Joint channel and phase noise estimation for mmWave full-duplex communication systems
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Abstract
Full-duplex (FD) communication at millimeter-wave (mmWave) frequencies suffers from a strong self-interference (SI) signal, which can only be partially canceled using conventional RF cancelation techniques. This is because current digital SI cancellation techniques, designed for microwave frequencies, ignore the rapid phase noise (PN) variation at mmWave frequencies, which can lead to large estimation errors. In this work, we consider a multiple-input multiple-output mmWave FD communication system. We propose an extended Kalman filter-based estimation algorithm to track the rapid variation of PN at mmWave frequencies. We derive a lower bound for the estimation error of PN at mmWave and numerically show that the mean square error performance of the proposed estimator approaches the lower bound. We also simulate the bit error rate performance of the proposed system and show the effectiveness of a digital canceler, which uses the proposed estimator to estimate the SI channel. The results show that for a 2×2 FD system with 64−QAM modulation and PN variance of 10^{−4}, the residual SI power can be reduced to − 25 dB and − 40 dB, respectively, for signal-to-interference ratio of 0 and 15 dB.
Keywords
Full-duplex Millimeter-wave Joint channel and PN estimation Residual self-interference power Synchronization1 Introduction
The next generation of wireless communication technologies, known as 5G, are expected to offer multi-gigabit data rates to mobile users [1, 2]. This has prompted wireless service providers to seek higher bandwidth at less crowded millimeter-wave (mmWave) frequencies. The short wave lengths of mmWave frequencies also allow for practical implementations of base stations with large number of antennas known as massive multiple-input multiple-output (MIMO) system, which is another promising technology for 5G networks [3]. Given these capabilities offered by mmWave communication, these systems have become increasingly popular in academia and industry. While MIMO systems can fully benefit from the capabilities offered by communication at mmWave frequencies [3], due to large peak-to-average power ratio, orthogonal frequency division modulation (OFDM) is not popular for mmWave communication [4]. Since there is still an open debate about the modulation type at mmWave frequencies [5], we do not consider OFDM in this work.
Full-duplex (FD) communication has also emerged recently as a promising wireless technology, which allows for efficient use of bandwidth by enabling in-band transmission and reception [6, 7, 8, 9]. The major obstacle in exploiting the full potential of FD communication is the self-interference (SI) signal, which is significantly stronger than the desired communication signal [10, 11]. The power of the SI signal can be reduced via two different suppression techniques: (i) passive suppression, where transmit and receive antennas are physically isolated to reduce the leakage of the transmit signal into the RF front end of the receiver chain, and (ii) active suppression, where the SI signal is suppressed via subtracting the analog replica of SI signal from the received signal [6]. The experimental results at microwave frequencies show that the successive combination of passive and active suppression can reduce the SI signal power to the receiver noise floor [12]. For this reason, in the majority of the radio architectures proposed for FD communication at microwave frequencies, the residual SI signal at baseband is treated as noise [6, 13, 14, 15]. The cancelation techniques that treat the SI signal as noise suffer from two fundamental problems: (i) they assume Gaussian distribution of the SI signal. However, as explained in [16, 17], the SI signal has a strong line of sight (LoS) component, and hence, it is not Gaussian, and (ii) treating SI as noise requires statistical knowledge of the SI channel, which might not be available.
Recently, SI channel measurements have been carried out for FD communication at mmWave frequencies [17, 18]. The measurements indicate that, as opposed to the microwave frequency band, the SI channel at mmWave has a non-line-of-sight (NLoS) component, which cannot be canceled using passive and active suppression techniques. This partial suppression of the SI signal results in a large residual SI signal at baseband, which is still significantly higher than the receiver noise floor [17]. Another challenge for mmWave FD communication systems is that the oscillator phase noise (PN) is large and rapidly changing [19]. Thus, the majority of the existing techniques for residual SI signal cancelation at baseband, which assume a very steady oscillator PN [20, 21, 22, 23] cannot be used for FD mmWave communication. We note that the important aspect of mmWave communication considered in this paper is the estimation of fast varying PN. This fast variation of PN is in the order of symbol time at mmWave [24].
- 1.
We construct a state vector for the joint estimation of the channel and PN and propose an algorithm based on extended Kalman filtering technique to track the fast PN variation at mmWave band.
- 2.
