- Open Access
Performance analysis of distributed ZF beamforming in the presence of CFO
© Lebrun et al; licensee Springer. 2011
Received: 18 March 2011
Accepted: 23 November 2011
Published: 23 November 2011
We study the effects of residual carrier frequency offset (CFO) on the performance of the distributed zero-forcing (ZF) beamformer. Coordinated transmissions, where multiple cells cooperate to simultaneously transmit toward one or multiple receivers, have gained much attention as a means to provide the spectral efficiency and data rate targeted by emerging standards. Such schemes exploit multiple transmitters to create a virtual array of antennas to mitigate the co-channel interference and provide the gains of multi-antenna systems. Here, we focus on a distributed scenario where the transmit nodes share the same data but have only the local knowledge of the channels. Considering the distributed nature of such schemes, time/frequency synchronization among the cooperating transmitters is required to guarantee good performance. However, due to the Doppler effect and the non-idealities inherent to the local oscillator embedded in each wireless transceiver, the carrier frequency at each transmitter deviates from the desired one. Even when the transmitters perform frequency synchronization before transmission, a residual CFO is to be expected that degrades the performance of the system due to the in-phase misalignment of the incoming streams. This paper presents the losses of the signal-to-noise ratio gain analytically and the diversity order semi-numerically of the distributed ZF beamformer for the ideal case and in the presence of a residual CFO. We illustrate our results and their accuracy through simulations.
Coordinated transmissions, where multiple cells cooperate to transmit simultaneously toward one or multiple receivers, have gained much attention recently as a means to provide the spectral efficiency and data rate targeted by emerging standards [1, 2]. Such schemes create a virtual array of antennas to provide the gains of multi-antenna systems and aid in mitigating the interference in cellular networks . They have the potential to improve the performance or the per-user capacity of the users at the cell edge. This benefits the overall network performance at a low cost, i.e., no need for new infrastructures or expensive devices.
In coordinated transmissions, the beamforming weights are chosen according to the level of knowledge available at each transmitter, i.e., the data and channel state information (CSI), and the degree of cooperation between the transmit cells. The exchange of full CSI and data information between the transmit cells enables the joint computation of the beamforming weights . However, even if this scheme achieves optimal performance, it requires a central coordinator to gather all CSI to jointly compute the beamforming weights and then to redistribute these weights to each transmit cell . The implementation of coordinated transmissions in a distributed network is hence challenging due to the complexity of the joint beamforming and the extensive sharing of information between the transmit cells where backhaul limitations and latency issues arise [6, 7]. In addition, considering a source broadcasting its symbol information to two relay stations, the symbol information is then readily available at both relays [8, 9]. However, in such a case, the sharing of the CSI is difficult, especially when the receivers are moving and their channels are varying.
Conversely, distributed (yet coordinated) beamforming schemes where each cell exploits the knowledge of the information data but only a limited knowledge about the channels and other transmitter weights are a more practical alternative . Additionally, distributed beamformers are computationally less intensive than their fully coordinated counterpart since they only require the local processing of the beamforming weights. Besides the difficult exchange of the data and CSI between the transmit cells, coordinated systems require perfect synchronization between different cells; this is also challenging to achieve.
Carrier frequency offset (CFO) is caused by the mobility of the wireless devices (Doppler effect) and by the non-ideality of the local oscillator embedded in each wireless transceiver. CFO is a major source of impairment in orthogonal frequency division modulation (OFDM) schemes and must be compensated to obtain acceptable performance . In point-to-point communication, the carrier frequency mismatch causes signal-to-noise ratio (SNR) loss, a phase rotation of the symbols and intercarrier interference (ICI). In coordinated communications, each stream originates from a distinct source, each with a different frequency error. As a result, the receiver needs to cope with multiple CFOs and the impacts of CFO in coordinated schemes are hence worse than for point-to-point communications. Because the frequency offset translates into the possibly destructive combination of the incoming streams, it is impossible to correct the multiple CFOs at the receiver. The primary method for mitigating the effects of CFO consists then in compensating the frequency offset before transmission, i.e., it must be corrected by each source [11, 12]. Methods to estimate the multiple CFOs at the receiver, which requires a different approach compared to point-to-point communications, have also been proposed [13–15].
In practical scenarios, the perfect synchronization of the wireless devices is very challenging and a residual CFO is to be expected even after synchronization . It is therefore of interest to understand the impacts of residual CFO on coordinated communications. Earlier work focused on the results of residual CFO on the bit error rate (BER) performance for cooperative space-frequency/block code systems [17, 18]. In addition, simulation results exist on the impacts of CFO in cooperative multi-user MIMO systems . Zarikoff also shows that in multiuser systems the CFOs degrade the accuracy of the beamformer, hence decreasing the capacity . Mudumbai et al., consider a cluster of single-antenna sensor nodes communicating with a distant receiver, where the sensor nodes share a consistent carrier signal . They identify the time-varying phase drift from the oscillator to dominate the performance degradation and study the resulting SNR loss. Works also include the study of the beamforming gain degradation caused by phase offset estimation errors . These results are complementary to the results presented here. While they study the impacts of phase noise and phase drift in distributed systems, we consider the negative impacts of the time- and CFO-dependent phase mismatch of the incoming streams on the SNR gain and diversity order. Derivations of the SNR and diversity gains without CFO are well known for single-user (SU) scenarios  and have also been proposed for amplify-and-forward scenarios [23–25], which are different scenarios to the one we consider in this work. To the best of our knowledge, literature does not evaluate the effects of residual CFO on the SNR gain and diversity order for distributed beamforming schemes where the transmitters share the same time and frequency resources for transmitting a common data toward multiple receivers. For the scenario of interest in this work, no analytical or simulation results exist.
