Modulation Methods/Spreading Sequences for CDMA - Revision history

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	For ergodic processes,  these are obtained as the  [[Signal_Representation/~~Fourier_Transform_and_its_Inverse~~#Fourier_transform\|$\text{Fourier transforms}$]]  of ACF and CCF:		For ergodic processes,  these are obtained as the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#Fourier_transform\|$\text{Fourier transforms}$]]  of ACF and CCF:
	:$${\it \Phi}_{xx}(f) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}\varphi_{xx}(\tau)\hspace{0.05cm} ,\hspace{0.3cm} {\it \Phi}_{xy}(f) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}\varphi_{xy}(\tau)\hspace{0.05cm}.$$		:$${\it \Phi}_{xx}(f) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}\varphi_{xx}(\tau)\hspace{0.05cm} ,\hspace{0.3cm} {\it \Phi}_{xy}(f) \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}\varphi_{xy}(\tau)\hspace{0.05cm}.$$

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	<br>		<br>
	For periodic signals,  the boundary transition can be omitted from the ACF and the CCF calculations,  and the period duration  $T_0$  (which must be the same for both signals) is used to obtain:		For periodic signals,  the boundary transition can be omitted from the ACF and the CCF calculations,  and the period duration  $T_0$  (which must be the same for both signals) is used to obtain:
	:$$\begin{align*}\varphi_{x}(\tau) & = \frac{1}{T_{\rm 0}}\cdot\int^{T_0}_{0}x(t)\cdot x(t+\tau)\,\,\rm d \it t\hspace{0.05cm} ,\\		:$$\begin{align}\varphi_{x}(\tau) & = \frac{1}{T_{\rm 0}}\cdot\int^{T_0}_{0}x(t)\cdot x(t+\tau)\,\,\rm d \it t\hspace{0.05cm} ,\\\varphi_{xy}(\tau) & = \frac{1}{T_{\rm 0}}\cdot\int^{T_0}_{0}x(t)\cdot y(t+\tau)\,\,\rm d \it t\hspace{0.05cm}.\end{align}$$
	\varphi_{xy}(\tau) & = \frac{1}{T_{\rm 0}}\cdot\int^{T_0}_{0}x(t)\cdot y(t+\tau)\,\,\rm d \it t\hspace{0.05cm}.\end{align*}$$
	In this case,  the ACF is also a periodic function,  and it is called the  "periodic auto-correlation function"  $\rm (PACF)$.  This shows the following:		In this case,  the ACF is also a periodic function,  and it is called the  "periodic auto-correlation function"  $\rm (PACF)$.  This shows the following:
	:$$\varphi_{x}(\pm T_0) = \varphi_{x}(\pm 2T_0) =\text{ ...} = \varphi_{x}(0)		:$$\varphi_{x}(\pm T_0) = \varphi_{x}(\pm 2T_0) =\text{ ...} = \varphi_{x}(0)\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$
	We now apply the above calculation rule to the spreading signal		We now apply the above calculation rule to the spreading signal
	:$$ c(t) = \sum\limits^{+\infty}_{\nu = -\infty}c_\nu\cdot g_c(t - \nu \cdot T_c)$$		:$$ c(t) = \sum\limits^{+\infty}_{\nu = -\infty}c_\nu\cdot g_c(t - \nu \cdot T_c)$$
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	Gold codes use this principle.  The rule for generating a  »'''Gold code family'''«   $\rm (GCF)$  is where a  "+"  denotes a modulo-2 addition:		Gold codes use this principle.  The rule for generating a  »'''Gold code family'''«   $\rm (GCF)$  is where a  "+"  denotes a modulo-2 addition:
	:$${\rm GCF}(C_1,\ C_2) = \{ C_1,\ C_2,\ C_1 + C_2,\ C_1 + D \cdot C_2,\ C_1 + D^2 \cdot C_2, \ \text{...} \ ,\		:$${\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}.$$
	C_1 + D^{P-1} \cdot C_2 \} \hspace{0.05cm}.$$
	[[File:EN_Mod_T_5_3_S6.png\|right\|frame\|Gold code generator  $(51, 75)$  of degree  $G = 5$]]		[[File:EN_Mod_T_5_3_S6.png\|right\|frame\|Gold code generator  $(51, 75)$  of degree  $G = 5$]]

