A Power Analysis Method for Self-Interference Signal Components in Full-Duplex Transceivers Under Constant/Nonconstant Modulus Signal Stimulation

Abstract

The existence of multiple self-interference (SI) signal components, particularly the nonlinear ones, seriously constrains the performance of self-interference cancellation (SIC) methods. To decrease the complexity of SIC methods in full-duplex devices, this article proposes a power analysis method for SI signal components in a full-duplex transceiver. The proposed method comprises a separate analysis algorithm and a system-level power model. Initially, the algorithm is conducted to obtain the spectrum of the linear and nonlinear components in the power amplifier (PA) output signal. Once the linear-to-nonlinear power ratio (LNPR) has been obtained, a system-level power model is constructed by taking both the transmitter noise and analog-to-digital converter (ADC) quantization noise into account. The proposed power model allows for the allocation of SIC method performance in multiple domains during the design of full-duplex transceivers at the top level, thereby reducing the overall system complexity. The simulation results demonstrate that in a full-duplex transceiver with only antenna isolation, the power of the SI signal component is susceptible to alterations due to the operating waveform and transmission power. Finally, the accuracy of the power analysis method is verified through measurement and Simulink.

1. Introduction

In-band full-duplex (IBFD) technology, also called simultaneous transmission and reception (STAR), allows users to transmit and receive signals on the same frequency simultaneously, which has significant research and practical value. Initially employed in continuous wave radar systems [1], it is currently being utilized in various scenarios [2,3]. In the field of electronic countermeasures, STAR jammers have the potential to achieve near real-time jamming effects, thereby rendering enemy radar’s intra-pulse and inter-pulse agile measures ineffective [4,5,6]. In radar applications, the full-duplex transceiver is capable of transmitting continuous wave signals, offering advantages such as low transmission power and the absence of working blind spots in comparison to traditional pulse systems [7,8]. In the field of communication, IBFD is one of the key emerging technologies in 6G because of its potential for significant savings in frequency spectrum resources and improvements in information transmission efficiency [9,10].

Self-interference cancellation (SIC) is a significant challenge when implementing a full-duplex transceiver. This issue arises because the self-interference (SI) signal comprises not merely the linear component of the intended transmission signal, but also additional signal components, including the nonlinear signal component emanating from the power amplifier (PA), transmitter noise, phase noise, IQ imbalance resulting from the local oscillator (LO), and quantization noise generated by the analog-to-digital converter (ADC) [11]. The phase noise and IQ imbalance can be effectively addressed through both a calibration mechanism and a common LO [11,12]. Increasing the effective quantization bit can help reduce ADC quantization noise in the receiver chain. Thus, the main signal components usually concerned are the linear component, nonlinear component, and transmitter noise.

The linear SI component can be represented as the multi-path channel response of the desired transmission signal, which is a weighted combination of transmit signals with varing time delays [13,14]. The nonlinear signal component originates from the compression of the signal’s highest peak value. As the input power of the PA approaches saturation, an increase in the instantaneous amplitude will result in a reduction in the PA’s instantaneous gain, thereby leading to the generation of nonlinear signal components. The existence of these nonlinear signal components gives rise to a phenomenon known as spectrum regeneration in the frequency domain, which serves to further complicate the task of SIC [3,11,15]. Despite the existence of several behavioral models designed to describe the output of a PA for SIC, pre-distortion [16,17], or post-distortion [18] purposes, no studies have yet been conducted on the clear analysis of the PA output signal components under different forms of signal stimulation.

The common solution for SIC is to achieve spatial isolation or signal cancellation in different domains: the propagation domain, analog domain, digital domain, or multi-domain joint [19,20]. In the propagation domain, the SIC approach typically designs antennas and antenna arrays with high directionality, spatial zeros, or polarization diversity [21,22,23]. This method can effectively suppress the effects of all concerned SI signal components, with an isolation performance of approximately 40+ dB in narrowband conditions [24,25]. There are two main architectures for analog domain SIC methods: active RF SIC architecture and digital-assisted RF SIC architecture [26,27]. Both methods can suppress the mentioned three SI components with an isolation performance of about 40 dB in narrowband conditions. The digital domain SIC mainly includes channel modeling and estimation, digital beamforming and filtering, and fingerprint signal design [28,29,30,31,32]. This type of SIC method effectively suppresses both linear and nonlinear SI signal components, achieving an isolation performance of approximately 60 dB under narrowband conditions [33]. Additionally, the aperture-level STAR digital phased array architecture can achieve transmit isolation of 140.5 dB within a 100 MHz instantaneous bandwidth at 2.45 GHz [34]. In addition, the multi-domain joint cancellation method faces challenges in achieving a true “1 + 1 = 2” effect, potentially compromising the overall isolation performance [35,36,37]. In summary, the existing literature aims to identify a method that can eliminate as many SI signal components as possible simultaneously, but there is a lack of prior quantification and analysis of these components. This limitation is the fundamental reason why existing SIC methods strive for complete cancellation of all SI signal components.

