Ensures and reliable methods of communication
With each passing day the world becomes a smaller place to live in as more techniques, methods and technologies are released that ensures quick and reliable methods of communication. Of all these, wireless communications has experienced rapid improvements and has thrust the world into a fast paced and impatient speed which leaves mobile communications one of the most desired methods.
With a growing demand for wireless communications, code division multiple access (CDMA) has been researched with great scrutiny. In a CDMA system it uses a ‘spread-spectrum' technology and a special coding scheme to allow multiple users to be multiplexed over the same physical channel. Random multi-access communication is one of the approaches to dynamic channel sharing. When a user has a message to transmit it transmits it as if it were the sole user of the channel.
Using this concept the optimal detector was created for multiple users to be able to send and receive information simultaneously over a single communications channel. However the optimal detector proved to be far too intricate to implement as the complexity of the system grew exponentially to the number of users. and hence a number of sub optimal detectors have been developed to address multi-user detection instead. Once the sub optimal detectors were made available to manufacturers, being in a competitive market; the question of which detector had the best performance arose.
The MMSE detector and the decorrelating detector are both linear detectors that are products of a quest to produce detectors that are less complex than the optimal detector and have acceptable rates of performance. The conventional detector was also designed with these objectives in mind and the objective of this experiment was to compare all three detectors to determine which was better in terms of performance.
Aim of the thesis:
The aim of this thesis is to compare the performances of the conventional, decorrelating and minimum mean squared error in terms of bit error rate and complexity. The study also shows the behavior of these detectors in the effect of three parameters, these parameters are:
- The effect of changing the number of users
- The effect of changing the value of σ.
One of the basic concepts in data communication is the idea of allowing several transmitters to send information simultaneously over a single communication channel. This allows several users to share a bandwidth of different frequencies. This concept is called multiplexing.
Multi access communication can also be referred to as ‘'multi point-to-point'' communication. For example, a base station transmitting to mobile units which imply that the receivers, which in this case are the mobile units, are only interested in one of the information sources being transmitted from the base station.
Initially multi user detection was used in either
- Frequency Division channels
- Time Division channels
Frequency Division Multiple Access (FDMA) assigns a different carrier frequency to each user so that the information signals do not get mixed up or interferes with one another. To make sure that each information package is received separately, they are demodulated separately using band pass filters.
Time Division Multiple Access (TDMA) separates the time into time slots that have been pre-determined for each incoming digital stream. Time Division Multiple Accessing is best used for multiplexing message sources that are collocated and for multiplexing message sources between the users who are spanned over a large geographical distance.
The advantages of using FDMA was that this method allowed for transmissions to be uncoordinated in the time domain which means that no time synchronization between the users is required whereas in TDMA the transmission needs to be time synchronized between the users with a common clock. Also for FDMA the users can transmit their information packages in separate channels that do not interfere with one another.
However both these methods were not ideal for multi user detection and still proved to have much interference and therefore in recent year's multi user detection is being used with Code Division Multiple Access (CDMA).
Code Division Multiple Access (CDMA) was used in multimedia applications which used numerous services such as voice, audio/video, graphics, data, internet and email. The goal behind the CDMA system was to improve the system capacity to deal with multiple users as compared to FDMA and TDMA. In CDMA each user is assigned an individual and separate code and the respective transmitter transmits its data stream by modulating its own waveform, unique to the transmitter, as is done in a single user communications system.
Because all users can transmit at the same time, and each is allocated the entire available frequency spectrum for transmission; CDMA is also known as spread-spectrum-multiple access (SSMA). 
When the information signal is modulated into a transmission signal; spread spectrum encoding is used by encoding the information-bit signal with a unique code signal which is independent of data and has a wider bandwidth than the information signal. The outcome of such modulation is
that the power of the original signal is spread over a wider bandwidth. At the receiver, the signal is detected and the original data is recovered.