We derive the lower bound on the estimation error of the proposed estimator and numerically show that the proposed estimator reaches the performance of the lower bound. We also show the effectiveness of a digital SI cancelation, which uses the proposed estimation technique to estimate the SI channel.
- 3.
We present simulation results to show the mean square error (MSE) and bit error rate (BER) performance of a mmWave FD MIMO system with different PN variances and signal-to-interference ratios (SIR). The results show that for a 2×2 FD system with 64−QAM modulation and PN variance of 10^{−4}, the residual SI power can be reduced to − 25 dB and − 40 dB, respectively, for signal-to-interference ratio of 0 and 15 dB.
Important symbols used in this paper
Symbol | Description |
---|---|
y(n) | The (N_{r}×1) vector of received symbols. |
x(n) | The (N_{t}×1) vector of transmitted symbols. |
x^{SI}(n) | The (N_{t}×1) vector of self-interfering (SI) symbols. |
w(n) | The (N_{r}×1) Gaussian noise vector. |
\(\theta _{i}^{[m]}(n)\) | The time-varying phase noise of ith oscillator and m∈{t=transmit,r=receive,SI}. |
H(n) | The (N_{r}×N_{t}) communication channel. |
H^{SI}(n) | The (N_{r}×N_{t}) self-interference (SI) channel. |
\(\bar {\mathbf {H}}(n)\) | The (N_{r}×2N_{t}) state transition matrix for joint PN and channel estimation. |
β(n) | The (2N_{t}×1) state vector for joint channel and PN estimation. |
2 System model
- 1.The same number of transmit and receive antennas for both nodes: We assume both nodes in the considered FD communication system have the same number of transmit and receive antennas.
- 2.
Modeling of RF impairments: RF impairments due to imperfect transmitter and receiver chain electronics have been shown to significantly degrade the performance of the analog cancelation techniques [25, 26]. Since the focus of this work is residual SI cancelation, we only include PN in our model and assume that the other hardware impairments are dealt by a RF canceler. Such an assumption is also made in [20, 21, 27, 28].
- 3.
Assumptions on oscillators: We make two assumptions about the oscillators. First, we assume that free-running oscillators are used. The assumption of using free-running oscillators for mmWave communications has also been made in [24, 29]. Second, we assume each transmit and receive antenna is equipped with an independent oscillator.
- 4.
Quasi-static flat-fading channel assumption: The SI measurement results of [17] show that even with omnidirectional dipole antennas, the delay spread of the channel does not exceed 800 ns. This delay is significantly smaller than the proposed symbol durations for 5G communication [30, 31], which are in order of μs. Hence, not only can the channel be assumed flat but it can also be assumed to remain constant over transmission of one block of data (quasi-static). Similarly, measurement results of the desired communication channel show that the channel delays are relatively small compared to the symbol durations [32].
- 5.
Synchronized transmission and reception: Although synchronizing transmission and reception of analog desired communication signal with the reception of analog SI signal is an important practical problem and requires its own detailed investigation, the synchronized FD communication assumption is widely used in the literature of channel and PN estimation for digital SI cancelation (DC) [20, 21, 33].
2.1 Mathematical representation of received vector
where δ(n) is Gaussian noise with mean 0 and variance \(\sigma ^{2}_{[m]}\), i.e., \(\delta (n)\sim \mathcal {N}\left (0,\sigma ^{2}_{[m]}\right)\).
Similarly, the element in the ith row and kth column of N_{r}×N_{t} SI channel matrix H^{SI}(n) is given by \(h^{\text {SI}}_{i,k} e^{j\left (\theta ^{[r]}_{i}(n)+\theta ^{[\text {SI}]}_{k}(n)\right)}\), where \(h^{\text {SI}}_{i,k}\) is the interference channel between the kth transmit antenna and the ith receive antenna of node a.
In addition, the kth elements of N_{t}×1 vectors x(n) and x^{SI}(n) are given by x_{k}(n) and \(x^{\text {SI}}_{k}(n)\), respectively, which are the transmitted symbols from the kth transmit antenna of nodes b and a, respectively.
Finally, \(\mathbf {w}(n) \triangleq [w_{1}(n),\cdots,w_{N_{r}}(n)]^{T}\), where w_{i}(n) is the complex Gaussian noise, i.e., \(w_{i}(n)\sim \mathcal {CN}(0,\sigma ^{2})\).
