Multiuser detection algorithm based on eigenvalue decomposition?

Jiang, Luo Huali, Feng Yumin

(Multimedia Laboratory, Communication Department, North Jiaotong University, Beijing 100044)

Abstract: Because multi-user detection can fully improve the capacity of CDMA system, a multi-user detection algorithm based on eigenvalue decomposition between sub-security checks is proposed. The simulation analysis of the algorithm shows that the algorithm has good convergence performance and steady-state performance, and can also work well in the case of low signal-to-noise ratio and the number of users close to spreading factor.

Keywords: code division multiple access; Multi-user detection; Subspace tracking; Blind adaptation

0 quotation

The proposal of CDMA technology solves the problem of small capacity of TDMA mobile communication system, and multi-user detection technology can fully improve the capacity of CDMA system. In the past ten years, various multi-user detection technologies [1] have been proposed. The main multi-user detection includes decorrelation detection, minimum mean square error (MMSE) detection, multi-level interference cancellation detection, decision feedback detection and detection based on neural network.

Literature [2] proposes a blind adaptive multiuser detection algorithm based on subspace. It uses PASTd algorithm for subspace.

The computational complexity of inter-frame tracking is O(NK), where n is spreading factor and k is the dimension of subspace. However, the simulation results show that the detection performance of Pasted algorithm is poor, which is due to the error accumulation caused by the approximate estimation of subspace, which makes the system performance worse, so Pasted algorithm is not suitable for blind multiuser detection algorithm [3]. In this paper, the fast approximate subspace tracking algorithm [4] is used to decompose the signal space, and its computational complexity is O(NK2).

Experiments show that the algorithm has good convergence performance, and it can work well in the case of low signal-to-noise ratio and the number of users close to spreading factor.

1 system model

Considering the synchronous DS-CDMA system with K users, the baseband received signal after AWGN channel can be described as:

Signal amplitude, information bits and characteristic waveform with unit power. Spread spectrum code adopts short code, that is, the length of code period is equal to spread spectrum gain n = t/Tc, and t and Tc are bit interval and chip interval respectively. Unless otherwise specified, the modulation waveform of the chip is a square wave, the number of users k ≤ n ... n (t) is zero-mean complex Gaussian white noise with unit power spectral density, and σ2 is the noise power in the channel.

The baseband signal first passes through a chip matching filter, and then is sampled at the chip rate to obtain a discrete signal rn. Let: rn = [r

Through the eigenvalue decomposition of r, we get:

Where ∧ s = diag (λ 1, …, λk) contains the k largest eigenvalues of R, and US = [U 1, …, UK] is its corresponding eigenvector: ∧ n = σ 2in-k and UN = [UK+65438+]. It can be seen that range (s) = range (Us), the value range of Us is called signal subspace, the subspace composed of Un is called noise subspace, and the two subspaces are orthogonal to each other.

Assuming that the demodulation vector of the kth user data demodulated by the linear multiuser detector is mk, the output of the decider is:

Here, in order to minimize the cost function value, we calculate the gradient of the above formula and get:

Because the signal subspace and the noise subspace are orthogonal, and because the decision result is only related to the direction of the demodulation vector, but not to its amplitude, it is obtained that:

mk=(Us∧- 1s UTs)sk

Adaptive algorithm for solving demodulation vector

In the above discussion, we attribute the subspace-based multiuser detection algorithm to subspace tracking. Traditional subspace tracking methods are eigenvalue decomposition (EVD) and singular value decomposition (SVD). Although their performance is better, their complexity is higher than O(N3), which is not conducive to engineering implementation. Therefore, fast subspace tracking algorithm has been widely studied. Literature [2] uses Pasted algorithm for subspace tracking, and the computational complexity is O(NK), but our simulation results show that Pasted algorithm has poor detection performance and is not suitable for blind multiuser detection algorithm.