Of the many spread spectrum techniques, direct sequence (DS) method is the most popular and easiest to use. It works by modulating the transmission signal with a digital signal code which has a higher bit rate than the information signal. The spreading sequence is generated when the code signal amplitude modulates the information waveform. The code consists of a number of code bits that are either ±1 and the code is called a Psuedo Noise Signal (PN) code. The PN code is responsible for the spreading and dispreading of the bas-band signal. The resulting signal modulates an RF carrier and then the signal will be transmitted.
At the receiver end of the system, the receiver should have prior information about the code sequence and be able to generate a local code sequence which can modulate with the received signal code sequence. The modulated data is then demodulated after dispreading to uncover the original transmitted data.
The objective of using a spread spectrum system for a multiple access system is to create spreading codes that will allow as many users as possible use of a band of frequencies with as little interference as possible.
Suppose a CDMA system of K users and for any number of users there are K-1 interferers. Each user is received at the same power level, P when power is applied to the system and therefore the signal to noise ratio is as follows:
According to the equation above, when there are a large number of users the SNR will be very low. My project does not focus on a large number of users and therefore the aforementioned phenomenon does not affect the outcomes of the conducted experiments.
There are two types of systems;
My system model is off the synchronous type. A synchronous channel allows all of the users to simultaneously send data bits through the channel and when these data bits are received the detection of each bit is independent of all the others.
When the system model is taken into consideration the assumption that the carrier phases are equivalent to zero to simplify the matter. It is also assumed that the transmitted signal is sent across a single path or channel. Below is a visual aid of a continuous synchronous system model:
In this chapter we will be focusing on two different detector types and they are as follows:
- The conventional detector
- Linear detectors
And off the linear detectors we will be focusing on :
- The decorellator
- The MMSE
1. The Conventional Detector
The conventional detector is also known as the matched filter detector and it is the simplest detector mainly because it treats each user individually and ignores any other users that may be transmitting signals at the same time. The detector simply correlates the received signal with the desired user's time reversed spreading waveform, and samples the output at the bit rate . This means that the transmitter of the conventional detector is not synchronous and multiple users therefore cannot transmit data bits simultaneously over the same channel. A model of the conventional detector is shown in
If there are K users using the system then the conventional detector will consist of K number of matched correlators allowing the received signal for each user to be filtered separately. Correlation occurs between the codes passing through the system and this occurrence is vital to the functioning of the conventional detector. Making the assumption that the above model is not subjected to power variations and has no channel dynamics the detector can be defined as.
and w(t) is the filtered additive white Gaussian noise (AWGN) and skis the product of the amplitude and the modulated data of the kth user. This ensures for normalized spreading codes that when i=k, Pk,k = 1, and when, i ≠ k, 0 ≤ p < 1.
This derivation shows us that the correlation of the kth user produces the recovered data, and that the mulitple access interferance (MAI) is produced by the correlation of the kth user with all the other users. The derivation and conditions of the equation also suggests that the MAI term can dominate the correlator output if the cross correlation between the spreading codes is not kept to a minimum. The MAI term is directly proportional to the number of users in the system and therefore an increase in the number of users will lead to an increase in the amount of the MAI term. A high MAI can cause high bit error rates and adecrease in system capacity.
Another factor to take into consideration is the near-far effect which does not ensure an equal MAI regardless of the position of the user in regards to the reciever. Users closer to the reciever will generate a larger MAI term and thus will effect users that are further away. Coupled with a signals characteristic to fade slightly across transmitted distances leads to weaker signals from users far away being overpowered by users closer by. A solution to this problem is to implement power control in the system.
All in all a conventional detector has one desirable quality and it is the low complexity of the system that attracts designers however it's inability to accomodate for multiple users and it's problems with power control have lead designers to look at other system models.
