View source for Modulation Methods/Error Probability of Direct-Sequence Spread Spectrum Modulation

 
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==The IS-95 CDMA system==
<br>
The properties of PN modulation are now given using the example of the American mobile radio standard &nbsp;[https://en.wikipedia.org/wiki/CdmaOne $\text{IS–95}$],&nbsp; which has resulted from the work of &nbsp;[https://en.wikipedia.org/wiki/Qualcomm $\text{Qualcomm Inc.}$]&nbsp; and in particular of &nbsp;[https://en.wikipedia.org/wiki/Andrew_Viterbi $\text{Andrew J. Viterbi}$].&nbsp;&nbsp; In a somewhat simplified representation – without convolutional encoder,&nbsp; interleaver and de-interleaver as well as the Viterbi decoder at the receiver - the following block diagram results.&nbsp; The following statements are valid:

[[File:EN_Mod_T_5_4_S1_v2.png |right|frame| Considered block diagram for this chapter]]

*The spreading signal &nbsp;$c(t)$&nbsp; causes &nbsp;"band spreading"&nbsp; by the spreading factor &nbsp;$J$,&nbsp; with&nbsp;[[Modulation_Methods/Spreading_Sequences_for_CDMA#Walsh_functions|$\text{Walsh functions}$]]&nbsp; or &nbsp;[[Modulation_Methods/Spreading_Sequences_for_CDMA#Pseudo-noise_sequences_of_maximum_length|$\text{M–Sequences}$]].&nbsp;  &nbsp;

*"Band compression"&nbsp; at the receiver uses the same spreading sequence in a phase-synchronous manner.

*The additional &nbsp;$±1$ signal &nbsp;$w(t)$&nbsp; provides additional &nbsp;"scrambling",&nbsp; but does not cause further band spreading.&nbsp; 

*The rectangular duration of &nbsp;$w(t)$&nbsp; is exactly as large &nbsp;$(T_c)$&nbsp; as the rectangular duration of &nbsp;$c(t)$.&nbsp; &nbsp;$T_c$&nbsp; is called the&nbsp; "chip duration". 

*Without band spreading and scrambling&nbsp; $($or with &nbsp;$J = 1)$,&nbsp; the transmission chain corresponds to &nbsp;[[Modulation_Methods/Lineare_digitale_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying|$\text{BPSK modulation}$]].&nbsp; 

*The matched filter is implemented by the &nbsp;[[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Optimal_binary_receiver_.E2.80.93_.22Integrate_.26_Dump.22_realization|$\text{Integrate & Dump}$]]&nbsp; variant,&nbsp; so it is an optimal system.

*With &nbsp;$H_{\rm K}(f) = 1$,&nbsp; the &nbsp;[[Modulation_Methods/Quality_Criteria#Some_remarks_on_the_AWGN_channel_model|$\text{AWGN channel model}$]]&nbsp; is obtained with the Gaussian distributed noise signal &nbsp;$n(t)$&nbsp; and the AWGN parameter &nbsp;$E_{\rm B}/N_0$.&nbsp; 

*The additional interference component&nbsp; $i(t)$&nbsp; summarizes the &nbsp;"interferences from other users".

*In the case of a &nbsp;[[Examples_of_Communication_Systems/General_Description_of_UMTS#Properties_of_the_UMTS_radio_channel|$\text{multipath channel}$]] &nbsp; (one main path and one or more secondary paths),&nbsp; the resulting intersymbol interference can be reduced by using a&nbsp;[[Modulation_Methods/Error_Probability_of_Direct-Sequence_Spread_Spectrum_Modulation#Examinations_of_the_rake_receiver|$\text{rake receiver}$]].&nbsp; 


