Aufgaben:Exercise 5.3: PACF of PN Sequences - Revision history

Guenter at 13:21, 13 December 2021

2021-12-13T13:21:17Z

Guenter at 12:51, 13 December 2021

2021-12-13T12:51:22Z

Guenter at 12:37, 13 December 2021

2021-12-13T12:37:16Z

Guenter at 12:36, 13 December 2021

2021-12-13T12:36:08Z

Hwang at 13:01, 10 December 2021

2021-12-10T13:01:48Z

Hwang at 13:01, 10 December 2021

2021-12-10T13:01:28Z

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Guenter at 10:24, 1 December 2021

2021-12-01T10:24:57Z

Guenter: Guenter moved page Aufgaben:Aufgabe 5.3: PAKF von PN–Sequenzen to Aufgaben:Exercise 5.3: PACF of PN Sequences

2021-12-01T10:19:15Z

Guenter moved page Aufgabe 5.3: PAKF von PN–Sequenzen to Exercise 5.3: PACF of PN Sequences

Javier: Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

2021-03-23T12:49:48Z

Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

Javier: Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

2020-07-09T12:44:09Z

Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

@@ Line 86: / Line 86: @@
 *Using the result of subtask&nbsp; '''(2)'''
 :$$ {\rm E}\left [ u_{\nu} \right ]= {\rm E}\left [ u_{\nu+\lambda} \right ]={8}/{15} \hspace{0.05cm},$$
-and the subtask&nbsp; '''(4)'''
+:and the subtask&nbsp; '''(4)'''
 :$${\rm E}\big [ u_\nu \cdot u_{\nu+\lambda} \big ] ={4}/{15} \hspace{0.05cm} \,\,{\rm{f\ddot{u}r}}\,\,\lambda = 0, \pm P, \pm 2P, \text{...}$$
-one thus arrives at the result&nbsp; $($if&nbsp; $λ$&nbsp; is not a multiple of&nbsp; $P)$:
+:one thus arrives at the result&nbsp; $($if&nbsp; $λ$&nbsp; is not a multiple of&nbsp; $P)$:
 :$${\it \varphi}_{c}(\lambda) = 1 + 4 \cdot \frac{4}{15} - 2 \cdot \frac{8}{15}- 2 \cdot \frac{8}{15}  = - \frac{1}{15} = - \frac{1}{P}\hspace{0.15cm}\underline {\approx - 0.067} \hspace{0.05cm}.$$

@@ Line 64: / Line 64: @@
+'''(3)'''&nbsp; In bipolar representation,&nbsp; it is always &nbsp;$c_ν^{\hspace{0.04cm}2} = 1$.&nbsp; Thus, also for the quadratic expectation value:
 :$${\rm E}\big [ c_\nu^{\hspace{0.04cm}2} \big ] \hspace{0.15cm}\underline {= 1}\hspace{0.05cm}.$$
+'''(4)'''&nbsp; <u>Solutions 1, 2 and 4</u>&nbsp; are correct:
 *The attached table makes clear that for the discrete PACF values with&nbsp; $λ = 1$, ... , $14$&nbsp; holds:
 :$${\it \varphi}_{u}(\lambda) = {\rm E}\big [ u_\nu \cdot u_{\nu+\lambda} \big ]= {4}/{15} \hspace{0.05cm}.$$
-*If we multiply&nbsp; 〈$u_ν$&nbsp;〉 with&nbsp; 〈$u_{ν+λ}$, where for the index &nbsp;$λ$&nbsp; again the values&nbsp; $1$, ... , $14$&nbsp; are to be used, then four ones occur in the product in each case.
+*If we multiply&nbsp; 〈$u_ν$〉 with 〈$u_{ν+λ}〉$,&nbsp; where for the index &nbsp;$λ$&nbsp; the values&nbsp; $1$, ... , $14$&nbsp; are to be used again,&nbsp; then four ones occur in the product in each case.
-*In contrast, for&nbsp; $λ = P = 15$:
+*In contrast,&nbsp; for&nbsp; $λ = P = 15$:
 :$${\it \varphi}_{u}(\lambda = 15) = {\rm E}\big [ u_\nu \cdot u_{\nu+P} \big ]= {8}/{15} \hspace{0.05cm}.$$
@@ Line 85: / Line 82: @@
 '''(5)'''&nbsp; The bipolar coefficients &nbsp;$c_ν$&nbsp; result from the unipolar coefficients &nbsp;$u_ν$&nbsp; according to the equation
 :$$c_\nu = 1 - 2 \cdot u_\nu \hspace{0.3cm} \Rightarrow \hspace{0.3cm} u_\nu = 0\text{:} \ \ c_\nu = +1\hspace{0.05cm},\hspace{0.3cm}u_\nu = 1\text{:} \ \ c_\nu = -1 \hspace{0.05cm}.$$
-*Thus, it follows from the calculation rules for expected values:
+*Thus,&nbsp; it follows from the calculation rules for expected values:
 :$${\it \varphi}_{c}(\lambda)  =  {\rm E} \big [ c_\nu \cdot c_{\nu+\lambda} \big ]= {\rm E} \big [ (1 - 2 \cdot u_\nu ) \cdot (1 - 2 \cdot u_{\nu+\lambda} ) \big ] =  1 + 4 \cdot {\rm E}\big [ u_\nu \cdot u_{\nu+\lambda} \big ] - 2 \cdot {\rm E}\big [ u_\nu \big ] - 2 \cdot {\rm E}\big [ u_{\nu+\lambda} \big ] \hspace{0.05cm}.$$
 *Using the result of subtask&nbsp; '''(2)'''

