Advertisement

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

  • Xin Liu
  • Athina P. Petropulu
  • H. Vincent Poor
  • Visa Koivunen
Open Access
Research Article
Part of the following topical collections:
  1. Interference Management in Wireless Communication Systems: Theory and Applications

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

The transmitted signal of user Open image in new window is of the form

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.

The continuous-time baseband received signal Open image in new window can be expressed as

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.

Sampling the received signal Open image in new window at times Open image in new window we obtain
where Open image in new window is the normalized CFO between the Open image in new window th user and the base station, Open image in new window "*" denotes convolution, and Open image in new window is defined as
The Open image in new window th polyphase component, Open image in new window , can be expressed as
Let us form the vector Open image in new window as Open image in new window . It holds that

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.

  1. (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.

     
  2. (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.

     
  3. (A3)

    The oversampling factor Open image in new window satisfies Open image in new window .

     
  4. (A4)

    The channel coefficients Open image in new window are nonzero.

     
  5. (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 .

     
  6. (A6)

    Either the CFOs are distinct, or the user delays are distinct.

     
  7. (A7)
     
Under assumption Open image in new window , it is easy to verify that the rotated input signals Open image in new window are also i.i.d. with zero mean and nonzero kurtosis. Also, the Open image in new window 's are mutually independent for different Open image in new window 's. Assumptions Open image in new window and Open image in new window are needed for blind MIMO estimation based on (7). Assumptions Open image in new window guarantee that the virtual MIMO channel matrix Open image in new window in (7) has full rank with high probability. Assumption Open image in new window can actually be relaxed. As will be discussed later, (see (18)), the contributions of low-value columns of Open image in new window in (7) can be viewed as noise. This effectively reduces the dimensionality of the problem. Open image in new window and Open image in new window guarantee that Open image in new window will be nonzero for all allowable values of Open image in new window , Open image in new window , and Open image in new window . To see the effect of Open image in new window , let us write the channel matrix Open image in new window as

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

One can apply to (7) any blind source separation algorithm (e.g., [16]) to obtain an estimate of the channel matrix, Open image in new window , which is related to the true matrix as
where Open image in new window is a column permutation matrix, and Open image in new window is a complex diagonal matrix. The method of [16] requires fourth-order cumulants of Open image in new window . Accordingly, the estimate of the decoupled signals Open image in new window within permutation and diagonal complex scalar ambiguities is
Denoting by Open image in new window the diagonal element of Open image in new window , which corresponds to the phase ambiguity of user Open image in new window with delay Open image in new window , the separated signals can be expressed as

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.

If the CFO is large, then a PLL does not suffice, with the severity of the problem depending on the modulation scheme. In this case, the phase of the estimated channel matrix Open image in new window can be used to obtain a CFO estimate. If Open image in new window for all Open image in new window , then it can be easily seen that Open image in new window with
where Open image in new window is a Open image in new window vector with all elements equal to one, and Open image in new window . The least-squares estimates of the CFO can be obtained as

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

On noting that the decoupled signals Open image in new window in (12) are permuted (see (11)) in the same manner as the estimated CFOs in (14), we can use the Open image in new window 's to compensate for the effect of CFO in the decoupled signals in (12) and obtain estimates of the input signals as

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 .

Let us partition the channel matrix Open image in new window into two sub-matrices Open image in new window and Open image in new window , containing, respectively, the columns of Open image in new window corresponding to the main lobe and those corresponding to the sidelobes of the pulse. We can rewrite (7) as follows:
with Open image in new window as defined in (9). Correspondingly, Open image in new window , and Open image in new window . If the sidelobes of the pulse are very low, then Open image in new window can be treated as noise and (16) can be written as

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.

Let Open image in new window be a Nyquist pulse that is positive within its main lobe, that is, Open image in new window for Open image in new window . We further assume Open image in new window is an even function with very low sidelobes. For Open image in new window and Open image in new window ( Open image in new window ) in Open image in new window , the absolute value of the correlation coefficient between Open image in new window and Open image in new window is upper bounded as follows:

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

Proof.

See the appendix.

When Open image in new window is large, the following approximation holds:

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 .

Because Open image in new window should be a Nyquist pulse with small sidelobes and Open image in new window for Open image in new window , it should hold that

where Open image in new window is small.

