\rangle$&nbsp; is denoted by &nbsp;$φ_{ij}(λ)$.&nbsp;&nbsp; Here:
 
:$${\it \varphi}_{ij}(\lambda) = {\rm E}\left [ w_{\nu}^{(i)} \cdot w_{\nu+ \lambda}^{(j)} \right ] \hspace{0.05cm}.$$
 
:$${\it \varphi}_{ij}(\lambda) = {\rm E}\left [ w_{\nu}^{(i)} \cdot w_{\nu+ \lambda}^{(j)} \right ] \hspace{0.05cm}.$$
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* Ist die PKKF $φ_{ij} \equiv 0$ (das heißt: $φ_{ij}(λ) = 0$ für alle Werte von $λ$), so stören sich die CDMA–Teilnehmer nicht, auch wenn zwei Teilnehmer unterschiedliche Laufzeiten aufweisen.
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* If &nbsp;$φ_{ij} \equiv 0$&nbsp; $($that is: &nbsp;$φ_{ij}(λ) = 0$&nbsp; for all values of &nbsp;$λ)$,&nbsp; the CDMA subscribers do not interfere with each other,&nbsp; even if they have different propagation times.
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* Gilt zumindest $φ_{ij}({\it λ} = 0) = 0$, so kommt es zumindest bei synchronem CDMA–Betrieb (keine oder gleiche Laufzeiten aller Teilnehmer) zu keinen Interferenzen.
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* If at least &nbsp;$φ_{ij}({\it λ} = 0) = 0$&nbsp; applies,&nbsp; then no interference occurs,&nbsp; at least in synchronous CDMA operation&nbsp; $($no or equal propagation times of all subscribers$).$&nbsp;
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* Die  [[Modulationsverfahren/Spreizfolgen_für_CDMA#Periodische_AKF_und_KKF|periodische  Autokorrelationsfunktion]] (PAKF) der Walsh–Funktion $ \langle w_\nu^{(i)}\rangle$ wird mit $φ_{ii}(λ)$ bezeichnet, und es gilt:
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* The &nbsp;[[Modulation_Methods/Spreading_Sequences_for_CDMA#Periodic_ACF_and_CCF|periodic auto-correlation function]]&nbsp; $\rm (PACF)$&nbsp; of the Walsh function &nbsp;$ \langle w_\nu^{(i)}\rangle$&nbsp; is denoted by &nbsp;$φ_{ii}(λ)$,&nbsp; and it holds:
 
:$${\it \varphi}_{ii}(\lambda) = {\rm E}\left [ w_{\nu}^{(i)} \cdot w_{\nu+ \lambda}^{(i)} \right ] \hspace{0.05cm}.$$
 
:$${\it \varphi}_{ii}(\lambda) = {\rm E}\left [ w_{\nu}^{(i)} \cdot w_{\nu+ \lambda}^{(i)} \right ] \hspace{0.05cm}.$$
  
  
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''Hinweise:''
 
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*Die Aufgabe gehört zum  Kapitel [[Modulationsverfahren/Spreizfolgen_für_CDMA|Spreizfolgen für CDMA]].
 
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*Bezug genommen wird insbesondere auf den Abschnitt [[Modulationsverfahren/Spreizfolgen_für_CDMA#Walsh.E2.80.93Funktionen |Walsh&ndash;Funktionen]] im Theorieteil.
 
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* Wir möchten Sie gerne auch auf das Interaktionsmodul [[Walsh-Funktionen]] hinweisen.
 
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*Sollte die Eingabe des Zahlenwertes „0” erforderlich sein, so geben Sie bitte „0.” ein.
 
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*Die Abszisse ist auf die Chipdauer $T_c$ normiert. Das bedeutet, dass $λ = 1$ eigentlich eine Verschiebung um die Verzögerungszeit $τ = T_c$ beschreibt.
 