2. Linear Detectors
One such system model is that of the linear detectors. The conventional detector does not regard information form any other user in the system and therefore is not viable to combat the effects of an overpowering MAI term. The decorrelating detector combats this problem by applying the inverse of the correlation matrix of the users spreading codes to the output of the conventional detector. This is shown in Figure 3:
a. Decorrelating Detector
The decorrelating detector is one of the two well-known linear detectors. The method a decorrelating detector uses is to apply the inverse of the correlation matrix to the vector of the matched filters of the conventional detector so as to attempt to recover the original transmitted data bits error-free. This method is only practical at a base station
because an unlimited power consumption will be required as well all all the information regarding each signal. The correlation matrix, R, is defined as
[R]i,j = pi,j
The soft decidion of the output of the vector-matrix formula would be
Y = Rs + w
Where s = [s1, s2, .... sk]T
and w = [w1, w2...wk]T
Now, by applying the inverse of the correlation matrix to the soft decision output vector , we can completely remove all Multiple Access Interference. The new soft decision of the combined detector is
d = s + R-1w
When the correlation matrix, R, is inversed the soft decision output vector y will only be affected by AWGN. However the noise of the decorrelating detector increases when the inverse of the correlation matrix is applied and therefore the power of the noise in the detector is sometimes greater than that of the matched filter. Also with the added matrix the complexity of the system increases, and though it's not as complex as the ideal detector, it increases the run time of the system.
b. Minimum Mean Square Error (MMSE) Detector
The minimum mean square error (MMSE) detector is another example of a linear detector. The MMSE detector, unlike the decorrelating detector, considers the background noise and also is aware of the received signal powers. The main objective of this detector is to reduce the mean-squared error between the actual data and the soft output of the conventional detector .
Verdu says that in the case for two users the following equation can be used instead of the cross correlation matrix R-1 in the decorrelating detector
The MMSE solution minimizes the squared error in the presence of channel noise, and becomes the zero-forcing solution when no noise is present. Performance is very similar to the zero forcing solution when the SNR is relatively high, but is improved at low SNR's.
The MMSE detector also has a problem of having the complexity of the system increase as the number of users increase however it eliminates the problem of an over bearing MAI term and because background noise is present in the minimum mean square error (MMSE) detector the MMSE detector has a better performance than that of the decorrelating detector when considering the probability of error.
The following report is based on the simulations I ran in order to test my hypothesis and come to a conclusion. The word simulate is defined by the Oxford dictionary as being an action that will produce a computer model or process of a situation or circumstance.
In order for the simulations to be carried out MATLAB was used as the programming software. The codes used a semi-analytical Monte Carlo simulation foundation. The need for model reduction is essential when performing system's reliability analysis with many components and high dimensionality. The increase in system complexity and the effect on the simulation time and convergence speed require reducing the models into less complex ones without the loss of the necessary information to perform an accurate analysis.
As the simulation is semi-analytic the noise had to be generated. This was done by generating noise in the form of white Gaussian noise that had a zero mean and a covariance, σ which was
then multiplied by the Cholesky factorization of the cross-correlation matrix R in order to achieve the correlated noise .
nCorrelated=White noise× U
Where U, is the upper triangular factor of the matrix R.
The probability of error is calculated for all the receivers and the equation is:
Pe=No.of errorsNo.of itrations
The signal to noise ratio is calculated using the equation shown below :
Where P is the power and σ is the noise power spectral density. It is necessary to find the average of the probability of error for a number of users for a large number of cycles of repetition. The larger the number of repetitions the better so as to acquire a more respectable set of results. The relative error has to be less that 0.1dB for each detector as otherwise it won't be feasible enough to consider the respective detectors. Therefore the cycles have been chosen to be repeated 80000 times. The formula used to calculate the relative error is:
Relative error=σrelative errorPe
σrelative error=Pe-Pe2No.of Iterations
And Pe is the probability of error.
The system runs for an unknown number of K users and each user has been assigned 1 bit. I have chosen to allow the system to run on 6 and 8 users to see what the responses are likening too.