2.2 Mathematical representation for joint channel and PN estimation
where \(\bar {\mathbf {H}}(n)\) is the state transition model matrix, f is a nonlinear function, and β(n) is the state vector to be estimated.
- The state vector:$$\begin{array}{*{20}l} \boldsymbol{\beta}(n)\triangleq[\boldsymbol{\bar{\beta}}_{1}(n),\cdots,\boldsymbol{\bar{\beta}}_{N_{r}}(n)]^{T} \end{array} $$(4)
- The state transition matrix:$$\begin{array}{*{20}l} \bar{\mathbf{H}}(n)\triangleq \left[ \begin{array}{ccc} \bar{\mathbf{h}}_{1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & \bar{\mathbf{h}}_{N_{r}} \\ \end{array} \right] \end{array} $$(5)
The principle idea behind the design of the state vector β(n) and the state transition matrix \(\bar {\mathbf {H}}(n)\) as given by (4) and (5), respectively, is the fact that the PN noise is the only random variable that varies from one symbol to another and needs to be tracked. On the other hand, because of the quasi-static nature of the communication and SI channels, they remain constant over transmission of a single data packet. Therefore, these channels need to be estimated only once at the beginning of data transmission. This initial channel estimation for the constant channels can be done using pilot transmission.
Furthermore, we note that at each receive antenna, there are 2N_{t} parameters that need to be estimated, N_{t} parameters for the communication channel, and N_{t} parameters for the SI channel. This explains the existence of index \(\bar {k}\in \{1,\cdots,2N_{t}\}\).
3 Joint channel and PN estimation
and \(\hat {\beta }_{i,k}(n|n-1)\) is the 2(i−1)N_{t}+k element of vector \(\boldsymbol {\hat {\beta }}(n|n-1)\).
Remark 1
We note that the state vector, as given by (8), is a real vector. This is because the state vector only contains the phases, which are real numbers. The complex channel coefficients are estimated using this estimated real vector and using the complex exponential function as given by (7). Since the states are all real, when updating the mean of the states in EKF, we can safely discard the imaginary part of the updated mean as in (11).
3.1 Symbol detection
The EKF Eq. (17) shows that z_{i} requires the knowledge of the constant channels h_{i,k}, \(h_{i,k}^{\text {SI}}\) and the transmitted symbols. Note that \(x^{\text {SI}}_{k}\), the SI symbol is perfectly known at the receiver.
3.2 Lower bound of estimation error
With the above definition of the MSE vector, we present the following proposition.
proposition 1
where Q is the state covariance matrix given by (9).
Proof
See Appendix 5. □
Remark 2
We note that (19) shows that the lower bound on the estimation error increases as the sum of diagonal elements of the covariance matrix of the states increases. Furthermore, (9) indicates that the diagonal elements of the state covariance matrix are the function of PN variance. Consequently, increasing the PN variance will result in worse estimation error. Since the residual SI cancelation is performed using the estimated SI channel, increasing the PN variance will result in worse SI cancelation performance. It is also worth to note that [34] shows that the PN variance is a monotonic increasing of function of carrier frequency. This means that the estimation error increases with increasing the carrier frequency and vice versa.
3.3 Complexity analysis of EKF
Remark 3
According to Table (2), the EKF has a polynomial complexity as a function of number of transmit N_{t} and receive N_{r} antennas. We can justify the increased complexity as follows. In [20], the authors propose an algorithm for channel estimation with linear complexity. However, the algorithm in [20] assumes a constant PN for a block of data. This could be an acceptable scenario in microwave communication but does not suit mmWave communication. Hence, the increased complexity of the proposed algorithm is justified because of fast variation of PN, i.e., PN variation over symbol time.
4 Simulation results
In this section, we present simulation results for MIMO FD systems at 60 GHz frequency, which corresponds to mmWave frequency band [3]. For each simulation run, we assume a communication packet is 40 symbol long, i.e., N=40. This communication packet is transmitted after the training packet, which is 2N_{t} symbols long, and is used for estimating the constant channels for EKF initialization as described in Section 3.1. We then use 10,000 simulation runs to obtain the desired simulation results.