  
 +
Notes:
 +
*The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Spreizfolgen_für_CDMA|Spreading Sequences for CDMA]].
 +
*Reference is made in particular to the section&nbsp; [[Modulation_Methods/Spreading_Sequences_for_CDMA#Walsh_functions|Walsh functions]]&nbsp; in the theory part.
 +
* We would also like to draw your attention to the interactive applet&nbsp; [[Applets:Generation_of_Walsh_functions|Generation of Walsh functions]].&nbsp;
 +
*The abscissa is normalized to the chip duration &nbsp;$T_c$.&nbsp; This means that &nbsp;$λ = 1$&nbsp; actually describes a shift by the delay time &nbsp;$τ = T_c$.&nbsp;
  
  
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===Fragebogen===
+
 
 +
===Questions===
  
 
<quiz display=simple>
 
<quiz display=simple>
−
{Wie lauten die Spreizfolgen für $J = 4$?
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{What are the spreading sequences for &nbsp;$J = 4$?
 
|type="[]"}
 
|type="[]"}
 
+ $ \langle w_\nu^{(1)}\rangle = +\hspace{-0.05cm}1 -\hspace{-0.15cm}1 +\hspace{-0.15cm}1 -\hspace{-0.15cm}1$,
 
+ $ \langle w_\nu^{(1)}\rangle = +\hspace{-0.05cm}1 -\hspace{-0.15cm}1 +\hspace{-0.15cm}1 -\hspace{-0.15cm}1$,
Line 44: Line 44:
 
+ $ \langle w_\nu^{(3)}\rangle = +\hspace{-0.05cm}1 -\hspace{-0.15cm}1 -\hspace{-0.15cm}1 +\hspace{-0.15cm}1$.
 
+ $ \langle w_\nu^{(3)}\rangle = +\hspace{-0.05cm}1 -\hspace{-0.15cm}1 -\hspace{-0.15cm}1 +\hspace{-0.15cm}1$.
  
−
{Welche Aussagen gelten bezüglich der PKKF–Werte $φ_{ij}(λ = 0)$?
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{Which statements are true regarding the PCCF values &nbsp;$φ_{ij}(λ = 0)$?
 
|type="[]"}
 
|type="[]"}
−
+ Für $J = 4$ ist $φ_{12}(λ = 0) = 0$.
+
+ For $J = 4$,&nbsp; &nbsp;$φ_{12}(λ = 0) = 0$.
−
+ Für $J = 4$ ist $φ_{13}(λ = 0) = 0$.
+
+ For $J = 4$,&nbsp; &nbsp;$φ_{13}(λ = 0) = 0$.
−
+ Für $J = 4$ ist $φ_{23}(λ = 0) = 0$.
+
+ For $J = 4$,&nbsp; &nbsp;$φ_{23}(λ = 0) = 0$.
−
- Für $J = 8$ kann durchaus $φ_{ij}(λ = 0) ≠ 0$ gelten $(i ≠ j)$.
+
- For $J = 8$,&nbsp; &nbsp;$φ_{ij}(λ = 0) ≠ 0$&nbsp; may well hold &nbsp;$(i ≠ j)$.
−
+ Bei synchronem CDMA stören sich die Teilnehmer nicht.
+
+ In synchronous CDMA,&nbsp; the subscribers do not interfere with each other.
  
−
{Welche Aussagen gelten für die PKKF–Werte mit $λ ≠ 0$?
+
{Which statements are true for the PCCF values with &nbsp;$λ ≠ 0$?
 
|type="[]"}
 
|type="[]"}
−
+ Für alle Werte von $λ$ ist die PKKF $φ_{12}(λ) = 0$.
+
+ For all values of &nbsp;$λ$,&nbsp; the PCCF is &nbsp;$φ_{12}(λ) = 0$.
−
+ Für alle Werte von $λ$ ist die PKKF $φ_{13}(λ) = 0$.
+
+ For all values of &nbsp;$λ$,&nbsp; the PCCF is &nbsp;$φ_{13}(λ) = 0$.
−
- Für alle Werte von $λ$ ist die PKKF $φ_{23}(λ) = 0$.
+
- For all values of &nbsp;$λ$,&nbsp; the PCCF is &nbsp;$φ_{23}(λ) = 0$.
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- Bei asynchronem CDMA stören sich die Teilnehmer nicht.
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- In asynchronous CDMA,&nbsp; the subscribers do not interfere with each other.
  
