# Multi User Detection For DS CDMA System Computer Science Essay

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2.1 Introduction of DS-CDMA. There are many techniques used in wireless communication systems, for example, each user which is in Time Division Multiple Access (TDMA) system is assigned to a specific time slot; each user, in Frequency Division Multiple Access (FDMA) system, is assigned to a specific frequency band. Nowadays, Code division multiple access (CDMA) technique becomes more and more popular. It is one of the major wireless multiplexing techniques. In order to improve the time and frequency efficiency, each user, in CDMA system, is assigned to a specific code. Thus, different user can transmit signals in same time and frequency.

Direct Sequence CDMA (DS-CMDA) which is applied widely is one of the CDMA technologies. In the DS-CDMA system, the data of each user can be sent by using high rate sequence. Each user is assigned to a unique code which has good auto-correlation property to recover the signal easily and poor cross-correlation property to minimize multi-user interference (MUI) [1]. The unique code is called as the spread code which can spreads the bandwidth of the original signal to the system bandwidth before the signal is transmitted. Pseudo-random or pseudo-noise (PN) sequences which can increase the capacity of a communication system are one kind of spread code used in the DS-CDMA system.

There is a conceptual block diagram of a wireless DS-CDMA system shown in the following figure.

## Figure 2.1 Conceptual block diagram of a wireless DS-CDMA system [2]

In the figure 2.1, the unmodulated users' is a binary bit stream (single user's case) or a multilevel data stream (multiple users' case). Then, this data stream is modulated with a higher bit code sequence which increases the bandwidth of the base band signal. Next, this base band spread spectrum signal is modulated with a radio frequency carrier before being transmitted. After received, the signal is demodulated to get the baseband spread spectrum signal . Finally, the signal is despread and recovered to the final signal [2].

2.2 Direct sequence spread spectrum (DS/SS) system

2.2.1 Spreading sequence

An important feature of DS-CMDA technique is the application of spreading sequence. There are three basic requirements of using the spreading sequence: one is a good auto-correlation property to make code synchronization and track the information signal easily; the other is a small cross-correlation value to minimize MUI [4]; in addition, the spreading sequence has to deterministic for the transmitter and the intended receiver. The pseudo-random or pseudo-noise (PN) sequences is one technique of spreading sequence [1].

PN sequences are periodic binary sequences with auto-correlation existing noise-like property. Let two sequences of period N be and . The auto-correlation of sequence is expressed in the following formula [4]:

(2.1)

The cross-correlation of sequence and is expressed in the following formula [4]:

(2.2)

2.2.1.1 M-sequences

An important class of binary sequence is the binary maximal-length shift register sequences, commonly known as m-sequence. It can be generated using the linear generator polynomials of degree m. The m-sequences are expressed in the following formula [1]:

(2.3)

In order to generate m-sequence, the generator polynomial g(x) must be from the class of polynomials which implies in simple terms, and g(x) can not be factorized into lower-order polynomials.

The auto-correlation of an m-sequence is expressed in the following formula [4]:

(2.4)

The property of auto-correlation is excellent. However, the property of cross-correlation is relatively poor. With the increase of MUI, the performance of code synchronization and tracking is better. Moreover, the size of m-sequence for a m is small (the value is usually much less than N), thus, the multi-user capacity is small.

2.2.1.2 Gold sequences

One of the best-known binary sequences having relatively good correlation values is the Gold sequence set. A set of Gold sequences is constructed from a preferred pair of m-sequence x and y having identical length Q. So, the period of gold sequences is also Q. Each gold sequences in a set is generated by a modulo-2 sum of x and cyclic shifts of y. The gold sequences are expressed in the following formula [1]:

(2.5)

Where for q = 0, 1, â€¦ , Q-1, represents a cyclic shift of y by q chip intervals, the symbol âŠ• represents modulo-2 addition.

There are certain pairs of a set of m-sequence have three value cross-correlation function (-1, -t(m), t(m)-2), and t(m) is expressed in the following formula [4]:

(2.6)

The maximum value of the cross-correlation of these pairs is smaller than those of the rest m-sequence. Thus, there pairs have smaller MUI and they are called as preferred pairs.