@@ Line 6: / Line 6: @@
 With a feedback shift register of degree &nbsp;$G$&nbsp; a spreading sequence &nbsp;$〈c_ν〉$&nbsp; with the&nbsp; (maximum)&nbsp; period length &nbsp;$P = 2^G - 1$&nbsp; can be generated if the feedback coefficients&nbsp; (taps)&nbsp; are chosen correctly.
-In this exercise,&nbsp; we consider the PN generator with octal identifier &nbsp;$(31)$&nbsp; shown in the left graph of&nbsp;[[Modulation_Methods/Spreizfolgen_für_CDMA#Pseudo.E2.80.93Noise.E2.80.93Folgen_maximaler_L.C3.A4nge|$\text{Example 1}$]]&nbsp; in the theory section,&nbsp; which yields a sequence with period length &nbsp;$P = 15$&nbsp; because of &nbsp;$G = 4$.&nbsp;
+In this exercise,&nbsp; we consider the PN generator with octal identifier &nbsp;$(31)$&nbsp; shown in the left graph of&nbsp;[[Modulation_Methods/Spreading_Sequences_for_CDMA#Pseudo-noise_sequences_of_maximum_length|$\text{Example 1}$]]&nbsp; in the theory section,&nbsp; which yields a sequence with period length &nbsp;$P = 15$&nbsp; because of &nbsp;$G = 4$.&nbsp;
 The graph for this exercise shows the unipolar sequence &nbsp;$〈u_ν〉$&nbsp; with &nbsp;$u_ν ∈ \{0, 1\}$&nbsp; and cyclic shifts &nbsp;$〈u_{ν+λ}〉$&nbsp; derived from it,&nbsp; where the shift parameter &nbsp;$λ$&nbsp; takes values between &nbsp;$1$&nbsp; and &nbsp;$15$.&nbsp;&nbsp; Here,&nbsp; a shift by &nbsp;$λ$&nbsp; means absolutely an offset by &nbsp;$λ · T_c$.&nbsp; Here,&nbsp; &nbsp;$T_c$&nbsp; denotes the chip duration.