−
{Welche Aussagen gelten für die PAKF–Kurven?
+
{Which statements are true for the PACF curves?
 
|type="[]"}
 
|type="[]"}
−
+ Alle $φ_{ii}(λ)$–Kurven sind periodisch.
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+ All&nbsp; &nbsp;$φ_{ii}(λ)$&nbsp; curves are periodic.
−
+ Es gilt $φ_{11}(λ = 0) = +\hspace{-0.05cm}1$ und $φ_{11}(λ = 1) = -\hspace{-0.05cm}1$.
+
+ &nbsp;$φ_{11}(λ = 0) = +\hspace{-0.05cm}1$&nbsp; and &nbsp;$φ_{11}(λ = 1) = -\hspace{-0.05cm}1$&nbsp; hold.
−
- Es gilt $φ_{22}(λ) = φ_{11}(λ)$.
+
- &nbsp;$φ_{22}(λ) = φ_{11}(λ)$&nbsp; holds.
−
+ Es gilt $φ_{33}(λ) = φ_{22}(λ)$.
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+ &nbsp;$φ_{33}(λ) = φ_{22}(λ)$&nbsp; holds.
 
</quiz>
 
</quiz>
  
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===Musterlösung===
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===Solution===
 
{{ML-Kopf}}
 
{{ML-Kopf}}
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'''1.''' Die Matrix $H_4$ ist die linke obere Teilmatrix von $H_8$. Die Spreizfolgen ergeben sich aus den Zeilen 2, 3 und 4 von $H_4$, und stimmen mit den angegebenen Folgen überein. Somit sind alle Vorschläge richtig.
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'''(1)'''&nbsp; <u>All solutions</u>&nbsp; are correct:
 +
*The matrix&nbsp; $ {\mathbf{H}_{4}}$&nbsp; is the upper left submatrix of&nbsp; $ {\mathbf{H}_{8}}$.  
 +
*The spreading sequences result from the rows 2,&nbsp; 3&nbsp; and 4&nbsp; of&nbsp; $ {\mathbf{H}_{4}}$,&nbsp; and agree with the given sequences.
  
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'''2.''' Entsprechend den Gleichungen im Angabenteil gilt:
 
−
$${\it \varphi}_{12}(\lambda = 0) = \frac{1}{4} \cdot \left [ (+1) \cdot (+1) + (-1) \cdot (+1) + (+1) \cdot (-1) + (-1) \cdot (-1) \right ] = 0\hspace{0.05cm},$$
 
−
$${\it \varphi}_{13}(\lambda = 0) = \frac{1}{4} \cdot \left [ (+1) \cdot (+1) + (-1) \cdot (-1) + (+1) \cdot (-1) + (-1) \cdot (+1) \right ] = 0\hspace{0.05cm},$$
 
−
$${\it \varphi}_{23}(\lambda = 0) = \frac{1}{4} \cdot \left [ (+1) \cdot (+1) + (+1) \cdot (-1) + (-1) \cdot (-1) + (-1) \cdot (+1) \right ] = 0\hspace{0.05cm}.$$
 
−
Auch für größere Werte von J ist der PKKF–Wert $φ_{ij}(λ = 0)$ für i ≠ j stets 0. Daraus folgt: Bei synchronem CDMA stören sich die Teilnehmer nicht. Richtig sind somit alle Aussagen mit Ausnahme von Lösungsvorschlag (4).
 