Currently, the literature has discussed the power of SI signal components. Cummings et al. proposed an aperture-level simultaneous digital array for transmitting and receiving signals [38]. They introduced observation chains to eliminate all SI signal components, achieving an effective isotropic isolation (EII) of 187.1 dB under narrowband conditions with a transmit power of 2500 W. They quantitatively analyzed the power of the transmit noise before and after SIC, but they did not analyze the nonlinear distortion power caused by the PA. Dani Korpi et al. quantitatively analyzed the PA nonlinear distortion, the distortion components in the receive chain, and the power of receiver thermal noise power [39]. They visually demonstrated the effectiveness of the SIC algorithm using a power model. However, the power model does not include transmit noise, and the PA nonlinear distortion power is only calculated directly by PA-related indicators without analyzing the impact of the actual working waveform on the PA nonlinear distortion power. Therefore, this article constructs the system-level power model based on the analysis of PA output signal components under different waveform stimulation to analyze the power of the received signal components.

The main contributions are summarized as follows:

• A theoretical analysis tool for PA nonlinearities is proposed, which is capable of observing in-band nonlinearities. While conventional communication and radar systems tend to focus on adjacent-band nonlinearities, in-band nonlinearities also do matter.
• It offers a theoretical basis for simplifying the design of SIC methods. As indicated in this paper, full-duplex devices in different application scenarios have different performance requirements for the SIC method, and thus their realization difficulties vary.
• The system-level power analysis methods can be employed to assess the effectiveness of all types of SICs. A verification of whether the SI has been suppressed below the receiver noise level can be conducted according to the system-level power model.

This paper initially employs the algorithm named separate analysis of linear and nonlinear components to accurately assess the power levels of various signal components in the behavior model output signal. After the linear-to-nonlinear power ratio has been obtained, a system-level power model is constructed. This model not only analyzes the nonlinear SI component, but it also considers the impacts of transmit noise and ADC quantization noise. The simulation results indicate that there are notable variations in the power levels of different signal components depending on the type of stimulation signal. Finally, the proposed algorithm and power model are confirmed through measurement and Simulink, respectively.

The following is a description of the structure of this article. The second section provides three signal forms and the PA behavior model that are employed in this paper. The third section details the principle of the proposed power analysis method, which comprises an algorithm and a system-level power model. The simulation analysis and verification are presented in the fourth section. Verification of the PA nonlinear component analysis is conducted through measurements, while the system-level power model is verified using the Simulink transceiver platform. Finally, the fifth section offers the conclusion.

2. Signal Model

As the PA introduces nonlinear distortion when stimulated by nonconstant modulus signals, whereas constant modulus signals do not, we discuss the three signal forms and the modeling of the PA output signal in this section.

For a nonconstant modulus signal, an orthogonal frequency division multiplexing (OFDM) that has N subcarriers is considered in this investigation. The k-th element of time domain signal $\mathbf{x}={[x\left(0\right),x\left(1\right),\dots ,x(N-1)]}^T$ can be obtained by

$$\mathit{x}\left(k\right)={\displaystyle \frac{1}{\sqrt{N}}}\sum _{n=0}^{N-1}X\left(n\right)e^{j2\pi kn/N}$$

(1)

with $\mathit{k}=0,1,\dots ,N-1$.

For a constant modulus signal, we choose a linear frequency modulation (LFM) signal with a bandwidth of B. In the digital domain, the signal can be represented as

$$\mathit{x}\left(n\right)=A{e}^{j\pi K_f{\left(n\right)}^2}$$

(2)

where $K_f=B/T_p$, A and $T_p$ define the amplitude and duration of the signal, respectively. If an LFM signal is transmitted without interruption, it is referred to as a linear frequency modulation continuous wave (LFMCW).

The following part discusses the modeling of the PA output signal. It is common practice to employ the amplitude modulation to amplitude modulation (AM/AM) and the amplitude modulation to phase modulation (AM/PM) techniques for analyzing PA nonlinearity by observing the gain and phase-shift variations of the PA input and output signals. When gain compression is present in the AM/AM curve, nonlinearity is considered to be present. This provides the theoretical foundation for existing linearization techniques. The behavior model of the PA is employed not only for output signal modeling but also as a compensator for nonlinear distortion. To enhance modeling accuracy and linearization performance, several models have been developed to capture the characteristics of the PA. Taking complexity and accuracy into account, in this paper, we adopt the MP model [17,19], which is represented as follows:

$$\mathit{y}\left(n\right)=\sum _{m=0}^M\sum _{p=1,odd}^P{\alpha }_{m,p}x\left(n-m\right){\left|x\left(n-m\right)\right|}^{p-1}$$

(3)

where ${\alpha }_{m,p}$ is the MP model coefficient, and P and M define the model order and memory depth, respectively. Because the nonlinear components with even orders are located far from the carrier frequency position, they can be effectively filtered out by using a filter. Therefore, we use an odd value for p.

