Modulation Methods/Spreading Sequences for CDMA - Revision history

Guenter at 16:37, 23 January 2023

2023-01-23T16:37:43Z

Guenter at 16:28, 23 January 2023

2023-01-23T16:28:30Z

Hwang at 15:36, 11 January 2023

2023-01-11T15:36:54Z

Hwang at 15:31, 11 January 2023

2023-01-11T15:31:41Z

Hwang at 15:29, 11 January 2023

2023-01-11T15:29:52Z

Hwang at 15:28, 11 January 2023

2023-01-11T15:28:46Z

Hwang at 15:28, 11 January 2023

2023-01-11T15:28:00Z

Hwang at 15:27, 11 January 2023

2023-01-11T15:27:13Z

Hwang at 15:24, 11 January 2023

2023-01-11T15:24:12Z

Hwang at 15:12, 11 January 2023

2023-01-11T15:12:23Z

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 The table shows that the number of PN sequences of maximum length is very small. For the degree &nbsp;$G = 5$ &nbsp;   ⇒ &nbsp;  $P = 31$,&nbsp; for example,&nbsp; there are just six M-sequences,&nbsp; namely the PN generators with the octal identifiers &nbsp;$(45), (51), (57), (67), (73)$&nbsp; and &nbsp;$(75)$.
-[[File:EN_Mod_T_5_3_S5neu_v2.png|right|frame| M-sequence code family &ndash; powerfullness '''KORREKTUR: Number M-sequences''']]
+[[File:EN_Mod_T_5_3_S5neu_v2.png|right|frame| M-sequence code family &ndash; powerfullness]]
 Furthermore, the term&nbsp; &raquo;'''powerfulness'''&laquo;&nbsp; of the code family can also be found in the literature.

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 The table shows that the number of PN sequences of maximum length is very small. For the degree &nbsp;$G = 5$ &nbsp;   ⇒ &nbsp;  $P = 31$,&nbsp; for example,&nbsp; there are just six M-sequences,&nbsp; namely the PN generators with the octal identifiers &nbsp;$(45), (51), (57), (67), (73)$&nbsp; and &nbsp;$(75)$.
-[[File:EN_Mod_T_5_3_S5neu.png|right|frame| M-sequence code family &ndash; powerfullness '''KORREKTUR: Number M-sequences''']]
+[[File:EN_Mod_T_5_3_S5neu_v2.png|right|frame| M-sequence code family &ndash; powerfullness '''KORREKTUR: Number M-sequences''']]
 Furthermore, the term&nbsp; &raquo;'''powerfulness'''&laquo;&nbsp; of the code family can also be found in the literature.

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-Gold codes are used for scrambling in UMTS,&nbsp; for example,&nbsp; as explained in the chapter&nbsp; [[Examples_of_Communication_Systems/Nachrichtentechnische_Aspekte_von_UMTS#Spreizcodes_und_Verw.C3.BCrfelung_bei_UMTS|"Spreading codes and scrambling in UMTS"]]&nbsp; in the book "Examples of Communication Systems".
+Gold codes are used for scrambling in UMTS,&nbsp; for example,&nbsp; as explained in the chapter&nbsp; [[Examples_of_Communication_Systems/Telecommunications_Aspects_of_UMTS#Spreading_codes_and_scrambling_with_UMTS|"Spreading codes and scrambling with UMTS"]]&nbsp; in the book "Examples of Communication Systems".
 *The two master code shift registers are each constructed with &nbsp;$G = 18$&nbsp; memory cells.
 *This results in the period length &nbsp;$P = 262\hspace{0.05cm} 143$.

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 To obtain larger code families than with M-sequences,&nbsp; the requirements for the periodic cross-correlation function&nbsp; $\rm (PCCF)$&nbsp; must be weakened.&nbsp; With this restriction,&nbsp; code families with a much greater powerfulness are obtained,&nbsp; so that more CDMA subscribers can also be supplied.
-Gold codes use this principle.&nbsp;  The rule for generating a&nbsp; &raquo;'''Gold code family'''&laquo; &nbsp; $\rm (GCF)$&nbsp; is where a&nbsp; „+”&nbsp; denotes a modulo-2 addition:
+Gold codes use this principle.&nbsp;  The rule for generating a&nbsp; &raquo;'''Gold code family'''&laquo; &nbsp; $\rm (GCF)$&nbsp; is where a&nbsp; "+"&nbsp; denotes a modulo-2 addition:
 :$${\rm GCF}(C_1,\ C_2) = \{ C_1,\ C_2,\ C_1 + C_2,\ C_1 + D \cdot C_2,\ C_1 + D^2 \cdot C_2, \ \text{...} \  ,\
 C_1 + D^{P-1} \cdot C_2 \} \hspace{0.05cm}.$$

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 The table shows that the number of PN sequences of maximum length is very small. For the degree &nbsp;$G = 5$ &nbsp;   ⇒ &nbsp;  $P = 31$,&nbsp; for example,&nbsp; there are just six M-sequences,&nbsp; namely the PN generators with the octal identifiers &nbsp;$(45), (51), (57), (67), (73)$&nbsp; and &nbsp;$(75)$.
-[[File:EN_Mod_T_5_3_S5neu.png|right|frame| M-sequence code family &ndash; powerfullness '''KORREKTUR: number M-sequences''']]
+[[File:EN_Mod_T_5_3_S5neu.png|right|frame| M-sequence code family &ndash; powerfullness '''KORREKTUR: Number M-sequences''']]
 Furthermore, the term&nbsp; &raquo;'''powerfulness'''&laquo;&nbsp; of the code family can also be found in the literature.

