Effects and suppression of signal correlation on radio frequency cancellation of full-duplex system

Abstract

In a full-duplex system where correlation exists between self-interference signal and useful signal, direct radio frequency (RF) coupling cancellation structure, which is based on the minimum mean square error (MMSE) criterion, is adopted to conduct self-interference cancellation; however, such cancellation damages the useful signal. In regards to this problem, this paper designs a new RF cancellation scheme: (1) On the basis of cancellation structure, phase shifter controlled by error signal power is thereby introduced to shift the phases of transmitted signal, decreasing the correlation of self-interference signal and useful signal; (2) New Time-varying Step-size Normalized Least Mean Square (NTVSSNLMS) algorithm, which utilizes the auto-correlation and time parameters t of real-time error signal to cooperatively control the step factor $μ (t)$ , is proposed to adjust the attenuator weight vectors, in order to assure a high interference cancellation ratio (ICR) and satisfy the requirements of phase shifter on algorithm convergence performance at the same time. Simulation results indicate that the new RF cancellation scheme can increase the ICR, while effectively reducing the damages on the useful signal simultaneously.

Keywords

Full-duplex radio frequency cancellation correlation NTVSSNLMS algorithm

1. Introduction

Full-duplex communication refers to the transmission of bidirectional data in the same frequency range simultaneously, which possesses various advantages, including the effective discovery of hidden terminals and the decrease of end-to-end delay. In comparison with the traditional frequency division duplex (FDD) and time division duplex (TDD), full-duplex communication is able to double the bandwidth efficiency theoretically, which makes it regarded as an important candidate technology [12,17]. It is difficult for self-interference signal and useful signal to be effectively separated from time domain and frequency domain, therefore, self-interference cancellation has become a major bottleneck in the application of full-duplex communication. Current self-interference cancellation methods include antenna cancellation, RF cancellation and digital cancellation; however, signals processed through RF cancellation must assure that analog-to-digital converter (ADC) never blocks, otherwise it will be unable to recover the useful signal after digital sampling [11,21].

Presently, the mainstream RF cancellation scheme can be divided into two schemes: direct RF coupling cancellation and digital-aided RF cancellation; the former scheme is more research worthy [23] when compared with the latter, because it does not consider thermal noise and different non-linear problems caused by the power amplifier. Ref. [7] adopts the disconnected single input single output (SISO) antenna to realize limited self-interference attenuation utilizing the intervals of antennas; besides, it also introduces Balun Active RF cancellation technology in order to realize 52 dB cancellation capacity of 5 MHz bandwidth signal. However, noise rejection unit adopted in such structure can cause non-linear interference, which will seriously interfere the digital cancellation afterwards; additionally, the design of double antenna may cause signal leakage interference. Ref. [4] applies circulator to the full-duplex system with single antenna and adopts passive RF cancellation structure instead of the active RF cancellation used in document [7], which effectively overcomes the faults of both leakage interference and non-linear interference. Concerning the adjustment of weight vectors of RF cancellation structures proposed in Refs. [4,7], Ref. [10] provides a detailed method of utilizing gradient descent algorithm to search the optimal weight vectors of RF cancellation structures and analyze the performances of the algorithm. Since Refs. [4,7,10] never fully utilizes the characteristics of self-interference signal when adopting direct RF coupling cancellation structure, Ref. [5] innovatively uses the self-hybrid RF self-interference cancellation structure to realize improved self-interference cancellation of Gaussian minimum shift keying (GMSK) narrow-band full-duplex signals based on lower hardware complexity; however, its application is limited and the improvement of RF cancellation capacity is based on its structure. RF cancellation structures proposed in Refs. [4,5,7,10] adopt the fixed-step least mean square (LMS) algorithm in order to adjust the weight vectors without exception, which may lead to several disadvantages, such as slow convergence velocity and being sensitive to condition number changes of tap input correlation matrices. Regarding the faults of weight vector adjustment algorithms proposed in Refs. [4,5,7,10], Ref. [8] establishes a non-linear relationship between the step factor of LMS algorithm and the modified time-varying Logistic Function, so as to realize a high ICR and maintain a high convergence velocity simultaneously. However, the update of its step factor is controlled by time parameters, which will continue to decrease during the signal intermittent period. Therefore, for full-duplex communication systems applying non-continuous wave signals, the step factor becomes very small when the next wave signal arrives, which makes it difficult to achieve fast convergence. Particularly, researches on direct RF coupling cancellation structure based on MMSE principles at the present stage mostly focus on the effect of algorithm characteristics and parameter estimation error on systematic RF cancellation performance, given the condition that self-interference signal and useful signal are not correlated [5,8]. Nevertheless, in actual application, it is inevitable that self-interference signal and useful signal are correlated in a full-duplex system. Due to such signal correlation, the application of the aforementioned direct RF coupling cancellation structure will cause serious damages to useful signal [13].

Addressing these problems, the author analyzes the main faults of direct RF coupling cancellation structure, which is based on MMSE criterion and applied to full-duplex systems with signal correlation theoretically, designs a phase shifter that is controlled by error signal power and finally shifts the phases of transmitted signals, so as to decrease the effects of signal correlation on the new cancellation structure. On the basis of aforementioned works, this paper proposes the NTVSSNLMS algorithm to adjust the attenuator weight vectors, whose ultimate purpose is to satisfy the requirement on algorithm performance of the new cancellation structure.

