Pseudo-Signal Interference Regularity of Single-Frequency Electromagnetic Radiation to Stepped-Frequency Radar

Abstract

When typical radar equipment is subjected to single-frequency electromagnetic radiation, the radar display interface forms a pseudo-signal, resulting in the misjudgment of real targets. Based on the working principle of stepped-frequency radar ranging, the effect mechanism of radar equipment pseudo-signal interference is revealed. Taking a Ku band stepped-frequency ranging radar as the test object, the pseudo-signal interference effect test of single-frequency electromagnetic radiation is carried out in this study. The pseudo-signal level value of 6 dBmV is selected as the sensitive criterion of the pseudo-signal interference effect. Through experiments, the variation curves of the pseudo-signal level values of the sensitive frequency bands and the typical frequency points inside and outside the band with the field strength of the single-frequency interference are obtained. Based on the nonlinear distortion analysis of the receiving circuit, the variation laws of the pseudo-signal level values inside and outside the band are explained, respectively. The experimental results show that there are at least seven pseudo-signal interference-sensitive bands in the tested radar, and the first pseudo-signal strength is only related to the interference signal strength. The essence of the second type of pseudo-signal interference is intermodulation interference, and the pseudo-signal level is related to the interference signal and the useful signal strength.

1. Introduction

The future information war is based on military electronic technology and information technology. It is a comprehensive confrontation between the warring parties in the field of information. The wave of information change has strongly impacted the traditional concept of war. Joint combat under the condition of information has become an inevitable trend of military development [1,2,3,4]. To win the war in the future, it is necessary to cooperate between different military types. In order to achieve a substantial increase in the overall combat capability and the combat effectiveness of weapons and equipment, it is required that the power of electromagnetic radiation sources such as early warning reconnaissance, detection and identification, communication command, navigation, and the positioning of various military equipment increase and the number is doubled. In addition, the emergence of directional energy weapons such as high-power microwave and ultra-wideband electromagnetic pulse, as well as electromagnetic pulse bombs and strong electromagnetic jammers, makes the frequency equipment on the modern battlefield highly intensive. The battlefield environment is deteriorating, and the security and reliability of the weapon system in the electromagnetic space are seriously threatened. The electromagnetic environment adaptability of equipment is becoming more and more prominent [5,6,7,8].

Radar refers to an electromagnetic sensor that uses radio to detect and measure targets, which is equivalent to the human “clairvoyance” and is widely used in military and civilian fields [9,10]. As a commonly used frequency equipment, radar equipment is responsible for important tasks such as battlefield reconnaissance, target monitoring, and artillery aiming, and plays a pivotal role on the battlefield [11,12]. The current research on radar interference by electromagnetic radiation mainly focuses on the establishment of single-frequency and multi-frequency interference blocking effect laws and effect evaluation models [13,14]. Reference [15] pointed out that the single-frequency interference frequency is located in the radar operating frequency band. The signal can cause spurious alarm interference to radar, but the target characteristics of spurious alarm interference are not analyzed. Reference [16] believes that single-frequency interference can make FM CW radar display interfaces produce false signal interference bands. However, the above literature does not involve the pseudo-signal interference of radar out-of-band single-frequency electromagnetic radiation.

The Ku-band CW radar antenna is small in physical size, has good portability and ranging functions, and is widely used in short-range tracking and guidance [17]. Based on the above background, this paper takes a Ku band stepped-frequency continuous-wave ranging radar as the experimental object to carry out the single-frequency electromagnetic radiation pseudo-signal interference effect test. For the first time, it is found that there are pseudo-signal interference sensitivity phenomena inside and outside the band of radar working frequency. The pseudo-signal interference effect law is obtained, and the mechanism of pseudo-signal interference effect inside and outside the band is revealed, which lays the foundation for the evaluation and protection of the pseudo-signal interference effect under the complex electromagnetic environment of radar equipment.

2. Interference Theory Analysis

Under normal circumstances, the interference of continuous-wave electromagnetic radiation in frequency equipment is mainly manifested as blocking interference, which means that when the interference signal is too strong, the sensitivity of the receiver of the frequency equipment decreases, which makes the gain of useful signals significantly decrease, and the normal working performance is lost. For the stepped-frequency continuous-wave ranging radar, when the single-frequency continuous-wave interference signal or the intermodulation signal generated by the interaction between the single-frequency continuous-wave interference signal and the useful signal is processed by the radar equipment as a useful signal, the radar display interface will generate a spurious signal response; this phenomenon is called the spurious signal interference effect.