Specifically, when the LFMCW is fed into the PA, (3) can be simplified as [40]

$$\mathit{y}\left(n\right)=\sum _{m=0}^M\sum _{p=1,odd}^P{\alpha }_{m,p}x\left(n-m\right)A^{p-1}=\sum _{m=0}^M{h}_{m}x\left(n-m\right)$$

(4)

where $h_m={\displaystyle \sum _{p=1,odd}^P}{\alpha }_{m,p}{A}^{p-1}$. It is widely acknowledged that (4) is the typical architecture of finite impulse response (FIR) filters, which constitute a linear model.

3. Power Analysis Method

3.1. Quantitative Analysis of PA Nonlinear Component

While the generation of PA nonlinearity and the design of linearization can be explained by the AM/AM and AM/PM curves, this approach does not allow for separate analysis of the spectra of linear and nonlinear components, and it is challenging to gain a comprehensive understanding of the in-band nonlinear distortion of the PA.

At first, we decompose the PA output signal spectrum $\mathit{Y}\left(f\right)$ into linear and nonlinear components using the following formula:

$$\mathit{Y}\left(f\right)=\mathit{H}\left(f\right)\mathit{X}\left(f\right)+\mathit{D}\left(f\right)$$

(5)

where $\mathit{X}\left(f\right)$ is the PA input signal spectrum, $\mathit{D}\left(f\right)$ is the nonlinear component’s spectrum, and $\mathit{H}\left(f\right)$ is the PA linear gain.

After multiplying the frequency spectrum of the output signal by the frequency spectrum of the input signal, multiple measurements are taken and averaged to obtain

$$\mathit{Y}\left(f\right)\mathit{X}*\left(f\right)=\mathit{H}\left(f\right)\mathit{X}\left(f\right)\mathit{X}*\left(f\right)+\mathit{D}\left(f\right)\mathit{X}*\left(f\right)$$

(6)

As the nonlinear component is not correlated with the input signal, the Formula (6) can be simplified as follows:

$$\mathit{Y}\left(f\right)\mathit{X}*\left(f\right)=\mathit{H}\left(f\right)\mathit{X}\left(f\right)\mathit{X}*\left(f\right)$$

(7)

Thus, the PA linear gain can be expressed as

$$\mathit{H}\left(f\right)={\displaystyle \frac{\mathit{Y}\left(f\right)\mathit{X}*\left(f\right)}{{\left|\mathit{X}\left(f\right)\right|}^2}}$$

(8)

The nonlinear component’s spectrum is

$$\mathit{D}\left(f\right)=\mathit{Y}\left(f\right)-\mathit{H}\left(f\right)\mathit{X}\left(f\right)$$

(9)

The detailed analysis steps are shown in Algorithm 1. The model coefficients are obtained in two ways. The first method is the most common and involves measuring the input and output signals of the PA and estimating them by adopting the 1-bit ridge regression least squares (LSs) algorithm [41]. The second method utilizes known model coefficients, whether previously measured or obtained from the existing literature. No matter how the coefficients are obtained, this has no impact on the validity of the proposed algorithm. Following this process, the spectrum of the linear and nonlinear components can be acquired.

Algorithm 1 Separate analysis of linear and nonlinear components

Input: ${\mathbf{x}}_1\in {\mathbb{C}}^{N\times 1}$, ${\mathbf{x}}_2\in {\mathbb{C}}^{N\times 1}$, $\mathbf{y}\in {\mathbb{C}}^{N\times 1}$. Output: $H\left(f\right)X\left(f\right)$, $D\left(f\right)$. 1: Calculate model coefficients ${\alpha }_{m,p}$ using 1-bit regression LSs estimation with ${\mathbf{x}}_1$, $\mathbf{y}$. 2: Calculate the estimation of PA output $\widehat{\mathbf{y}}$ according to (3). 3: Calculate the NMSE. 4: When PA input is ${\mathbf{x}}_2$, calculate the estimation of PA output ${\widehat{\mathbf{y}}}_2$ according to (3). 5: FFT for ${\mathbf{x}}_2$ and ${\widehat{\mathbf{y}}}_2$ to obtain $\mathit{X}\left(f\right)$ and $\mathit{Y}\left(f\right)$. 6: Calculate the inter-correlations $\mathit{Y}\left(f\right)\mathit{X}*\left(f\right)$ and ${\mathit{X}}^2\left(f\right)$. 7: Calculate the $\mathit{H}\left(f\right)$ and $\mathit{D}\left(f\right)$. 8: return $\mathit{H}\left(f\right)\mathit{X}\left(f\right)$, $\mathit{D}\left(f\right)$.

Generally, the normalized mean squared error (NMSE) is used to evaluate in-band nonlinear distortion in the time domain, and it can be expressed as

$$\mathrm{NMSE}=10lg\left({\displaystyle \frac{{\displaystyle \sum _{n=1}^N}{\left|\mathit{y}\left(n\right)-\widehat{\mathit{y}}\left(n\right)\right|}^2}{{\displaystyle \sum _{n=1}^N}{\left|\mathit{y}\left(n\right)\right|}^2}}\right)$$

(10)

where $\mathit{y}\left(n\right)$ and $\widehat{\mathit{y}}\left(n\right)$ represent the PA measured output data and the estimation of the output data, respectively. Nevertheless, it is challenging for the NMSE to characterize the spectral distribution of in-band nonlinearity.