2.2.2 DS/SS system

Spread spectrum (SS) is defined as a communication technique in which the data signal is modulated to a much larger bandwidth than the minimum requirement of transmitted signal. It can improve the interference immunity of communication system. A DS/SS technique is performed by multiplying a radio frequency carrier and a PN digital signal. There is a basic DS/SS system shown in the following figure 2.2. The difference between a normal digital communication system is that DS/SS system has two PN sequence generators (spreading code generator) in the transmitter and receiver.

## Figure 2.2 Basic DS/SS System [3]

Firstly, a PN code is modulated to a digital data signal by modulation. Then, the PN digital data signal is modulated with (using multiplication) a RF carrier in order to obtain a very wide bandwidth signal. This is the DS/SS process which is shown in the figure 2.3. The spreading code in the transmitter is multiplied with the data signal to make the signal appearing noise-like to all other users. The spreading code generator in the receiver generates the same code with the spreading code generator of spreading code generator, and uses it to extract the information signal.

## Figure 2.3 Generation a DS/SS signal [4]

The multiplication in the time domain of the digital data signal by the PN code sequence results in a signal with a frequency spectrum which is similar to the spectrum of the PN code signal. Therefore, the effects of increasing the data rate from R (symbol level) to Rc (chip level) are a reduction in the amplitude spectrum and an expansion of the signal in the frequency domain. Since the wide bandwidth of the PN codes allows us to reduce the amplitude spectrum to noise levels (without loss information), the generated signals appears as background noise in the frequency domain [3].

Next, the mathematical operation is used for describing the modulation and demodulation of DS/SS system. Hypothetically, the digital data signal is modulated by BPSK. According to the figure 2.1 and figure 2.2, the data rate is R bits/second and the bit duration Tb = 1/R seconds. The channel bandwidth of this system is Bc Hz, and the value of Bc is much larger than the value of R. The bandwidth of data signal is expanded to Bc Hz by superimposing the PN sequence on the data signal at the modulator. The resulting modulated signal is called as a DS/SS signal.[5]

The digital data signal is expressed in the following formula:

(2.7)

Where , and is a rectangular pulse of duration . The transmission power is normalized to 1 here.

The output of the PN sequence generator in the transmitter is expressed in the following formula:

(2.8)

Where is the binary PN code sequence of ±1, and is a the other rectangular pulse of duration .

The DS/SS signal is expressed in the following formula:

(2.9)

The transmitted signal is expressed in the following formula:

(2.10)

The received signal is expressed in the following formula:

(2.11)

Where is a sinusoidal interfering signal.

The received signal after despreading is expressed in the following formula:

(2.12)

Where .

2.3 Multi-user detection technology

In DS-CDMA system, the received signal is a sum of all users' signals which overlap in time and frequency [6]. If nonorthogonal codes are used or the orthogonal codes are destroyed in wireless channel, MUI will occur. Hence, a multi-user detection is applied to improve detection performance with the present of MUI caused by the correlation among the received signal of multiple users.

2.3.1 Correlation detection

2.3.1.1 Correlation detector (the matched filter)

The correlation detector is an optimum multi-user detector when we have orthogonal waveforms. Therefore, this detector is not robust to asynchronism, fading channels, or PN sequences with substantial cross-correlation which can not be made orthogonal. The structure of correlation detector is shown in the following figure:

## Figure 2.4 Correlation detector (the matched filter)

The correlation detector which is known as conventional detector or matched filter correlates the received signal with the original user's time reversed spreading waveform. This method of detection is exploited from single user designs. The output of the correlation detector (Figure 2.4) for user k and time I is expressed in the following formula: [7]

(2.13)

Where and are estimated time delay and phase of user k. The estimates of the transmitted bits are expressed in the following formula:

(2.14)

Next, the mathematical method is used to explain a process of correlation detection. According to the figure 2.1 and figure 2.2 firstly, the every user's data is in a column , where k is the number of data. The every user's spreading spectrum sequences is in the column . The simple received signal y is expressed in the following formula:

(2.15)

Where n is the noise. Then, the received signal y is despread by multiplying the transpose of the spreading spectrum sequences C. So, the detected signal passing the correlation detector is expressed in the following formula:

(2.16)

Where the correlation matrix R is equal to .