There are additional constraints that the pulse should satisfy, the most important of which is a bandwidth constraint. Most commercial systems, for example, the IEEE 802.11a, IEEE 802.11b, and IEEE 802.11g wireless local-area networks (WLANs) [18], are equipped with a spectral mask that dictates the maximum allowable spectrum, or equivalently, the maximum symbol rate. This leads to a constraint of the form

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

Based on the above constraints and assuming that Open image in new window satisfies the conditions of Proposition 1, the pulse design problem can be expressed as
The problem of (24a)-(24e) is not easy to solve. Next, we will take steps towards reformulating it into a convex optimization problem. Let Open image in new window be a vector containing samples of Open image in new window taken in Open image in new window , with sampling interval Open image in new window , in which case Open image in new window ( Open image in new window is an integer representing the number of samples in each symbol interval). The objective function (24a) is equivalent to
As Open image in new window is an even symmetric function, the Fourier transform of Open image in new window can be represented as Open image in new window , where Open image in new window , with power spectral density (PSD) equal to Open image in new window . Hence, the constraint (24b) is equivalent to
Because (27) involves an infinite number of constraints, we sample Open image in new window in the frequency domain:

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].

In the discrete-time domain, (24c) is equivalent to

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

Define Open image in new window , with the Open image in new window th element equal to Open image in new window . Equation (24d) is equivalent to
with Open image in new window . Hence the problem of (24a)-(24e) can be reformulated as
Since it involves 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
However, the constraint of (32g) is not a convex constraint. By dropping it, we obtain a semidefinite relaxation of the primal problem [20]. The resulting convex optimization problem is
As we drop the constraint Open image in new window , the resulting Open image in new window might not be of unit rank. In this case, we apply eigen-decomposition to Open image in new window . Let
where Open image in new window is the largest eigenvalue of Open image in new window , and Open image in new window is the corresponding eigenvector. As Open image in new window , its eigenvalues Open image in new window for Open image in new window . If
then Open image in new window can result in a good pulse-shape. If Open image in new window , then it holds that
which indicates that Open image in new window in the problem of (24a)–(24e) is maximized. Moreover, Open image in new window can guarantee the validity of (31b) and (31c). Also, if Open image in new window and Open image in new window , then
This indicates that the PSD of Open image in new window will be under the IEEE 802.11 mask. In the same way, we can prove that
which further indicates that Open image in new window has small sidelobes. Moreover, Open image in new window is small and the validity of (38) implies that

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

In this section, we demonstrate the performance of a pulse designed as described in Section 4.2. We take Open image in new window samples per symbol, that is, Open image in new window , and set Open image in new window . Then we obtain Open image in new window and Open image in new window samples in the time and frequency domains, respectively. We take Open image in new window to be Open image in new window . In Figure 1, we show the ratio Open image in new window of the resulting matrix Open image in new window at different symbol rates, where Open image in new window is the largest eigenvalue of Open image in new window . One can see that the smallest Open image in new window is above Open image in new window , which means that the condition of (35) is satisfied. Therefore, Open image in new window is a good choice of pulse-shape.
Figure 1

Comparison of Open image in new window for different symbol rates.

For symbol rate Open image in new window , or equivalently, Open image in new window , the designed time-domain pulse is shown in Figure 2. For comparison, the Isotropic Orthogonal Transform Algorithm (IOTA) pulse [21] is also shown in the same figure. The corresponding PSDs, along with the IEEE 802.11 spectral mask are given in Figure 3. From the figures we can see that the proposed pulse decreases faster than the IOTA pulse within Open image in new window . The larger the value of Open image in new window , the faster Open image in new window decreases. In Figure 3, one can see that the PSD of the proposed pulse is under the 802.11 mask, while the PSD of the IOTA pulse violates the mask at Open image in new window  MHz.
Figure 2

Pulse-shapes in the time domain for symbol rate 10 M/sec.

Figure 3

Pulse-shapes in the frequency domain for symbol rate 10 M/sec.

For symbol rate Open image in new window , or, Open image in new window , the obtained pulse is given in Figures 4 and 5. We also plot the raised cosine pulse with roll-off factor being equal to Open image in new window . One can see that, in the frequency domain, the proposed pulse is under the 802.11 mask, while in the time domain the proposed pulse is narrower. Note that at this symbol rate, the IOTA pulse cannot meet the mask constraint.
Figure 4

Pulse-shapes in the time domain for symbol rate 12. 19 M/sec.

Figure 5

Pulse-shapes in the frequency domain for symbol rate 12. 19 M/sec.

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.

Eliminating CFO effects from the decoupled users can be done via a PLL, if the CFO is small, or a PLL initialized with a good CFO estimate, if the CFO is large as the PLL by itself would not converge in this case. For the latter case, since we sample around the peak of one user, the CFO estimation formula of (14) requires a small modification before it is applied. Let the Open image in new window sampling points occur at Open image in new window , and let Open image in new window be the phase of the channel matrix corresponding to these sampling points. The least-squares estimate of the CFO Open image in new window can be obtained as

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].

The symbol error rate (SER) performance at Open image in new window  sec, that is, at symbol rate Open image in new window  M/sec, using the waveform of Figure 2, is shown in Figure 6 along with the performance corresponding to the IOTA pulse. We can see that the SER of the proposed pulse is lower; there is an approximate Open image in new window  dB SNR advantage over the IOTA result.
Figure 6

SER performance for different pulse-shapes for symbol rate 10 M/sec, with CFOs randomly chosen within the range Open image in new window .