==System configurations for minimum error probability==
<br>
[[File:EN_Mod_T_5_4_S2.png|right|frame| Bit error probability curves for the AWGN channel]]
The diagram shows the bit error probability at BPSK as a function of the logarithmized AWGN parameter &nbsp;$E_{\rm B}/N_0$ &nbsp; (signal energy per bit related to the noise power density) &nbsp; as a solid blue curve.&nbsp; With the&nbsp; [[Applets:Complementary_Gaussian_Error_Functions|$\text{complementary Gaussian error function}$]]&nbsp; ${\rm Q}(x)$:
:$$p_{\rm B} = {\rm Q} \left ( \sqrt{{2\cdot E_{\rm B}}/{N_{\rm 0}}}\hspace{0.05cm}\right ) \hspace{0.05cm}.$$

For example,&nbsp; with &nbsp;$10 · \lg \ (E_{\rm B}/N_0) = 8 \ \rm dB$&nbsp; the bit error probability is &nbsp;$p_{\rm B} \approx \rm 2 · 10^{–4}$.&nbsp; The same minimum value is provided by a system simulation&nbsp; (red dots)&nbsp; under the following conditions:
#&nbsp; PN modulation,&nbsp; whether with an&nbsp; "M-sequence"&nbsp; or with a&nbsp; "Walsh function",&nbsp; at any spreading factor&nbsp; $J$,&nbsp; if only one user is active.
#&nbsp; Synchronous CDMA operation with Walsh functions,&nbsp; even if other users&nbsp; $($maximum&nbsp; $J -1)$&nbsp; are active in the same frequency band.
#&nbsp; The rake receiver can be dispensed with,&nbsp; since no intersymbol interference occurs with the AWGN channel.

==Two users with M-sequence spreading==
<br>
We consider the interfering influence of a second&nbsp; (clock-synchronous)&nbsp; user on the error probability of user 1.

[[File:EN_Mod_T_5_4_S3a.png|right|frame| Bit error probability with band spreading&nbsp; $(J = 15)$&nbsp; with M-sequences]]

*Spreading is done using M-sequences which,&nbsp; unlike Walsh functions,&nbsp; are not orthogonal to each other. 
*The spreading factor in each case is &nbsp;$J = 15$.&nbsp; Let the octal identifiers of the two PN spreading sequences involved be &nbsp;$(23)$&nbsp; and &nbsp;$(31)$. 


For example,&nbsp; from this diagram can be read:

*With only one user,&nbsp; it results in the blue solid curve.&nbsp; 
*The second user increases the error probability enormously:&nbsp;  At&nbsp; $10 · \lg \ (E_{\rm B}/N_0) = 8 \ \rm dB$&nbsp; from &nbsp;$p_{\rm B} = 2· 10^{–4} = 0.02\%$&nbsp;  to &nbsp;$p_{\rm B} =1.5\%$&nbsp;  $($brown markings, &nbsp;$τ = 0)$. 
*By phase-shifting the PN sequences with respect to each other by multiples of the chip duration,&nbsp; one can achieve large improvements.&nbsp; 
*For example,&nbsp; if the PN sequence &nbsp;$(31)$&nbsp; of the interfering user is shifted by &nbsp;$τ = 2T_c$&nbsp; to the right&nbsp; (red markings),&nbsp; instead of &nbsp;$p_{\rm B} =1.5\%$&nbsp; only &nbsp;$p_{\rm B} = 0.034\%$&nbsp; is obtained.
<br clear=all>
[[File:P_ID1896__Mod_T_5_4_S3b_ganz_neu.png |right|frame| PCCF of the PN sequences &nbsp;$(23)$&nbsp; and &nbsp;$(31)$]] 