This paper is organized as follows: Section 2 analyzes the faults of traditional RF cancellation structure. Section 3 gives the new cancellation scheme, which includes new cancellation structure and NTVSSNLMS algorithm. Then, experimental results and analysis are provided in Section 4. Finally, Section 5 gives the conclusions of the work.

2. Faults of traditional cancellation structure

In this paper, traditional cancellation structures refer to the direct RF coupling cancellation structures containing in-phase and quadrature branches. Since the main focuses of this paper are effects and suppression of signal correlation on full-duplex communication system, therefore, the model of full-duplex communication system given in this paper does not involve antenna cancellation and digital cancellation structures, as shown in Fig. 1 [22].

Fig. 1.

Model of traditional full-duplex communication system.

Taking communication node A as an example, digital baseband signal passes through the modulator, the digital-to-analog converter (DAC), the up-converter and the power amplifier (PA) successively, in order to obtain the transmitted signal $s (t)$ ; some parts of $s (t)$ enter the RF cancellation structure to achieve RF cancellation through reconstructing the self-interference signal, while other parts enters the circulator [15] for outgoing antenna radiation. Since the buffer function of an actual circulator is unsatisfactory, signals used for antenna radiation are disclosed to the receiver, thereby forming the self-interference. Transmitted signal $s (t)$ of communication node A is given as the equation below: $\begin{matrix} (1) & s (t) = ℜ {A (t) exp (j 2 π f_{c} t)} \end{matrix}$

In the equation, $A (t) = a (t) exp [j φ (t)]$ is the low pass complex signal of $s (t)$ , while $f_{c}$ is the carrier frequency. If the transmitted signal $s (t)$ passes through a circulator whose leakage function is $h_{I} = k_{I} exp (- j 2 π f_{c} τ_{I})$ , then the self-interference signal $s_{I} (t)$ is obtained as: $\begin{matrix} (2) & s_{I} (t) = \int_{0}^{+ \infty} s (t - τ) h_{I} (τ) d τ = k_{I} ℜ {A (t - τ_{I}) exp [j 2 π f_{c} (t - τ_{I})]} \end{matrix}$

In the equation, $k_{I}$ and $τ_{I}$ represent amplitude attenuation and time delay respectively. Useful signal from communication node B to node A is recorded as $s_{U} (t)$ , referring to Equation (1), the expression of $s_{U} (t)$ is given by: $\begin{matrix} (3) & s_{U} (t) = ℜ {A_{U} (t) exp (j 2 π f_{c} t)} \end{matrix}$

Where $A_{U} (t) = a_{U} (t) exp [j θ (t)]$ represents the equivalent low pass of $s_{U} (t)$ . Thermal noise in the frequency band is a band-limited white Gaussian random process [3], which can be expressed as: $\begin{matrix} (4) & n (t) = ℜ {A_{n} (t) exp (j 2 π f_{c} t)} \end{matrix}$

In the equation, $A_{n} (t) = a_{n} (t) exp [j ϕ (t)]$ is the equivalent low pass form of thermal noise. Combining Equations (2)–(4), it can be obtained that the general received signals that enter into communication node A are expressed as $r (t)$ : $\begin{matrix} (5) & r (t) = s_{U} (t) + s_{I} (t) + n (t) = ℜ {[A_{U} (t) + h_{I} A (t - τ_{I}) + A_{n} (t)] exp (j 2 π f_{c} t)} \end{matrix}$

Transmitted signal $s (t)$ is processed successively through group delay τ and quadrature power splitter, in order to obtain the reference vector signal $S (t) = {[s_{i} (t) s_{q} (t)]}^{T}$ : $\begin{matrix} (6) & S (t) = \{\begin{matrix} ℜ {A (t - τ) exp (j 2 π f_{c} (t - τ))} \\ ℜ {A (t - τ) exp (j 2 π f_{c} (t - τ) - j π / 2)} \end{matrix}\} \end{matrix}$

In the equation $s_{i} (t)$ and $s_{q} (t)$ represent the component of the in-phase and the quadrature branch respectively. Suppose the weight vector of attenuator is $W (t) = {[w_{i} (t) w_{q} (t)]}^{T}$ , then the reconstructed self-interference signal can be expressed as: $\begin{matrix} (7) & {\hat{s}}_{I} (t) = {[W (t)]}^{T} \cdot S (t) = w_{i} (t) \cdot s_{i} (t) + w_{q} (t) \cdot s_{q} (t) \end{matrix}$

Error signal $e (t)$ after RF cancellation is given by: $\begin{matrix} (8) & e (t) = r (t) - {\hat{s}}_{I} (t) = s_{I} (t) + s_{U} (t) + n (t) - {[W (t)]}^{T} \cdot S (t) \end{matrix}$

The LMS algorithm and its improved algorithm adopted in the adjustment of attenuator weight vector are based on MMSE criterion [9], i.e.: $\begin{array}{l} min {E ({| e (t) |}^{2})} \\ = min {E ({| r (t) - {\hat{s}}_{I} (t) |}^{2})} = min {E (| s_{U} (t) + s_{I} (t) + n (t) - {\hat{s}}_{I} (t) |)} \\ (9) & = min {E ({| s_{U} (t) + n (t) |}^{2}) + E (| (s_{I} (t) - {\hat{s}}_{I} (t)) \cdot (s_{I} (t) - {\hat{s}}_{I} (t) + 2 s_{U} (t) + 2 n (t)) |)} \end{array}$