When the data signal is over the Rayleigh fading channel, the simple received signal y is expressed in the following formula:

(2.17)

Where the Rayleigh fading channel , and are the amplitude and phase of un-correlation Rayleigh fading channel. So, the detected signal passing the correlation detector is expressed in the following formula:

(2.17)

Where the correlation matrix is equal to .

2.3.1.2 BER performance of correlation detector

The following figures are all modulated by BPSK.

## Figure 2.5 BER verses SNR of DS-CDMA over AWGN channel with correlation detection

## (7 users, m-sequence)

## Figure 2.6 BER verses SNR of DS-CDMA over AWGN channel with correlation detection

## (15 users, m-sequence)

Figure 2.5 and figure 2.6 show the various BER vs. SNR curves of performance of BPSK passing over the AWGN channel with correlation detection respectively. The Difference between these two figures is the number of users. We can find that the performance of the BER curves becomes worse when the number of users is increasing. The reason is the correlation detector does not cancel the MUI. Therefore, the interference from multiple-user increase, the performance becomes worse.

If we still use the method of m-sequence to detect, increasing the length of m-sequence is the way to improving the performance of the BER curve because the cross-correlation value of m-sequence is proportional to its length, or just change another sequence which has better cross-correlation property, for example gold sequence. The performance of correlation detector using gold sequence over the AWGN channel is shown in the following figure.

## Figure 2.7 BER verses SNR of DS-CDMA over AWGN channel with Correlation detection

## (31 users, gold sequence)

Figure 2.7 shows the BER vs. SNR curves of performance of BPSK passing over the AWGN channel, and the method of gold sequence is used for plotting this figure. Compare figure 2.7 with figure 2.5, we can find that the value of SNR is equal to 15.5dB in figure 2.5 and the value of SNR is equal to 10.5dB in figure 2.7 when the number of users is 5 and the value of BER is equal to 10-5. Therefore, the BER performance by using the method of gold sequence is better than the method of m-sequence.

The performance of correlation detector over Rayleigh fading channel is shown in the following figure:

## Figure 2.8 BER verses SNR of DS-CDMA over Rayleigh channel with correlation detection

## (7 users, m-sequence)

Figure 2.8 shows the BER vs. SNR curves of performance of BPSK passing over the Rayleigh fading channel. According to the figure 2.8, we can find that the BER performance is not very good comparing with the previous curves. The reason is the magnitude attenuation and phase rotation is generated in the Rayleigh fading channel. More importantly, the correlation detector can not combat against the magnitude attenuation and phase rotation, and also can not remove MUI. With the increasing of the number of users, the performance degrades badly.

2.3.2 Decorrelation detection

2.3.2.1 Decorrelating detector [7]

The decorrelating detector implements a linear transformation on the outputs of the conventional correlation detectors. This transformation removes the interference between the users and the performance is therefore independent of the different users' received powers. The disadvantage is that decorrelating detector causes noise enhancement, which is same as the zero-forcing equalizer. The structure of decorrelating detector is shown in the following figure:

## Figure 2.8 Decorrelating detector

The correlation detector does not use any information about the other users in the systems. Therefore, it can not combat against MUI. The decorrelating detector essentially applies the inverse of the correlation matrix of users' spreading codes to generate the outputs which is shown in the figure 2.8. This method of decorrelation is only suit for the base-station practically, where all the information is readily available, and there are no power consumption constraints.

The decorrelating estimates of the transmitted bits can be expressed in the following formula:

(2.18)

Where R is the correlation matrix which is shown in formula 2.16. As shown in figure 2.8, the method of decorrelation detection multiplies the output of correlation detector with the inverse correlation matrix R.

After correlation detector, the output is shown in the following formula (same as the formula 2.16):

Then, the output of correlation detector is despread by multiplying with the inverse correlation matrix R. So, the detected signal passing the decorrelating detector is expressed in the following formula:

(2.19)

As shown in formula 2.19, the is different with the x. The reason is is not be equal to one, but . And, the interference among the received signals of different users is removed. However, the noise is enhanced by the term . Decorrelating detector does not need to know the user' powers and its performance is independent of the powers of the interfering users. When users' energies are unknown and the objective is to optimize the performance for the strongest MUI, the decorrelating detector is the optimal method [8].

