# Impact on Antijamming Performance of Channel Mismatch in GNSS Antenna Arrays Receivers

Authors: Zukun Lu; Junwei Nie; Feiqiang Chen; Gang Ou
Journal: International Journal of Antennas and Propagation
 (2016)

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## Abstract

The performance of antijamming is limited by channel mismatch in global navigation satellite system (GNSS) antenna arrays receivers. Only when the amplitude and phase characteristics of each array channels are the same is the interference likely to be completely suppressed. This paper analyzes the impact on antijamming performance of channel mismatch. We built the model of channel mismatch and derived the impact on transfer function with space-time adaptive processor (STAP) of channel mismatch in theory. The impact factor of channel mismatch is proposed by the fuzzy transfer function, which could directly reflect the antijamming performance under channel mismatch. In addition, every characteristic in the channel mismatch model is analyzed. The analysis results show that the greatest impacts on antijamming performance are the range of amplitude wave and the group delay bias, while the influence of the number of amplitudes is next. As for the effect of group delay fluctuation is the smallest.


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## Body

## 1. Introduction

Since 1970s, global navigation satellite system (GNSS), whose representative is global position system (GPS), has a rapid development. GNSS has brought great convenience to people in traffic, and it is also widely used in military field [1]. The distance between earth surface and navigation satellite is about 27,000 km. The navigation signal is very weak when it reaches the ground. The navigation receivers get the signals with the signal/noise ratio (SNR) is about −30 dB, which makes receivers vulnerable to all kinds of intentional or unintentional interference. The performance of antijamming has become a key indicator in military and some civilian applications [2, 3]. According to the number of antennas, the antijamming technology of receivers can be divided into single antenna and antenna arrays, in which the single antenna could only suppress narrowband interference and the antenna arrays could suppress both narrowband and wideband interference [4]. Under the condition of navigation war, the receivers would face the strong wideband interference. The antenna arrays equipped with space-time adaptive processor (STAP) would be an optimal choice in the electromagnetic countermeasure [5]. The fundamental principle of STAP is array weight in digital baseband, which could suppress interference effectively when the amplitude and phase characteristics of each channel are unanimous [6]. However, each array channel is composed of the low noise amplifier (LNA), analog filter, inverter, and other analog devices. The manufacturing errors and environmental and other factors could lead to inconsistencies of amplitude and phase in each channel, which would cause channel mismatch [7, 8].Channel mismatch has attracted the attention of scholars. The impact of channel mismatch on array gain is studied by space-only processor (SOP) in [9], which notes that the amplitude and phase mismatch would affect the performance of interference cancellation, which would lead to the degradation of antijamming performance. Literature [10] gives a preliminary analysis to the impact of channel mismatch on antenna arrays equipped with STAP. The estimated carrier phase would be biased due to the channel mismatch, and the greater mismatch would bring greater deviation. A correction method of channel mismatch based on the correction table is proposed in [11], which needs to measure the frequency response in all directions and estimate the bias of pseudocode and carrier phase which is introduced by antenna arrays, and it also needs to make the correction table bind to the receivers, whereas the nonideal characteristic of the channel might be changed in some time, whose method lacks certain robustness.This paper further analyzes the impact of channel mismatch on antijamming performance with antenna arrays. First of all, we establish a model of channel mismatch and deduce the impact of channel mismatch on transfer function with STAP in theory. Then, the impact on antijamming performance is analyzed. We propose a concept of channel mismatch impact factor and point out the largest impact characteristic of channel mismatch to antijamming performance. Finally, simulation experiments are implemented to validate the correctness of the analytical results.


## 2. Channel Mismatch Model

Nonideal characteristics of the channel are mainly reflected in the nonflatness of the amplitude and nonlinearity of the phase. The impact of the interference suppression in antenna arrays is not the absolute value of each channel, but the different between each channel, which can be called channel mismatch [12].Suppose the transfer function in the channeln is Hn(ω), and literature [13] gives a channel mismatch model which can be written as(1)Hnω=Aωejφωω≤πB0else,where B is the bandwidth of receivers. A(ω) and φ(ω) are the response of amplitude-frequency and phase-frequency, respectively. In the ideal case, A(ω) and φ(ω) satisfy the following relationship:(2)Aω≡1φω≡0.A more accurate channel mismatch model has been demonstrated in [14] which can be expressed as(3)Hnω=1+Racos⁡ωRanejRpcos⁡ωRpn+ωRbω≤πB0else,where Ra and Rp are the range of amplitude and group delay wave, respectively. Ran and Rpn are the number of amplitude and group delay waves, separately. Rb is the group delay bias.As for the antenna arrays withN elements, the channel mismatch model can be written as(4)Hω=H1ω⋯HNωT.