In the equation above, $| \cdot |^{2}$ represents the square of module value. Since $n (t)$ is uncorrelated with neither $s_{U} (t)$ nor $s_{I} (t)$ , if $s_{U} (t)$ and $s_{I} (t)$ are uncorrelated with each other, i.e. $n (t)$ , $s_{U} (t)$ and $s_{I} (t)$ are mutually uncorrelated, then Equation (9) can be further expressed as: $\begin{matrix} (10) & min {E ({| e (t) |}^{2})} = min {E ({| s_{U} (t) |}^{2}) + E ({| n (t) |}^{2}) + E ({| (s_{I} (t) - {\hat{s}}_{I} (t)) |}^{2})} \end{matrix}$

According to Equation (10), if the weight vector of attenuator $W (t)$ converges to its optimal value $W^{*} (t)$ , i.e. $E (| e (t) |^{2})$ obtains its minimal value, then the self-interference signal $s_{I} (t)$ in received signal $r (t)$ will be fully canceled; therefore, only useful signal $s_{U} (t)$ and in-band thermal noise $n (t)$ will exist in error signal $e (t)$ .

From aforementioned analysis, it can be concluded that various RF domain self-interference cancellation algorithms based on MMSE criterion basically adjust the weight vector $W (t)$ by utilizing signal correlation [16] to realize the cancellation of components correlated to reference signal in received signals. Despite possible improvements that can be made to the RF cancellation algorithms based on MMSE criterion, the problem of correlation effect on RF cancellation performance cannot be solved; additionally, the better the algorithms perform, the more thorough the correlated components in both $s_{U} (t)$ and $s_{I} (t)$ are canceled.

3. New RF cancellation scheme

3.1. System model

Based on the full-duplex communication system model demonstrated in Fig. 1, on one hand, this paper introduces a phase shifter controlled by error signal power to decrease the correlation between self-interference signal $s_{I} (t)$ and useful signal $s_{U} (t)$ through shifting the phases of transmitted signal $s (t)$ ; on the other hand, this paper proposes the better-performed NTVSSNLMS algorithm to adjust the weight vector of attenuator and to assure that every measurement is able to rapidly obtain the instantaneous power of error signal in convergence state to control the phase adjustment of phase shifter, thereby improving the working efficiency of phase shifter. Both of them coordinate effectively to simultaneously increase the ICR in RF domain and decrease the damages on useful signal $s_{U} (t)$ [19]. A model of the new full-duplex communication system is shown in Fig. 2.

Fig. 2.

Model of the new full-duplex communication system.

Assuming the step interval of parameter adjustment in RF cancellation structure is $Δ t$ , then the instantaneous power of error signal $e (t)$ at moment t is: $\begin{matrix} (11) & P_{e} (t) = \frac{1}{Δ t} \int_{t - Δ t}^{t} {| e (t) |}^{2} d t \end{matrix}$

According to Equation (10), when adopting the direct RF coupling cancellation scheme based on MMSE principles, component $ζ_{c} (t)$ of useful signal $s_{U} (t)$ , which is correlated with self-interference signal $s_{n} (t)$ , will be canceled together with self-interference signal $s_{n} (t)$ ; only component $ζ_{u} (t)$ of $s_{U} (t)$ which is uncorrelated with $s_{I} (t)$ will be retained, i.e. the steady-state error signal power $P_{est} (t)$ obtained through RF cancellation is given as: $\begin{matrix} (12) & P_{est} (t) = σ_{n}^{2} + P_{u} (t) \end{matrix}$

In the equation above, $σ_{n}^{2}$ is the average power of in-band thermal noise $n (t)$ , while $P_{u}$ is the power of component $ζ_{u} (t)$ in $s_{U} (t)$ which is uncorrelated with $s_{I} (t)$ .

Given the condition that RF cancellation converges to the steady-state, instantaneous power $P_{e} (t)$ of error signal which is obtained from measurement and calculation approximates the power of steady-state error signal $P_{est} (t)$ ; thus, $P_{e} (t)$ can replace $P_{est} (t)$ and the value of $P_{e} (t)$ can be utilized to judge whether the correlation between useful signal and self-interference signal is strong or weak. Assume that the threshold of instantaneous power is κ, if $P_{e} (t) ⩾ κ$ , the damages on useful signal during RF cancellation are relatively small and acceptable, indicating a weak correlation with self-interference signal, thus, it is unnecessary to shift the phases of transmitted signal $s (t)$ ; if $P_{e} (t) < κ$ , the damages on useful signal during RF cancellation are relatively big and unacceptable, indicating a strong correlation with self-interference signal, thus, it would be necessary to shift the phases of transmitted signal $s (t)$ so as to obtain the phase shift signal $s_{1} (t)$ : $\begin{matrix} (13) & s_{1} (t) = ℜ {A (t) exp (j 2 π f_{c} t - j Δ φ)} \end{matrix}$