In this paper, we study the effects of a residual CFO on the performance of the distributed zero-forcing (ZF) beamforming scheme, i.e., where both transmitters simultaneously transmit a shared data toward both receivers while suppressing the co-channel interference. We first introduce the system model and derive analytically the SNR gain and the diversity order numerically in the ideal case, i.e., assuming perfect synchronization. Next, we propose the derivations of the SNR gain and diversity order when residual CFO is present. We show that the performance decreases with time as the residual CFO introduces a misalignment of the incoming streams. Finally, simulation results confirm the analytical derivations.
The outline is as follows: In Section 2, the system model of the considered coordinated transmission scheme with perfect synchronization is introduced. The derivations of the average SNR gain and the diversity order are given in Section 3. In Section 4, the system model is defined for multiple CFOs from different transmitters and the derivations for the average SNR and diversity gains with CFO are presented. Simulations in Section 5 show the performance of the cooperative scheme for both the ideal case and when residual CFO is present. These results are discussed together with the proposed analytical derivations. Section 6 concludes our work.
Notations: The following notations are used: The vectors and matrices are in boldface letters, vectors are denoted by lower-case and matrices by capital letters. The superscript (·) H denotes the Hermitian transpose operator, and (·)† denotes the pseudo-inverse. E[·] is the expectation operator, I N is an identity matrix of size (N × N) and ℂN × 1denotes the set of complex vectors of size (N × 1). The definition x ~ ℂN(0, σ2I N ) means that the vector x of size N × 1 has zero-mean Gaussian distributed independent complex elements with variance σ2. We define a n as the n th element of the vector a.
2 System model
We consider that a prior-to-transmission frequency synchronization is performed so that only a residual CFO is present at the receivers. The initial phase error of the local oscillator at the transmitter side creates a phase error when down (up) converting the receive (transmit) signal. However, this phase error is included in the channel response when estimating the channel. Since the beam-forming weights are computed based on the channel estimates, the beamformer compensates also for this phase error. As a result, this initial phase error can be omitted [15, 26].
where the term n ∈ ℂ1 × 1 is the zero-mean circularly symmetric complex additive white Gaussian noise (AWGN) with variance .
We consider a ZF beamformer. Such a beamformer exploits the knowledge of the channels from its own antennas to choose the beamforming vector that maximizes the energy while placing the nulls in the direction of the non-targeted user. The computation of the beamforming weights can be decomposed into two steps: null beamforming and maximal energy beamforming. We focus on the computation of the weights for Tx1, and a similar approach can be done for Tx2.
2.0.1 Null beamforming
2.0.2 Maximum-ratio combining
which fulfills the power constraint in (2). Since the ZF beamforming weights lay in the null space of the non-targeted user, the received signal is interference free.
We have expressed the transmit and received signals and defined the beamforming weights for the considered scheme. In the next section, we derive the resulting SNR and diversity gains assuming perfect synchronization, i.e., no CFO.
3 SNR and diversity gains
The SNR gain comes from the (coherent) addition of the incoming streams at the receiver antennas. It is obtained by averaging the instantaneous SNR over the channel realizations and indicates the SNR gain over the single-user (SU) single-input-single-output (SISO) case. We derive the resulting average SNR (Section 3.1) and to compare it to the SNR gain in SU scenarios. The diversity gain is obtained by combining the multiple replicas of the signal collected at the receiver. The diversity order is calculated by evaluating the resulting slope of the average bit error rate curve, and the derivation of the diversity order is proposed in Section 3.2.
3.1 Average SNR gain
From these results, with P = P1 + P2 (P1 = P2) and N t = 2, the SNR of the ZF coordinated and EGC schemes is equal. This is expected since the two cells transmit with equal power and because one degree of freedom is used by the ZF scheme to cancel interference. However, with N t ≥ 3, the SNR gain of the ZF coordinated scheme outperforms the EGC and MRC beamformers, i.e., G = 8.77 dB while GMRC = 7.78 dB and GEGC = 7.1 dB.
3.2 Diversity order
The diversity gain is obtained by combining the multiple replicas of the signal collected at the receiver. The diversity order is calculated by evaluating the resulting slope of the average bit error rate curve.
However, no general closed form of the equivalent PDF can be obtained. Therefore, we compute the integral in Equation (28) numerically and for a varying number of antennas. These numerical approximations of the error probability Pe are then used to compute the diversity order of the considered cooperative scheme. With two antennas at each transmitter, each chi-random variable has 2(N t - 1) = 2 degrees of freedom. Because a chi-RV with two degrees of freedom has a Rayleigh distribution, the diversity order is then equivalent to a single-user EGC scenario with two antennas and provides then a diversity order of 2 . From the numerical analysis and simulation results in Section 5.1, we recognize a diversity order of 2(N t - 1). This result should be expected as one degree of freedom cancels the interference toward the non-intended receiver. A similar approach can be employed for the second receiver.