The power output 1 dB compression point (PO1dB), 2nd-order intermodulation distortion (IMD2), 3rd-order intermodulation distortion (IMD3), and adjacent channel power ratio (ACPR) are also crucial parameters for assessing the nonlinearity of the PA. PO1dB represents the output power when PA generates a 1 dB gain compression. According to PO1dB, this paper examines the output signal component power when the PA operates at the linear region and 1 dB compression point, respectively. IMD2 is typically filtered out due to its distance from the carrier frequency and is thereby excluded from the analysis presented in this paper. In two-tone or multi-tone tests, IMD3 is commonly employed to assess the PA’s nonlinearity. In contrast, ACPR is typically utilized in band-limit signal testing to evaluate the PA’s adjacent-band nonlinearity.

Given the focus on in-band nonlinearity, we define the in-band linear-to-nonlinear power ratio (LNPR) which can be expressed as

$$\mathrm{LNPR}={\displaystyle \frac{min\left({\mathit{P}}_{\mathrm{lin}}\left(f\right)\right)}{max\left({\mathit{P}}_{\mathrm{nonlin}}\left(f\right)\right)}},f\in f_c+\left[-B/2,B/2\right]$$

(11)

where ${\mathit{P}}_{\mathrm{lin}}\left(f\right)$ and ${\mathit{P}}_{\mathrm{nonlin}}\left(f\right)$ represent the power spectrum values of the linear component and nonlinear component, respectively.

3.2. The System-Level Power Model

Based on the quantitative analysis of the PA’s nonlinear power magnitude, a system-level power analysis model can be constructed next.

The full-duplex transceiver consists of multiple devices, including a digital-to-analog converter (DAC), a lowpass filter (LPF), an IQ mixer, a variable gain controller (VGA), the PA, a bandpass filter (BPF), a lower noise amplifier (LNA), an attenuator, and an ADC, as illustrated in Figure 1. The antenna section comprises either a dual antenna with separate transmission and reception capabilities or a single antenna with a circulator. It is assumed that there are no analog or digital cancellations in the system, and the receiver operates correctly.

Assume that the power of the SOI reaching the receiving end is ${\mathit{p}}_{\mathit{soi}}$, and the system transmission power is ${\mathit{p}}_{\mathit{tx}}$. Using the receiving antenna as a reference point, the power difference between the two can be denoted as

$$\eta ={\displaystyle \frac{{\mathit{p}}_{\mathit{tx}}-a_{ant}}{{\mathit{p}}_{\mathit{soi}}}}$$

(12)

where $a_{ant}$ is the isolation of Tx-Rx.

If the power of the SI nonlinear component introduced by PA is ${\mathit{p}}_{\mathit{PA}},\mathit{non}$, the power of transmitter noise is ${\mathit{p}}_{\mathit{tx}},\mathit{noise}$. Thus, the transmit power satisfies

$${\mathit{p}}_{\mathit{tx}}={\mathit{p}}_{\mathit{tx}},\mathit{lin}+{\mathit{p}}_{\mathit{PA}},\mathit{non}+{\mathit{p}}_{\mathit{tx}},\mathit{noise}={\mathit{p}}_{\mathit{tx}},\mathit{lin}+{\displaystyle \frac{{\mathit{p}}_{\mathit{tx}}}{\mathrm{LNPR}}}+{\displaystyle \frac{{\mathit{p}}_{\mathit{tx}}}{{\mathit{snr}}_{\mathit{tx}}}}$$

(13)

where ${\mathit{snr}}_{\mathit{tx}}$ is the SNR of the transmit signal, and ${\mathit{p}}_{\mathit{tx}},\mathit{lin}$ is the power of the linear component in the transmit signal. Since the nonlinear component and the transmitter noise are several orders of magnitude weaker than the linear component, thus, we have ${\mathit{p}}_{\mathit{tx}},\mathit{lin}\approx {\mathit{p}}_{\mathit{tx}}$.

The signal power of the receiving antenna is denoted as

$${\mathit{p}}_{\mathit{rx}}={\displaystyle \frac{{\mathit{p}}_{\mathit{tx}}}{a_{ant}}}+{\mathit{p}}_{\mathit{soi}}+{\mathit{p}}_{\mathit{rx}},\mathit{noise}$$