In the equation, $Δ φ$ is the phase deviation of phase shifter, its equation is given as: $\begin{matrix} (14) & Δ φ = \{\begin{matrix} 0 & P_{e} (t) ⩾ κ \\ \pm (\frac{π}{2} - arcsin (\sqrt{(P_{e} (t) - σ_{n}^{2}) / P_{U}})) & P_{e} (t) < κ \end{matrix} \end{matrix}$

Where $P_{U}$ is the power of useful signal $s_{U} (t)$ , whether the value of $Δ φ$ is positive or negative depends on the signal modulation methods. Therefore, the reference vector signal $S (t)$ in Equation (6) can be re-written as: $\begin{matrix} (15) & S (t) = \{\begin{matrix} ℜ {A (t - τ) exp (j 2 π f_{c} (t - τ) - j Δ φ)} \\ ℜ {A (t - τ) exp (j 2 π f_{c} (t - τ) - j Δ φ - j π / 2)} \end{matrix}\} \end{matrix}$

Substitute Equation (15) into Equation (7) and Equation (8) successively to obtain the expression of the reconstructed self-interference signal ${\hat{s}}_{I} (t)$ and error signal $e (t)$ in the new system model.

3.2. New RF cancellation algorithm

It can be inferred from the phase shifting strategies demonstrated in Section 3.1 that in order to ensure the normal operation of phase shifter, the measured and calculated instantaneous power of error signal $P_{e} (t)$ should be as close as possible to the power of steady-state error signal $P_{est} (t)$ , which requires faster convergence velocity and smaller steady-state error of the RF cancellation algorithm. The new scheme copes with statistical characteristics of digital modulation signals in the general cyclostationary stochastic process [18], although RF cancellation algorithms proposed in documents [2,6,8,14] is applicable to the new RF cancellation scheme proposed in this paper. From the perspective of ensuring the effectiveness of the new scheme, the new RF cancellation algorithm proposed in this paper is the better choice when compared with reference algorithms, because it not only possesses faster convergence velocity and smaller steady-state error, but also has stronger tracking ability. The NTVSSNLMS algorithm proposed in this paper establishes the non-linear relationship between step factor of MMSE $μ (t)$ and modified time-varying Sigmoid function, and adjusts $μ (t)$ through the coordination between real time error signal and time parameter t. The updated equation for weight vector of attenuator in the new cancellation scheme is given below: $\begin{matrix} (16) & \{\begin{matrix} W (t + Δ t) = W (t) + [(2 μ (t)) / (η + P_{c})] e (t) S (t) \\ μ (t) = γ [1 - exp ((- α | e (t) e (t - Δ t) |) / (t - t_{0}))] \end{matrix} \end{matrix}$

In the equation above, η is a small normal number to ensure that the denominator of the fraction is not 0; $P_{c}$ is the power of equal reference signal $s_{i} (t)$ and $s_{q} (t)$ output by quadrature power splitter; parameter γ controls the value ranges of $μ (t)$ and parameter α controls the shape of $μ (t)$ . As long as $0 < μ (t) < 1$ , it can be ensured that through enough iterations, the aforementioned normalized equation of weight iteration is able to converge, because the step factor $μ (t)$ is controlled by the auto-correlation of the error signal and the time parameter. The error signal contains the useful signal, the residual self-interference signal, as well as the noise signal. Therefore, the adjustment of the step factor $μ (t)$ must be affected by the statistical characteristics of the transmitted signal, which is related to the type of transmitted signal.

In NTVSSNLMS algorithm, self-correlation of error signals can simultaneously increase the tracking ability of the system and decrease the effects of burst pulse interference effectively; given the condition of steady-state, the time parameter t can assure that, as time passes, $μ (t)$ can converge till it is small enough to make $P_{e} (t)$ approximate to $P_{est} (t)$ , so as to promote the working efficiency of phase swifter; by setting the power threshold and checking the difference between the before-and-after instantaneous power of error signals obtained from measurement and calculation $P_{e} (t)$ , the initial time $t_{0}$ can be used to decide whether sudden changes exist or not, thus, the update value of $t_{0}$ can be determined to ensure that the affecting weight of time parameter t on $μ (t)$ maintains in a reasonable range; the update of $t_{0}$ is given as: $\begin{matrix} (17) & t_{0} = \{\begin{matrix} t & | P_{e} (t) - P_{e} (t - Δ t) | ⩾ κ_{1} \\ 0 & | P_{e} (t) - P_{e} (t - Δ t) | < κ_{1} \end{matrix} \end{matrix}$

Where $κ_{1}$ is the power threshold of initial time $t_{0}$ . Combining the work flow of phase shifter, the specific steps of full-duplex system RF domain self-interference cancellation with signal correlation are given as follows:

Utilize the training sequence to estimate the group delay component τ of signals, then determine the value of component α and γ in NTVSSNLMS algorithm based on the detailed modulation parameters of digitally modulated signals in the full-duplex system.

Give the attenuator weight vector $W (t) = {[w_{i} (t) w_{q} (t)]}^{T}$ the initial value $W (0) = {[0 0]}^{T}$ , the feedback control step interval of $W (t)$ is represented as $Δ t$ .

According to Equation (11), calculate the instantaneous power of error signals $P_{e} (t)$ , then substitute $P_{e} (t)$ into Equation (13) to adjust the phases of transmitted signals $s (t)$ and obtain the reference vector signal $S (t)$ of Equation (15);

Substitute $P_{e} (t)$ into Equation (17) to obtain the update value of initial time $t_{0}$ .