<br>The results can be understood by considering the periodic cross-correlation function&nbsp; $\rm (PCCF)$ &nbsp; $φ_\text{23, 31}(λ \cdot T_c)$&nbsp; between the sequences &nbsp;$(23)$&nbsp; and &nbsp;$(31)$.&nbsp; 
*The smaller the PCCF magnitude at &nbsp;$\tau$,&nbsp; the smaller &nbsp;$p_{\rm B}$.
*Thus,&nbsp; one could also shift the second PN sequence to the right by six or eight chip durations or to the left by five,&nbsp; six,&nbsp; or seven chip durations&nbsp; (red dots). 
*In all these cases,&nbsp;  the PCCF magnitude &nbsp;$|φ_\text{23, 31}(2 T_c)| = 1/15$&nbsp; is smaller compared to
:#$\ \ |φ_\text{23, 31}(0)| = 7/15$&nbsp; (ochre dots), 
:#$\ \ |φ_\text{23, 31}(3 T_c)| = 5/15$&nbsp; (green dots), and
:#$\ \ |φ_\text{23, 31}(T_c)| = 3/15$&nbsp; (purple dots).
<br clear=all>
==Asynchronous CDMA operation with Walsh functions==
<br>
In the section &nbsp;[[Modulation_Methods/Error_Probability_of_Direct-Sequence_Spread_Spectrum_Modulation#System_configurations_for_minimum_error_probability|"System configurations for minimum error probability"]],&nbsp; it was shown that when orthogonal Walsh functions are used,&nbsp; the bit error probability &nbsp;$p_{\rm B}$&nbsp; is not changed compared with the BPSK comparison curve&nbsp; (system without band spreading)&nbsp; even in the presence of other CDMA subscribers,&nbsp; as long as all subscribers operate synchronously.

[[File:EN_Mod_T_5_4_S4neu.png|right|frame| Bit error probability for asynchronous CDMA operation with Walsh spreading functions ]]
 
*In mobile communications, this condition is generally fulfilled for the downlink&nbsp; (the transmitter is a single base station),
*but not in the uplink&nbsp; (transmitters are many mobile terminals). 


The diagram shows bit error probability curves for the most unfavorable phase offset of the spreading sequences of the considered user and the interfering user,&nbsp; each with spreading factor &nbsp;$J = 16$.&nbsp; The first user always used Walsh function no.&nbsp; $15$. 

The result can be summarized as follows:
*If the second participant uses the Walsh function no.&nbsp; $1$,&nbsp; a phase shift does not have a negative effect,&nbsp; since for all &nbsp;$λ$–values the PCCF &nbsp;$φ_\text{1, 15}(λ · T_c) = 0$.&nbsp; 
*On the other hand,&nbsp; if the second participant uses Walsh function no.&nbsp; $14$&nbsp; $($or any other with identification number greater than/equal to&nbsp; $8)$, a phase shift by one chip duration results in a huge degradation.
*Although &nbsp;$φ_\text{14, 15}(\lambda=0) = 0$&nbsp; holds,&nbsp; for &nbsp;$λ = 1$&nbsp; this PCCF has a very large value with &nbsp;$φ_\text{14, 15}(T_c) = 3/4$.&nbsp; 


==Bit error probability for the two-way channel==
<br>
For the rest of this chapter&nbsp; "Error Probability of Direct-Sequence Spread Spectrum Modulation"&nbsp; we assume:
*Walsh functions are used for band spreading. &nbsp; The spreading factor in each case is &nbsp;$J = 16$.&nbsp; In particular, we consider the functions:
:$$ \langle w_\nu^{\hspace{0.12cm}(1)}\rangle  =  
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm} 
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm} 
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.05cm},$$
:$$ \langle w_\nu^{\hspace{0.12cm}(2)}\rangle   =  
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{+\hspace{-0.05cm}1} \hspace{0.15cm} 
{-\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{+\hspace{-0.05cm}1} \hspace{0.15cm}
{-\hspace{-0.05cm}1}\hspace{0.15cm} 
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{+\hspace{-0.05cm}1} \hspace{0.15cm}
{-\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{+\hspace{-0.05cm}1} \hspace{0.15cm}
{-\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.05cm},$$
:$$ \langle w_\nu^{(12)}\rangle  =  
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{+\hspace{-0.05cm}1} \hspace{0.15cm} 
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{+\hspace{-0.05cm}1} \hspace{0.15cm}
{-\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{-\hspace{-0.05cm}1}\hspace{0.15cm} 
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{-\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{-\hspace{-0.05cm}1}\hspace{0.15cm}
{-\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{+\hspace{-0.05cm}1} \hspace{0.15cm}
{+\hspace{-0.05cm}1}\hspace{0.15cm}
{+\hspace{-0.05cm}1} \hspace{0.05cm}.$$