The step-frequency continuous-wave ranging radar has a high working frequency band and a large working bandwidth, and usually undergoes at least one frequency conversion process. In order to ensure the universality of pseudo-signal analysis, this paper takes the zero-IF receiver of double-frequency conversion as an example, and the schematic block diagram of the stepped-frequency radar is shown in Figure 1. The frequency synthesizer generates the first local oscillator fixed-frequency signal and the second local oscillator stepped-frequency signal, which are sent out by the transmitting antenna after being mixed, filtered and amplified. After the echo signal returned from the target is received by the receiving antenna, it is filtered and amplified; first, it is mixed with the local oscillator fixed-frequency signal, and the received signal frequency is down-converted to the intermediate frequency, then selected by the band-pass filter, and amplified again. Then, it is mixed with the second local oscillator step-frequency signal. Finally, it is filtered by a low-pass filter to further amplify the low-frequency echo signal containing time delay information, and after signal acquisition and processing in the later stage, a spike signal containing useful target distance information can be observed on the radar, and the radar ranging function can be realized.

According to Figure 1, set the single-frequency continuous-wave signal as the interference source. Based on the signal processing circuit of the stepped-frequency continuous-wave ranging radar receiver, the analysis is carried out in the form of complex signals. The effect law of radar equipment pseudo-signal interference is explored theoretically. The signal of the true target echo signal and the interference signal coupled to the receiver can be expressed as:

$$u_r(t)={\displaystyle \sum _{i=-N/2}^{N/2}{U}_{si}rect\left[\frac{t-T_R/2-\left(i+N/2\right)T_R}{T_R}\right]\cos \left[2\pi \left(f_0+i\Delta f\right)(t-\tau )\right]}+U_j\cos \left(2\pi f_{j}t\right)$$

(1)

Let $u_{rs}(t)$ represent the useful signal and $u_{rj}(t)$ represent the interference signal. Then, Equation (1) can be decomposed into:

$$u_{rs}(t)={\displaystyle \sum _{i=-N/2}^{N/2}{U}_{si}rect\left[\frac{t-T_R/2-\left(i+N/2\right)T_R}{T_R}\right]\cos \left[2\pi \left(f_0+i\Delta f\right)(t-\tau )\right]}$$

(2)

$$u_{rj}(t)=U_j\cos \left(2\pi f_{j}t\right)$$

(3)

Among them, $U_{si}$ is the sub-cycle voltage amplitude of the useful signal coupled to the receiver, $N$ is the number of sub-cycles, $\Delta f$ is the step ladder, and the bandwidth of the radar operating frequency band is $2f_H$; then, $N=2f_H/\Delta f$, $T_R$ are the sub-cycle frequency hopping time, $f_0$ is the center frequency of the radar operating frequency band, $\tau$ is the time delay from the useful signal sent to the return, $U_j$ is the voltage amplitude of the interference signal coupled to the receiver, and $f_j$ is the frequency of the interference signal.

Assuming that the IO signal of the radar is a fixed-frequency signal, and $f_a$ is the IO frequency of the radar receiver, the IO signal of the radar can be expressed as:

$$u_a(t)=\cos (2\pi f_{a}t)$$

(4)

Assuming that the radar’s second local oscillator signal is a stepped-frequency signal with the same form as the echo signal, and $f_b$ is the radar receiver’s second local oscillator center frequency, the radar’s second local oscillator signal can be expressed as:

$$u_{{}_b}(t)={\displaystyle \sum _{i=-N/2}^{N/2}rect\left[\frac{t-T_R/2-\left(i+N/2\right)T_R}{T_R}\right]\cos \left[2\pi \left(f_b+i\Delta f\right)t\right]}$$

(5)

In the experiment assessing the single-frequency pseudo-signal interference effect of step-frequency radar, it is found that when the single-frequency signal interference frequency is close to the working frequency band and the local oscillator frequency of the tested radar, the pseudo-signal interference is most obvious. Assuming that the receiving antenna receives two single-frequency electromagnetic radiation interference signals of different frequencies, let the frequency of the interference signal close to the operating frequency of the radar be $f_1$, and let the frequency of the interference signal close to the local oscillator frequency of the radar be $f_2$; considering only the first- and second-order components and ignoring the influence of high-order components, the frequencies of each component of the output signal after the interference signal enters the structure of the tested radar are shown in

Table 1. The output component frequency of the single-frequency interference signal after entering each structure of the step-frequency radar.