## 3. Impact on Transfer Function

STAP has obvious advantages compared to the simple time domain, frequency domain, and SOP. STAP could suppress a variety of interferences. Without increasing the antenna elements, the degree of freedom (DoF) of interference suppression can greatly increase, and its interference suppression capability has a quality improvement [15]. In addition, STAP could mitigate the impact of channel mismatch to interference suppression. The taps in STAP can compensate the channel mismatch.The STAP equipped withN elements and M taps is shown in Figure 1 [16].Figure 1
The STAP equipped withN elements and M taps.According to the STAP model shown in Figure1, the received data in different sampling time could form a new data vector which could be expressed as(5)x=XTtXTt-T⋯XTt-M-1TT,where Xt is the received data vector at the time of t.Array weightings could be written as(6)w=w11,…,wN1,…,w1M,…,wNMTIn order to conveniently perform analysis, assume the number of interferences is one and its power spectral density (PSD) is smooth. In addition, STAP is equipped with 2 elements and 2 taps.The steering vector of time and space domain can be, respectively, denoted as(7)vtω=1⋯e-jωM-1TTvsθ=1⋯e-jωRN-1τθT,where T is the sampling period, ωR is the radio frequency (RF), and τθ is the delay of the two adjacent antenna elements and the signal or interference direction is θ and it can be further expressed as [17](8)τθ=dsin⁡θc,where d is the distance of the adjacent antenna elements with half wavelength and c is the propagation speed of signal or interference.According to (7), the steering vector of space-time domain can be written as(9)aω,θ=vtω⊗vsωR,θ.Suppose channel 1 is desirable, and channel 2 represents the nonideal characteristics, and its transfer function isH that is related to the frequency with ω. According to (4), the model of channel mismatch could be expressed as(10)Hω=H0HT.As for all frequency,H0 in the above equation would satisfy the following relation:(11)H0≡1.Then, the steering vector of space-time domain could be improved as(12)aω,θ=vtω⊗vsωR,θ⊙Hω,where ⊗ stands for Kronecker product and ⊙ indicates Hadamard product.The power of signal is about 30 dB lower than the noise, so the signal can be negligible in calculating the correlation matrix. Thus, the correlation matrix of the received data could be expressed as(13)Rxx=ExtxHt=EjtjHt+EntnHt=EajωI,θjjtajωI,θjjtH+EntnHt=EajωI,θjjtjHtajHωI,θj+EntnHt=σj2EajωI,θjajHωI,θj+σn2I,where jt and nt stand for the interference and noise, separately. σn2 and σj2 are the power of noise and interference, respectively. I is a unit matrix, θj is the incident direction of interference, and ωI is the intermediate frequency.Then,(14)Eajω,θjajHω,θj=ζH022H0,Hejπsin⁡θJH022ejωITH0,HejωIT+πsin⁡θJH,H0e-jπsin⁡θJH22H,H0ejωIT-jπsin⁡θJH22ejωITH022e-jωITH0,He-jωIT-πsin⁡θJH022H0,Hejπsin⁡θJH,H0e-jωIT+jπsin⁡θJH22e-jωITH,H0e-jπsin⁡θJH22,where ζ is related to the receiver bandwidth which is B, and it could be written as(15)ζ=12πB.H n 2 2 is the 2-norm of Hn and Hm,Hn represents the inner product with