*Only one subscriber is active at a time &nbsp; &rArr; &nbsp; interference from other users does not occur.
*The channel contains a multipath component in addition to the AWGN noise: &nbsp; The main path is attenuated by a factor of &nbsp;$0.8$.&nbsp;
*In addition,&nbsp; there is an echo at a distance of &nbsp;$2T_c$&nbsp; with a weight of &nbsp;$0.6$.&nbsp; The channel impulse response in this case is:
[[File:EN_Mod_T_5_4_S5a_neu.png |right|frame| CDMA error probability with two-way channel]]
:$$h_{\rm K}(t) = 0.8 \cdot \delta (t) +   0.6 \cdot \delta (t - 2 T_c).$$

The diagram shows bit error probability curves.&nbsp; These show:
*The Walsh function no.&nbsp; $2$ &nbsp;(brown dots)&nbsp; is conceivably unsuitable for the two-way channel defined above,&nbsp; since here the band spreading signal &nbsp;$b(t)$&nbsp; almost cancels itself out due to the echo at the distance &nbsp;$2T_c$.&nbsp;&nbsp; This can also be seen from the PACF value &nbsp;$φ_{22}(τ = 2T_c) = -1$. 
*The Walsh function no.&nbsp; $1$ &nbsp;(yellow dots)&nbsp; is very well suited for this channel.&nbsp; The echo signal is constructively superimposed on the signal on the main path and &nbsp;$b(t)$&nbsp; is almost doubled.&nbsp; The good result can be explained by the PACF value &nbsp;$φ_{11}(τ = 2T_c) = +1$.&nbsp;
*With Walsh function no.&nbsp; $12$  &nbsp;(red dots),&nbsp; the constructive and destructive superpositions almost balance out,&nbsp; so that the error probability is approximately on the BPSK curve.&nbsp; All other Walsh functions also lie between the brown and yellow boundary curves.




{{BlaueBox||TEXT=
$\text{Conclusion:}$&nbsp;
*The user with Walsh function no.&nbsp; $1$&nbsp; has significantly better conditions than the user with Walsh function no.&nbsp; $2$&nbsp; at the considered channel with echo delay &nbsp;$2 \cdot T_c$.&nbsp;
*For a channel with echo delay &nbsp;$T_c$&nbsp; the user with Walsh function no.&nbsp; $2$&nbsp; would have significantly better conditions than the user with Walsh function no.&nbsp; $1$.
* However,&nbsp; since a network operator must provide the same conditions for all users with any channel, this configuration is unsuitable for practical operation. 
*In the following,&nbsp; it is shown how to provide approximately equal conditions for all subscribers.}} 


==Influence of additional scrambling of the spreading sequence==
<br>
[[File:EN_Mod_T_5_4_S5b_neu.png|right|frame | CDMA error probability with two-way channel,&nbsp; additional scrambling ]]
One way to equalize the quality for the individual subscribers even for the two-way channel is to add scrambling with &nbsp;$w(t)$&nbsp; according to the &nbsp;[[Modulation_Methods/Error_Probability_of_Direct-Sequence_Spread_Spectrum_Modulation#The_IS-95_CDMA_system|$\text{block diagram}$]]&nbsp; shown at the front.&nbsp; For the diagram shown on the right it is assumed:
*The participating subscribers use different Walsh functions for band spreading,&nbsp; all with spreading factor &nbsp;$J = 16$.
*Each user is additionally scrambled by an M-sequence of period length &nbsp;$P = 63$&nbsp; $($register degree &nbsp;$G = 6)$.&nbsp; 
*The plotted points apply to Walsh function no.&nbsp; $12$&nbsp; as spreading signal &nbsp;$c(t)$&nbsp; and scrambling &nbsp;$w(t)$&nbsp; by the M-sequence with octal identifier &nbsp;$\rm (163)_{octal}$. 