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 :$$G_{\rm 2}(D) = D^4 \cdot  (D^{-4}  + D^{-3} +1) =D^4 + D +1 \hspace{0.05cm}.$$
-Since both &nbsp;$G_1(D)$&nbsp; and &nbsp;$G_{\rm 2}(D)$&nbsp; are&nbsp; [[Channel_Coding/Erweiterungskörper#Bin.C3.A4re_Erweiterungsk.C3.B6rper_.E2.80.93_Primitive_Polynome|$\text{primitive generator polynomials}$]]&nbsp; &nbsp; (the proof for this is not easy),&nbsp; both output sequences have the maximum period length &nbsp;$P = 15$&nbsp; for &nbsp;$G = 4$.
+Since both &nbsp;$G_1(D)$&nbsp; and &nbsp;$G_{\rm 2}(D)$&nbsp; are&nbsp; [[Channel_Coding/Extension_Field#Binary_extension_fields_.E2.80.93_Primitive_polynomials|$\text{primitive generator polynomials}$]]&nbsp; &nbsp; (the proof for this is not easy),&nbsp; both output sequences have the maximum period length &nbsp;$P = 15$&nbsp; for &nbsp;$G = 4$.
 As shown in  &nbsp;[[Aufgaben:Exercise_5.3:_PACF_of_PN_Sequences|"Exercise 5.3"]],&nbsp; the PACF in unipolar representation &nbsp; ⇒ &nbsp; $c_ν ∈ \{0, 1\}$&nbsp; results in

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 After conversion to bipolar coefficients &nbsp;$(0 \ ⇒  +1$, &nbsp; &nbsp; $1 \ ⇒  -1)$,&nbsp; we obtain:
 :$${\it \varphi}_{c{\rm ,\hspace{0.15cm}bipolar} }(\lambda \cdot T_c) = \left\{ \begin{array}{c}1 \\ -P \\  \end{array} \right. \begin{array}{*{10}c}  \ \  {\rm{for} }\ \lambda = 0, \pm P, \pm 2P, \text{...} \hspace{0.05cm} \\  {\rm{otherwise} } \hspace{0.05cm}.   \\ \end{array}$$
-*One can see from the bottom diagram the desired distinct PACF peaks at intervals of periode &nbsp;$P$.
+*One can see from the bottom diagram the desired distinct PACF peaks at intervals of period &nbsp;$P$.
 *The PCCF properties of PN sequences are less good,&nbsp; as will be shown in chapter &nbsp;[[Modulation_Methods/Fehlerwahrscheinlichkeit_der_PN–Modulation|"Error Probability of Direct-Sequence Spread Spectrum Modulation"]].&nbsp;}}

← Older revision		Revision as of 15:24, 11 January 2023
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−	When selecting code families for a CDMA system,  care must also be taken to ensure that as many ~~code~~ sequences as possible with favorable properties with respect to these three criteria can be found for a given spreading factor  $J = P$,  so that as many subscribers as possible can be supplied simultaneously in the same frequency band.	+	When selecting code families for a CDMA system,  care must also be taken to ensure that as many encoded sequences as possible with favorable properties with respect to these three criteria can be found for a given spreading factor  $J = P$,  so that as many subscribers as possible can be supplied simultaneously in the same frequency band.

	==Pseudo-noise sequences of maximum length==		==Pseudo-noise sequences of maximum length==

← Older revision		Revision as of 15:12, 11 January 2023
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−	For ergodic processes,  these are obtained as the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#~~Fouriertransformation~~\|$\text{Fourier transforms}$]]  of ACF and CCF:	+	For ergodic processes,  these are obtained as the  [[Signal_Representation/Fourier_Transform_and_its_Inverse#Fourier_transform\|$\text{Fourier transforms}$]]  of ACF and CCF:
	:$${\it \Phi}_{xx}(f) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}\varphi_{xx}(\tau)\hspace{0.05cm} ,\hspace{0.3cm} {\it \Phi}_{xy}(f) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}\varphi_{xy}(\tau)\hspace{0.05cm}.$$		:$${\it \Phi}_{xx}(f) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}\varphi_{xx}(\tau)\hspace{0.05cm} ,\hspace{0.3cm} {\it \Phi}_{xy}(f) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}\varphi_{xy}(\tau)\hspace{0.05cm}.$$