  
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'''3.''' Die PKKF $φ_{12}(λ)$ ist für alle Werte von λ gleich 0, wie die folgenden Zeilen zeigen:
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'''(2)'''&nbsp; <u>Solutions 1, 2 and 3</u>&nbsp; are correct:
−
$$\langle w_\nu^{(1)}\rangle  = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$ $$\langle w_\nu^{(2)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm}, \hspace{0.3cm}{\rm Produkt\hspace{0.1cm} mit \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},$$  
+
*According to the equations in the data section,&nbsp; the following holds:
−
$$\langle w_{\nu+1}^{(2)}\rangle  = {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm}, \hspace{0.3cm}{\rm Produkt\hspace{0.1cm} mit \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},$$
+
:$${\it \varphi}_{12}(\lambda = 0) = 1/4 \cdot \left [ (+1) \cdot (+1) + (-1) \cdot (+1) + (+1) \cdot (-1) + (-1) \cdot (-1) \right ] = 0\hspace{0.05cm},$$  
−
$$\langle w_{\nu+2}^{(2)}\rangle  =  {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},\hspace{0.3cm}{\rm Produkt\hspace{0.1cm} mit \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$  
+
:$${\it \varphi}_{13}(\lambda = 0) = 1/4\cdot \left [ (+1) \cdot (+1) + (-1) \cdot (-1) + (+1) \cdot (-1) + (-1) \cdot (+1) \right ] = 0\hspace{0.05cm},$$  
−
$$\langle w_{\nu+3}^{(2)}\rangle  = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},\hspace{0.3cm}{\rm Produkt\hspace{0.1cm} mit \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
+
:$${\it \varphi}_{23}(\lambda = 0) =1/4 \cdot \left [ (+1) \cdot (+1) + (+1) \cdot (-1) + (-1) \cdot (-1) + (-1) \cdot (+1) \right ] = 0\hspace{0.05cm}.$$
−
$$\langle w_{\nu+4}^{(2)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} = \langle w_\nu^{(2)}\rangle \hspace{0.05cm}.$$
+
*Also,&nbsp; for larger values of&nbsp; $J$,&nbsp; for&nbsp; $i ≠ j$&nbsp; the PCCF value is always&nbsp; $φ_{ij}(λ = 0)= 0$.  
−
Das gleiche gilt für die PKKF $φ_{13}(λ)$. Dagegen erhält man für die PKKF zwischen den Folgen 〈$w_ν^{(2)}$〉 und 〈$w_ν{(3)}$〉:
+
*It follows: &nbsp; In synchronous CDMA,&nbsp; the subscribers do not interfere with each other.
−
$${\it \varphi}_{23}(\lambda ) = \left\{ \begin{array}{c}0 \\+1\\ -1 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ \end{array} \begin{array}{*{20}c} \lambda = 0, \pm 2, \pm 4,\pm 6, ... \hspace{0.05cm}, \\ \hspace{0.14cm} \lambda = ... \hspace{0.05cm} , -3, +1, +5, ... \hspace{0.05cm}, \\ \hspace{0.14cm} \lambda = ... \hspace{0.05cm} , -5, -1, +3, ... \hspace{0.05cm}. \\ \end{array}$$
 
  
−
Das bedeutet: Wird das Signal von Teilnehmer 3 gegenüber Teilnehmer 2 um ein Spreizchip verzögert oder umgekehrt, so lassen sich die Teilnehmer nicht mehr trennen und es kommt zu einer signifikanten Erhöhung der Fehlerwahrscheinlichkeit. Richtig sind also nur die Lösungsvorschläge 1 und 2.
 
  
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In der nachfolgenden Grafik sind die PKKF–Kurven gestrichelt eingezeichnet (violett und rot).
 