Compared with the pure AWGN channel,&nbsp; the two-way channel considered with coefficients &nbsp;$0.8$&nbsp; and &nbsp;$0.6$&nbsp; gives a degradation of about &nbsp;$2 \ \rm dB$ to $3 \ \rm dB$ &nbsp; &rArr; &nbsp; horizontal distance between the plotted points and the blue comparison curve.

'''Note:''' &nbsp; The results for other Walsh functions, e.g. no.&nbsp; $1$&nbsp; or no.&nbsp; $2$&nbsp; differ only insignificantly from this diagram&nbsp; $($valid for Walsh function no.&nbsp; $12)$&nbsp; within the character accuracy.
<br clear =all>
{{BlaueBox||TEXT=
$\text{Conclusion:}$&nbsp; 
*The additional scrambling with &nbsp;$w(t)$&nbsp; serves only to create equal conditions for all participants in a channel with echo distortions.
*Thus,&nbsp; $w(t)$&nbsp; does not cause any additional band spreading.


All participants thus experience the same degradation due to an echo,&nbsp; compared to the ideal channel.&nbsp; Otherwise&nbsp; (without this scrambling),&nbsp; the customers with poor conditions would complain to the operator and possibly claim recourse,&nbsp; while the other customers would accept the constructive echo overlay,&nbsp; which is favorable to them,&nbsp; perhaps delightedly but surely tacitly.}} 


==Examinations of the rake receiver==
<br>
Finally,&nbsp; let us consider the improvement by using a rake receiver,&nbsp; which is illustrated by the following diagram.&nbsp; For example,&nbsp; for &nbsp;$10 · \lg \ E_{\rm B}/N_0 = 8 \ \rm dB$ &nbsp; this reduces the bit error probability from &nbsp;$\rm 6 · 10^{–3}$&nbsp; to &nbsp;$\rm 5 · 10^{-4}$. 

[[File:EN_Mod_T_5_4_S6a_neu.png|right|frame  |CDMA error probability curves with and without rake receiver]]

Note that the exact same assumptions apply to this diagram as to the&nbsp; [[Modulation_Methods/Error_Probability_of_Direct-Sequence_Spread_Spectrum_Modulation#Influence_of_additional_scrambling_of_the_spreading_sequence|$\text{diagram in the last section}$]]: 
*Band spreading with &nbsp;$J =16$&nbsp; and Walsh function no.&nbsp; $12$, 
*additional scrambling by the M-sequence no. &nbsp;$\rm (163)_{octal}$, 
*two-way channel with &nbsp; $h_{\rm K}(t) = 0.8 · δ(t) + 0.6 · δ(t - 2T_c)$,
*SNR: &nbsp; $10 · \lg \ E_{\rm B}/N_0 = 8 \ \rm dB$. 


The reason for this improvement is the smaller variance &nbsp;${σ_d}^2$&nbsp; of the detection signal samples,&nbsp; as can be seen from the probability density functions &nbsp;$f_d(d)$&nbsp; shown on the right.&nbsp; Here, &nbsp;${σ_d}^2 = {σ_{\rm I}}^2 + {σ_{\rm N}}^2$&nbsp; is composed of two components:
*The fraction &nbsp;${σ_{\rm N}}^2$&nbsp; of the AWGN noise &nbsp;$n(t)$&nbsp; in the total variance &nbsp;${σ_d}^2$&nbsp; depends solely on the abscissa &nbsp;$(E_{\rm B}/N_0)$&nbsp; and is the same with and without rake.
*The smaller &nbsp;$σ_d^2$&nbsp; with rake is solely due to the fact that the rake receiver combats the intersymbol interference.&nbsp; Thus, &nbsp;${σ_{\rm I}}^2$&nbsp; is not zero for the (red) receiver with rake, but it is significantly smaller than for the (black) receiver without rake.