(14)

where ${\mathit{p}}_{\mathit{rx}},\mathit{noise}$ is the power of the receiver thermal noise, which is calculated by

$${\mathit{p}}_{\mathit{rx}},\mathit{noise}=k{T}_{0}BF$$

(15)

where k is the Boltzmann constant, and $T_0$ and F are the temperature and noise figure of the receiver. The thermal noise of the receiver is typically smaller than the receiver’s sensitivity and can be ignored. Therefore, the total signal power reaching the ADC is

$${\mathit{p}}_{\mathit{adc}}=g_{rx}\left({\displaystyle \frac{{\mathit{p}}_{\mathit{tx}}}{a_{ant}}}+{\mathit{p}}_{\mathit{soi}}\right)={\displaystyle \frac{g_{rx}{\mathit{p}}_{\mathit{tx}},\mathit{lin}}{a_{ant}}}+{\displaystyle \frac{g_{rx}{\mathit{p}}_{\mathit{PA}},\mathit{non}}{a_{ant}}}+{\displaystyle \frac{g_{rx}F{\mathit{p}}_{\mathit{tx}},\mathit{noise}}{a_{ant}}}+g_{rx}{\mathit{p}}_{\mathit{soi}}$$

(16)

where $g_{rx}$ represents the receiver gain.

Assuming that the receiver ADC employs a full voltage configuration, wherein ${\mathit{p}}_{\mathit{adc}}$ is the full input power of the ADC, the quantization noise of the ADC [42] is ${\mathit{p}}_{\mathit{qua}}$. Since quantization noise is only determined by the PAPR of the input signal and the number of bits of the ADC and is not influenced by other system parameters, the power of the ADC output can be expressed as

$$\begin{array}{cc}\hfill {\mathit{p}}_{\mathit{final}}=& {\displaystyle \frac{g_{rx}{\mathit{p}}_{\mathit{tx}},\mathit{lin}}{a_{ant}}}+{\displaystyle \frac{g_{rx}{\mathit{p}}_{\mathit{PA}},\mathit{non}}{a_{ant}}}+{\displaystyle \frac{g_{rx}F{\mathit{p}}_{\mathit{tx}},\mathit{noise}}{a_{ant}}}\hfill \\ \hfill & +g_{rx}{\mathit{p}}_{\mathit{soi}}+{\mathit{p}}_{\mathit{qua}}\hfill \end{array}$$

(17)

where $g_{rx}={\mathit{p}}_{\mathit{ADC}}-{\mathit{p}}_{\mathit{rx}}$. Thus, the system-level power model of various SI signal components at the receiver end are as follows:

$${\mathit{p}}_{\mathit{rx}},\mathit{lin}\approx {\displaystyle \frac{g_{rx}{\mathit{p}}_{\mathit{tx}}}{a_{ant}}}$$

(18)

$${\mathit{p}}_{\mathit{rx}},\mathit{non}={\displaystyle \frac{g_{rx}{\mathit{p}}_{\mathit{tx}}}{a_{ant}·\mathrm{LNPR}}}$$

(19)

$${\mathit{p}}_{\mathit{tx}},\mathit{noise}={\displaystyle \frac{g_{rx}F{p}_{tx}}{a_{ant}{\mathit{snr}}_{\mathit{tx}}}}$$

(20)

4. Simulation Analysis and Verification

4.1. The PA Nonlinear Component

4.1.1. Signal Parameter and Experimental Setup

To verify the components of the PA output signal, this section compares the measured results and model analysis results of the PA output signal components under three forms of signal stimulation. The three signals are the OFDM, LFM, and LFMCW. The parameters of these signals are shown in

Table 1. Signal parameters setup.

Parameter	OFDM	LFM	LFMCW
Signal bandwidth	20 MHz	20 MHz	20 MHz
Signal duration	32 μs	10 μs	10 μs
Subcarriers	64	-	-
Modulation type	QPSK	LFM	LFM
Pulse repetition interval	-	70.8 μs	-

. It is noted that the duty cycle of pulse modulation in the LFM was set to $14\%$ to yield a PAPR of 8.5 dB for both.

This experiment adopted a well-established PA output signal component testing scheme from Keysight. Figure 2 shows the measurement framework design. The signal source provides signal excitation through the rear panel interface of the vector network analyzer (VNA). The provided modulation distortion (MOD) function in the VNA can be used to measure the power of the output signal component of the PA while it is in operation. The PA is the device-under-test (DUT), while the coupler, attenuator, and the matched load are test accessories.

The signal generator used is the Keysight N5182B, with an analysis bandwidth of 160 MHz. The VNA used is the Keysight N5245B. The DUT used is the DBPA3500700300A, which is a commercial PA whose datasheet is shown in [43]. It has an output power of 39 dBm and operates within a frequency range of 700 MHz to 3 GHz. The coupler used operates within a frequency range of DC to 18 GHz, and its attenuation is 6 dB. To protect the VNA, a 40 dB attenuator needed to be added. For this experiment, the 40 dB attenuation was achieved by using three attenuators with attenuation levels of 20 dB, 10 dB, and 10 dB. Additionally, the signal carrier frequency for this experiment was set to 2.4 GHz.

4.1.2. Simulation and Measurement Analysis

In Figure 3, the horizontal axis displays ten power values ranging from the linear region to the saturation region. This arrangement allowed for a clearer examination of the NMSE variations within the same signal under different operating states of the PA. The results indicate that both the measured PA data and the model output data exhibited an NMSE better than −28 dB for all three signal stimulations. Notably, the model fitting performance was superior for constant modulus signals compared to nonconstant modulus signal, and the NMSEs for the LFM and LFMCW signals were both better than −41 dB.