  
−
[[File:P_ID1890__Mod_A_5_4c.png]]
+
'''(3)'''&nbsp; <u>Solutions 1 and 2</u>&nbsp; are correct:
 +
*For all values of&nbsp; $λ$,&nbsp; the PCCF is&nbsp; $φ_{12}(λ) = 0$,&nbsp; as shown by the following lines:
 +
:$$\langle w_\nu^{(1)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$ $$\langle w_\nu^{(2)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm}, \hspace{0.3cm}{\rm product\hspace{0.1cm} with \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},$$
 +
:$$\langle w_{\nu+1}^{(2)}\rangle  =  {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm}, \hspace{0.3cm}{\rm product\hspace{0.1cm} with \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},$$
 +
:$$\langle w_{\nu+2}^{(2)}\rangle  =  {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},\hspace{0.3cm}{\rm product\hspace{0.1cm} with \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
 +
:$$\langle w_{\nu+3}^{(2)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},\hspace{0.3cm}{\rm product\hspace{0.1cm} with \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
 +
:$$\langle w_{\nu+4}^{(2)}\rangle  =  {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} = \langle w_\nu^{(2)}\rangle \hspace{0.05cm}.$$
 +
[[File:P_ID1890__Mod_A_5_4c.png|right|frame|Some PCCF and PACF curves]]
 +
*The same is true for the PCCF&nbsp; $φ_{13}(λ)$.
 +
*In contrast,&nbsp; for the PCCF between the sequences&nbsp; $ \langle w_\nu^{(2)}\rangle$&nbsp; and&nbsp; $ \langle w_\nu^{(3)}\rangle$&nbsp; we obtain:
  
 +
:$${\it \varphi}_{23}(\lambda ) = \left\{ \begin{array}{c}0 \\+1\\ -1 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array} \begin{array}{*{20}c} \lambda = 0, \pm 2, \pm 4,\pm 6, ... \hspace{0.05cm}, \\ \hspace{0.14cm} \lambda = ... \hspace{0.05cm} , -3, +1, +5, ... \hspace{0.05cm}, \\ \hspace{0.14cm} \lambda = ... \hspace{0.05cm} , -5, -1, +3, ... \hspace{0.05cm}. \\ \end{array}$$
 +
*This means: &nbsp; If the signal from subscriber&nbsp; $3$&nbsp; is delayed by one spreading chip with respect to subscriber&nbsp; $2$&nbsp; or vice versa,&nbsp; the subscribers can no longer be separated and there is a significant increase in the error probability.
 +
*In the diagram,&nbsp; the PCCF curves are drawn in dashed lines&nbsp; (violet and red).
  
−
'''4.'''  Richtig sind die Aussagen 1, 2 und 4. Da die Walsh–Funktion Nr. 1 periodisch ist mit $T_0 = 2T_c$, ist auch die PAKF periodisch mit λ = 2.
 
  
−
Die zweite Aussage ist richtig, wie die folgende Rechnung zeigt (grüner Kurvenzug):
 
−
$${\it \varphi}_{11}(\lambda = 0)  =  \frac{1}{4} \cdot \left [ (+1) \cdot (+1) + (-1) \cdot (-1) + (+1) \cdot (+1) + (-1) \cdot (-1) \right ] = 1\hspace{0.05cm},$$
 
−
$${\it \varphi}_{11}(\lambda = 1)  =  \frac{1}{4} \cdot \left [ (+1) \cdot (-1) + (+1) \cdot (-1) + (+1) \cdot (-1) + (+1) \cdot (-1) \right ] = -1\hspace{0.05cm}.$$
 
−
Da sich die beiden Walsh–Funktionen Nr. 2 und 3 nur durch eine Verschiebung um $T_c$ unterscheiden und sich eine Phase in der PAKF prinzipiell nicht auswirkt, ist tatsächlich entsprechend dem letzten Lösungsvorschlag $φ_{33}(λ) = φ_{22}(λ)$. Diese beiden PAKF–Funktionen sind blau eingezeichnet.
 