Structure	Frequency
Antenna reception	$F_0$$,f_1$	$F_0$$,f_2$
First-level magnification	$F_0$$,f_1$$,F_0\pm f_1$,	$F_0$$,f_2$$,F_0\pm f_2$
First-stage mixing ($f_a$)	$F_0$$,f_1$$,F_0\pm f_1$$,F_0\pm f_a$$,f_1\pm f_a$$,F_0\pm f_1\pm f_a$	$F_0$$,f_2$$,F_0\pm f_2$$,F_0\pm f_a$$,f_2\pm f_a$$,F_0\pm f_2\pm f_a$
Band-pass filtering ($f_b\pm f_H$)	$F_0-f_a$$,f_1-f_a$	$F_0-f_a$$,F_0-f_2$
Secondary mixing ($F_b$)	$F_0-f_a$,$f_1-f_a$$,F_0-f_a\pm F_b$$,f_1-f_a\pm F_b$	$F_0-f_a$$,F_0-f_2$$,F_0-f_a\pm F_b$$,F_0-f_2\pm F_b$
Low-pass (narrowband) filtering	$F_0-f_a-F_b$ (True target echo signal), $f_1-f_a-F_b$ (Type 1 spurious signal)	$F_0-f_a-F_b$ (True target echo signal), $F_0-f_2-F_b$ (Type 2 spurious signal)

. Among them, we set the working step-frequency signal of the radar as $F_0$, and set $f_{io}=f_0+i\Delta f,i=-\frac{N}{2},-\frac{N}{2}+1,\cdot \cdot \cdot ,\frac{N}{2}-1,\frac{N}{2}$, then $F_0=\left\{f_{-\frac{N}{2}0},f_{\left(-\frac{N}{2}+1\right)0},\cdot \cdot \cdot ,f_{\left(\frac{N}{2}-1\right)0},f_{\frac{N}{2}0}\right\}$; we set the radar two local oscillator step-frequency signal as $F_b$, and set $f_{ib}=f_b+i\Delta f,i=-\frac{N}{2},-\frac{N}{2}+1,\cdot \cdot \cdot ,\frac{N}{2}-1,\frac{N}{2}$, then $F_b=\left\{f_{-\frac{N}{2}b},f_{\left(-\frac{N}{2}+1\right)b},\cdot \cdot \cdot ,f_{\left(\frac{N}{2}-1\right)b},f_{\frac{N}{2}b}\right\}$.

It can be seen from Table 1 and the above theoretical analysis that the stepped-frequency radar equipment can form two types of pseudo-signal interference under the single-frequency continuous-wave interference. The reason for the first type of spurious signal interference is that the interference signal whose interference frequency is in the working frequency band of the radar is difficult to be filtered by the radar radio frequency front-end, which directly affects the signal processing of the back-end; the reason for the second type of spurious signal interference is that the interference frequency is close to the intermodulation signal generated by the intermodulation between the interference signal of the radar’s local oscillator frequency, and the useful signal is difficult to be filtered by the intermediate frequency band-pass filter after the local oscillator and the low-frequency generated after mixing with the radar’s second local oscillator step-frequency signal. The signal is difficult to be filtered by the low-pass filter after the two local oscillators, which eventually affects the signal processing at the rear end of the radar.

2.1. Linear Workspace

When the input signal $u_r(t)$ is weak, the radar receiver is in the linear working area, and the echo signal $u_{rs}(t)$ is amplified by mixing and filtering, and is finally expressed as:

$$u_{rs}(t)u_a(t)u_b(t)\stackrel{filteramplification}{\to }{U}_{si}\cos \left[2\pi \left(f_b+i\Delta f\right)\tau \right]$$

(6)

When the frequency $f_j$ of the interfering signal $u_{rj}(t)$ falls within the working frequency range of the useful signal, that is, $f_0-f_H\le f_j\le f_0+f_H$, the interfering signal is also amplified by the mixing filter, which can cause misjudgment in the radar system. The signal can finally be expressed as:

$$u_{rj}(t)u_a(t)u_b(t)\stackrel{filteramplification}{\to }{U}_j\cos \left[2\pi \left(f_j-f_0-i\Delta f\right)t\right]$$

(7)

The signal represented in Equation (7) is the first type of pseudo-signal.