Figure 4 shows the simulation and experimental results of the signal components of the PA operating in the linear region. The blue curves represent the simulation result, and the red curves refer to the measurement result. The solid and dotted curves represent the spectrum of the in-band linear component and nonlinear component, respectively. Figure 4a displays the results when PA was stimulated by the OFDM. At this point, the nonlinear component was observed within the operating bandwidth and the adjacent bands. This was caused by gain compression of the signal with high amplitude due to inadequate power back-off. “SL” and “ML” denote the $min\left(P_{lin}\left(f\right)\right)$ of the simulation and measurement, respectively, while “SN” and “MN” denote the $max\left(P_{nonlin}\left(f\right)\right)$, respectively. Then, the LNPR simulation result of the PA’s output signal components under OFDM stimulation was calculated equal to 27.8 dB (37.0593 dB − 9.24659 dB); the measured value was 26.7 dB (−43.7826 dB + 70.4714 dB). Figure 4b illustrates the results when the PA was stimulated by the LFM. It is difficult to observe the impact of adjacent bands and in-band nonlinearity directly from the spectrum of the output signal. In that instance, the distribution of nonlinearity across the entire frequency band could only be observed through the proposed algorithm. The LNPR of the simulation and measurement came out to 47.3 dB and 51.4 dB, respectively. Compared to the OFDM case, the nonlinear component power introduced by the PA under LFM stimulation was significantly weaker. Figure 4c shows the results when the PA was stimulated by the LFMCW. In this case, no notable nonlinear features were observed either by the spectrum of the output signal or by the proposed algorithm. The spectrum of the nonlinear component exhibited a greater similarity to the spectrum of the noise. At that time, the LNPR of the simulation and measurement came out to 52.8 dB and 52.9 dB, respectively.

Figure 5 displays the simulation and experimental results of the signal components of the PA operating at a 1 dB compression point. From Figure 5a, there is a clear nonlinear distortion evident in the adjacent bands. Compared to Figure 4a, the nonlinearity in the adjacent bands of this case was significantly stronger. In Figure 5b, it is also difficult to evaluate the nonlinear distortion through the adjacent channel power ratio. The power of the nonlinear component still needs the LNPR to be characterized. According to Figure 5c, there are no notable nonlinear features, and the spectrum of the nonlinear component is similar to noise. In addition, the LNPR values of the simulation results, in this situation, came out to 16.3 dB, 45.1 dB, and 51.1 dB. The experimental measurements came out to 15.3 dB, 47.7 dB, and 51.7 dB. Compared to Figure 4, the LNPR value in Figure 5 is much higher. This is consistent with the conventional understanding that the nonlinear distortion introduced when the PA operates in the linear region is less than that introduced when the PA operates in the saturation region.

In summary, we can draw the following conclusions:

• The method of analyzing PA output signal components through behavioral models is universally significant for analysis in different application scenarios.
• Regardless of the state in which the PA operates, it does not generate nonlinear distortion when excited by the LFMCW signal.
• If the constant modulus signal contains rectangular window modulation in the time domain, in-band nonlinear distortion will be introduced by the PA but will be smaller than the nonconstant modulus signal.

4.2. The System-Level Power Model

4.2.1. The Transceiver Parameter Setup

To provide actual numerical results with the derived equations,

Table 2. STAR system parameters setup.

Parameter	Symbol	Unit	Value
Signal bandwidth	B	MHz	20
Receiver noise figure	F	dB	4.1
Antenna isolation	$a_{ant}$	dB	30
Receiver sensitivity	s	dBm	−86.9
Receiver dynamic range		dB	70
Minimum SNR	${\mathit{snr}}_{\mathit{min}}$	dB	10
Transmitter SNR	${\mathit{snr}}_{\mathit{tx}}$	dB	60
ADC max input voltage	$V_{max}$	V	4.5
ADC bit	b		12

specifies the parameters for the full-duplex transceiver. Although we only measured the power of the signal components of the DBPA3500700300A under constant/nonconstant modulus signal stimulation, this did not affect the simulation analysis of the power of the PA output signal component at different output powers. In practical scenarios, the behavior model coefficients of the PA used can be extracted and updated based on the proposed algorithm described in Section 3. For convenience, we assumed that the distortion characteristics of the PA at different output powers were consistent with the DBPA3500700300A in subsequent simulations.