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	:$${\mathbf{H}_{2}} = \left[ \begin{array}{ccc} +1 & +1 \\ +1 & -1 \end{array} \right] \hspace{0.5cm} \Rightarrow \hspace{0.5cm}{\mathbf{H}_{2J}} = \left[ \begin{array}{ccc} \mathbf{H}_J & \mathbf{H}_J \\ \mathbf{H}_J & -\mathbf{H}_J \end{array} \right] $$		:$${\mathbf{H}_{2}} = \left[ \begin{array}{ccc} +1 & +1 \\ +1 & -1 \end{array} \right] \hspace{0.5cm} \Rightarrow \hspace{0.5cm}{\mathbf{H}_{2J}} = \left[ \begin{array}{ccc} \mathbf{H}_J & \mathbf{H}_J \\ \mathbf{H}_J & -\mathbf{H}_J \end{array} \right] $$
	:$$\hspace{0.5cm} \Rightarrow \hspace{0.5cm}		:$$\hspace{0.5cm} \Rightarrow \hspace{0.5cm}{\mathbf{H}_{4}} = \left[ \begin{array}{cccc} +1 & +1 & +1 & +1 \\ +1 & -1 & +1 & -1 \\+1 & +1 & -1 & -1 \\+1 & -1 & -1 & +1 \end{array} \right] .$$
	{\mathbf{H}_{4}} = \left[ \begin{array}{cccc} +1 & +1 & +1 & +1 \\ +1 & -1 & +1 & -1 \\
	+1 & +1 & -1 & -1 \\+1 & -1 & -1 & +1 \end{array} \right] .$$
	The  $J$  rows of such a matrix describe the  $J$  possible spreading sequences  $($each of length  $J)$,  which are numbered  $w_0(t)$  to  $w_{J–1}(t)$.		The  $J$  rows of such a matrix describe the  $J$  possible spreading sequences  $($each of length  $J)$,  which are numbered  $w_0(t)$  to  $w_{J–1}(t)$.

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	{{Display}}		{{Display}}
			[[de:Modulationsverfahren/Spreizfolgen_für_CDMA]]

Guenter at 16:37, 23 January 2023

2023-01-23T16:37:43Z

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	The table shows that the number of PN sequences of maximum length is very small. For the degree  $G = 5$   ⇒   $P = 31$,  for example,  there are just six M-sequences,  namely the PN generators with the octal identifiers  $(45), (51), (57), (67), (73)$  and  $(75)$.		The table shows that the number of PN sequences of maximum length is very small. For the degree  $G = 5$   ⇒   $P = 31$,  for example,  there are just six M-sequences,  namely the PN generators with the octal identifiers  $(45), (51), (57), (67), (73)$  and  $(75)$.

	[[File:EN_Mod_T_5_3_S5neu_v2.png\|right\|frame\| M-sequence code family – powerfullness ~~'''KORREKTUR: Number M-sequences'''~~]]		[[File:EN_Mod_T_5_3_S5neu_v2.png\|right\|frame\| M-sequence code family – powerfullness]]

	Furthermore, the term  »'''powerfulness'''«  of the code family can also be found in the literature.		Furthermore, the term  »'''powerfulness'''«  of the code family can also be found in the literature.

Guenter at 16:28, 23 January 2023

2023-01-23T16:28:30Z

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	The table shows that the number of PN sequences of maximum length is very small. For the degree  $G = 5$   ⇒   $P = 31$,  for example,  there are just six M-sequences,  namely the PN generators with the octal identifiers  $(45), (51), (57), (67), (73)$  and  $(75)$.		The table shows that the number of PN sequences of maximum length is very small. For the degree  $G = 5$   ⇒   $P = 31$,  for example,  there are just six M-sequences,  namely the PN generators with the octal identifiers  $(45), (51), (57), (67), (73)$  and  $(75)$.

	[[File:~~EN_Mod_T_5_3_S5neu~~.png\|right\|frame\| M-sequence code family – powerfullness '''KORREKTUR: Number M-sequences''']]		[[File:EN_Mod_T_5_3_S5neu_v2.png\|right\|frame\| M-sequence code family – powerfullness '''KORREKTUR: Number M-sequences''']]

	Furthermore, the term  »'''powerfulness'''«  of the code family can also be found in the literature.		Furthermore, the term  »'''powerfulness'''«  of the code family can also be found in the literature.

Hwang at 15:36, 11 January 2023

2023-01-11T15:36:54Z

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

Hwang at 15:31, 11 January 2023

2023-01-11T15:31:41Z

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	To obtain larger code families than with M-sequences,  the requirements for the periodic cross-correlation function  $\rm (PCCF)$  must be weakened.  With this restriction,  code families with a much greater powerfulness are obtained,  so that more CDMA subscribers can also be supplied.		To obtain larger code families than with M-sequences,  the requirements for the periodic cross-correlation function  $\rm (PCCF)$  must be weakened.  With this restriction,  code families with a much greater powerfulness are obtained,  so that more CDMA subscribers can also be supplied.