  
−
Dagegen unterscheidet sich $φ_{22}(λ)$ von $φ_{11}(λ)$ durch eine andere Periodizität: $φ_{22}(λ) = φ_{33}(λ)$ ist doppelt so breit wie $φ_{11}(λ)$.
+
'''(4)'''&nbsp; <u>Statements 1,&nbsp; 2&nbsp; and 4</u>&nbsp; are correct:
 +
* Since the Walsh function no.&nbsp; $1$&nbsp; is periodic with&nbsp; $T_0 = 2T_c$,&nbsp; the PACF is also periodic with&nbsp; $λ = 2$.
 +
*The second statement is correct,&nbsp; as shown by the following calculation&nbsp; (green curve):
 +
:$${\it \varphi}_{11}(\lambda = 0)  =  1/4 \cdot \big [ (+1) \cdot (+1) + (-1) \cdot (-1) + (+1) \cdot (+1) + (-1) \cdot (-1) \big ] = +1\hspace{0.05cm},$$
 +
:$${\it \varphi}_{11}(\lambda = 1)  =  1/4 \cdot \big [ (+1) \cdot (-1) + (+1) \cdot (-1) + (+1) \cdot (-1) + (+1) \cdot (-1) \big ] = -1\hspace{0.05cm}.$$
 +
*Since the two Walsh functions no.&nbsp; $2$&nbsp; and&nbsp; $3$&nbsp; differ only by a shift around&nbsp; $T_c$&nbsp; and a phase in the PACF has no effect in principle,&nbsp; in fact,&nbsp; according to the last statement,&nbsp; $φ_{33}(λ) = φ_{22}(λ)$.&nbsp; These two PACF functions are plotted in blue.
 +
*In contrast,&nbsp; $φ_{22}(λ)$&nbsp; differs from&nbsp; $φ_{11}(λ)$&nbsp; by a different periodicity: &nbsp; $φ_{22}(λ) = φ_{33}(λ)$&nbsp; is twice as wide as&nbsp; $φ_{11}(λ)$.
  
 
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[[Category:Aufgaben zu Modulationsverfahren|^5.3 Spreizfolgen für CDMA^]]
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[[Category:Modulation Methods: Exercises|^5.3 Spread Sequences for CDMA^]]

Latest revision as of 16:31, 13 December 2021

Hadamard matrix  ${\mathbf{H}_{8}}$

The so-called  "Walsh functions",  which can be constructed by means of the Hadamard matrix,  are often used for band spreading and band compression.  Starting from the matrix

$${\mathbf{H}_{2}} = \left[ \begin{array}{ccc} +1 & +1 \\ +1 & -1 \end{array} \right] $$

the further Hadamard matrices  $ {\mathbf{H}_{4}}$,  $ {\mathbf{H}_{8}}$,  etc. can be derived by the following recursion:

$$ {\mathbf{H}_{2J}} = \left[ \begin{array}{ccc} \mathbf{H}_J & \mathbf{H}_J \\ \mathbf{H}_J & -\mathbf{H}_J \end{array} \right] \hspace{0.05cm}.$$

The diagram shows the matrix  $ {\mathbf{H}_{8}}$  for the spreading factor  $J = 8$.  From this we can derive the spreading sequences

$$ \langle w_\nu^{(1)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
$$ \langle w_\nu^{(2)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm}{-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
$$...$$
$$\langle w_\nu^{(7)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm}{-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm}{+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm}$$

for seven CDMA subscribers.  The spreading sequence  $ \langle w_\nu^{(0)}\rangle$  corresponding to the first row in the Hadamard matrix is usually not assigned because it does not spread.

The questions mostly refer to the spreading factor  $J = 4$.  Thus,  correspondingly,  a maximum of three CDMA subscribers can be supplied with the spreading sequences  $ \langle w_\nu^{(1)}\rangle$,  $ \langle w_\nu^{(2)}\rangle$  and  $ \langle w_\nu^{(3)}\rangle$,  which result from the second, third and fourth rows of the matrix $ {\mathbf{H}_{4}}$.

Regarding the correlation functions, the following nomenclature shall apply in this exercise:

  • The  periodic cross-correlation function  $\rm (PCCF)$  between the sequences  $ \langle w_\nu^{(i)}\rangle$  and  $ \langle w_\nu^{(j)}\rangle$  is denoted by  $φ_{ij}(λ)$.   Here:
$${\it \varphi}_{ij}(\lambda) = {\rm E}\left [ w_{\nu}^{(i)} \cdot w_{\nu+ \lambda}^{(j)} \right ] \hspace{0.05cm}.$$
  • If  $φ_{ij} \equiv 0$  $($that is:  $φ_{ij}(λ) = 0$  for all values of  $λ)$,  the CDMA subscribers do not interfere with each other,  even if they have different propagation times.
  • If at least  $φ_{ij}({\it λ} = 0) = 0$  applies,  then no interference occurs,  at least in synchronous CDMA operation  $($no or equal propagation times of all subscribers$).$ 
  • The  periodic auto-correlation function  $\rm (PACF)$  of the Walsh function  $ \langle w_\nu^{(i)}\rangle$  is denoted by  $φ_{ii}(λ)$,  and it holds:
$${\it \varphi}_{ii}(\lambda) = {\rm E}\left [ w_{\nu}^{(i)} \cdot w_{\nu+ \lambda}^{(i)} \right ] \hspace{0.05cm}.$$


Notes:

  • The exercise belongs to the chapter  Spreading Sequences for CDMA.
  • Reference is made in particular to the section  Walsh functions  in the theory part.
  • We would also like to draw your attention to the interactive applet  Generation of Walsh functions. 
  • The abscissa is normalized to the chip duration  $T_c$.  This means that  $λ = 1$  actually describes a shift by the delay time  $τ = T_c$. 


Questions

1

What are the spreading sequences for  $J = 4$?

$ \langle w_\nu^{(1)}\rangle = +\hspace{-0.05cm}1 -\hspace{-0.15cm}1 +\hspace{-0.15cm}1 -\hspace{-0.15cm}1$,
$ \langle w_\nu^{(2)}\rangle = +\hspace{-0.05cm}1 +\hspace{-0.15cm}1 -\hspace{-0.15cm}1 -\hspace{-0.15cm}1$,
$ \langle w_\nu^{(3)}\rangle = +\hspace{-0.05cm}1 -\hspace{-0.15cm}1 -\hspace{-0.15cm}1 +\hspace{-0.15cm}1$.

2

Which statements are true regarding the PCCF values  $φ_{ij}(λ = 0)$?

For $J = 4$,   $φ_{12}(λ = 0) = 0$.
For $J = 4$,   $φ_{13}(λ = 0) = 0$.
For $J = 4$,   $φ_{23}(λ = 0) = 0$.
For $J = 8$,   $φ_{ij}(λ = 0) ≠ 0$  may well hold  $(i ≠ j)$.
In synchronous CDMA,  the subscribers do not interfere with each other.

3

Which statements are true for the PCCF values with  $λ ≠ 0$?

For all values of  $λ$,  the PCCF is  $φ_{12}(λ) = 0$.
For all values of  $λ$,  the PCCF is  $φ_{13}(λ) = 0$.
For all values of  $λ$,  the PCCF is  $φ_{23}(λ) = 0$.
In asynchronous CDMA,  the subscribers do not interfere with each other.

4

Which statements are true for the PACF curves?

All   $φ_{ii}(λ)$  curves are periodic.
 $φ_{11}(λ = 0) = +\hspace{-0.05cm}1$  and  $φ_{11}(λ = 1) = -\hspace{-0.05cm}1$  hold.
 $φ_{22}(λ) = φ_{11}(λ)$  holds.
 $φ_{33}(λ) = φ_{22}(λ)$  holds.


Solution

Solution

(1)  All solutions  are correct:

  • The matrix  $ {\mathbf{H}_{4}}$  is the upper left submatrix of  $ {\mathbf{H}_{8}}$.
  • The spreading sequences result from the rows 2,  3  and 4  of  $ {\mathbf{H}_{4}}$,  and agree with the given sequences.


(2)  Solutions 1, 2 and 3  are correct:

  • According to the equations in the data section,  the following holds:
$${\it \varphi}_{12}(\lambda = 0) = 1/4 \cdot \left [ (+1) \cdot (+1) + (-1) \cdot (+1) + (+1) \cdot (-1) + (-1) \cdot (-1) \right ] = 0\hspace{0.05cm},$$
$${\it \varphi}_{13}(\lambda = 0) = 1/4\cdot \left [ (+1) \cdot (+1) + (-1) \cdot (-1) + (+1) \cdot (-1) + (-1) \cdot (+1) \right ] = 0\hspace{0.05cm},$$
$${\it \varphi}_{23}(\lambda = 0) =1/4 \cdot \left [ (+1) \cdot (+1) + (+1) \cdot (-1) + (-1) \cdot (-1) + (-1) \cdot (+1) \right ] = 0\hspace{0.05cm}.$$
  • Also,  for larger values of  $J$,  for  $i ≠ j$  the PCCF value is always  $φ_{ij}(λ = 0)= 0$.
  • It follows:   In synchronous CDMA,  the subscribers do not interfere with each other.


