# Blind Separation of Two Users Based on User Delays and Optimal Pulse-Shape Design

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## Abstract

A wireless network is considered, in which two spatially distributed users transmit narrow-band signals simultaneously over the same channel using the same power. User separation is achieved by oversampling the received signal and formulating a virtual multiple-input multiple-output (MIMO) system based on the resulting polyphase components. Because of oversampling, high correlations can occur between the columns of the virtual MIMO system matrix which can be detrimental to user separation. A novel pulse-shape waveform design is proposed that results in low correlation between the columns of the system matrix, while it exploits all available bandwidth as dictated by a spectral mask. It is also shown that the use of successive interference cancelation in combination with blind source separation further improves the separation performance.

### Keywords

Channel Matrix Blind Source Separation Successive Interference Cancellation Symbol Error Rate Symbol Rate## 1. Introduction

We consider the problem of multiuser separation in wireless networks via approaches that do not use scheduling. This problem is of interest, for example, when traffic is generated in a bursty fashion, in which case fixed bandwidth allocation would result in poor bandwidth utilization. Lack of scheduling results in collisions, that is, users overlapping in time and/or frequency. To separate the colliding users, one could enable multiuser separation via receive antenna diversity, or code diversity, as in code-division multiple-access (CDMA) systems. However, the former requires expensive hardware since multiple transceiver front ends involve significant cost. Further, the use of multiple antennas might not be possible on small-size terminals or devices. CDMA systems require bandwidth expansion, which requires greater spectral resources, and also introduces frequency-selective fading. In the following, we narrow our field of interest to random-access systems that for the aforementioned reasons cannot exploit antenna diversity, and that are inexpensive in terms of bandwidth. In such systems, the use of different power levels by the users can enable user separation by exploiting the capture effect [1], or successive interference cancellation (SIC) [2]. Different power levels can result from different distances between the users and the destination, or could be intentionally assigned to users in order to facilitate user separation. While the former case, when it arises, makes the separation problem much easier, the latter approach might not be efficient, as low-power users suffer from noise and channel effects. In the following, we focus on the most difficult scenario of separating a collision of equal-power users. Almost equal powers would also result from power control. Power control is widely used, hence this scenario is of practical interest.

A delay-division multiple access approach was proposed in [3], which exploits the random delays introduced by transmitters. The approach of [3] considers transmissions of isolated frames. It requires that users have distinct delays, assumes full channel knowledge at the receiver and exploits the edges of a frame over which users do not overlap. Pulse-shape waveform diversity was considered in [4] to separate multiple users in a blind fashion. In [4], the received signal is oversampled and its polyphase components are viewed as independent mixtures of the user signals. User separation is achieved by solving a blind source separation problem. Although no specifics on waveform design are given in [4], the examples used in the simulations of [4] consider wideband waveforms for the users. However, if large bandwidth is available, then CDMA would probably be a better alternative to blind source separation. Pulse-shape diversity is also employed in [5, 6], addressing situations in which the pulse-shape waveforms have bandwidth constraints.

In this paper we follow the oversampling approach of [4], with the following differences. First, we introduce an intentional half-symbol delay between the two users. Second, both users use the same optimally designed pulse-shape waveform. Third, we use successive interference cancelation in combination with blind source separation to further improve the separation performance.

The paper is organized as follows. In Section 2, we describe the problem formulation. The proposed blind method is presented in Section 3. The Pulse-shape design is derived in Section 4. Simulation results validating the proposed method are presented in Section 5, while concluding remarks are given in Section 6.

Notation 1.

Bold capitals denote matrices. Bold lower-case symbols denote vectors. The superscript Open image in new window denotes transposition. The superscript Open image in new window denotes the pseudoinverse. Open image in new window denotes the diagonal matrix with diagonal elements the elements of Open image in new window . Open image in new window denotes rounding down to the nearest integer. Open image in new window denotes the trace of its argument. Open image in new window denotes the phase of its argument.