When the frequency $f_j$ of the interference signal $u_{rj}(t)$ is close to the local oscillator frequency $f_a$ of the radar receiver, and the radar RF front-end cannot effectively filter the signal, the signal generated by the intermodulation of the interference signal and the useful echo signal cannot be detected by the local oscillator. The latter filter is filtered out, and then directly mixed with the two local oscillators. After filtering and amplification, the radar system can also be misjudged. The signal can finally be expressed as:

$$u_{rj}(t)u_{rs}(t)u_b(t)\stackrel{filteramplification}{\to }{U}_j{U}_{si}\cos \left[2\pi \left(f_j-f_a\right)t+2\pi \left(f_0+i\Delta f\right)\tau \right]$$

(8)

The signal represented in Equation (8) is the second type of pseudo-signal.

In the subsequent signal processing, the distance to the real target can be calculated through the time delay $\tau$ in the useful signal component, and $U_{si}$ determines the intensity of the echo of the real target. It is basically unchanged. The strength of the first type of spurious signal is determined by $U_j$, and the level of the spurious signal is proportional to the strength of the interference signal; the strength of the second type of spurious signal is determined by $U_j{U}_{si}$, and the level of the spurious signal is proportional to the product of the strength of the interference signal and the strength of the useful signal.

2.2. Weak Nonlinear Region

With the increase in the interference signal $u_{rj}(t)$, the radar receiver gradually works in the weak nonlinear region, and the power series analysis method can be used for analysis at this stage. When the system input signal is shown in Equation (1), the output signal can be expressed as:

$$u_0\left(t\right)=B_0+B_1{u}_r\left(t\right)+B_2{u}_r^2\left(t\right)+B_3{u}_r^3\left(t\right)+\cdot \cdot \cdot$$

(9)

In Equation (9), $B_i\left(i=0,1,2,3\dots \right)$ is the nonlinear coefficient related to the characteristics of the radar receiver circuit. Combining Equation (1) with Equation (9), the output components of each signal in Equations (6)–(8) can be expressed as:

$$\left\{\begin{array}{l}\left(B_1{U}_{si}+\frac{3}{4}{B}_3{U}_{si}^3+\frac{3}{2}{B}_3{U}_{si}{U}_j^2\right)\cos \left[2\pi \left(f_b+i\Delta f\right)\tau \right]\\ \left(B_1{U}_j+\frac{3}{4}{B}_3{U}_j^3+\frac{3}{2}{B}_3{U}_j{U}_{si}^2\right)\cos \left[2\pi \left(f_j-f_0-i\Delta f\right)t\right]\\ \left(B_1{U}_j+\frac{3}{4}{B}_3{U}_j^3+\frac{3}{2}{B}_3{U}_j{U}_{si}^2\right)\left(B_1{U}_{si}+\frac{3}{4}{B}_3{U}_{si}^3+\frac{3}{2}{B}_3{U}_{si}{U}_j^2\right)\cos \left[2\pi \left(f_j-f_a\right)t+2\pi \left(f_0+i\Delta f\right)\tau \right]\end{array}\right.$$

(10)

When the interference signal is strong enough, the nonlinear effect of the useful signal on the circuit can be neglected, and the gain of the useful signal and the two types of pseudo-signals can be deduced from Equation (10) as:

$$\left\{\begin{array}{l}{K}_{si}=\left(B_1+\frac{3}{2}{B}_3{U}_j^2\right)\\ K_{j1}=\left(B_1+\frac{3}{4}{B}_3{U}_j^2\right)\\ K_{j2}=\left(B_1+\frac{3}{2}{B}_3{U}_j^2\right)\left(B_1+\frac{3}{4}{B}_3{U}_j^2\right)\end{array}\right.$$

(11)

In Equation (11), $K_{si}$ is the gain of the useful signal, and $K_{j1}$ and $K_{j2}$ are the gains of the first type and the second type of pseudo-signal, respectively, which can be obtained from Equations (10) and (11):

(1)

(1) Due to $B_3<0$, when the interference signal is too strong and the radar receiver works in the weak nonlinear region, the gain of both the useful signal and the spurious signal is suppressed. With the increase in the interference signal strength, the gain gradually decreases. Blocking interference is caused by the circuit nonlinear coefficient $B_3$.