Hm and Hn, which could be denoted as(16)Hm2=∫-πBπBHmω2dωHm,Hn=∫-πBπBHmωHnHωdω.Consequently, the correlation matrix could be expressed as(17)Rxx=ζσj2H022+σn2ζσj2H0,Hejπsin⁡θJζσj2H022ejωITζσj2H0,HejωIT+πsin⁡θJζσj2H,H0e-jπsin⁡θJζσj2H22+σn2ζσj2H,H0ejωIT-jπsin⁡θJζσj2H22ejωITζσj2H022e-jωITζσj2H0,He-jωIT-πsin⁡θJζσj2H022+σn2ζσj2H0,Hejπsin⁡θJζσj2H,H0e-jωIT+jπsin⁡θJζσj2H22e-jωITζσj2H,H0e-jπsin⁡θJζσj2H22+σn2.The inverse of correlation matrix could be denoted as(18)Rxx-1=1RxxRxx∗,where Rxx and Rxx∗ are the determinant and adjoint matrix of Rxx, separately.R x x is a Hermite matrix, so its determinant is a constant value. We suppose that υ=Rxx.R x x ∗ can be simplified as(19)Rxx∗=σn42σj2ζH22+σj2ζH022+σn2-σj2ζH0,Hejπsin⁡θJ-σj2ζH022ejωIT-σj2ζH0,Hejπsin⁡θJ+ωIT-σj2ζH,H0e-jπsin⁡θJσj2ζH22+2σj2ζH022+σn2-σj2ζH,H0e-jπsin⁡θJ-ωIT-σj2ζH22ejωIT-σj2ζH022e-jωIT-σj2ζH0,Hejπsin⁡θJ-ωIT2σj2ζH22+σj2ζH022+σn2-σj2ζH0,Hejπsin⁡θJ-σj2ζH,H0e-jπsin⁡θJ+ωIT-σj2ζH22e-jωIT-σj2ζH,H0e-jπsin⁡θJσj2ζH22+2σj2ζH022+σn2.Power inversion (PI) is a typical blind criterion to suppress interference, which does not require any a priori information. It can form nulls in the direction of strong interference with superior adaptive performance, and its suitable application is in the condition of strong interference and weak signal, especially in GNSS receivers [18]. Its constraint can be written as(20)minwwHRxxws.t.wHb=1,where b=10⋯0T is the constraint vector.The weights can be further expressed as(21)w=μRxx-1b=μ1RxxRxx∗b=μυ2σj2ζH22+σj2ζH022+σn2σn4-σn4σj2ζH,H0e-jπsin⁡θJ-σn4σj2ζH022e-jωIT-σn4σj2ζH,H0e-jπsin⁡θJ+ωIT=ς2σj2ζH22+σj2ζH022+σn2-σj2ζH,H0e-jπsin⁡θJ-σj2ζH022e-jωIT-σj2ζH,H0e-jπsin⁡θJ+ωIT≈ςσj2ζ2H22+H022-H,H0e-jπsin⁡θJ-H022e-jωIT-H,H0e-jπsin⁡θJ+ωIT.The above approximate condition is that the jammer power is much higher than the noise, which is suitable for the actual situation.The weight vector could be normalized as(22)w=α2H22+H022-H,H0e-jπsin⁡θJ-H022e-jωIT-H,H0e-jπsin⁡θJ+ωIT,where α is the factor of normalization and it could be denoted as(23)α=12H22+H0222+2H,H02+H024.According to Figure1, (5), and (8), the array output signal in time domain can be represented as(24)yt=wHx=∑i=1N∑m=1Mx0t-i-1τθ-m-1Twim∗,where x0t is the sample value of reference array element in the time of t. The Fourier transform of (24) can be expressed as(25)Yω=∑i=1N∑m=1MXωe-jπi-1sin⁡θe-jωm-1Twim∗=Xω∑m=1Me-jωm-1T∑i=1Ne-jπi-1sin⁡θwim∗.Therefore, the transfer function of STAP can be obtained as(26)Hω,θ=∑m=1Me-jωm-1T∑i=1Ne-jωi-1τθwim∗.According to the space-time steering vector in (9), the transfer function of STAP in the direction of interference can be expressed as(27)Hω,θJ=wHaω,θJ=α2H22+H022-H0,H-H022+H0,He-jω-ωIT=α2∫-πBπBAω2dω+2πB-∫-πBπBAωe-jφωdω-2πB+∫-πBπBAωe-jφωdωe-jω-ωIT.The mismatched steering vector is used in (17), because the received data are all from the mismatched channel. However, the ideal steering vector is used in (27), because the mismatched parameters are unknown when we estimate the transfer function. If the same steering vector is used in (17) and (27), the transfer function would be meaningless.