There are some obvious advantages of this detector comparing with the correlation detector. For one, it removes all the multiple access interference. This also implies that the power of each user does not have to be estimated or controlled. One disadvantage to this detector is noise enhancement caused by . Also, this detector has no way of eliminating any ISI caused by channel dynamics. Even so, this detector will outperform the correlation detector when the user spreading spectrum codes are not orthogonal.

2.3.2.2 BER performance of decorrelating detector

The following figures are all modulated by BPSK.

## Figure 2.9 BER verses SNR of DS-CDMA over AWGN channel with decorrelation detection

## (7 users, m-sequence)

## Figure 2.10 BER verses SNR of DS-CDMA over AWGN channel with decorrelation detection

## (15 users, m-sequence)

The performance of decorrelating detector over Rayleigh fading channel is shown in the following figure:

## Figure 2.11 BER verses SNR of DS-CDMA over Rayleigh channel with decorrelation detection

## (7 users, m-sequence)

Figure 2.9 and figure 2.10 show the various BER vs. SNR curves of performance of BPSK passing over the AWGN channel with decorrelating detection respectively. The Difference between these two figures is the number of users. We also can find that the value of SNR is larger with same BER when the number of users is increasing. That means there is more noise added into the signal when the number of users is increasing. However, comparing with figure 2.5 and figure 2.6, the BER performance of decorrelation detection is better than correlation detection. The reason is the decorrelation detector can cancel the MUI.

Figure 2.11 shows the BER vs. SNR curves of performance of BPSK passing over the Rayleigh fading channel with decorrelating detection. As discussed before, the theory, the decorrelating detector can remove MUI by multiplying with the output of the correlation detector, is proved by comparing figure 2.11 with figure 2.8.

2.3.3 Minimum mean squared error (MMSE) detection.

2.3.3.1 MMSE detector

Minimum mean squared error (MMSE) detector which is same as the decorrelating detector applies a linear transformation to the output of the correlation detector. This transformation is chosen to minimise the mean-square error between its output and the data. At the same time, MMSE detector takes into account the background noise and need the information of the received signal power. The MMSE detector always attempts to strike a balance between the residual interference and the noise enhancement.

For the MMSE scheme, there is a weight matrix W. An estimated signal is generated in a processing of multiplying the output of the correlation detector with the weight matrix, which is expressed in the following formula:

(2.20)

The weight matrix W is chosen such that the mean-square between each element of the transmitted vector x and its estimate is minimized, which is expressed in the following formula [5]:

(2.21)

The optimal weight matrix is expressed in the following formula:

(2.22)

Where is the auto-correlation matrix of the received signal which is shown in formula 2.17, and is the cross-corrlation matrix between the received signal and the desired response of x.

Assuming that the data of the different users are independent, such as (k and u is the different user), then, is expressed in the following formula:

(2.23)

Where I is the identity matrix, and is variance of the AWGN samples. On the other hand, is equal to for kth user. So, the optimal weight matrix can be calculated.

(2.24)

Thus, the estimates of the transmitted bits are expressed in the following formula [7] [3]:

(2.25)

When the MMSE detector minimises the mean-squared error before added channel noise, it approaches the decorrelating detector when no noise is present. Performance is same as the solution of decorrelating detector when the SNR is relatively high. However, the performance of MMSE detector is better at low SNR's. On the other hand, the MMSE detector approaches the correlation detector when the MUI is smaller than the noise. This theory will be proved by figure 2.15.

The disadvantages of this detector are that it need estimate the received amplitudes, and its performance depends on the power of the interfering users. In terms of complexity, the MMSE detector faces, like the decorrelating detector, the problem of implementing matrix inversion.

2.3.3.2 BER performance of MMSE detector

The following figures are all modulated by BPSK.