(2)

With the increase in the interference signal, the level of the radar target echo signal gradually decreases, and the amplitude of the decrease gradually increases. The first type of spurious signal level does not change with the change in the useful signal strength; the second type of spurious signal level is proportional to the useful signal strength.

2.3. Strong Nonlinear Region

If the interference signal $u_{rj}(t)$ continues to increase, the radar system will work in the cut-off, saturation and even breakdown region, which is called the strong nonlinear region. Different from the weak nonlinear region, the power series analysis method is used in this stage, and the number of expansion terms will not be exhausted. At this time, the vector method can be used to carry out the analysis, and its working mechanism can be explained by the limiter. When the input signal is expressed as Equation (1), the vector analysis diagram of the received signal can be drawn, as shown in Figure 2, where ${\omega }_{d}t=2\pi \left(f_0+i\Delta f\right)(t-\tau )-2\pi f_{j}t$.

It can be seen from Figure 2 that the input component of the input signal before the nonlinear device of the receiver circuit can be expressed as:

$$S(t)\cos \left[2\pi \left(f_0+i\Delta f\right)(t-\tau )+\theta \right]$$

(12)

Let the limiting level of the radar system be $U_x$. When the radar receiver works in a strong nonlinear region, the input signal level is greater than the limiting level, and the amplitude of the interference signal is much greater than the amplitude of the useful signal, that is, $U_j>>U_{si}$. At this time, there is $U_j\approx S\left(t\right),\theta \approx 0$. In the derivation process of reference [15], combined with Figure 2, it can be seen that the output components of each signal in Equations (6)–(8) can be expressed as:

$$\left\{\begin{array}{l}\frac{U_x{U}_{si}}{2U_j}\cos \left[2\pi \left(f_b+i\Delta f\right)\tau \right]\\ U_x\cos \left[2\pi \left(f_j-f_0-i\Delta f\right)t\right]\\ \frac{U_x^2{U}_{si}}{2U_j}\cos \left[2\pi \left(f_j-f_a\right)t+2\pi \left(f_0+i\Delta f\right)\tau \right]\end{array}\right.$$

(13)

From Equation (13), it can be deduced that the gains of the useful signal and the two types of pseudo-signals are:

$$\left\{K_{si}=\frac{U_x}{2U_j};K_{j1}=\frac{U_x}{U_j};K_{j2}=\frac{U_x^2}{2U_j^2}\right.$$

(14)

From Equations (13) and (14), the following conclusions can be drawn:

(1)

When the interference signal is too strong and the radar receiver works in a strong nonlinear region, the gain of the useful signal and the first type of pseudo-signal is inversely proportional to the voltage value of the interference signal, and the second type of pseudo-signal is proportional to the second voltage value of the interference signal. The square is inversely proportional.

(2)

As the interference signal increases, the true echo signal level of the radar target decreases linearly; the level of the first type of pseudo-signal remains unchanged and does not change with the change in the interference signal strength; the level of the second type of pseudo-signal echo signal is a linear decrease (in logarithmic coordinates, the same below).

3. Test Preparation

3.1. Build a Single-Frequency Electromagnetic Radiation Test Platform

A certain type of Ku-band stepped-frequency continuous-wave ranging radar is selected. The operating frequency of the radar is $\left(f_0-0.1\right)$ GHz~$\left(f_0+0.1\right)$ GHz, the frequency hopping time is 0.05 ms, the step-frequency step is 10 kHz, and the maximum detection distance can reach 5000 m. The radar transmitting antenna and receiving antenna are placed to ensure the physical isolation of the transceiver branch. The horn antenna with a distance of about 8.3 m is set as the radar detection target. In most cases, the target distance to be detected will be farther than the distance set in the test during the actual working process of the radar, and the echo intensity of the short-range target is stronger. In order to make the test results universal, attenuators of different degrees are added at the connection port of the radar transmitting antenna to simulate the echo strength of the detection target at a distance. In order to avoid the influence caused by the existence of interfering antennas in the radiation method test and ensure the good repeatability and stability of the test, and at the same time make up for the shortcoming of insufficient maximum power of the interference source, refer to GJB8848-2016 [18] to adopt the equivalent differential mode. The injection method was used instead of the irradiation method for testing. The interference source is a single-frequency signal source, generating a continuous-wave interference signal. Through the directional coupling module with the monitoring function, the interference signal is directly injected into the radar receiving port. The spectrum analyzer monitors the injection power value of the single-frequency electromagnetic radiation, and then converts it into the interference field strength value of single-frequency electromagnetic radiation at the radar receiving antenna. Select the interference frequency and turn on the signal source; if the radar has pseudo-signal interference, the interference signal power should be reduced by 6 dB, and then the interference power should be gradually increased until the pseudo-signal level value reaches the critical interference state, and the data can be recorded. Then, change the frequency of single-frequency signal source and repeat the test. The characteristics and laws of the pseudo-signal interference of single-frequency electromagnetic radiation on the radar under test are studied, and the effect test platform is shown in Figure 3.

