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Prof. C. Regazzoni


  • R. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of Spread-Spectrum Communications – A Tutorial”, IEEE Transactions on Communications, Vol. COM-30, No. 5, Maggio 1982, pp. 855-884.

  • K. Pahlavan, A.H. Levesque, “Wireless Information Networks”, Wiley: New York 1995.

  • A.J. Viterbi, “CDMA: Principles of Spread Spectrum Communications”: Addison Wesley: 1995.

  • J.G. Proakis, “Digital Communications”, (Terza Edizione), McGraw-Hill: 1995.

  • M.B. Pursley, “Performance Evaluation for Phase-Coded Spread-Spectrum Multiple Access Communications – Part I: System Analysis”, IEEE Trans. on Comm., Vol. 25, No. 8, pp. 795-799, Agosto 1977.

  • A. Lam, F. Olzluturk, “Performance Bounds of DS/SSMA Communications with Complex Signature Sequences”, IEEE Trans. on. Comm, vol. 40, pp. 1607-1614, Ottobre 1992.

  • D. Sarwate, M. B. Pursley, “Correlation Properties of Pseudorandom and Related Sequences”, Proceedings of the IEEE, Vol. 68, No. 5, pp. 593-619, Maggio 1980.

  • F.M. Ozluturk, S. Tantaratana, A.W. Lam: “Performance of DS/SSMA Communications with MPSK Signalling and Complex Signature Sequences”, IEEE Trans. on Comm. Vol. 43, No. 2/3/4, Febbraio 1995, pp.1127-1133.


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.

Multi User DS-CDMA

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

Perfomances – AWGN hp

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:

Perfomances– AWGN hp

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.

BER Evaluation - Gaussian hp

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

Perfomances - Gaussian hp

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

Perfomances - Gaussian hp

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:

MUI Variance

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.

Reference User





Intereference User





MUI Variance

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

MUI Variance

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

where .

MUI Variance

By substituting the integral with the summation of integrals and

with the values obtained in slide 12, the variance becomes:


MUI Variance - Conclusion

The last formula allow us to conclude:

  • The higher the process gain N, the lower the MUI variance. This means that by increasing the SS bandwidth the power of the Multi-User interference will be reduced.

  • A fundamental parameter is the cross-correlation function among PN sequences. With low correlation the MUI will be reduced and the interference can have weak effects.

    In the following section these aspects will be analyzed in details

Optimal Receiver: Asynchronous Transmission

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.:

Optimal Receiver: Asynchronous Transmission

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

Optimal Receiver: Asynchronous Transmission

First integral:

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:

Optimal Receiver: Asynchronous Transmission

Indeed can be written:

fork  l

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.

Optimal Receiver: Asynchronous Transmission

By using vectorial notation can be shown that NK outputs of correlators or matched filters can be expressed in form:


Optimal Receiver: Asynchronous Transmission

is a KxK matrix which elements are:

Optimal Receiver: Asynchronous Transmission

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

Optimal Receiver: Alternative Approach






Considering maximization of L(b) like a problem of forward dynamic programming can be possible by using Viterbi algorithm after matched filters bench.

Viterbi algorithm

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

Optimal Receiver: Alternative Approach

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

Sub-optimal Receivers: Conventional Receiver

The conventional receiver for single user is a demodulator which:

  • Correlates received signal with user’s sequence.

  • Connect matched filter output to a detector which implements a decision rule.

Conventional receiver for single user suppose that the overall noise (channel noise and interference) is white Gaussian

Sub-optimal Receivers: Conventional Receiver

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

Sub-optimal Receiver: De-correlating Detector

The correlator output is:

Likelihood function is:


Sub-optimal Receiver: De-correlating Detector

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.

Sub-optimal Receiver: De-correlating Detector

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:

Sub-optimal Receiver: Minimum Mean Square Error Detector

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.

Sub-optimal Receiver: Minimum Mean Square Error Detector

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

Conclusions: BER Evaluation

For an asynchronous DS/CDMA system, BER expression can be written (partially reported in slide 14)[5] 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

Conclusions: BER Evaluation

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:

Conclusions: BER Evaluation

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.

Example: numerical results

BER Gaussian evaluation for DS/CDMA systems

BPSK modulation

Gold sequences

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.

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