## Figure 2.12 BER verses SNR of DS-CDMA over AWGN channel with MMSE detection

## (7 users, m-sequence)

## Figure 2.13 BER verses SNR of DS-CDMA over AWGN channel with MMSE detection

## (15 users, m-sequence)

Figure 2.12 and figure 2.13 show the various BER vs. SNR curves of performance of BPSK passing over the AWGN channel with MMSE detection respectively. The Difference between these two figures is the number of users. We also can find that the value of SNR is larger with same BER when the number of users is increasing. That means there is more noise added into the signal when the number of users is increasing.

## Figure 2.14 BER verses SNR of DS-CDMA over Rayleigh channel with MMSE detection

## (7 users, m-sequence)

## Figure 2.15 BER verses SNR of DS-CDMA over AWGN channel with decorrelation and MMSE detection (15 users, m-sequence)

Figure 2.15 shows the BER vs. SNR curves of performance of BPSK passing over the AWGN channel with decorrelation and MMSE detection. According to figure 2.15, we can find that MMSE detector performs better than decorrelating detector when SNR is low. With the increasing of SNR, the performance of both detectors is more and more similar because the influence of noise is negligible. In this case, the MMSE detector is equivalent to decorrelating detector. The information which is observed from figure 2.15 can prove the previous theory.

2.3.4 Maximum Likelihood (ML) detection

2.3.4.1 ML detector

Maximum Likelihood (ML) detector is the different detector with correlation detector, decorrelating detector and MMSE detector. The optimal ML detector occurs when the transmitted signal is independently and identically distributed. As introduced before, the every user's data is in a column , where k is the number of data. And, define as the sat of all possible transmitted information vectors of x. The every user's spreading spectrum sequences is in the column . So, the received signal . Then, one form of ML detector is expressed in the following formula [10]:

(2.26)

Assume that all the information vectors are equiprobable. Thus, another form of ML detector can be expressed in the following formula:

(2.27)

The ML detector is optimum when the noise n in y is additive Gaussian noise.

According to the formula 2.27, the ML detector represents a discrete optimization problem over candidate vectors . Unfortunately, this problem does not have several efficient algorithms available for the solution or approximation of ML detector. The other disadvantage of ML detector is that it requires knowledge of the received amplitudes and phases. These values are not known from a priori, but must be estimated.

2.3.4.2 BER performance of ML detector

The following figures are all modulated by BPSK.

## Figure 2.16 BER verses SNR of DS-CDMA over AWGN channel with ML detection

## (7 users, m-sequence)

## Figure 2.17 BER verses SNR of DS-CDMA over Rayleigh channel with ML detection

## (7 users, m-sequence)

According to figure2.16 and figure 2.17, we can find that the performance is not change when the number of users is increasing. All the curves of BER is same as the signal user's curve in the AWGN channel or Rayleigh fading channel. The reason is ML detector is a non-linear detector, the performance does not have the limit of linear structure. Comparing with the performance of correlation detector, decorrelating detector or MMSE detector, the ML detector can improve the BER performance. But, ML detector also has some disadvantages. The main one is the complex for practical implementation. If there are K users using the ML detector, each user has to calculate 2K times, which makes ML detector harder to implement practically.

2.4 Conclusion

The spread spectrum, general DS-CDMA systems and several multi-user detection have been introduced in the part. Firstly, the block diagram of DS-CDMA is shown, the structure of transmitter and receiver is shown too. Secondly, there are two forms of spread spectrum, one is m-sequence, and another is gold sequence. The BER performance by using the gold sequence is better than m-sequence. Next, the correlation detector is introduced. It is suit for the signal user to detect because it can not remove MUI. Moreover, decorrelating detector and MMSE detector are introduced. As a result of removing MUI, these two detectors have better performance than correlation detector. Finally, ML detector is shown. The BER performance of ML detector is the best than all other detectors. But, it is too complex for practical implementation.

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[7] 10.1.1.65.2302

[8] Shimon Moshavi Bellcore. Multi-user detection for DS-CDMA communications. IEEE Communication Magazine, October 1996.

[9] S. Verdu. Computational complexity of multi-user detection. Algorithmica, 4:303{312, 1989.

[10] IR-S3-SB-0443