Here we use a fast subspace tracking algorithm with computational complexity of O(NK2): fast approximate subspace tracking algorithm [4]. Because we are only interested in the performance of this algorithm in subspace-based multiuser detection, we only briefly introduce their basic principles here, and refer to reference [4] for details.

Let x (t- 1) = [x 1, x2, …, XJ] be the data matrix of N×J, and j be the length of the sampling data window. When updating data, X (t) = [x2, …, xJ, XJ+ 1] is the new data matrix, and U (t- 1) = [U 1, …, UK] is the matrix X (t- 1).

1) construct a low-rank (rank k+ 1) matrix a similar to X(t), so that

2) Construct a (k+ 1) × (k+ 1) matrix F, so that it contains all the information of matrix A, and SVD decomposes F to get the eigenvectors and eigenvalues of A. ..

Three numerical results

In this part, we simulate and analyze the performance of the algorithm. The measure of simulation performance is the output RSI (Signal to Interference Ratio (RSI), defined as r}. The RSI in the experiment is the result of 100 simulations, and the window length of the algorithm is 64.

In the simulation experiment, we use 3 1 bit Gold code to spread spectrum. In figure 1, we use five interfering users, of which the first four users have the same power, 3, 4, 5; The interference power of the fifth jamming user is dB, and the signal-to-noise ratio is 20 dB. As can be seen from the figure 1, FAST reaches a stable state after 30 iterations. The steady-state performance of FAST is about 14dB. From the simulation results of figure 1, it can be seen that PASTd algorithm is not convergent, so it is not suitable for blind multiuser detection.

In Figure 2, we analyze the dynamic performance of the algorithm. The signal-to-noise ratio of the system is 20 dB, and the initial state is 4 interfering users, and = = 10 dB, I = 2, 3, 4, 5. When the number of iterations is 500, MAI is 20 dB users (that is, cut into the system); When the iteration number is 1000, the user with MAI of 20 dB and two users with MAI of 10 dB leave the system. When the number of iterations is 1500, three users with a MAI of 20dB enter the system. The simulation results show that the fast algorithm can adapt to the dynamic environment well, but its performance decreases in the dynamic environment.

In Figure 3, we test the performance of the fast algorithm in the case of low signal-to-noise ratio and a large number of users, in which the system signal-to-noise ratio is 7 dB, the number of users is 29, the interference power of interfering users = = 10 dB, I = 2, …, 29. The simulation results show that the fast algorithm has good convergence performance and steady-state performance in the case of low SNR and large number of users.

4 conclusion

In this paper, a new multi-user detection algorithm based on subspace tracking is proposed by using adaptive eigenvalue decomposition algorithm, and its performance is analyzed. The complexity of the algorithm is O(N2k). The simulation results show that the algorithm has good convergence performance and steady-state performance, can realize adaptive convergence in dynamic environment, and has fast convergence speed.

refer to

[1] verdu S. Multi-user detection [m] Cambridge University. Press, 1998.

Wang X, H.V. Blind multiuser detection: a subspace method [J]. Journal of Information Theory, 1998, (2):677-690.

Wang Yi. Research on linear multiuser detection technology in DS-CDMA system [D]. Beijing University of Posts and Telecommunications, 2000.

[4] Realec, TUFTSD W, Cooley JW. Two fast approximate subspace tracking algorithms [J] IEEE Transon Signal Proc.1999,47 (7):1936-1945.

journal of chongqing university of posts and telecommunications

This paper introduces a real-time channel simulator for simulating the basic characteristics of mobile channels, such as Rayleigh fading, multipath propagation, radio wave propagation path loss, Doppler frequency shift, etc. ), including the digital principle of the simulator and its implementation scheme. The fading rate of the simulator is adjustable in the range of 8 ~ 80 Hz, the simulated fading depth exceeds 20dB, and the maximum multipath delay is10.2 μ s. ..

