SISTEMI DI RADIOCOMUNICAZIONE. ASYNCHRONOUS DIRECT SEQUENCE SPREAD SPECTRUM. Prof. C. Regazzoni. References.
ASYNCHRONOUS DIRECT SEQUENCE SPREAD SPECTRUM
Prof. C. Regazzoni
In the previous session “TECNICHE DI TRASMISSIONE-DATI DIGITALI BASATE SUL CONCETTO DI SPREAD SPECTRUM” a Direct Sequence Spread Spectrum system with two or more users using the same band (as usual in CDMA) but different spreading codes has been partially analyzed.
The users involved in other communications are considered as interference called Crosstalk Interference whose power is related to Process Gain N. By modifying and choosing particular spreading code, their effects can be reduced.
The previous instances are main features of Code Division Multiple Access, which uses the strength of Spread Spectrum techniques to transmit, over the same band and with no temporal limitation (Asynchronous) information provided by several users.
In Multi-user DS-CDMA each transmitter is identified by its PN sequence.
It is possible to detect the information transmitted through a receiver based on a conventional matched filter. The other users, different by the transmitting one, will be considered as Multi User Interference, MUI, generally non Gaussian distributed.
The received signal after the sampling can be considered as the contribution of three components:
First Term is the tx signal
ηis the AWGN
I is the MUI
Usually, real systems are composed by several users, so due to the central limit theorem the overall interference (MUI) can be considered as Gaussian distributed.
This hypothesis is reflected in BER computation where its Gaussian approximation is considered.
Considering (as first case) a very simple situation where (k-1) DS-SS users are Gaussian, their power in the transmission band B is (k-1)P, where P is the transmitted power, considered equal for all users.
Its spectral density is :
The powerof overall noise (MUI and AWGN) is:
With previous data it is possible to obtain the Signal to Noise Ratioat the receiver:
By using a BPSK modulator the transmission bandwidth is and the BER is with Gaussian hypothesis we have:
Where is the Gaussian Error Function
In a single user (k=1) and Gaussian (AWGN) scenario the DS-CDMA has the same performance of a narrow band BPSK modulation.
In the last two slides a particular and usually wrong hypothesis has been considered: the MUI is modeled as white noise. In real case its spectral density is NOT flat, thus the Multi User Interference can not be considered as white noise.
To carry out a deeper analysis, the first and second order statistics of random variables (considered Gaussian) have to be computed.
Being η and I Gaussian distributed,the pdf of ng is Gaussian with zero mean and variance given by:
because I and η are independent random variables with zero mean.
η is the output of the receiver when n(t) (the AWGN) is the input:
whose variance is N0T/4
I, as already explained, is the interference generated by other users.
It can be defined as out at the receiver as:
where k is the phase delay and is the time delay for user k
The symbols have the same probability and
the error probability is :
where is the gaussian pdf of ng
is the multi-user interference I normalized with respect to
is the spectral density of AWGN
is the signal to noise ratio in the transmitter
From the previous formula the error probability becomes:
where the SNR for the considered user at the receiver is:
The variance of I, var(I), or the mean square value of , , has to be computed to obtain the final formula of Pe.. It is sufficient the mean square value because .
Note: time delay and phase delay are uniformly distributed variables in [0,T) and [0,2p) and the transmitted symbols have the same probability.
In the figures an example of asynchronous transmission with delay is presented.
The previous quantities can be defined considering the a-periodic cross-correlation between PN sequence of reference user and PN sequence of user K.
The integrals of slide 10 can be computed as:
for lk such as
Using the previous values the variance of normalized MUI has been reduced to:
This integral can be divided in a summation of all integrals in the interval
By substituting the integral with the summation of integrals and
with the values obtained in slide 12, the variance becomes:
The last formula allow us to conclude:
In the following section these aspects will be analyzed in details
We assume that transmitted signal is corrupted by AWGN in the channel; received signal can be so expressed as:
where s(t) is transmitted signal and n(t) is noise with spectral density .
Optimal receiver is, for definition, receiver which select bit sequence:
Which is the most probable, given received signal r(t) observed during a temporal period 0 t NT+2T, i.e.:
Two consecutive symbols from each user interfere with desired signal.
Receiver knows energies of signals and their transmission delays.