The results have been displayed as graphs of the probability of error versus the signal to noise ratio (SNR) of each of the detectors displayed collectively on one graph and also of the relative error versus the SNR to see how accurate the simulations are.
The characteristics of the detectors that I will be investigating are:
- The effect of the number of users
- The effect of increasing Rho
The effect of the number of users:
To determine what the effect of an increase in users resulted in, the rest of the variables had to remain fixed for this segment of the simulations. The value of ρ was fixed at a value of 0.2 and σ was fixed at the three values off 0.5, -0.0375 and 0.2375. We compare the case of two active users with six active users and eight active users, knowing that all the simulations were carried out in the same environment mentioned before.
The Figures (4), (5) and (6) show plots of the average bit-error probability of the three receivers as function of the signal to noise ratio for K = 2, K=6 and K=8 respectively in a synchronous user system, by comparing the three graphs we see various changes occur between them. The conventional detector drops from -0.05dB to -0.0004dB while both the MMSE and decorrelating detector degradation is from -0.04dB to -0.00004dB, and a general deterioration occurs for all of the receivers when increasing the number of users from two to six to eight. It is also noticed, especially for the conventional receiver that as the number of users increases, the drop in the curve is lesser. This is due to the fact that the increment in number of users means increasing the
interference powers and hence increasing multiple access interference (MAI), which is a major limiting factor the receivers face.
It's also noticeable that the gaps between the conventional detector and the other two detectors sgrows significantly as the number of users increases as well, and this could only mean that the decorrelating detector and the MMSE detector have a better performance rate.
The Steadiness of the conventional detectors curve in Figure (5) and Figure (6) where it does not seem affected by the increase in the signal to noise ratio is because when there are just two users the interfering power is rather small compared to the noise and therefore the MAI term is more dominant and the curve changes as the signal to noise ratio changes. However with the increase of users the interfering power has risen dramatically and has become larger than the noise and therefore the signal is far less sensitive to the signal to noise ratio changes and hence the steadiness in the two latter graphs.
The effect of increasing σ:
By increasing the value of Rho from 0.2 to 0.4 whilst still maintaining a constant value of users (K = 6) and a constant bit number of 1, we calculated and plotted the probabilities of error of the users for the three detectors and the results are shown in Figure (7).
For the conventional detector, comparing Figure (5) and Figure (7), it has been affected ever so slightly. As it can be seen, as the σ increases from 0.2 to 0.4, the probability of error for the conventional detector shifts slightly lower from a steady -0.75dB to -0.4dB. The other two detectors are also affected by the increase in σ as their curves also shift slightly lower when the value of σ increases.
The reason for the slight shift in the curves of the probability of error graphs is because it indicates that the signal to noise ratio is relatively stronger as the σ increases from 0.2 to 0.4 and hence the conventional detector curve also becomes more stable and steady as once again the interfering power of the system is comparatively stronger than the noise and therefore dominates and is not affected by an increase in signal to noise ratio.
Three different multiuser detectors have been analysed in the past few chapters and these were the conventional detector, the minimum mean square error detector, and the decorrelating detector. The initial pages covers the background theory of CDMA communications and the importance of modulation and transmission and the needs for a suitable multiuser detector.Each detector was subjected to a few chapters of background information so that the reader would understand the functioning's of these detectors even better. The emphasis was placed on the number of users and the possible changes each detector could have if parameters were changed.
From the results it can be seen that the Decorrelator and the MMSE detector always lay very close to each other in the graphs and it was difficult to differentiate which was the better one. However after much analyzing it can be seen that the MMSE detector sometimes performs ever so slightly better than the decorrelating detector. It can also be seen that when the MAI term is high we see only very slight changes taking place on the output of the receievers whereas when the MAI term is very low the detectors are very greater affected.
It would be easy to say the conventional detector, based on the results, is the worst performer however this was expected as it is a much less complex detector as stated in the background information and lacks the ability to over the effect of the MAI term.