## 2. Problem Formulation

We consider a distributed antenna system, in which Open image in new window users transmit simultaneously to a base station. Although much of this paper studies the case Open image in new window , for reasons that will be explained later, we will keep the Open image in new window user notation throughout. Narrow-band transmission is assumed here, in which the channel between any user and the base station undergoes flat fading. In addition, quasi-static fading is assumed, that is, the channel gains remain fixed during several symbols.

where Open image in new window is the Open image in new window th symbol of user Open image in new window ; Open image in new window is the symbol period; Open image in new window is a pulse-shaping function with support Open image in new window , where Open image in new window is an integer.

where Open image in new window denotes the complex channel gain between the Open image in new window th user and the base station; Open image in new window denotes the delay of the Open image in new window th user; Open image in new window is the carrier frequency offset (CFO) of the Open image in new window th user, arising due to relative motion or oscillator mismatch between receive and transmitter oscillators, and Open image in new window represents noise.

Our objective is to obtain an estimate of each user sequence, Open image in new window , up to a complex scalar multiple that is independent of Open image in new window . The estimation will be based on the received signal only, while channel gains, CFOs and user delays are assumed to be unknown. During the recovery process, there is permutation ambiguity, that is, the order of the users may be lost and again the user signals will be recovered up to a scalar multiple. However, these are considered to be trivial ambiguities and are inherent in blind estimation problems.

We should note that typically, in high-speed communication systems, the main lobes of the pulse-shape functions overlap by Open image in new window [7]. This extended time support allows for better frequency concentration, or equivalently, lower spectral occupancy for the transmission of each symbol. However, it introduces intersymbol interference (ISI). Examining Open image in new window for Open image in new window (see (1)), we note the contribution of the Open image in new window th symbol, the contribution of symbol Open image in new window due to the main lobe of Open image in new window , and also contributions of symbols Open image in new window due to the sidelobes of Open image in new window , respectively. If Open image in new window is a Nyquist pulse and samples are taken at times Open image in new window , the overlap does not play any role. However, when we obtain more than one sample during the symbol interval, we expect ISI effects.

where Open image in new window is a Open image in new window matrix whose Open image in new window th row equals Open image in new window ; Open image in new window ; and Open image in new window . This is a Open image in new window instantaneous multiple-input multiple-output (MIMO) problem. Under certain assumptions, to be provided in the following section, the channel matrix Open image in new window is identifiable, and the vector Open image in new window can be recovered up to certain ambiguities. In particular, for each Open image in new window , we get Open image in new window different versions of Open image in new window , that is, Open image in new window within a scalar ambiguity. The effects of the CFO on the separated signals can be mitigated by using any of the existing single-CFO estimation techniques (e.g., [8, 9, 10, 11, 12, 13]), or a simple phase-locked loop (PLL) device [14].

## 3. Blind User Separation

### 3.1. Assumptions

The following assumptions are sufficient for user separation.

- (A1)
Each of the elements of Open image in new window , as a function of Open image in new window , is a zero-mean, complex Gaussian stationary random process with variance Open image in new window , and is independent of the inputs.

- (A2)
For each Open image in new window , Open image in new window is independent and identically distributed (i.i.d.) with zero mean and nonzero kurtosis, that is, Open image in new window The Open image in new window 's are mutually independent, and each user has unit transmission power.

- (A3)
The oversampling factor Open image in new window satisfies Open image in new window .

- (A4)
The channel coefficients Open image in new window are nonzero.

- (A5)
The user delays, Open image in new window , Open image in new window in (3) are randomly distributed in the interval Open image in new window .

- (A6)
Either the CFOs are distinct, or the user delays are distinct.

- (A7)
Open image in new window for Open image in new window ; and Open image in new window only for Open image in new window and Open image in new window .

and consider the case in which all users have the same delays, that is, Open image in new window . If the CFOs are different, **A** has full column rank. Even if the CFOs are not distinct, the columns of the channel matrix can be viewed as having been drawn independently from an absolutely continuous distribution, and thus the channel matrix has full rank with probability one [15].

### 3.2. Channel Estimation and User Separation

At this point, the users' signals have been decoupled, and all that is left is to mitigate the CFO in each recovered signal. This can be achieved with any of the existing single CFO estimation methods, such as [8, 9, 10, 11, 12], or [13]. Alternatively, if the CFO is very small, then we can estimate it and at the same time mitigate its effects using a PLL. We should note here that even a very small CFO needs to be mitigated in order to have good symbol recovery. For example, for 4-ary quadrature amplitude modulation (4QAM) signals and without CFO compensation, even if the normalized CFO Open image in new window is as small as Open image in new window , the constellation will be rotated to a wrong position after Open image in new window samples.

where Open image in new window is the Open image in new window th element of Open image in new window .

where Open image in new window with Open image in new window . In order to resolve user permutation and shift ambiguities, one can use user IDs embedded in the data [17].