Optimal receiver evaluates the following likelihood function:
Where b represents the data sequences received from K users
doesn’t depend on K, so can be ignored in maximization while the second integral:
represents correlator o matched filter outputs for K-th user in each signal interval.
Third integral can be easily decomposed in terms regarding cross-correlation:
Indeed can be written:
fork > l
can be expressed as a correlation measure (one for each K identifier sequences) which involves the outputs:
of K correlators or matched filters.
By using vectorial notation can be shown that NK outputs of correlators or matched filters can be expressed in form:
is a KxK matrix which elements are:
Gaussian noise vector n(i) is zero mean and its autocorrelation matrix is:
Vector r constitutes a set of statistics which are sufficient for estimation of transmitted bits .
The maximum likelihood detector has to calculate 2NK correlation measures to select the K sequences of length N which correspond to the best correlation measures.
The computational load of this approach is too high for real time usage
Considering maximization of L(b) like a problem of forward dynamic programming can be possible by using Viterbi algorithm after matched filters bench.
Each transmitted symbol is overlapped with no more than 2(K-1) symbols
When the algorithm uses a finite decision delay (a sufficient number of states), the performances degradation becomes negligible
The previous consideration points out that there is not a singular methodto decompose .
Some versions of Viterbi algorithm for multi-user detection, proposed in the state of the art, are characterized by 2K states and computational complexity O(4K/K) which is still very high.
This kind of approach is so used for a very little number of users (K<10 ).
When number of users is very high, sub-optimal receivers are considered
The conventional receiver for single user is a demodulator which:
Conventional receiver for single user suppose that the overall noise (channel noise and interference) is white Gaussian
The conventional receiver is more vulnerable to MUI because is impossible to design orthogonal sequences, for each couple of users, for any time offset.
The solution can be the use of sequences with good correlation properties to contain MUI (Gold, Kasami).
The situation is critical when other users transmit signals with more power than considered signal (near-far problem).
Practical solutions require a power control method by using a separate channel monitored by all users.
The solution can be multi-user detectors
The correlator output is:
Likelihood function is:
It can be proved that the vector b which maximize maximum likelihood function is:
This ML estimation of b is obtained transforming matched filters bench outputs.
(see slide 27)
So is an unbiased estimation of b.
The interference is so eliminated.
The solution is obtained by searching linear transformation:
Where matrix A is computed to minimize the mean square error (MSE)
It can be proved that the optimal value A to minimize J(b) in asynchronous case is:
The output of detector is:
When is low compared to other diagonal elements in , minimum MSE solution approximate ML solution of de-correlating receiver.
When noise level is high with respect to signal level in diagonal elements in matrix approximate identical matrix (under a scale factor ).
So when SNR is low, detector substantially ignore MUI because channel noise is dominant.
Minimum MSE detector provides a biased estimation of b, then there is a residual MUI.
To obtain b a linear system is to be computed:
An efficient solving method is the square factorization(*) of matrix:
With this method 3NK2 multiplications are required to detect NK bits.
Computational load is 3K multiplications per bit and it is independent from block length N and increase linearly with K.
* Proakis, appendix D
For an asynchronous DS/CDMA system, BER expression can be written (partially reported in slide 14) as:
It leads to:
If stochastic PN sequences are considered:
This formulation is wrong for “few users”
whereas can be used for large number
of users. It is useful for a simple evaluation of DS/CDMA system performances
From PE expression can be derived an evaluation of CDMA system capacity, in terms of number simultaneous users served with a certain Quality of Service (QoS)
For high values of x:
Considering admissible PE 10-3 (sufficient for vocal applications)
Considering the right side of equation as upper bound:
For high values of signal-to-noise ratio an approximation is possible:
A simple guidance, about a DS/CDMA system, to estimate system capacity is that more than N/3 asynchronous users can’t be served, where N is the process gain, with a probability error lower than 10-3.
BER Gaussian evaluation for DS/CDMA systems
K = number of users
BER Gaussian evaluation is only an approximation of real BER.
For SNR < 10 dB, Gaussian noise is predominant and BER is barely influenced by new users.
For very high SNR MUI is predominant and the higher the number of users, the lower are performances, if process gain is low.
Increasing SNR over a certain threshold, BER saturates: this is the bottle-neck given by MUI presence.
To increase performances, a higher process gain is needed; this fact involves an expansion of transmission band, at the equal bit-rate.