In Figure 7, we show the SER versus SNR at different symbol rates. First, by taking Open image in new window  sec, or equivalently symbol rate Open image in new window  M/sec, we compare the SER performance of the proposed pulses and the raised cosine pulse with roll-off factor 1. As we can see, the performance of the proposed pulse is better. For example, the proposed pulse can achieve Open image in new window at Open image in new window  dB SNR, while the raised cosine pulse needs Open image in new window  dB SNR to achieve the same SER. In the same figure we show the SER performance of the proposed pulse at symbol rate Open image in new window  M/sec. At this rate, the proposed pulse can achieve an SER of Open image in new window at Open image in new window  dB SNR.
Figure 7

SER performance for different pulse-shapes and different symbol rates, with CFOs randomly chosen within the range Open image in new window .

In Figure 8, we show SER performance for different values of the oversampling factor, Open image in new window , at different symbol rates. For Open image in new window , the sampling occurs evenly within the interval Open image in new window of each received symbol with sampling period Open image in new window . One can see that, for symbol rate Open image in new window  M/sec, when the SNR is higher than Open image in new window  dB, the SER performance improves by increasing Open image in new window from Open image in new window to Open image in new window . For symbol rates equal to Open image in new window  M/sec and Open image in new window  M/sec the SER performance remains almost the same with increasing Open image in new window .
Figure 8

SER performance comparison for different oversampling factors Open image in new window , with CFOs randomly chosen within the range Open image in new window .

In order to demonstrate the effect of the proposed pulse on the condition number of the system matrix, we show in Figure 9 the condition number of Open image in new window corresponding to the proposed and IOTA pulses, averaged over Open image in new window random channels realizations and with Open image in new window . In order to make a fair comparison, the CFOs and random delays were set to be the same for both pulses. No noise was added in the data. The estimated Open image in new window 's were collected from the JADE output, and their condition numbers were calculated. One can see that the proposed pulse results consistently in lower condition number than the IOTA pulse.
Figure 9

Condition number comparison for different pulses, with CFOs randomly chosen within the range Open image in new window .

Next, we show the effect of user delays on performance. As before, one user is delayed by a half-symbol interval, and in addition, a random delay Open image in new window is added to both users to model random delays introduced at the transmitter. In this experiment, the range for the random delay Open image in new window is increased from Open image in new window to Open image in new window . For random delays within Open image in new window , in order to prevent the delay difference of two users from being too small, we select the delays so that their difference is no less than a threshold Open image in new window . The resulting SER performance is shown in Figure 10. When the range of Open image in new window increases from Open image in new window to Open image in new window the performance becomes worse. This is because by increasing the range for the random delay, the signals of the two users overlap by a larger amount, which results in high condition number for the channel matrix Open image in new window . The best performance would be obtained with just the half-symbol delay and no random delays; however, this is not a realistic case.
Figure 10

SER performance comparison for different amounts of random delays, with CFOs randomly chosen within the range Open image in new window .

Figure 11

SER performance comparison for random delay only at different symbol rates, with CFOs randomly chosen within the range Open image in new window .

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.

Finally, we show the effect of CFOs on performance (see Figure 12). In order to highlight the effect of the CFOs, SER results were obtained without intentional delay, with random delays taken in the interval Open image in new window and by setting the delay difference of the two users to be no less than Open image in new window . The normalized CFOs were chosen randomly within the range Open image in new window for Open image in new window , and Open image in new window . For Open image in new window we restricted the smallest difference between two CFOs to be no less than Open image in new window , and for Open image in new window , we set no threshold on the CFO difference of the two users. For Open image in new window , the CFO is quite large, and the PLL by itself is not enough to remove the CFO in the decoupled users. Therefore, we first used the method described in Section 3.2 to estimate the CFOs and then used the PLL to compensate for the residual CFO.
Figure 12

SER performance comparison for random delay only and different amounts of CFO.

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.