Keywords: Rayleigh fading channel simulation delay multipath propagation

1 development background of mobile communication channel simulator

Mobile communication is a very rapid communication mode in recent years. In the future public future public land mobile telecommunication system, the envelope and phase of the received signal will change randomly due to the complex terrain of the mobile station and the movement of the mobile station itself.

In order to evaluate the performance of mobile communication equipment, it is necessary to conduct repeated experiments in the actual communication environment, which will inevitably consume a lot of manpower and material resources. In order to shorten the development cycle and save the development cost, various channel simulators are widely used in the development of mobile communication equipment.

This paper introduces a mobile communication channel simulator with a signal frequency of 70MHz and a base station antenna height of 18m. The simulator can simulate the main characteristics of mobile communication channels, such as Rayleigh fading, multipath propagation, battery propagation path loss, Doppler frequency shift and so on.

2 the development basis of mobile communication channel simulator

2. 1 Rayleigh fading

Due to the influence of terrain and environment, the fading mechanism of land mobile communication is very complicated. However, Rayleigh fading with frequency selectivity is dominant among many channel parameters simulated by mobile communication channel simulator. That is to say, the realization of Rayleigh distribution of signal envelope and uniform distribution of phase is the core of channel simulation.

2. Mathematical principle of1.1Rayleigh fading

Let the stochastic process ξ(t) be expressed as:

In the equation (1), ξc(t) and ξs(t) are in-phase components and orthogonal components of ξ(t), respectively.

It can be proved that the in-phase component ξc(t) and the orthogonal component ξs(t) of a narrow-band stationary Gaussian process with zero mean are also stationary Gaussian processes with the same variance. In addition, ξc(t) and ξs(t) obtained at the same time are uncorrelated or statistically independent. It can also be proved that the one-dimensional distribution of the envelope and the one-dimensional distribution of the phase of the stationary Gaussian narrowband process with zero mean and variance σ2ξ obey Rayleigh distribution, and they are statistically independent in one-dimensional distribution.

To sum up, the one-dimensional distribution of the envelope of the stationary Gaussian narrowband process with zero mean value obeys Rayleigh distribution, and its phase obeys uniform distribution, which are statistically independent. At the same time, the in-phase and orthogonal components of two stationary Gaussian processes can also be combined into a narrow-band stationary Gaussian process with zero mean.

2. 1.2 single-path Rayleigh fading

Assume that the input of the single-path fading channel is:

In formula (2), A(t) and θ(t) are the actual amplitude modulation and phase modulation of the carrier signal with frequency ωc, respectively. Modulated by two independent Gaussian random variables X(t) and Y(t), the output signal So(t) can be expressed as:

Therefore, the random envelope R(t) is Rayleigh distribution, and the random phase φ(t) is uniformly distributed in the range of 0 ~ 2 л.

It can be seen from the above derivation that the quadrature modulation of the input signal is a single-channel frequency-selective Rayleigh fading simulation, and the random interference of the amplitude and phase of the input signal can be realized as required, thus realizing the mathematical model shown in Equation (3).

2. 1.3 multipath Rayleigh fading

To simplify the analysis, it is assumed that the input is a single-frequency sinusoidal signal.

After multipath transmission, the output is:

In Formula (7), αi is the main weighting coefficient of amplitude, τi is the time delay, φi is the random phase, and n is the number of paths.

In the case of only two paths, the output amplitude is:

That is, there is a time delay difference △τ≠0 between the two paths, and the field strength of the synthesized signal varies with the frequency ω. In the actual mobile communication channel, due to multipath transmission, the delay of each path is different, and the relative delay difference is also different, thus causing frequency selective fading.

2.2 Multipath propagation

2.2. 1 multipath propagation path number selection

In mobile communication, when there are more than two scatterers, the received signal will inevitably have frequency selective fading. This simulator uses three paths, that is, it can generate three independent fading paths, thus simulating the actual communication environment more truly.