Although in theory, under the above stated conditions, the matrix Open image in new window has full rank for any number of users, Open image in new window , the matrix condition number may become too high when CFOs or delay differences between users become small. As Open image in new window increases, the latter problem will escalate. Further, for large Open image in new window , the oversampling factor, Open image in new window , must be large. However, as Open image in new window increases, neighboring pulse-shape function samples will be close to each other, and the condition number of Open image in new window will increase. Therefore, the shape of the pulse-shape function sets a limit on the oversampling factor one can use and thus on the number of users one can separate. Recognizing that the above are difficult issues to deal with, we next focus on the two-user case. Further, we propose to introduce an intentional delay of Open image in new window between the two users, in addition to any small random delays there exist in the system.

The performance of user separation depends on the pulse-shape function and also on the location of the samples. Although uniform sampling was described above, non-uniform sampling can also be used, in which case the expressions would require some straightforward modifications. If the samples correspond to a low-value region of the pulse, the corresponding polyphase components will suffer from low signal-to-noise ratio. Also, if the sampling points are close to each other, then the condition number of Open image in new window will increase. Therefore, one should select the sampling points so that the corresponding samples are all above some threshold and the sampling points are as separated as possible. The effect of pulse-shape and optimal shape design will be discussed in the following section.

## 4. Pulse-Shape Design

In this section, we first investigate the effects of pulse-shape on the condition number of Open image in new window . Since the condition number of a matrix increases as the column correlation increases, we next look at the correlation between the columns of Open image in new window .

### 4.1. Pulse Effects

In order to maintain a well-conditioned Open image in new window , the correlation coefficient between its columns should be low. Let us further divide the matrix Open image in new window into Open image in new window and Open image in new window . The elements of Open image in new window are samples from the decreasing part of the main lobe of the pulse. On the other hand, the elements of Open image in new window are from the increasing part of the main lobe of the pulse. Thus, the correlation coefficient of Open image in new window and Open image in new window is smaller than the correlation coefficient of Open image in new window and Open image in new window , or that of Open image in new window and Open image in new window . Thus, we focus on the effects of the pulse on the column correlations within Open image in new window and Open image in new window .

Proposition 1.

where Open image in new window denotes the first-order derivative of Open image in new window .

Proof.

See the appendix.

Thus, for fixed Open image in new window and Open image in new window , the correlation coefficient between Open image in new window and Open image in new window decreases with increasing Open image in new window . It can be shown that the same holds for the correlation coefficient between Open image in new window and Open image in new window .

where Open image in new window is small.

where Open image in new window is the Fourier transform of Open image in new window , and Open image in new window denotes the spectral mask.

### 4.2. Optimum Pulse Design

*Proposition 1*, the pulse design problem can be expressed as

where Open image in new window is the number of samples in Open image in new window . In order for (28) to be a good approximation of (27), Open image in new window should be on the order of Open image in new window [19].

where Open image in new window is small and Open image in new window with Open image in new window leading zeros.

*maximization*of a convex function, (31a) is not a convex optimization problem. Letting Open image in new window , Open image in new window should be a positive semidefinite matrix of rank Open image in new window . The problem of (31a)–(31e) is equivalent to

which indicates that, if we sample at intervals Open image in new window , the interference from neighboring symbols can be neglected.

If Open image in new window , then it holds that Open image in new window . Also, (33e) requires that the Open image in new window th element of Open image in new window be greater than zero for Open image in new window . Hence, Open image in new window or Open image in new window for Open image in new window . Thus, within its mainlobe, Open image in new window is greater than zero, or its amplitude becomes very small.

## 5. Simulation Results

### 5.1. Pulse Design Examples

### 5.2. SER Performance

In this section, we demonstrate the performance of the proposed user separation approach via simulations. We consider a two-user system. The channel coefficients Open image in new window and Open image in new window are taken to be zero-mean complex with unit amplitude and phase that is randomly distributed in Open image in new window . The CFOs are chosen randomly in the range Open image in new window . The input signals are Open image in new window -QAM containing Open image in new window symbols. The estimation results are averaged over Open image in new window independent channels, and Open image in new window Monte-Carlo runs for each channel. One user is intentionally delayed by half a symbol and in addition, small delays, taken randomly from the interval Open image in new window , are introduced to each user.