(3)  Solutions 1 and 2  are correct:

  • For all values of  $λ$,  the PCCF is  $φ_{12}(λ) = 0$,  as shown by the following lines:
$$\langle w_\nu^{(1)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$ $$\langle w_\nu^{(2)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm}, \hspace{0.3cm}{\rm product\hspace{0.1cm} with \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},$$
$$\langle w_{\nu+1}^{(2)}\rangle = {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm}, \hspace{0.3cm}{\rm product\hspace{0.1cm} with \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},$$
$$\langle w_{\nu+2}^{(2)}\rangle = {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},\hspace{0.3cm}{\rm product\hspace{0.1cm} with \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
$$\langle w_{\nu+3}^{(2)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},\hspace{0.3cm}{\rm product\hspace{0.1cm} with \hspace{0.1cm}}\langle w_\nu^{(1)}\rangle: {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
$$\langle w_{\nu+4}^{(2)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} = \langle w_\nu^{(2)}\rangle \hspace{0.05cm}.$$
Some PCCF and PACF curves
  • The same is true for the PCCF  $φ_{13}(λ)$.
  • In contrast,  for the PCCF between the sequences  $ \langle w_\nu^{(2)}\rangle$  and  $ \langle w_\nu^{(3)}\rangle$  we obtain:
$${\it \varphi}_{23}(\lambda ) = \left\{ \begin{array}{c}0 \\+1\\ -1 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array} \begin{array}{*{20}c} \lambda = 0, \pm 2, \pm 4,\pm 6, ... \hspace{0.05cm}, \\ \hspace{0.14cm} \lambda = ... \hspace{0.05cm} , -3, +1, +5, ... \hspace{0.05cm}, \\ \hspace{0.14cm} \lambda = ... \hspace{0.05cm} , -5, -1, +3, ... \hspace{0.05cm}. \\ \end{array}$$
  • This means:   If the signal from subscriber  $3$  is delayed by one spreading chip with respect to subscriber  $2$  or vice versa,  the subscribers can no longer be separated and there is a significant increase in the error probability.
  • In the diagram,  the PCCF curves are drawn in dashed lines  (violet and red).


(4)  Statements 1,  2  and 4  are correct:

  • Since the Walsh function no.  $1$  is periodic with  $T_0 = 2T_c$,  the PACF is also periodic with  $λ = 2$.
  • The second statement is correct,  as shown by the following calculation  (green curve):
$${\it \varphi}_{11}(\lambda = 0) = 1/4 \cdot \big [ (+1) \cdot (+1) + (-1) \cdot (-1) + (+1) \cdot (+1) + (-1) \cdot (-1) \big ] = +1\hspace{0.05cm},$$
$${\it \varphi}_{11}(\lambda = 1) = 1/4 \cdot \big [ (+1) \cdot (-1) + (+1) \cdot (-1) + (+1) \cdot (-1) + (+1) \cdot (-1) \big ] = -1\hspace{0.05cm}.$$
  • Since the two Walsh functions no.  $2$  and  $3$  differ only by a shift around  $T_c$  and a phase in the PACF has no effect in principle,  in fact,  according to the last statement,  $φ_{33}(λ) = φ_{22}(λ)$.  These two PACF functions are plotted in blue.
  • In contrast,  $φ_{22}(λ)$  differs from  $φ_{11}(λ)$  by a different periodicity:   $φ_{22}(λ) = φ_{33}(λ)$  is twice as wide as  $φ_{11}(λ)$.

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