2.2.2 Determination of multipath propagation delay value

The typical measured maximum multipath delay is 20μs[ 1], the domestic test result is 15μs, and the root mean square delay is about 10μs [1, 2,3]. The scheme adopts flexible selection of multiple delays to accept the root mean square delay of the actual channel. The minimum total delay is 0.2μs and the maximum delay is 10.2μs, including the direct path (delay is 0).

2.3 Determination of Radio Wave Propagation Path Loss

At present, people generally use Okura empirical model to predict the propagation path loss of land mobile communication. However, the application scope of the big warehouse model is: frequency 100 MHz ~ 1500 MHz, base station antenna height 30 ~ 200 m, mobile station antenna height 1 m ~ 10 m, and transmission distance 1 km ~ 20 km. The signal frequency of simulator is 70MHz, and the antenna height of base station is 18m. This is inconsistent with the application scope of the big warehouse model, so the model can not be directly applied to this scheme.

Mr. Li Jianye, a Chinese-American communication expert, put forward the Lee model of radio wave propagation prediction. The model does not specifically limit the antenna height of the base station, and its idea is to calculate the signal transmission loss between regions first, and then calculate the point-to-point transmission loss at a specific location.

Because this simulator simulates the typical path loss in general environment, there is no need to accurately simulate point-to-point transmission to a certain area. Therefore, the calculation of inter-regional radio wave loss by Lee model is suitable for the simulation scheme, and no error correction is needed.

In order to calculate the propagation loss with Lee model, it is necessary to know in advance the determined loss value at the propagation distance of 1 mile (or 1 km) in each environment. However, the simulator simulates the general environment, so it is not necessary to measure one by one. Therefore, in general, the typical broadcast distance of communication room is calculated by Okun model first, and then converted to Lee model. In other words, the developed simulator uses Okun model and Lee model to calculate the radio wave propagation loss.

See table 1 for the specific propagation loss.

Table 1 radio wave propagation path loss

Propagation distance 1km, 8km, 15km, propagation loss linear path 69dB 87dB 9 1dB 93dB, urban environment 98dB134dB145dB154dB, and suburban environment 9. +038dB 147dB open environment 75 db11db122 db131db.

2.4 Doppler frequency shift

Doppler frequency shift is a common phenomenon in mobile communication.

fd=v/λ (9)

In equation (9), v is the speed of the mobile station and λ is the wavelength of the signal. For an actual channel with uniformly distributed channel paths in azimuth, the shape of the spectrum is:

Ω d in equation (10) is the angular frequency corresponding to the maximum Doppler shift caused by the movement of the mobile station, that is:

In order to generate this spectrum, the Gaussian noise used for modulation must have a low-pass spectrum, as shown in equation (12):

Realization method of three-channel simulator

As can be seen from the previous discussion, the main functions of this mobile communication channel simulator are Rayleigh fading, multipath propagation, radio wave propagation path loss, Doppler frequency shift and so on.

3. Implementation method of1Rayleigh fading

According to the formula (1), Rayleigh fading is realized by modulating the input signal with two unrelated low-frequency Gaussian noises to simulate Rayleigh fading with Rayleigh distribution envelope and uniform phase distribution, and the function spectrum of the output signal is determined by the spectrum of low-frequency Gaussian noise. Multipath Rayleigh fading can be synthesized by delayed single-path Rayleigh fading.

3. 1. 1 Generation of low-frequency Gaussian noise

The band-pass Gaussian process spectrum determined by equation (10) is shown in figure 1.

The corresponding low-pass Gaussian process spectrum is shown in Figure 2.

Considering that the frequency response of the filter expressed by the formula (12) is not a rational fraction, it cannot be directly constructed, so it can only be approximated by numbers. According to reference [2], the frequency response of the required filter is:

h(s)= 1/[(0.897 S2+0.3 1s+ 1)(0.897 S2+0.3 1s+ 1)(0.3 1s+ 1)]

Fig. 3 shows the difference between the frequency response of H(s) and that of an ideal filter.