Update the weight vector of attenuator $W (t)$ in Equation (16), then modify the gains of in-phase and quadrature branches.

Calculate the error signal $e (t)$ at current instant according to Equation (8).

Return and perform Step 3.

3.3. Performance analysis of the algorithm

The performance of NTVSSNLMS RF cancellation algorithm is analyzed by combining the new RF self-interference cancellation structure. To facilitate this analysis, time delay error is assumed as 0, a mean square calculation within feedback control step interval is made using Equation (9): $\begin{matrix} (18) & E [e^{2} (t)] = P_{c} [{(w_{i}^{*} - w_{i})}^{2} + {(w_{q}^{*} - w_{q})}^{2}] + P_{u} + σ_{n}^{2} \end{matrix}$

In the equation above, $σ_{n}^{2}$ is the average power of thermal noise. Simplify the weight values in Equation (16): $\begin{matrix} (19) & E [e^{2} (t)] = P_{c} [{(w_{i}^{*} - w_{i})}^{2} + {(w_{q}^{*} - w_{q})}^{2}] + P_{u} + σ_{n}^{2} \end{matrix}$

Substitute Equation (19) into (18) and simplify as given below: $\begin{array}{rcl} E [e^{2} (t)] & = & P_{c} E {{[W (t - Δ t) - W^{*}]}^{T} {[I - \frac{2 μ (t) P_{c}}{η + P_{c}}]}^{2} [W (t - Δ t) - W^{*}]} \\ (20) & + {(\frac{2 μ (t) P_{c}}{η + P_{c}})}^{2} P_{c}^{2} (P_{u} + σ_{n}^{2}) + P_{u} + σ_{n}^{2} \end{array}$

The initial value of attenuator weight vector $W (t)$ is $W (t = 0) = {[w_{i} (t = 0) w_{q} (t = 0)]}^{T} = {[0 0]}^{T}$ ; if $t = n Δ t$ , then according to Equation (20), the correlation between iterative mean square error (MSE) and initial MSE at the nth step will be: $\begin{array}{rcl} E [e^{2} (t)] & = & [{(w_{i}^{*})}^{2} + {(w_{q}^{*})}^{2}] P_{c} \prod_{k = 0}^{n - 1} {[1 - \frac{2 μ (k Δ t) P_{c}}{η + P_{c}}]}^{2} + (P_{u} + σ_{n}^{2}) [1 + {(\frac{2 μ (k Δ t) P_{c}}{η + P_{c}})}^{2}] \\ (21) & + (P_{u} + σ_{n}^{2}) \sum_{k = 0}^{n - 2} {P_{c}^{2} {(\frac{2 μ (k Δ t) P_{c}}{η + P_{c}})}^{2} \prod_{i = k + 1}^{n - 1} {[1 - \frac{2 μ (i Δ t) P_{c}}{η + P_{c}}]}^{2}} \end{array}$

The condition of MSE convergence can be obtained from Equation (21): $\begin{matrix} (22) & | 1 - (2 μ (k Δ t) P_{c} / η + P_{c}) | < 1 \end{matrix}$

Since η is a small normal number, the convergence condition of MSE can be further simplified into $| 1 - 2 μ (k Δ t) | < 1 \Rightarrow | μ | < 1$ , i.e. the same with that of the normalized LMS algorithm. According to Equation (21), the convergence factor of NTVSSNLMS algorithm $χ_{NTVSS} (n Δ t)$ is expressed as: $\begin{array}{rcl} χ_{NTVSS} (n Δ t) & = & \prod_{k = 0}^{n - 1} {[1 - \frac{2 μ (k Δ t) P_{c}}{η + P_{c}}]}^{2} \\ = & \prod_{k = 0}^{n - 1} {[1 - \frac{2 γ [1 - exp ((- α | e (k Δ t) e (k Δ t - Δ t) |) / (k Δ t - t_{0}))] P_{c}}{η + P_{c}}]}^{2} \\ (23) & \approx & \prod_{k = 0}^{n - 1} {[1 - 2 γ [1 - exp ((- α | e (k Δ t) e (k Δ t - Δ t) |) / (k Δ t - t_{0}))]]}^{2} \end{array}$

According to the equation of convergence factor χ_NTVSS obtained from Equation (23), the power of reference input signal $P_{c}$ has no effect on convergence velocity of the algorithm, indicating the effective avoidance of interference generated through the amplification of gradient noise. Besides, the introduction of error signal $e (t)$ can effectively solve the problem that just relying on time parameter t alone cannot cope with intermittent communication or the interference of sudden pulse noise.