==Principle of the rake receiver==
<br>
[[File:EN_Mod_T_5_4_S6b.png |right|frame| Two-way channel and associated rake receiver]] 
The rake principle is illustrated by the diagram.&nbsp; The two-way channel consists of

*the direct path with delay time &nbsp;$τ_0$&nbsp; and weight &nbsp;$h_0$, 
*an echo with delay &nbsp;$τ_1 > τ_0$&nbsp; and weight &nbsp;$h_1$. 


Let both amplitude coefficients be real.&nbsp; As a normalization condition is valid in the following: 
:$${h_0}^2 + {h_1}^2 = 1.$$ 

The task of the rake receiver is to concentrate the signal energies of the two paths&nbsp; (in general: &nbsp; of all paths)&nbsp; to a single point in time.&nbsp; It therefore works like a&nbsp; "rake"&nbsp; for the garden. 

If one applies a Dirac delta pulse at time &nbsp;$t = 0$&nbsp; to the channel input &nbsp;  ⇒ &nbsp;  $s(t) = δ(t)$,&nbsp; there are three Dirac delta pulses at the output of the rake receiver:
:$$b(t) = \big [ h_0 \cdot h_1\cdot \delta (t - 2 \tau_0)\big ] + \big [(h_0^2 +  h_1^2) \cdot \delta (t - \tau_0- \tau_1) \big ]+
\big [h_0 \cdot h_1\cdot \delta (t - 2 \tau_1) \big ] \hspace{0.05cm}.$$
*The signal energy is concentrated in the output signal at time &nbsp;$τ_0 + τ_1$,&nbsp; and two of the total four paths contribute to this.
*The Dirac delta functions at &nbsp;$2τ_0$&nbsp; and &nbsp;$2τ_1$&nbsp; do cause intersymbol interference.
*However,&nbsp;  their weights &nbsp;$(h_0 · h_1)$&nbsp; are much smaller than the weight of the main path &nbsp;$({h_0}^2 + {h_1}^2)$. 


{{GraueBox|TEXT=
$\text{Example 1:}$&nbsp; 
*With parameter values &nbsp;$h_0 = 0.8$&nbsp; and &nbsp;$h_1 = 0.6$,&nbsp; the main path&nbsp;  $($with weight &nbsp;$h_0)$&nbsp; contains only &nbsp;$\rm 0.8^2/(0.8^2 + 0.6^2) = 64\%$&nbsp; of the total signal energy.
*With rake receiver and the same weights the above equation is
:$$b(t) = \big [ 0.48 \cdot \delta (t - 2 \tau_0)\big ] + \big [1.0 \cdot \delta (t - \tau_0- \tau_1)\big ] +
\big [ 0.48 \cdot \delta (t - 2 \tau_1)\big ] \hspace{0.05cm}.$$
*The contribution of the main path to the total energy is now &nbsp;$\rm 1^2/(1^2 + 0.48^2 + 0.48^2) ≈ 68\%$,&nbsp; slightly larger than without rake. }}


{{BlaueBox|TEXT=
$\text{Conclusion:}$&nbsp; 
*Rake receivers are preferred for implementation in mobile devices,&nbsp; but have limited performance with many active subscribers.
*In a multipath channel with many &nbsp;$(M)$&nbsp; paths,&nbsp; the rake also has&nbsp; $M$&nbsp; fingers.
*The main finger&nbsp; (called the&nbsp; "searcher")&nbsp;  is responsible for identifying and classifying the individual paths of multiple propagation in most mobile radio systems. }}


==Exercises for the chapter==
<br>
[[Aufgaben:Exercise_5.5:_Multi-User_Interference|Exercise 5.5: Multi-User Interference]]

[[Aufgaben:Exercise_5.5Z:_About_the_Rake_Receiver|Exercise 5.5Z: About the Rake Receiver]]



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