Moreover, we use the assumptions presented in Section 2 to generate the random noise and PN. As summarized in [38], there are many mmWave channel models available for mmWave systems. In this work, similar to a large number of existing works in [24, 29, 39, 40, 41], we adopt a general Rician model. Note that the proposed estimator is independent of the adopted model. A performance comparison of the different mmWave channel models is outside the scope of this work.
where K is the Rician distribution K-factor; H_{LoS} is the LoS component of the channel, and is generated assuming uniform distribution for angle of arrival, using the approach presented in [24]; H_{NLoS} is the NLoS component of the channel; and for both SI and communication channel is generated assuming Rayleigh fading. Furthermore, for both the SI and communication channel, we set the K-factor to 2 dB.
where \(\sigma ^{2}_{\text {COM}}\) and \(\sigma ^{2}_{\text {SI}}\) are the variances of NLoS components of the communication and SI channels, respectively.
where E_{s} is the symbol energy, \(\mathbb {E}[E_{s}]=1\), and σ^{2} is the noise variance.
In what follows, we first present the MSE results for different FD MIMO communication systems. We then investigate the residual SI power after digital cancelation and the bit error rate (BER) performance of these systems with the proposed PN estimation technique.
4.1 MSE performance
In this section, we investigate the MSE performance of the proposed PN estimation technique for a 2×2 FD MIMO system and assume that SIR=0 dB, i.e., the SI signal is as strong as the desired communication signal.
In Fig. 3, we also plot the MSE result of the state-of-the-art pilot-based phase noise estimator in [20, 23] for microwave frequency. As expected, this estimator does not perform well compared to our proposed estimator. This is because it assumes that the PN variations are small, which is not applicable for the case for mmWave frequency. Note that we only show the MSE result of the estimator in [20, 23] for 64−QAM modulation since the MSE performance is invariant with respect to the modulation order (the estimator uses pilots and does not require detection).
4.2 Comparison with unscented Kalman filter
4.3 Residual SI power
The numerical result of Fig. 5 shows that the performance of digital canceler depends on the residual SI power after passive and analog cancelation stages. As the residual SI power after passive and analog cancelation decreases, so does the residual SI power after the digital cancelation. The results show that the residual SI power can be reduced to − 25 and − 40 dB for SIR of 0 and 15 dB, respectively. This is important as it shows the effectiveness of digital SI cancelation after passive and analog cancelation.
4.4 BER performance
5 Conclusion
In this paper, we considered a MIMO FD system for mmWave communication and proposed a joint channel and PN estimation algorithm^{1}. We also derived a lower bound on the estimation error and numerically showed that the MSE of the proposed estimator approaches the error bound. Furthermore, we investigated the residual SI power after the digital cancelation and showed that the digital canceler, which uses the estimated SI channel can reduce the SI power to − 25 to − 40 dB. These results indicate the effectiveness of digital cancelation after passive and analog cancelation stages.
6 A lower bound of the estimation error
Next, we show that the last two terms of (25) are zero. We do this by showing only \(\mathbb {E}\left [\boldsymbol {\beta }(n)\boldsymbol {\hat {\beta }}^{T}(n)\right ]=0\) as a similar approach can be used to show that \(\mathbb {E}\left [\boldsymbol {\hat {\beta }}(n)\boldsymbol {\beta }^{T}(n)\right ]=0\).
Finally, using (29) and the definitions of Q and MSE in (9) and (18), we can establish the proof of the proposition.
7 B Complexity analysis of EKF
8 C Unscented Kalman filter (UKF)
Algorithm 1 summarizes the UKF joint channel and PN estimation algorithm.
Footnotes
- 1.
Indeed, the main focus of this work is to correctly estimate the channel and PN for effective SI cancelation. In case of inter-node interference [45], the proposed estimator would need to be modified. However, in the special case, if the inter-node interference can be treated as Gaussian, then the system model given by (1) can capture the effect of the inter-node interference by including an additional Gaussian noise term due to inter-node interference.
Notes
Acknowledgements
The authors would like to thank Professor Taneli Rihhonen for his insightful comments on this work.
Funding
The work of Abbas Koohian was supported by an Australian Government Research Training Program (RTP) Scholarship. The work of Hani Mehrpouyan was partially funded by the NSF ERAS grant award number 1642865.
Authors’ contributions
All authors contributed equally to this work. The final manuscript has been read and approved by all authors for submission.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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