## 4. Impact on Interference Suppression

If the channel is ideal, (27) could be expressed as(28)Hω,θJ=α4πB-4πBe-jω-ωIT≈0.Hence, the interference could be suppressed perfectly. However, the channels are always mismatched, and the performance of interference suppression is also limited.Suppose the interference is the smooth Gaussian white noise, and its bandwidth has covered the entire bandwidth of signal. According to (27), the output power of interference can be expressed as(29)PJ-OUT=PJ-IN2πB∫-πBπB2∫-πBπBAω2dω+2πB-∫-πBπBAωe-jφωdω-2πB+∫-πBπBAωe-jφωdωe-jω-ωIT2dω=σJ2α22πB∫-πBπB2∫-πBπBAω2dω+2πB-∫-πBπBAωe-jφωdω-2πB+∫-πBπBAωe-jφωdωe-jω-ωIT2dω.For convenience, we only analyze the above equation in IF, which is supposed to be suitable, because the antijamming performance in other frequencies is close to the performance in IF. Constantly, the output power of interference could be simplified as(30)PJ-OUT=2σJ2α2πB∫-πBπB∫-πBπBAω2dω-∫-πBπBAωe-jφωdω2dω.Assume that(31)HJω=∫-πBπBAω2dω-∫-πBπBAωe-jφωdω2=∫-πBπBAω2-Aωe-jφωdω2≈∫-πBπBAω2-Aωe-jφω2dω.Suppose the noise power isPN. The interference cancellation ratio (ICR) can be expressed as(32)ICR=PJ-IN+PNPJ-OUT+PN=σJ2+σn22σJ2α2/πB∫-πBπBHJωdω+σn2.According to (32), the ICR is limited by channel mismatch and its limits could be written as(33)ICRlim=limσJ2→∞πB2α2∫-πBπBHJωdω.The fundamental principle of interference suppression in antenna arrays is to achieve interference cancellation, and (31) can be further simplified as(34)HJω=∫-πBπBA2ω-2Aωcos⁡φω+1dω.The above equation is the main factor to affect the performance of interference suppression. According to (34), we define a channel mismatch impact factor that can be expressed as(35)F=HJω.The smaller the value of the impact factor, the better the performance of interference suppression.We define another parameter which can also reflect the performance of interference suppression and it can be expressed as(36)HFω=A2ω-2Aωcos⁡φω+1.According to (3), we can obtain that(37)φω=Rpcos⁡ωRpn+ωRb≈ωRb,where Rp and Rb are in the same order of magnitude, but cos⁡ωRpn is less than 1 and also much less than ω, so the above equation can be approximated.According to (36) and (37), the main factors to affect the performance of interference suppression are the range of amplitude wave, the number of amplitude waves, and the group delay bias.In the above analysis, channel 2 is nonideal and its mismatch parameters have not been taken into consideration, and we can recognize they are random. But, in the next plotting, the specific mismatch parameters are necessary, and the model in (3) is used and its parameters are listed in Table 1.Table 1
Mismatch parameters.
Parameters Value Range of amplitude wave 0.5 dB Number of amplitude waves 2 cycle Range of group delay 3 ns Number of group delays 2 cycle Group delay bias 4 nsFigure2 shows the impact factors to HFω in (36). The curve of reference is with the mismatch in Table 1. The curves marked with Ra=1dB, Ran=3cycle, and Rb=10ns stand for resetting Ra, Ran, and Rb, respectively, whose reset value is listed in the figure, and the other parameter is the same as Table 1 when the parameter has reset. Figure 2 further presents that the main factors to impact the performance of interference suppression are the range of amplitude wave and the group delay bias.Figure 2
Impact factor toHFω.