Assume that a positive integer m exists; if $t ⩾ m Δ t$ , MSE tends to be stable, i.e. weight vector of attenuator $W (t) \to W^{*} (t)$ , at this moment, $\begin{matrix} (24) & μ (t ⩾ m Δ t) \approx γ [1 - exp ((- α | (r (t) - s_{I} (t)) (r (t - Δ t) - s_{I} (t - Δ t)) |) / (k Δ t - t_{0}))] \end{matrix}$

Combining Equation (24) to further simplify Equation (21): $\begin{array}{rcl} E [e^{2} (t)] & \approx & 2 [{(w_{i}^{*})}^{2} + {(w_{q}^{*})}^{2}] P_{c} {(1 - μ^{'})}^{2 (n - m)} \\ (25) & + 2 (P_{u} + σ_{n}^{2}) {(μ^{'})}^{2} \cdot \frac{1 - {(1 - μ^{'})}^{2 (n - m)}}{1 - {(1 - μ^{'})}^{2}} + (P_{u} + σ_{n}^{2}) \end{array}$

Where $μ^{'} = \frac{2 μ (t ⩾ m Δ t) P_{c}}{η + P_{c}}$ . It can be inferred from Equation (22) that ${lim}_{n \to \infty} {(1 - μ^{'})}^{2 (n - m)} \approx 0$ , therefore, Equation (25) can be further simplified into: $\begin{matrix} (26) & E [e^{2} (\infty)] \approx (P_{u} + σ_{n}^{2}) \cdot [(2 + μ^{'}) / (2 - μ^{'})] \end{matrix}$

According to the ICR formula: $ICR = 10 {log}_{10} (P_{b} / P_{a})$ , in which $P_{b}$ and $P_{a}$ represents the power of self-interference signal at proximal end before-and-after the RF cancellation respectively. Combining Equation (26), under the convergence condition of NTVSSNLMS algorithm, ICR can be expressed as: $\begin{matrix} (27) & ICR = 10 {log}_{10} (\frac{P_{b}}{E [e^{2} (\infty)] - (P_{u} + σ_{n}^{2})}) \approx 10 {log}_{10} (\frac{P_{b} [η + (1 - μ (t ⩾ m Δ t)) P_{c}]}{2 μ (t ⩾ m Δ t) P_{c} (P_{u} + σ_{n}^{2})}) \end{matrix}$

4. Experiment and analysis

4.1. Settings of experimental parameters

In the practical application of full-duplex, both the transmitting and receiving antennas of the same communication node are close in distance and the main path power of self-interference signal is much larger than its reflection path power. Thus, the effects of reflection multipath are not concluded. Simulation adopts the 2FSK Full-duplex System Model with signal correlation [24]. According to Equation (2) and Equation (13), the equivalent lowpass form of self-interference signal $s_{I} (t)$ is expressed as: $\begin{array}{rcl} s_{I - low} (t) & = & {(\int_{0}^{+ \infty} s_{1} (t - τ) h_{I} (τ) d τ)}_{low - pass} \\ (28) & = & ℜ {k_{I} k_{f} g (t - τ_{I}) exp [j 2 π m Δ f (t - τ_{I}) + j φ_{m} - j Δ φ - j 2 π f_{c} τ_{I}]} m \in \{\begin{matrix} 0 & 1 \end{matrix}\} \end{array}$

In the equation above, $k_{f}$ is the amplitude of $s_{1} (t)$ , m is the binary code element sequence in the probability distribution, $Δ f = k / T_{s}$ is the frequency interval, k is a positive integer, $T_{s}$ is the time duration of code element, $g (t)$ is the rectangular pulse signal with its time duration being $T_{s}$ and $φ_{m}$ is the corresponding initial phase of base-band modulation of code element m. Similarly, the equivalent lowpass form of useful signal $s_{U} (t)$ from communication node B to node A is expressed as follow: $\begin{matrix} (29) & s_{U - low} (t) = ℜ {k_{U} g (t) exp [j 2 π m Δ f t + j ϕ_{m}]} \end{matrix}$

In the equation above, $k_{U}$ is the amplitude of $s_{U} (t)$ received, while $ϕ_{m}$ is the corresponding initial phase of base-band modulation of code element m. Based on the calculation method of correlation demonstrated in Ref. [20], the normalized correlation coefficient of self-interference signal $s_{I} (t)$ and useful signal $s_{U} (t)$ given the condition of code element alignment can be expressed as: $\begin{matrix} (30) & ρ = \frac{cov (s_{I - low}, s_{U - low})}{\sqrt{D (s_{I - low}) D (s_{U - low})}} = \frac{1}{4} [cos (φ_{0} - Δ φ - 2 π f_{c} τ_{I} - ϕ_{0}) + cos (φ_{1} - Δ φ - 2 π f_{c} τ_{I} - ϕ_{1})] \end{matrix}$

The waveform phase difference of code elements on the same frequency points of $s_{I} (t)$ and $s_{U} (t)$ is recorded as $θ_{1} = φ_{0} - Δ φ - 2 π f_{c} τ_{I} - ϕ_{0}$ and $θ_{2} = φ_{1} - Δ φ - 2 π f_{c} τ_{I} - ϕ_{1}$ respectively. According to Equation (14), phase shifter adjusts the phase deviation $Δ φ$ to make the normalized correlation coefficient ρ become 0.

At the early stage, some scholars carried out some research on various RF cancellation algorithms. Time-varying Step-size LMS (TVSSLMS) proposed in Ref. [8] shortens the change of Logistic function value from large to small and further establishes the non-linear relationship between step factor $μ (t)$ and time parameter t; Fixed Step-size NLMS (FXSSNLMS) proposed in Ref. [14] uses constant step size, constant filter length and a ratio of energy spectral density (ESD) of useful signal and interference signal in order to update filter weights. Depending on the ratio, weights are adjusted automatically; Variable Step-size LMS (SVSLMS) proposed in Ref. [2] builds a non-linear function relationship between step factor $μ (t)$ and interference signal and presents a novel variable step size LMS adaptive algorithm, by improving Sigmoid function based on translational transformation; since the algorithm proposed in Ref. [6] is inferior to that of Ref. [8], it will not be discussed herein.