3.2. Selection of Sensitive Criterion for Spurious Signal Interference

After receiving the echo signal of the detected target, the radar under test can obtain the one-dimensional range image of the detected target after processing. The echo intensity of the detected target is represented by the normalized level. The nature of the normalized level displayed at different distances is the difference between the level of each position and the peak level. When the tested radar equipment is not disturbed, the one-dimensional image is shown in Figure 4. It can be observed that there is a clutter signal caused by electronic noise in the radar display interface. The point marked in Figure 4 is the maximum clutter signal level value. After many tests, the maximum absolute level value of the clutter signal is lower than 0 dBmV. In order to avoid the influence of the clutter signal and increase the test contrast, this paper increases 6 dB on the basis of 0 dBmV, and it is considered that when the absolute level of the pseudo-signal reaches 6 dBmV, the single-frequency electromagnetic radiation causes spurious signal interference to the tested radar equipment.

4. The Law of Spurious Signal Interference Effect

4.1. Spurious Signal Critical Interference Field Strength Test

In the single-frequency electromagnetic radiation effect test platform constructed according to Figure 3, the attenuator of the radar receiving port was set to 30 dB, and the electromagnetic radiation pseudo-signal interference effect test was carried out on the radar under test. When the tested radar was in the critical interference state of spurious signal, that is, when the spurious signal level reached 6 dBmV, the critical interference power value of the spectrum analyzer was recorded and converted into the critical interference field strength value. A total of seven pseudo-signal interference-sensitive frequency bands were measured in the test, and the variation curves of the pseudo-signal single-frequency critical interference field strength of the seven sensitive frequency bands are shown in Figure 5a–f and Figure 6.

It can be seen from Figure 5 and Figure 6 that:

• The radar under test is not only subject to spurious signal interference in the working frequency band ($f_0$ GHz ± 60 MHz), but also in the out-of-band working frequency band ($f_0-1.2$ GHz ± 60 MHz, $f_0-0.3$ GHz ± 30 MHz, $f_0-0.9$ GHz ± 30 MHz, $f_0-0.8$ GHz ± 15 MHz and $f_0-0.4$ GHz ± 15 MHz). This will also be interfered by pseudo-signals, as the first type of pseudo-signal interference occurs in the above-mentioned six sensitive frequency bands, and the out-of-band critical interference field strength is much higher than in-band. The second type of spurious signal interference occurs in the sensitive frequency band $f_0-0.6$ GHz ± 4 MHz.
• The sensitive frequency bands in Figure 5 and Figure 6 are “U”-shaped curves. The most sensitive points of each sensitive frequency band appear at the symmetry axis of the “U”-shaped curve.

The critical interference field strength of the pseudo-signal interference in the sensitive frequency bands $f_0$ GHz ± 60 MHz and $f_0-0.6$ GHz ± 4 MHz is the lowest. The appearance of these two sensitive frequency bands conforms to the above theoretical analysis. The causes of other pseudo-signal interference-sensitive frequency bands are further analyzed as follows:

• When the frequency of the interfering signal is in the working frequency band of the radar under test, it will generate a difference frequency signal with a frequency of $f_j-f_a$ after mixing with the radar local oscillator signal. If the passband range of the filter is $\left(f_0-f_a\right)\pm f_H$, the difference frequency signal can pass through the bandpass filter to form spurious signal interference in subsequent signal processing. The interference signal in the sensitive frequency band $f_0-1.2$ GHz ± 60 MHz is mixed with the local oscillator signal, which can generate a difference frequency signal with a frequency of $f_a-f_j$. This signal can also pass through the band-pass filter, so it will still cause spurious signal interference. The critical interference field strength of the pseudo-signal in the in-band sensitive frequency band is low, and the generated beat frequency signal is an up-conversion signal, while the critical interference field strength of the pseudo-signal in the out-of-band sensitive frequency band of the radar is higher, and the generated beat frequency signal is a down-conversion signal. The variation law of the spurious signal interference level with the interference field strength should be the same for the sensitive frequency band $f_0$ GHz ± 60 MHz and the sensitive frequency band $f_0-1.2$ GHz ± 60 MHz.
• The reason for the generation of other sensitive frequency bands may be that the interference signal is first mixed with the local oscillator signal to generate a difference frequency signal, and the high frequency multiplier signal of the signal falls within the range of the band-pass filter after the local oscillator of the radar. Take the signal as an example: combined with Equation (7), it can be expressed as:

  $${\left[u_{rj}(t)u_a(t)\right]}^2{u}_b(t)\stackrel{filteramplification}{\to }{U}_j^2\cos \left[2\pi \left(2\left(f_j-f_a\right)-f_b-i\Delta f\right)t\right]$$

  (15)

It can be known from Equation (15) that when the radar works in the linear region, the signal strength is determined by $U_j^2$, and the spurious signal level is proportional to the interference signal strength. If a graph of the variation of the spurious signal level with the interference field strength under the logarithmic coordinate axis is drawn, the change slope of the signal level in the linear region should be twice the change slope of the spurious signal level in the sensitive frequency band within the band. From this, it can be inferred that, in addition to the above sensitive frequency bands, there may be other out-of-band sensitive frequency bands, but in the case of the same interference signal strength, the higher the multiplier of the difference frequency signal, the lower the signal strength and the narrower the sensitive frequency band range.

4.2. Variation in Spurious Signal Level with Interference Field Strength

In order to verify the correctness of the above theoretical analysis, the center frequency of each sensitive frequency band is selected, the power of the single-frequency interference source is changed, and the variation in the pseudo-signal interference level of the center frequency of each sensitive frequency band of the first type of pseudo-signal with the critical interference field strength is obtained. The curves are shown in Figure 7a–c.

From Figure 7, the following conclusions can be drawn:

• Under the single-frequency interference of different frequencies, the absolute level value of the first type of spurious signal is basically the same as the overall change trend of the interference field strength: when the interference field strength is low, the tested radar equipment works in the linear region, and the spurious signal level value increases approximately linearly with the increase in the interference field strength, indicating that the gain of the first type of spurious signal is constant; when the interference field strength continues to increase and the tested radar equipment works in the weak nonlinear region, the value of the spurious signal level increases with the interference field. The intensity increases, but the increase decreases gradually, indicating that the gain of the first type of pseudo-signal is gradually decreasing; when the interference field strength increases to the point where the tested radar equipment works in the strong nonlinear region, the pseudo-signal level value basically does not follow. The increase in the interference field strength changes, indicating that the gain of the first type of spurious signal has dropped to a constant value. The variation law of the first type of spurious signal level with the interference field strength conforms to the theoretical analysis in Section 2.
• When the tested radar works in the linear region, the slope of the pseudo-signal level value of the interference frequency $f_0-1.2$ GHz increases, and the interference field strength is basically the same as that of the interference frequency $f_0$ GHz; the pseudo-signal level value of the interference frequency is $f_0-0.3$ GHz and $f_0-0.9$ GHz. The slope of the average value with the increase in the interference field strength is twice that of the interference frequency $f_0$ GHz; the slope of the spurious signal level value of the interference frequency of $f_0-0.4$ GHz and $f_0-0.8$ GHz with the increase in the interference field strength is three times the interference frequency $f_0$ GHz. The maximum value of the interference level of the first type of spurious signal is basically between 11 dBmV and 16 dBmV. The reason for the generation of the five sensitive frequency bands outside the band is that the interference signal is mixed with the local oscillator signal of the radar, and the first, second and third frequency multiplication signals of the difference frequency signal are not filtered by the bandpass filter, thus causing spurious signals to the radar interference, which proves the correctness of the aforementioned theoretical analysis.