## 5. Simulation Experiment

GNSS software receiver is used in this simulation. The analysis in this paper concerns the performance of interference suppression, and the PI criterion is adapted, so the navigation signals are unnecessary to be generated. The parameters utilized in simulation experiments are listed in Table2.Table 2
Utilized parameters.
Parameters type Parameters value Array type 2-element linear array Interelement spacing 0.5 wavelength Time-domain taps 2 Interference type White Gaussian noise Interference bandwidth 20 MHz Interference DOA 15 degrees INR 35 dB Sampling frequency 124 MHz IF 15 MHzWe assume channel 1 is ideal and the characteristic of channel 2 is nonideal whose mismatch parameters are shown in Table1.Figure3 shows the relationship between input INR and ICR. Whether the channel is ideal or mismatch, with the increase of ICR, ICR shows a rising trend. When the channel is ideal, the ICR is approximately equal to the INR and their relationship is linear, and the interference could be completely suppressed. But, in the case of channel mismatch, ICR has different limits in different degree of mismatch. The curve marked with reference shows the ICR in different INR under the channel mismatch whose parameters are set in Table 1. The curve of Ra=1dB expresses the impact of the range of amplitude wave reset to 1 dB and the other parameters are the same as reference, whose limit of ICR has decreased. The curve marked with Rb=10ns stands for the ICR in different INR under the group delay bias with 10 ns, whose limit of ICR has also been reduced. Nevertheless, when the range of amplitude is reset to 3 cycles, the ICR has little influence compared with the reference.Figure 3
ICR in different INR.Figure4 shows the impact of mismatch extent on ICR. According to the mismatch model in (3), the channel mismatch is related to the range of amplitude wave, the number of amplitude waves, the range of group delay wave, the number of group delay waves, and the group delay bias. In order to put the impact of these parameters on ICR to a diagram, the limits of the parameters are set which are shown in figure, and x-axis in figure represents the percentage of those limits. As can be seen from Figure 4, the main influence factors to ICR are the range of amplitude wave and the group delay bias, and the impact of the number of amplitude waves is weaker, and the group delay waves express a little effect on ICR, which is consistent with the conclusion of theoretical analysis.Figure 4
ICR in different mismatch degree.In (21), (28), (31), (34), and (37), the approximate processing is used. The simulation has proved that the approximate processing is right.The above simulation is executed only in 2 elements and 2 taps, but these STAP structure is scarce. In the following, we will do simulation with different STAP structure. In all STAP, the distance of the adjacent elements is half wavelength. Regardless of STAP structure, the reference element is ideal, and the other element is nonideal, which would meet the channel mismatch. The degree of channel mismatch is the same as Figure4. The ICRs in 3 elements’ linear array with 2 taps and 4 elements’ square array with 2 taps are shown in Figures 5(a) and 5(b). Different elements have different ICRs, but the impact of channel mismatch on ICR is almost the same. The ICRs in 2 elements’ linear array with 3 taps and 5 taps are shown in Figures 5(c) and 5(d). The taps are more complex on ICR compared to elements. With the increase of taps, the impact of channel mismatch on ICR would become weaker and weaker, because STAP can equalize part of channel mismatch, but its equalization effect is also limited. In addition, the amplitude mismatch is harder to be equalized compared to group delay mismatch.Figure 5
ICR in different STAP structure.
(a)
Three elements’ uniform linear array with 2 taps (b)
Four elements’ uniform square array with 2 taps (c)
Two elements’ uniform linear array with 3 taps (d)
Two elements’ uniform linear array with 5 taps


## 6. Conclusion

In this paper, we have analyzed the impact on antijamming performance of channel mismatch in GNSS antenna arrays receivers. The characteristics of channel mismatch and impact of mismatch level are studied, and the study has been developed through theoretical analysis and simulation experiment.The outcome of the study is the following:( 1 ) Under the condition of channel mismatch, the antijamming performance of antenna arrays is limited which is connected to the degree of channel mismatch.( 2 ) A concept of channel mismatch impact factor is proposed, whose value could directly determine the antijamming performance of antenna arrays.( 3 ) When the STAP is equipped with 2 elements and 2 taps, the range of amplitude wave and group delay bias are the main factors to impact the antijamming performance, and the number of amplitude waves is next, and the effect of group delay wave is the smallest.( 4 ) STAP can equalize part of channel mismatch. More taps would bring better equalization effect, and the group delay mismatch is easier to be equalized compared to amplitude mismatch. With the increase of taps, the range and number of amplitude waves have become the main factors to impact the antijamming performance, and the impact of group wave and bias on antijamming performance would become weaker and weaker.Although the effect of group delay wave is the smallest, the group delay wave might affect the measurement in pseudocode and carrier phase seriously, which is the future work for us.



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