Based on the 2FSK full-duplex system model with signal correlation, NTVSSNLMS algorithm proposed in this paper is compared with the above three RF cancellation algorithms. According to the principles in Refs. [8,14,16], the detailed parameters of all four algorithms are listed in Table 1. The settings of parameters of system model are: transmission rates of both useful source and self-interference source are 2 kbps, the corresponding carrier frequencies of code element “0” and “1” in equal possibility distribution are 4 KHz and 8 KHz respectively; after modulation, the sampling rate is 64 KHz, the step interval $Δ t = 1 / 32$ ms, the power threshold of update of initial time $t_{0}$ $κ_{1} = 4 / 5 (P_{U}) + σ_{n}^{2}$ , the threshold of step factor $μ_{max} = 20 μ_{min}$ , the signal-to-noise ratio takes $E_{b} / N_{0} = 10$ dB, coherent demodulation [1] method is applied to useful signal and the experiment results are obtained from 100 Monte Carlo Simulations.

Table 1
Parameter values of each algorithm

Algorithm Update formula of $μ (t)$ Parameter group 1 Parameter group 2

NTVSSNLMS $μ (t) = γ [1 - exp ((- α | e (t) \cdot e (t - Δ t) |) / (t - t_{0}))]$ $γ = 0.26$ , $α = 14$ $γ = 0.32$ , $α = 13$

TVSSLMS $μ (t) = μ_{min} + μ_{max} [1 - 2 / exp [{(\frac{κ}{t - t_{0} + 1})}^{3} + 1]]$ $κ = 23$ , $μ_{min} = 0.05$ $κ = 27$ , $μ_{min} = 0.04$

SVSLMS $μ (t) = β [1 / (1 + exp (- ξ | e (t) |)) - 0.5]$ $β = 0.23$ , $ξ = 15$ $β = 0.28$ , $ξ = 11$

FXSSNLMS $μ_{Fix}$ $μ_{Fix} = 0.08$ $μ_{Fix} = 0.08$

Algorithm	Update formula of $μ (t)$	Parameter group 1	Parameter group 2
NTVSSNLMS	$μ (t) = γ [1 - exp ((- α \| e (t) \cdot e (t - Δ t) \|) / (t - t_{0}))]$	$γ = 0.26$ , $α = 14$	$γ = 0.32$ , $α = 13$
TVSSLMS	$μ (t) = μ_{min} + μ_{max} [1 - 2 / exp [{(\frac{κ}{t - t_{0} + 1})}^{3} + 1]]$	$κ = 23$ , $μ_{min} = 0.05$	$κ = 27$ , $μ_{min} = 0.04$
SVSLMS	$μ (t) = β [1 / (1 + exp (- ξ \| e (t) \|)) - 0.5]$	$β = 0.23$ , $ξ = 15$	$β = 0.28$ , $ξ = 11$
FXSSNLMS	$μ_{Fix}$	$μ_{Fix} = 0.08$	$μ_{Fix} = 0.08$

Fig. 3.

Effects of code element waveform phase difference on signal correlation.

4.2. Experiment results

Figure 3 shows the effects of waveform phase difference of code elements on the same frequency point on their correlation, from which it can be inferred that the normalized correlation coefficient ρ of the two signals, together with the waveform phase difference of code elements vary between $[- 0.5 0.5]$ , the simulation results agree with Equation (30), indicating the accuracy of theoretical analysis. Figure 4 illustrates the ICR curves given different combinations of code element waveform difference in NTVSSNLMS algorithm proposed in this paper, the parameters take the values listed in group 1. It can be discovered that if the power of useful signal $P_{U}$ is far less than that of self-interference signal $P_{b}$ and the reference signal power $P_{c}$ is close to thermal noise power $σ_{n}^{2}$ , then the ICR value is not sensitive to the changes of correlation of $s_{I} (t)$ and $s_{U} (t)$ ; this is because ICR index mainly evaluates the cancellation of self-interference signal of a certain system, rather than indicating the damages to useful signal during the process of cancellation.

Fig. 4.

Effects of code element waveform phase difference on ICR.

In order to test the convergence and anti-interference performance of NTVSSNLMS algorithm, a strong pulse interference is injected into the system at 7 ms; given the condition that the waveform phase difference of code elements on the same frequency point of $s_{I} (t)$ and $s_{U} (t)$ is set as $θ_{1} = θ_{2} = π / 2$ , ICR curves of two groups of parameter values of the algorithm are obtained, as shown in Fig. 5 and Fig. 6.

Fig. 5.

ICR curve of parameter group 1.

Fig. 6.

ICR curve of parameter group 2.