Exchange the above analog filters to get the tap coefficients corresponding to the FIR filter.

Gaussian white noise is generated by MATLAB software, and this white noise is input into the FIR filter above, and the filter output is the required narrow-band Gaussian process.

The output of narrow-band Gaussian process is set to DA, filtered, amplified and impedance matched, and input to the next stage for processing.

3. Realization of1.2 quadrature modulation

There are many ways to realize orthogonal modulation. I/Q modulator of Mini company is adopted in this mobile channel simulator. Its structure is shown in Figure 4.

3.2 Realization of Multipath Propagation

In order to realize the simulation of multipath propagation, the power divider of Mini company is used to split the input signal. Firstly, the input signal is divided into two power channels: an analog direct channel; The other path is divided into three paths of power. After these different delays and narrowband Gaussian orthogonal modulation, the power is synthesized, and the output signal simulates multipath propagation.

In this channel simulator, the propagation path and delay are selected by controlling the analog switch.

3.3 Realization of Simulated Path Loss

In order to simulate the path loss of propagation, the channel simulator is realized by the joint control of fixed attenuator and numerical control attenuator. The basis of attenuation control is table 1.

3.5 Realization method of Doppler frequency shift

According to the conclusion of 3. 1, Doppler frequency shift can be realized by controlling the spectrum of narrow-band Gaussian process. In this simulator, the spectrum control of narrowband Gaussian process can be realized by changing the DA conversion rate of narrowband Gaussian process, so as to realize the simulation of Doppler frequency shift.

3.6 Realization of system control and man-machine interface

The embedded operating system based on single-chip microcomputer AT89C52 is used for system control, which can control digital attenuator, analog switch and so on. And realize a good man-machine interface through keyboard and LCD.

4 conclusion

4. 1 general introduction

The overall structure of this channel simulator is shown in Figure 5.

After the signal is input, it is divided into two paths: one is used as a direct branch; After the delay, the other path is divided into two paths. One path I/Q modulator is modulated by two independent low-frequency Gaussian noises, and the output signal envelope is Rayleigh distribution, and the phase is evenly distributed, thus realizing single-path frequency selective Rayleigh fading. The other path is sent to the next delay unit, which is very important for the above process. The outputs of various I/Q modulators are added in the combiner, and the amplitude envelope of their output signals is Rayleigh distribution and the phase is uniform distribution. With the initial direct signal, Rice channel can also be simulated. The actual path loss is simulated by controlling the numerical control attenuator. In the direct path and delay path, adjustable white noise is superimposed respectively to realize the adjustable output signal-to-noise ratio.

4.2 Function indicator

4.3 Description of test methods for main indicators

4.3. 1 Rayleigh fading test method

Observe the output waveform of the simulator with TEKTRONIX oscilloscope TDS3052. As shown in Figure 6, it can be seen that its envelope is Rayleigh distribution.

4.3.2 Testing method of phase distribution of fading waveform

Li Shayu's graph was tested by LC584A oscilloscope of Lecroy Company, and Figure 7 is the image of 10s point scanning accumulated by this storage oscilloscope. The graph is obtained by controlling the horizontal and vertical deflection of the oscilloscope with two orthogonal low-frequency Gaussian noises. Because the deflection control of noise is 90 relative orientation, the display obtained is equivalent to the polar coordinates of the random variable vector of Rayleigh fading signal output by this simulator. On the arc with any fixed radius about the origin in fig. 7, the uniformity of spot intensity indicates that the phase is uniformly distributed.

4.3.3 Test methods of other indicators

This paper introduces the design and implementation of a mobile communication channel simulator. In this simulator, the signal is modulated in the I/Q modulator to simulate the Rayleigh distribution in the actual channel. Low frequency Gaussian noise data are generated by digital method and Matlab software and stored in EPROM. When the simulator works