	Gold codes use this principle.  The rule for generating a  »'''Gold code family'''«   $\rm (GCF)$  is where a  „+”  denotes a modulo-2 addition:		Gold codes use this principle.  The rule for generating a  »'''Gold code family'''«   $\rm (GCF)$  is where a  "+"  denotes a modulo-2 addition:
	:$${\rm GCF}(C_1,\ C_2) = \{ C_1,\ C_2,\ C_1 + C_2,\ C_1 + D \cdot C_2,\ C_1 + D^2 \cdot C_2, \ \text{...} \ ,\		:$${\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}.$$		C_1 + D^{P-1} \cdot C_2 \} \hspace{0.05cm}.$$

Hwang at 15:29, 11 January 2023

2023-01-11T15:29:52Z

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	The table shows that the number of PN sequences of maximum length is very small. For the degree  $G = 5$   ⇒   $P = 31$,  for example,  there are just six M-sequences,  namely the PN generators with the octal identifiers  $(45), (51), (57), (67), (73)$  and  $(75)$.		The table shows that the number of PN sequences of maximum length is very small. For the degree  $G = 5$   ⇒   $P = 31$,  for example,  there are just six M-sequences,  namely the PN generators with the octal identifiers  $(45), (51), (57), (67), (73)$  and  $(75)$.

	[[File:EN_Mod_T_5_3_S5neu.png\|right\|frame\| M-sequence code family – powerfullness '''KORREKTUR: ~~number~~ M-sequences''']]		[[File:EN_Mod_T_5_3_S5neu.png\|right\|frame\| M-sequence code family – powerfullness '''KORREKTUR: Number M-sequences''']]

	Furthermore, the term  »'''powerfulness'''«  of the code family can also be found in the literature.		Furthermore, the term  »'''powerfulness'''«  of the code family can also be found in the literature.

Hwang at 15:28, 11 January 2023

2023-01-11T15:28:46Z

← Older revision		Revision as of 17:28, 11 January 2023
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	The table shows that the number of PN sequences of maximum length is very small. For the degree  $G = 5$   ⇒   $P = 31$,  for example,  there are just six M-sequences,  namely the PN generators with the octal identifiers  $(45), (51), (57), (67), (73)$  and  $(75)$.		The table shows that the number of PN sequences of maximum length is very small. For the degree  $G = 5$   ⇒   $P = 31$,  for example,  there are just six M-sequences,  namely the PN generators with the octal identifiers  $(45), (51), (57), (67), (73)$  and  $(75)$.

	[[File:EN_Mod_T_5_3_S5neu.png\|right\|frame\| M-sequence code family – powerfullness]]		[[File:EN_Mod_T_5_3_S5neu.png\|right\|frame\| M-sequence code family – powerfullness '''KORREKTUR: number M-sequences''']]

	Furthermore, the term  »'''powerfulness'''«  of the code family can also be found in the literature.		Furthermore, the term  »'''powerfulness'''«  of the code family can also be found in the literature.

Hwang at 15:28, 11 January 2023

2023-01-11T15:28:00Z

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	:$$G_{\rm 2}(D) = D^4 \cdot (D^{-4} + D^{-3} +1) =D^4 + D +1 \hspace{0.05cm}.$$		:$$G_{\rm 2}(D) = D^4 \cdot (D^{-4} + D^{-3} +1) =D^4 + D +1 \hspace{0.05cm}.$$

	Since both  $G_1(D)$  and  $G_{\rm 2}(D)$  are  [[Channel_Coding/~~Erweiterungskörper~~#~~Bin.C3.A4re_Erweiterungsk.C3.B6rper_~~.E2.80.~~93_Primitive_Polynome~~\|$\text{primitive generator polynomials}$]]    (the proof for this is not easy),  both output sequences have the maximum period length  $P = 15$  for  $G = 4$.		Since both  $G_1(D)$  and  $G_{\rm 2}(D)$  are  [[Channel_Coding/Extension_Field#Binary_extension_fields_.E2.80.93_Primitive_polynomials\|$\text{primitive generator polynomials}$]]    (the proof for this is not easy),  both output sequences have the maximum period length  $P = 15$  for  $G = 4$.

	As shown in  [[Aufgaben:Exercise_5.3:_PACF_of_PN_Sequences\|"Exercise 5.3"]],  the PACF in unipolar representation   ⇒   $c_ν ∈ \{0, 1\}$  results in		As shown in  [[Aufgaben:Exercise_5.3:_PACF_of_PN_Sequences\|"Exercise 5.3"]],  the PACF in unipolar representation   ⇒   $c_ν ∈ \{0, 1\}$  results in

Hwang at 15:27, 11 January 2023

2023-01-11T15:27:13Z

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