From Fig. 5, Fig. 6 and Table 2, it can be concluded that: given the condition of two parameter groups, the convergence velocity of NTVSSNLMS algorithm is 1.87 ms and 1.78 ms successively, all of which are higher than the convergence velocity of reference algorithms, the average ICR value under convergence condition is 88.25 dB and 88.11 dB successively, which is slightly less than the calculation result of Equation (27), indicating the accuracy of theoretical analysis; although the steady-state average ICR value of NTVSSNLMS algorithm is 1.5 dB higher than that of TVSSLMS algorithm, the convergence velocity and anti-interference performance of the former are significantly increased; in terms of performance, TVSSLMS algorithm is better than FXSSNLMS algorithm because the former possesses faster convergence velocity than the latter with equal ICR values; although SVSLMS algorithm possesses better convergence velocity, it adjusts $μ (t)$ only through error signals with Gaussian white noise, which leads to the consequence that $μ (t)$ is not small enough, thus, its steady-state ICR value is the lowest among the four algorithms. Specifically, as time passes, the step factor $μ (t)$ proposed in Ref. [10] may be small enough to ensure a higher ICR value of the algorithm; however, within the power threshold of error signal, the continuous decrease of $μ (t)$ cannot adjust in real-time as error signals may vary, leading to a poorer tracking ability for time-varying systems and faults of accumulated errors within power threshold. Hence, the algorithm proposed in Ref. [10] possesses lower steady-state ICR value than the algorithm proposed in this paper; besides, the more severe the system time-varying characteristics are, the larger the performances differences are.

Table 2

Comparison of algorithm performance

		NTVSSNLMS	TVSSLMS	SVSLMS	FXSSNLMS
Parameter group 1	Convergence time (ms)	1.87	4.68	2.62	3.84
Parameter group 1	ICR of steady state (dB)	88.25	86.83	82.48	83.51
Parameter group 2	Convergence time (ms)	1.78	4.59	2.25	3.93
Parameter group 2	ICR of steady state (dB)	88.11	86.56	82.06	83.13

Fig. 7.

Demodulation BER of useful signal ( $θ_{1} = θ_{2} = π / 3$ ).

Figure 7 indicates the demodulation BER of useful signal for different RF cancellation algorithms, given the condition that the waveform phase difference of code elements on the same frequency point of $s_{I} (t)$ and $s_{U} (t)$ is set as $θ_{1} = θ_{2} = π / 3$ . Based on the new full-duplex communication system model, it can be observed that the algorithm proposed in this paper possesses lower demodulation BER; furthermore, it has stronger adaptability toward insensitive changes of interference-to-signal ratio (ISR). Figure 8 indicates the demodulation BER curve of useful signal of different RF cancellation algorithms, given the condition that the waveform phase difference of code elements on the same frequency point of $s_{I} (t)$ and $s_{U} (t)$ is set as $θ_{1} = θ_{2} = π / 2$ . It can be observed that when self-interference signal and useful signal are exactly in quadrature, excluding the effects of ADC quantifying dynamic range on useful signal, through direct demodulation of received signals, the demodulation BER of useful signal will be lower than that which was calculated through the algorithm proposed in this paper and reference algorithms; the reason is that, during the process of RF cancellation, useful signal will inevitably suffer from losses; however, in order to avoid the block of ADC caused by strong self-interference signal, it is tolerable to lose certain useful damages in order to obtain a higher ICR value. Therefore, given the special condition where $θ_{1} = θ_{2} = π / 2$ , demodulation BER without RF cancellation is the lowest. Specifically, the ideal line in Fig. 7 and Fig. 8 refer to the demodulation BER under the condition that self-interference signal is completely canceled, while useful signal has no damages.

Fig. 8.

Demodulation BER of useful signal ( $θ_{1} = θ_{2} = π / 2$ ).

From Figs 5–8, it can be inferred that although the new RF cancellation scheme proposed in this paper is better than the reference scheme in terms of ICR and demodulation BER of useful signals, it still has a relatively large gap between the ideal demodulation BER; this is because the step factor of NTVSSNLMS algorithm is controlled by error signals, which include remnant self-interference signal and components in Gaussian noise signal, as well as the useful signal that is uncorrelated with self-interference signal. Therefore, these self-interference signal and statistical characteristics of Gaussian noise will significantly affect the demodulation BER. In practical application, adaptive reduction of thermal noise may be applied to further decrease the damages to useful signal which are caused during the process of RF cancellation.

5. Conclusion

This paper proposes a full-duplex RF cancellation scheme which is applicable to the condition where a correlation exists between self-interference signal and useful signal. Based on the direct RF coupling cancellation structure, the scheme introduces a phase shifter controlled by error signal power, in order to decrease the correlation between self-interference signal and useful signal through shifting the phases of transmitted signal. To obtain the power of error signal under the convergence condition rapidly and accurately, as well as to improve the working efficiency of the phase shifter, this paper proposes the NTVSSNLMS algorithm which utilizes the self-correlated parameter and time parameter t of real time error signal, to collaboratively control the step factor $μ (t)$ and to adjust the weight vector of attenuator, finally satisfying the requirements of the new scheme on the convergence performance of this new algorithm. Theoretical analysis and Monte Carlo Simulations indicate that the new RF cancellation scheme proposed in this paper can improve the ICR of full-duplex system with signal correlation, while decreasing the damages to useful signal caused by RF cancellation simultaneously.

Footnotes

Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (Grant Number: 61472442), Shanxi Provincial Natural Science Foundation (Grant Number: 2014JM2-6106) and Aviation Science Foundation of China (Grant Number: 20155896025).

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