In order to find out the relationship between the interference level of the second type of spurious signal and the strength of the useful signal, on the basis of the test platform in Figure 3, by replacing the attenuator at the radar transmitting port, the radar transmitting signal is increased by 10 dB and reduced by 10 dB as the control group. From the carried out experiments, the variation curve of the pseudo-signal interference level at the center frequency of the second type of pseudo-signal sensitive frequency band with the critical interference field strength is shown in Figure 8.

From Figure 8, the following conclusions can be drawn:

• Under the single-frequency interference with different useful signal strengths, the absolute level value of the second type of spurious signal is basically the same as the overall variation trend of the interference field strength: when the interference field strength is low, the tested radar equipment works in the linear region, and the spurious signal level value increases approximately linearly with the increase in the interference field strength, indicating that the gain of the second type of spurious signal is constant; when the interference field strength continues to increase, and the tested radar equipment works in the weak nonlinear region, the spurious signal level value first increases. It increases with the increase in the interference field strength, then reaches the maximum value, and then gradually decreases, indicating that the gain of the second type of spurious signal first gradually decreases, then drops to 0, and finally the gain decreases. When the equipment works in the strong nonlinear region, the pseudo-signal level value decreases approximately linearly with the increase in the interference field strength, indicating that the gain of the second type of pseudo-signal is constant and negative. The variation law of the second type of spurious signal level with the interference field strength conforms to the above theoretical analysis.
• Under the same condition of interference field strength, the stronger the useful signal strength, the greater the level value of the second type of spurious signal. Regardless of the strength of the useful signal, the maximum value of the second type of spurious signal interference level appears around −22 dBV/m.

5. Conclusions

In this paper, the pseudo-signal interference effect of single-frequency continuous-wave electromagnetic radiation on a stepped-frequency ranging radar is studied. Based on the working principle of radar ranging, the single-frequency continuous-wave interference signal can cause two types of spurious signal interference to the radar through theoretical deduction. Through the experiment, two types of pseudo-signal interference-sensitive frequency bands and the variation law of pseudo-signal level with the interference field strength are obtained. The specific conclusions are as follows:

• Single-frequency electromagnetic radiation with different interference frequencies can cause two types of spurious signal interference to radar equipment. There are at least six typical sensitive frequency bands for the first type of pseudo-signal interference. The reasons for the occurrence of several sensitive frequency bands are: the difference frequency signal or the frequency multiplied signal of the difference frequency signal generated by mixing the interference signal and the local oscillator signal of the radar receiver does not exist. It is filtered out by the filter after the local oscillator, causing the first type of spurious signal to appear on the radar display interface. There is only one sensitive frequency band for the second type of spurious signal interference. The reason for this sensitive frequency band is: it is close to the radar local oscillator. The frequency interference signal is mixed with the radar echo signal to generate an intermodulation signal, and the intermodulation signal is mixed with the radar two local oscillator signal to generate a low frequency signal, which is difficult to be filtered by the low-pass filter, resulting in the appearance of a second signal on the radar display interface (pseudo-signal).
• The bandwidth of the pseudo-signal interference-sensitive frequency band is smaller than the radar operating bandwidth, and the bandwidth of the second type of pseudo-signal interference-sensitive frequency band is much smaller than that of the first type of pseudo-signal interference-sensitive frequency band. The two types of pseudo-signal interference-sensitive curves are all U-shaped, and the critical interference field strength of pseudo-signal increases with the increase in the frequency offset of each center frequency point.
• With the increase in the interference field strength, when the radar is in the linear working area, the level values of the two types of pseudo-signals increase linearly; when the radar works in the weak nonlinear area, the level values of the two types of pseudo-signals continue to increase, but the growth rate decreases gradually; when the radar works in a strong nonlinear region, the level of the first type of spurious signal is a constant value, and the level of the second type of spurious signal decreases linearly.
• The first type of spurious signal level value has nothing to do with the useful signal strength—the maximum value is only 16 dBmV—but when the interference field strength is large enough, the spurious signal always exists; the second type of spurious signal level value is proportional to the useful signal strength. The maximum value is the value can reach more than 40 dBmV, which is much higher than the maximum value of the first type of spurious signal level. With the increase in the interference field strength or the decrease in the useful signal, the level of the second type of spurious signal will gradually decrease. When the interference field strength is large enough, the spurious signal will be covered by the clutter signal.