From the perspective of SIC, the most challenging situation is when the SOI power is close to the receiver’s sensitivity level. The reference sensitivity can be written as [44]

$$\mathit{s}=k{T}_{0}BF{\mathit{snr}}_{\mathit{min}}$$

(21)

We only need to focus on the SI component that exceeds the receiver’s noise level, as it will worsen the SNR of the received signal. In this research, we only considered the noise power of receiver and did not look at distortion in the receiver chain. The noise power of receiver after ADC sampling can be denoted as

$${\mathit{p}}_{\mathit{RX}},\mathit{noise}={\mathit{p}}_{\mathit{rx}},\mathit{noise}{g}_{rx}$$

(22)

Generally, the transmit power is determined when designing a full-duplex transceiver at the top level, while the power of the SOI varies depending on the target distance. Consequently, $\eta$ is a variable when $a_{ant}$ is a fixed value. To ensure the normal operation of the full-duplex transceiver even under the most challenging conditions, the proposed power model can be used to analyze the power of the SI components in the received signal. This analysis helps determine the power levels of the SI signal components that need to be suppressed and allocate isolation performance based on the characteristics of different domain SIC methods.

Considering certain special scenarios, such as communication relay nodes or forwarding jamming, $\eta$ is a fixed value. This signifies that the transmit power of the full-duplex transceiver fluctuates with the power of the SOI. Consequently, the power level of the SI signal components in the received signal will also vary, leading to differing performance requirements for the SIC method.

Because of the significant impact of transmission power on the SI signal component in the received signal, we analyzed the changes in the performance requirements of SIC methods under different transmission powers in the most challenging scenarios. In this analysis, we assumed that the power of the SOI remained fixed at the receiver sensitivity while adjusting the transmit power of the full-duplex transceiver.

4.2.2. Transceiver Platform Establishment

To further validate the proposed power model of the received signal components, this paper utilized the Simulink and RF Blockset libraries to construct a full-duplex transceiver simulation platform. The platform, as depicted in Figure 6, mainly includes the SignalGenerator module, the SOI module, and the Transceiver module. The SignalGenerator module was used to generate the transmitted digital baseband signal. The SOI module was used to generate a 2.405 GHz signal as the SOI. After IQ demodulation, the SOI’s baseband emitted a 5 MHz single frequency complex tone signal. The transceiver simulated the entire transmit and receive chain, and its detailed block diagram is shown in Figure 7.

4.2.3. Constant Modulus Signal Stimulation

Figure 8a illustrates the simulation results of different signal components under constant modulus signal stimulation when $\eta$ was a fixed value. For that case, the transmit power was set to −20 dBm, while the power of the SOI varied from −85 dBm to −20 dBm, while the receiver sensitivity was set to −86.9 dBm. According to the PA measurement data in Section 4.1, it is evident that the LNPR under LFM stimulation was 47.7 dB. To ensure that the received signal power would fall within the dynamic range of the ADC, the receiver needed to adjust the gain of the chain dynamically. However, this adjustment is affected by the power of the SOI. When the power of the SOI is low, the received signal consists mostly of SI. Since the transmission power is fixed, the power of the SI component remains constant, and the receiver link gain stays relatively stable. On the other hand, when the power of the SOI is high, the received signal is a combination of the SOI and SI signals. In this case, the higher the power of the SOI, the smaller the gain of receiver chain becomes. The changes in the variable signal components in Figure 8a illustrate this influence.

As seen in Figure 8a, the linear SI components needed to be suppressed by −45 dBm or more, and the nonlinear SI components reached the receiver noise that could be ignored. The transmit noise could also be overlooked, because it was lower than the receiver noise, as well as the quantization noise of the ADC. Significantly, the requirement for SIC methods did not vary with the power of the SOI.

Figure 8b shows the simulation results when $\eta$ was a fixed value. We assumed that the power difference between ${\mathit{p}}_{\mathit{tx}}$ and ${\mathit{p}}_{\mathit{soi}}$ was 30 dB; then, $\eta$ was equal to 0 dB according to Equation (12). Consequently, the power of the SOI and SI signal was equal, and the gain of the chain varied with the transmit power. As seen in Figure 8b, the full-duplex transceiver suppressed linear SI across the entire dynamic range of reception effectively, while the nonlinear SI and transmit noise were different. When ${\mathit{P}}_2>{\mathit{p}}_{\mathit{tx}}\ge {\mathit{P}}_1$, the SIC method not only needed to suppress the linear SI, but it also needed to suppress the nonlinear SI. When ${\mathit{P}}_{max}\ge {\mathit{p}}_{\mathit{tx}}\ge {\mathit{P}}_2$, additional suppression of transmitter noise is required. According to (15), (19), and (22), it can be obtained when ${\mathit{P}}_1$ is −19.2 dBm. Based on (15), (20), and (22), it can be calculated that ${\mathit{P}}_2$ is −11 dBm. This is consistent with simulation results.

Finally, the simulation results of different transmit power values with a fixed power of the SOI are shown in Figure 8c. As the power of the SOI was equal to the receiver sensitivity, and the transmit power changed from −58 dBm to −3 dBm, the gain of the receiver chain adjusted dynamically. When the transmit power was low, the SOI and SI signals both mattered. As the transmit power increased, the SI became the main signal component of the received signal, and the gain of the receiver chain decreased. The changes in the variable signal components also reflect this influence. It is worth noting that the performance requirements of the SIC methods in Figure 8a are equal to the situation in Figure 8c when the transmit power was −20 dBm. Thus, Figure 8c can be used to analyze the changes in the performance requirements of SIC methods under different transmit powers. Compared to Figure 8b, there was no change in the $P_1$ and $P_2$ values, and the power values of various signal components had increased.