@@ Line 4: / Line 4: @@
 [[File:P_ID1884__Mod_A_5_3.png|right|frame|M–sequence&nbsp;$(P = 15)$&nbsp; plus cyclic permutations]]
-With a feedback shift register of degree &nbsp;$G$&nbsp; a spreading sequence &nbsp;$〈c_ν〉$&nbsp; with the (maximum) period length &nbsp;$P = 2^G - 1$&nbsp; can be generated if the feedback coefficients (taps) are chosen correctly.
+With a feedback shift register of degree &nbsp;$G$&nbsp; a spreading sequence &nbsp;$〈c_ν〉$&nbsp; with the&nbsp; (maximum)&nbsp; period length &nbsp;$P = 2^G - 1$&nbsp; can be generated if the feedback coefficients&nbsp; (taps)&nbsp; are chosen correctly.
-In this exercise, we consider the PN generator with octal identifier &nbsp;$(31)$&nbsp; shown in the left graph of&nbsp;[[Modulation_Methods/Spreizfolgen_für_CDMA#Pseudo.E2.80.93Noise.E2.80.93Folgen_maximaler_L.C3.A4nge|$\text{example 1}$]]&nbsp; in the theory section, which yields a sequence with period length &nbsp;$P = 15$&nbsp; because of &nbsp;$G = 4$.&nbsp;
+In this exercise,&nbsp; we consider the PN generator with octal identifier &nbsp;$(31)$&nbsp; shown in the left graph of&nbsp;[[Modulation_Methods/Spreizfolgen_für_CDMA#Pseudo.E2.80.93Noise.E2.80.93Folgen_maximaler_L.C3.A4nge|$\text{Example 1}$]]&nbsp; in the theory section,&nbsp; which yields a sequence with period length &nbsp;$P = 15$&nbsp; because of &nbsp;$G = 4$.&nbsp;
-The graph for this exercise shows the unipolar sequence &nbsp;$〈u_ν〉$&nbsp; with &nbsp;$u_ν ∈ \{0, 1\}$&nbsp; and cyclic shifts &nbsp;$〈u_{ν+λ}〉$&nbsp; derived from it, where the shift parameter &nbsp;$λ$&nbsp; takes values between &nbsp;$1$&nbsp; and &nbsp;$15$.&nbsp;&nbsp; Here, a shift by &nbsp;$λ$&nbsp; means absolutely an offset by &nbsp;$λ · T_c$.&nbsp; Here, &nbsp;$T_c$&nbsp; denotes the chip duration.
+The graph for this exercise shows the unipolar sequence &nbsp;$〈u_ν〉$&nbsp; with &nbsp;$u_ν ∈ \{0, 1\}$&nbsp; and cyclic shifts &nbsp;$〈u_{ν+λ}〉$&nbsp; derived from it,&nbsp; where the shift parameter &nbsp;$λ$&nbsp; takes values between &nbsp;$1$&nbsp; and &nbsp;$15$.&nbsp;&nbsp; Here,&nbsp; a shift by &nbsp;$λ$&nbsp; means absolutely an offset by &nbsp;$λ · T_c$.&nbsp; Here,&nbsp; &nbsp;$T_c$&nbsp; denotes the chip duration.
-For use in a CDMA system, however, one uses the bipolar (antipodal) sequence &nbsp;$〈c_ν〉$&nbsp; with &nbsp;$c_ν ∈ \{+1, -1\}$, which is to be investigated starting in subtask&nbsp; '''(5)'''.&nbsp;&nbsp; We are looking for its periodic auto-correlation function (PACF)
+For use in a CDMA system,&nbsp; however,&nbsp; one uses the bipolar (antipodal) sequence &nbsp;$〈c_ν〉$&nbsp; with &nbsp;$c_ν ∈ \{+1, -1\}$,&nbsp; which is to be investigated starting in subtask&nbsp; '''(5)'''.&nbsp;&nbsp; We are looking for its periodic auto-correlation function&nbsp; $\rm (PACF)$
 :$${\it \varphi}_{c}(\lambda) = {\rm E} \big [ c_\nu \cdot c_{\nu+\lambda} \big ] \hspace{0.05cm}.$$
-In order to derive it, first the PACF
+In order to derive it,&nbsp; first the PACF
 :$${\it \varphi}_{u}(\lambda) = {\rm E}\big [ u_\nu \cdot u_{\nu+\lambda} \big ]$$
 with the unipolar coefficients &nbsp;$u_ν ∈ \{0, 1\}$,&nbsp; will be calculated.&nbsp; The conversion of the coefficients is given by &nbsp;$c_ν = 1 - 2u_ν$.&nbsp;
@@ Line 18: / Line 18: @@
 Note:
 *The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Spreizfolgen_für_CDMA|Spreading Sequences for CDMA]].
@@ Line 43: / Line 40: @@
 {Which statements are true for the expected value &nbsp;${\rm E}\big[u_ν · u_{ν+λ}\big]$?
 |type="[]"}
-+ &nbsp;${\rm E}\big[u_ν · u_{ν+1}\big] = 4/15$ is valid.
++ &nbsp;${\rm E}\big[u_ν · u_{ν+1}\big] = 4/15$&nbsp; is valid.
-+ &nbsp;${\rm E}\big[u_ν · u_{ν+2}\big] = 4/15$ is valid.
++ &nbsp;${\rm E}\big[u_ν · u_{ν+2}\big] = 4/15$&nbsp; is valid.
-- &nbsp;${\rm E}\big[u_ν · u_{ν+15}\big] = 4/15$ is valid.
+- &nbsp;${\rm E}\big[u_ν · u_{ν+15}\big] = 4/15$&nbsp; is valid.
 + The PACF values &nbsp;$φ_u(λ = 1)$, ... , $φ_u(λ = 14)$&nbsp; are all equal.