References

  1. 1.
    Onozato Y, Liu J, Noguchi S: Stability of a slotted ALOHA system with capture effect. IEEE Transactions on Vehicular Technology 1989, 38(1):31-36. 10.1109/25.31132CrossRefGoogle Scholar
  2. 2.
    Tse D, Viswanath P: Fundamentals of Wireless Communication. Cambridge University Press, Cambridge, UK; 2005.CrossRefMATHGoogle Scholar
  3. 3.
    Brandt-Pearce M: Signal separation using fractional sampling in multiuser communications. IEEE Transactions on Communications 2000, 48(2):242-251. 10.1109/26.823557CrossRefGoogle Scholar
  4. 4.
    Zhang Y, Kassam SA: Blind separation and equalization using fractional sampling of digital communications signals. Signal Processing 2001, 81(12):2591-2608. 10.1016/S0165-1684(01)00155-4CrossRefMATHGoogle Scholar
  5. 5.
    Petropulu AP, Olivieri M, Yu Y, Dong L, Lackpour A: Pulse-shaping for blind multi-user separation in distributed MISO configurations. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '08), March-April 2008, Las Vegas, Nev, USA 2741-2744.Google Scholar
  6. 6.
    Liu X, Oymak S, Petropulu AP, Dandekar KR: Collision resolution based on pulse shape diversity. Proceedings of the IEEE International Workshop on Signal Processing Advances for Wireless Communications (SPAWC '09), June 2009, Perugia, Italy 409-413.Google Scholar
  7. 7.
    Sklar B: Digital Communications: Fundamentals and Applicatons. Prentice Hall, Upper Saddle River, NJ, USA; 2001.Google Scholar
  8. 8.
    Ciblat P, Loubaton P, Serpedin E, Giannakis GB: Performance analysis of blind carrier frequency offset estimators for noncircular transmissions through frequency-selective channels. IEEE Transactions on Signal Processing 2002, 50(1):130-140. 10.1109/78.972489CrossRefGoogle Scholar
  9. 9.
    Ghogho M, Swami A, Durrani T: On blind carrier recovery in time-selective fading channels. Proceedings of the 33rd Asilomar Conference on Signals, Systems, and Computers, 1999, Pacific Grove, Calif, USA 1: 243-247.Google Scholar
  10. 10.
    Gini F, Giannakis GB: Frequency offset and symbol timing recovery in flat-fading channels: a cyclostationary approach. IEEE Transactions on Communications 1998, 46(3):400-411. 10.1109/26.662646MathSciNetCrossRefGoogle Scholar
  11. 11.
    Scott KE, Olasz EB: Simultaneous clock phase and frequency offset estimation. IEEE Transactions on Communications 1995, 43(7):2263-2270. 10.1109/26.392969CrossRefGoogle Scholar
  12. 12.
    Wang Y, Ciblat P, Serpedin E, Loubaton P: Performance analysis of a class of nondata-aided frequency offset and symbol timing estimators for flat-fading channels. IEEE Transactions on Signal Processing 2002, 50(9):2295-2305. 10.1109/TSP.2002.801919CrossRefGoogle Scholar
  13. 13.
    Roman T, Visuri S, Koivunen V: Blind frequency synchronization in OFDM via diagonality criterion. IEEE Transactions on Signal Processing 2006, 54(8):3125-3135.CrossRefGoogle Scholar
  14. 14.
    Wolaver DH: Phase-Locked Loop Circuit Design. Prentice Hall, Englewood Cliffs, NJ, USA; 1991.Google Scholar
  15. 15.
    Sidiropoulos ND, Giannakis GB, Bro R: Blind PARAFAC receivers for DS-CDMA systems. IEEE Transactions on Signal Processing 2000, 48(3):810-823. 10.1109/78.824675CrossRefGoogle Scholar
  16. 16.
    Cardoso JF, Souloumiac A: Blind beamforming for non-Gaussian signals. IEE Proceedings F 1993, 140(6):362-370.Google Scholar
  17. 17.
    Liu X, Kountouriotis J, Petropulu AP, Dandekar KR: ALOHA with collision resolution (ALOHA-CR): theory and software defined radio implementation. IEEE Transactions on Signal Processing 2010, 58(8):4396-4410.MathSciNetCrossRefGoogle Scholar
  18. 18.
    IEEE standard 802.11 http://www.ieee802.org/11/
  19. 19.
    Wu S, Boyd S, Vandenberghe L: FIR filter design via spectral factorization and convex optimization. In Applied and Computational Control, Signals and Circuits. Edited by: Datta BN. Birkhauser, Basel, Switzerland; 1998:215-245.Google Scholar
  20. 20.
    Sidiropoulos ND, Davidson TN, Luo Z-Q: Transmit beamforming for physical-layer multicasting. IEEE Transactions on Signal Processing 2006, 54(6):2239-2251.CrossRefGoogle Scholar
  21. 21.
    Le Floch B, Alard M, Berrou C: Coded orthogonal frequency division multiplex [TV broadcasting]. Proceedings of the IEEE 1995, 83(6):982-996. 10.1109/5.387096CrossRefGoogle Scholar

Copyright information

© Xin Liu et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Xin Liu
    • 1
  • Athina P. Petropulu
    • 1
  • H. Vincent Poor
    • 2
  • Visa Koivunen
    • 3
  1. 1.Electrical and Computer Engineering DepartmentDrexel UniversityUSA
  2. 2.School of Engineering and Applied SciencePrinceton UniversityPrincetonUSA
  3. 3.Signal Processing LaboratoryAalto UniversityAaltoFinland

Personalised recommendations