In our simulations, we combine blind source separation method with SIC [2]. For blind source separation the Joint Approximate Diagonalization of Eigenmatrices (JADE) algorithm was used, which was downloaded from http://perso.telecom-paristech.fr/~cardoso/Algo/Jade/jade.m. We first apply JADE to decouple the users, and then correct the decoupled users' CFOs. Subsequently, the strongest user, that is, the one which shows the best concentration around the nominal constellation is deflated from the received polyphase components to detect the other user. SIC requires that the first user should be detected very well. To achieve this, the sampling points are chosen around the peak of one user signal, so that ISI and interuser interference effects are minimized.

where Open image in new window is the Open image in new window th element of Open image in new window .

In this experiment, the pulse has time support Open image in new window . We take Open image in new window polyphase components of the received symbols, each consisting of samples taken evenly over the interval Open image in new window , with sampling period Open image in new window . In order to sample around the peak of one user, we used the true shift values. However, in a realistic scenario this information would be obtained via synchronization pilots [17].

Next, to show the advantage of the intentional half-symbol delay, we consider a case without intentional delay, with random user delays only. The random delays of both users are taken within Open image in new window . In order to prevent worsening of performance we restricted the smallest delay difference between two users to be no less than Open image in new window . We compare the SER performance of the proposed pulse with IOTA and raised cosine pulses at different symbol rates. Firstly, comparing the corresponding curves in Figure 10, one can first see that without the intentional delay the SER performance decreases. In particular, for the proposed pulse in order to achieve SER Open image in new window , we need an SNR of Open image in new window dB and Open image in new window dB for symbol rates Open image in new window M/sec and Open image in new window M/sec, respectively. Secondly, the SER performance of the proposed pulse is still better than that of IOTA and raised cosine pulses at the corresponding symbol rate.

The quality of the CFO estimates depends on the accuracy of the channel matrix estimate. Since low-magnitude elements of the channel matrix correspond to low values of the pulse, and as such are susceptible to errors, we set a threshold, Open image in new window , defined as Open image in new window , and for CFO estimation, we only use elements of Open image in new window whose amplitudes are greater than Open image in new window . In this experiment, we took Open image in new window . The CFO effects were eliminated via a PLL initialized with the CFO estimate of (40). One can see that the larger Open image in new window gives better performance. It is important to note that the large CFOs involve bandwidth expansion. The percentage of bandwidth expansion can be calculated as Open image in new window , where Open image in new window MHz is the bandwidth of the pulse. For Open image in new window and Open image in new window , the percentages of bandwidth expansion for symbol rates Open image in new window M/sec, Open image in new window M/sec and Open image in new window M/sec are, respectively, Open image in new window , Open image in new window , and Open image in new window .

## 6. Conclusions

A blind Open image in new window -user separation scheme has been proposed that relies on intentional user delays, optimal pulse-shape waveform design, and also combines blind user separation with SIC. The proposed approach achieves low SER at a reasonable SNR level. Simulation results for the Open image in new window case have confirmed that the proposed pulse design leads to SER performance better than that of conventional pulse-shape waveforms. The intentional delay was equal to half a symbol interval, which means that the users still overlap significantly during their transmissions. The use of intentional delay is necessitated by the fact that, although small user delay and CFO differences help preserve the identifiability of the problem, in practice, they may not suffice to separate the users. Also, although the proposed approach can work for any number of users, as the number of users increases, the CFO and delay differences become smaller, which makes the separation more difficult. Based on our experiments, small CFO differences did not affect performance. Although introducing large intentional CFO differences among users could help, that would increase the effective bandwidth. A new ALOHA-type protocol that separates second-order collision based on the ideas described in this paper, along with a software-defined radio implementation can be found in [17].

## Notes

### Acknowledgment

This research was been supported by the National Science Foundation under Grants CNS-09-16947 and CNS-09-05398 and by the Office of Naval Research under Grants N00014-07-1-0500 and N00014-09-1-0342.

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