Here, we take the LFM as an example to analyze the correctness of the received signal power model. Firstly, subtract the SOI from the received signal, and then analyze the power of the remaining signal components. The reference input signal is a digital transmit baseband signal. For convenience of comparison, the value of the transmission power is consistent with Figure 8b. After calculation, the power of each signal component is shown in Figure 9.

In the Simulink simulation results, the linear SI component was close to the power of the SOI and aligned with the results obtained from the power model. The difference in the power of nonlinear SI components between the proposed model and the Simulink is due to the way the LNPR was calculated. When the transmission power is low, the nonlinear component power is lower than the receiver noise, and the receiver noise power is more noticeable. Therefore, when calculating the LNPR, the most powerful point generated by the receiver noise within the frequency band was used as a reference, resulting in the nonlinear component power in the Simulink simulation results reflecting the changes in the receiver noise. Conversely, when the transmission power is high, the nonlinear component power exceeds the receiver noise and approaches a convergence value. At this point, the receiver noise power is relatively small and difficult to estimate accurately from the power spectrum, which makes it impossible to plot a receiver noise curve. Overall, except for the difficulty in calculating transmit noise, the power variation trend of other signal component is consistent with the model results.

4.2.4. Nonconstant Modulus Signal Stimulation

Figure 10a illustrates the simulation results of different signal components under nonconstant modulus signal stimulation when $\eta$ was set to a fixed value. For this case, the transmit power and the power of the SOI were same as the constant modulus signal. According to the PA measurement data in Section 4.1, it is evident that the the LNPR under OFDM stimulation yielded a value of 15.3 dB. In the case of nonconstant modulus signal stimulation, the observed trends of all signal components with the SOI power were consistent with those observed in the constant modulus signal case. The discrepancy between the two cases can be attributed to the different power levels of the nonlinear SI components. In the case of nonconstant modulus signals, the designed SIC method must cancel not only the linear SI above 45 dBm but also the nonlinear SI at a level of at least 31 dBm. It can be seen that the performance requirements of the full-duplex transceiver under nonconstant modulus signal stimulation for the SIC method are higher than those of constant modulus signals.

Figure 10b displays the results when $\eta$ was 0 dB. According to (15), (19), and (22), it can be obtained that ${\mathit{P}}_1$ is −51.6 dBm. Based on (15), (20), and (22), it can be calculated that ${\mathit{P}}_2$ is −11 dBm. This is consistent with simulation results. Compared to Figure 8b, ${\mathit{P}}_1$ decreased rapidly. This means that under the stimulation of nonconstant modulus signals, the full-duplex transceiver needed to suppress the nonlinear SI component at a lower transmit power. Because of the unchanged SNR of the transmit chain, there would be no influence on ${\mathit{P}}_2$ with changes in form of the stimulation signal.

At last, Figure 10c shows the simulation results of different transmit powers with a fixed power of the SOI. The observed trends of all signal components with the SOI power under nonconstant modulus signal case were consistent with those observed in the constant modulus signal case. The distinction between the two cases is ${\mathit{P}}_1$. It is obvious that the designed SIC methods needed to suppress the nonlinear SI components at lower transmit powers. The SNR of the transmitter remained unaltered, as did the ${\mathit{P}}_2$. Likewise, the performance requirements of the SIC methods in Figure 10a are equal to the situation in Figure 10c when the transmit power was set to −20 dBm.

Using the OFDM signal as an example, the correctness of the received signal power model is analyzed. The processing is consistent with the situation described above, and the simulation results are presented in Figure 11. The SOI power and linear SI component power remained mostly unchanged, being consistent with the model results. There was a slight discrepancy between the simulation results obtained from Simulink, which were approximately 1–2 dB higher. In cases of low transmission power, the nonlinear component power reflected variations in the receiver noise similarly. Conversely, at high transmission power, the nonlinear component power surpassed the receiver noise and approached the convergence value. The convergence value was 1 dB higher than the model result.

5. Conclusions

This paper proposes a power analysis method for SI signal components to guide the design of SIC methods in a full-duplex transceiver under different forms of signal stimulation, and a simplified strategy for SIC methods is provided. First, the output signal components of the PA were analyzed based on the proposed algorithm. Then, the power model of various signal components at the digital end of the receiver was derived. The results of the model simulations and those obtained using Simulink demonstrate that full-duplex transceivers stimulated by nonconstant modulus signals have higher performance requirements for the SIC method than those stimulated by constant modulus signals, particularly in suppressing strong nonlinear SI components. In addition, by synthesizing the characteristics of different domain SIC methods and allocating the isolation performance of each method, this reduces the overall complexity of the system. Building on this work, future research will focus on investigating unique SIC methods for full-duplex transceivers excited by constant modulus signals.