4 Distributed ZF beamforming: impact of CFO
From the results given in the Section 2, good performance is expected from the distributed ZF beamformer thanks to the added SNR and diversity gains. In this section, we discuss the effects of the residual CFO on those gains. In 4.1, we extend the system model given in Section 2 to the case where CFO is present. Then, the average SNR gain and diversity order are derived for the general case (N t transmit antennas) in Sections 4.2 and 4.3.
4.1 System model with CFOs
4.2 Average SNR gain with CFO
4.2.1 Fixed CFO
We can observe that the average SNR gain degrades, following a sinc function with the parameters f Δ and T p . As a result, for a long enough T p , the sinc function produces a zero, i.e., no SNR gain is obtained.
4.2.2 Uniform CFO
4.3 Diversity order
4.3.1 Fixed CFO
However, similar to the results in Section 3.2, the third term in Equation (47) is not independent from the two others and the expectation operator cannot be separated, obtaining the equivalent PDF (and hence P e ) in a general closed form is difficult.
We then use these numerical approximations of the error probability Pe to compute the diversity order of the considered cooperative scheme when CFO is present. The Section 5.2 presents the resulting diversity order.
4.3.2 Uniform CFO
We study the effects of a random and uniformly distributed CFO on the diversity order. In such a case, the diversity order is obtained by numerically approximating the PDF of the equivalent channel for a random variable Δ f uniformly distributed over the interval [-f c , f c ]. Section 5 presents the results from the diversity order with an uniformly distributed CFO.
5 Simulation results
This section aims at comparing the SNR and diversity gains of the distributed ZF beamforming scheme without synchronization errors (Section 2) with respect to the case with residual CFO (Section 4) and verifying the proposed derivations.
As already mentioned in the text, we consider a distributed transmission scenario where two independent cells transmit simultaneously a same data toward two receivers. The simulations are performed for the IEEE802.11 n system  with a 5 GHz carrier frequency and a 20 MHz bandwidth. We consider an uncoded OFDM scheme with 64 subcarriers. A power of 1 is allocated from a receiver to each transmitter, i.e., P1 = P2 = 1. The multiple CFOs are assumed known at the receiver where a zero-forcing frequency domain equalizer is applied for synchronization. Presynchronization of the frequency offset is performed at the transmitters so that only residual CFO f Δ is left. The f Δ is expressed in part per million (ppm) with respect to the system carrier frequency. We assume the network protocol to guarantee the transmitters to be time synchronized and assume no initial phase offset between the transmitters. Each scenario can be described as NTX(N t ) × NRX(N r ), where NTX denotes the number of transmitters and NRX the number of receivers, N t is the number of transmit antennas at each transmitter, and N r is the number of antennas at each receiver.
5.1 Performance of cooperative beamforming: ideal case
5.2 Performance of coordinated beamforming with frequency offset
Here, we study the effects of residual CFO on the SNR and diversity gains. In the simulations, we assume two transmitters each equipped with two or more antennas, i.e., N t ≥ 2.
5.2.1 SNR gain with residual CFO
5.2.2 Diversity order with residual CFO
5.2.3 Tightness of the analytical results
5.3 Scenario study for 802.11 and 3GPP-LTE systems
Scenario study of the performance degradation due to CFO in coordinated beamforming for the IEEE802.11n and 3GPP-LTE systems
CFO (f Δ )
SNR loss (dB)
Diversity loss (%)
240μ s + 50μ s
500μ s + 400μ s
Both systems experience a delay between the synchronization stage and the transmission stage, e.g., the synchronization is based on a feedback from the receiver terminal . Therefore, we consider a delay of 50μ s for the IEEE802.11 n, e.g., equivalent to the DCF interframe space (DIFS). And a delay of 400μ s for the 3GPP-LTE scenario, e.g., because the synchronization occurs within the third OFDM symbol and due to timing advance requirements and control signaling between the transmit cells.
The Table 1 indicates, for representative values of transmit duration, time offset and range of frequency offsets, the SNR and diversity losses in both systems evaluated with Monte Carlo simulations. It shows that the SNR and diversity gains of coordinated beamforming schemes degrade significantly even when the CFO is precompensated before transmission and is worse than a single-user scheme.
This paper proposes the study of the effects of residual CFO on the ZF distributed beamforming scheme, where both transmitters simultaneously transmit toward two receivers. Analytical derivations and numerical approximations of the SNR and the diversity loss introduced by a fixed and uniform residual CFO, as well as for perfect synchronization, were developed. Results show that, even with a prior-to-transmission synchronization, the residual CFO degrades significantly the SNR and diversity gains. As a result, additional efforts for the estimation and correction of the frequency offset and for reducing the latency between the synchronization stage and the transmission stage are necessary to achieve the gain promised by coordinated transmissions.
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