← Older revision		Revision as of 13:01, 10 December 2021
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← Older revision		Revision as of 10:24, 1 December 2021
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−	[[Category:Modulation Methods: Exercises\|^5.3 ~~Spreizfolgen für~~ CDMA^]]	+	[[Category:Modulation Methods: Exercises\|^5.3 Spread Sequences for CDMA^]]

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← Older revision		Revision as of 12:49, 23 March 2021
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−	[[Category:~~Aufgaben zu Modulationsverfahren~~\|^5.3 Spreizfolgen für CDMA^]]	+	[[Category:Modulation Methods: Exercises\|^5.3 Spreizfolgen für CDMA^]]

@@ Line 6: / Line 6: @@
 Mit einem rückgekoppelten Schieberegister vom Grad &nbsp;$G$&nbsp; lässt sich eine Spreizfolge &nbsp;$〈c_ν〉$&nbsp; mit der (maximalen) Periodenlänge &nbsp;$P = 2^G - 1$&nbsp; erzeugen, wenn die Rückführungskoeffizienten (Anzapfungen) richtig gewählt sind.
-In dieser Aufgabe wird der in der linken Grafik von &nbsp;[[Modulationsverfahren/Spreizfolgen_für_CDMA#Pseudo.E2.80.93Noise.E2.80.93Folgen_maximaler_L.C3.A4nge|$\text{Beispiel 1}$]]&nbsp; im Theorieteil dargestelle PN–Generator mit der Oktalkennung &nbsp;$(31)$&nbsp; betrachtet, der wegen &nbsp;$G = 4$&nbsp; eine Folge mit der Periodenlänge &nbsp;$P = 15$&nbsp; liefert.
+In dieser Aufgabe wird der in der linken Grafik von &nbsp;[[Modulation_Methods/Spreizfolgen_für_CDMA#Pseudo.E2.80.93Noise.E2.80.93Folgen_maximaler_L.C3.A4nge|$\text{Beispiel 1}$]]&nbsp; im Theorieteil dargestelle PN–Generator mit der Oktalkennung &nbsp;$(31)$&nbsp; betrachtet, der wegen &nbsp;$G = 4$&nbsp; eine Folge mit der Periodenlänge &nbsp;$P = 15$&nbsp; liefert.
 In der Grafik zu dieser Aufgabe sind die unipolare Folge &nbsp;$〈u_ν〉$&nbsp; mit &nbsp;$u_ν ∈ \{0, 1\}$&nbsp; und daraus abgeleitete zyklische Verschiebungen &nbsp;$〈u_{ν+λ}〉$&nbsp; dargestellt, wobei der Verschiebungsparameter &nbsp;$λ$&nbsp; Werte zwischen &nbsp;$1$&nbsp; und &nbsp;$15$&nbsp; annimmt.&nbsp; Eine Verschiebung um &nbsp;$λ$&nbsp; bedeutet dabei absolut einen Versatz um &nbsp;$λ · T_c$.&nbsp; Hierbei bezeichnet &nbsp;$T_c$&nbsp; die Chipdauer.
@@ Line 23: / Line 23: @@
 ''Hinweis:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulationsverfahren/Spreizfolgen_für_CDMA|Spreizfolgen für CDMA]].
+*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Spreizfolgen_für_CDMA|Spreizfolgen für CDMA]].