Principle, method and previous work
According to Doppler effect, when the radar moves away from the stationary receiving antenna, the frequency of the receiving signal decreases and the pulse broadens. Equations of frequency \({f_r}\), doppler frequency \({f_d}\) and pulse width \({\tau _r}\)of the receiving signal are illustrated as follows.
$${f_r}=\sqrt {{{\left( {c - v} \right)} \mathord{\left/ {\vphantom {{\left( {c - v} \right)} {\left( {c+v} \right)}}} \right. \kern-0pt} {\left( {c+v} \right)}}} {f_0}$$
1
$${f_d}={f_r} - {f_0}=\left[ {\sqrt {{{\left( {c - v} \right)} \mathord{\left/ {\vphantom {{\left( {c - v} \right)} {\left( {c+v} \right)}}} \right. \kern-0pt} {\left( {c+v} \right)}}} - 1} \right]{f_0}$$
2
$${\tau _r}=\sqrt {{{\left( {c+v} \right)} \mathord{\left/ {\vphantom {{\left( {c+v} \right)} {\left( {c - v} \right)}}} \right. \kern-0pt} {\left( {c - v} \right)}}} {\tau _0}$$
3
where \({f_0}\) and \({\tau _0}\) are frequency and pulse width of the radar signal, and are velocities of the radar and light.
Reference 8 enables the composite signal of the antenna array to be equivalent to the receiving signal when the radar leaves through setting waveform, phase and time series of RE signals. Figure 1(a) shows the discrete process of the radar moving and emitting signal. The initial distance between the receiving antenna and the radar is \({R_0}\), the radar moves away from the receiving antenna at the speed of , and the process is decomposed by time interval \(\Delta t\). Figure 1(b) introduces the equivalent antenna array: the radiating element spacing is \(v\Delta t\), radiating elements from the near end to the far end of the array transmit signals at time interval \(\Delta t\) sequentially, and initial phases of RE signals are set specifically. With time interval \(\Delta t\)smaller than carrier period of RE signal, the composite signal in space is modulated multiple times in a carrier period, thus the LF signal is generated.
On the basis of Reference 8, the staggered array and periodic pulse train signal are proposed in Reference 10.
Structure of the staggered array is shown in Fig. 1(c), where \({\lambda _0}\) is the carrier wavelength of the RE signal, and is radiating element spacing. Compared with the traditional antenna array, which limits radiating element spacing with half-wavelength of the carrier, the staggered array reduces radiating element spacing through arranging multiple arrays. When radiating elements are small enough, multiple arrays can form one antenna array. Through shortening element spacing, the staggered array reduces the time interval and improves the performance of the composite signal.
Waveform of the periodic pulse train signal with 50% duty cycle is presented in Fig. 1(d), where \({T_0}\) and are period and broadcast period of the signal. Bandwidth of the signal is determined by the pulse width, and the initial phase of each pulse is set according to Doppler effect. Multiple carrier periods are included in \({T_0}\), and the signals in different broadcast period are the same. RE signal is the periodic pulse train signal, the resting period of the signal can be filled by the broadened pulse generated by the array, and the pulse width of the composite signal can be increased through increasing the number of periods of the signal, which lays foundation for generation of VLF signals.
As previous work of this paper, Reference 8,10 analyze and derive the principle and method of LF and VLF signal generation under conditions of the antenna array and the staggered array respectively, and the corresponding simulations are completed.
When the receiving antenna and the array form a 45° angle, 400 MHz LF composite signal based on 105 m antenna array and phase-modulated RE signals at 1 GHz carrier frequency is simulated in Reference 8, 10 kHz VLF composite signal generated by 120 m staggered array and RE signals at 100 MHz carrier frequency is simulated in Reference 10. Parameters, such as peak sidelobe ratio (PSLR) and integral sidelobe ratio (ISLR)13, are applied to evaluate the performance of the composite signal.
The simulation parameters in Reference 8,10 are listed in Table 1, and simulation results are shown in Fig. 2. It is necessary to explain that the emission signal in figure is product of one RE signal and the number of radiating elements. Due to the same spectrum of RE signals, the emission signal spectrum can indicate the energy distribution of all RE signals in each frequency component. Random phase composite signal is generated by RE signals with random phase, and its comparison with the composite signal illustrates the necessity of RE signal phase setting.
System composition and equipment parameters
The scheme is divided into two parts, including 8-element short-array experiment and 64-element long-array experiment.
Radiating elements are fed by the 8-channel digital module directly in the 8-element short-array experiment. As displayed in Fig. 3(a), 1×8 power splitters, cables and the 8-channel digital module are combined to feed the radiating elements in the 64-element long-array experiment, and the array structure refers to the staggered array structure in Fig. 1(c).
The RE signals are generated by the digital module, which bases on an 8-channel Digital Analogue Converter (DAC). The clock frequency of the 14-bit DAC is 1 GHz, the signal generation is controlled by the server, the waveform is determined by the data stored in Random Access Memory (RAM), and channels of the digital module work independently and circularly. Photos of the digital module, the server and waveform of the signal output are presented in Fig. 3(b)-(c).
TX170 whip antenna is selected to be the radiating element. In the 8-element short-array experiment, radiating elements are connected to the digital module by cables. In the 64-element long-array experiment, radiating elements are connected to the 1×8 power splitter by cables, and length of cables connecting to the same power splitter vary. In the direction along the array, a log-periodic antenna is set to receive the composite signal, and the signal is analyzed by a spectrometer. Parameters of the experimental equipment and the short/long-array experiment based on the 8-channel digital module are listed in Table 2, where the equivalent radar speed and the frequency of the composite signal are determined by the parameters of unequal-length cables.
Design and test results of unequal-length cables
Radiating elements in the antenna array emit periodic pulse train signals, and formulae of the signals are illustrated in Reference 10. The delay and phase difference of signals emitted by adjacent radiating elements are as follows.
$$\Delta t={d \mathord{\left/ {\vphantom {d v}} \right. \kern-0pt} v}$$
4
$$\Delta \varphi ={\varphi _{n+1}} - {\varphi _n}=2\pi {f_0}\sqrt {1 - {{\left( {{v \mathord{\left/ {\vphantom {v c}} \right. \kern-0pt} c}} \right)}^2}} {d \mathord{\left/ {\vphantom {d v}} \right. \kern-0pt} v}$$
5
where is the number of radiating elements, and \(n=0,1, \cdots ,N - 1\) is the radiating element serial number.
In order to simplify the experimental prototype and feed 64 radiating elements with the 8-channel digital module at the same time, radiating elements are divided into groups, with 8 radiating elements in a group. On the foundation that each channel of the digital module generates one RE signal, whose phase and delay are certain, unequal-length cables are applied to generate delay and phase changes for other RE signals in the group14,15.
Supposing that the length difference between unequal-length cables connected to adjacent radiating elements is \({l_0}\), then the delay and phase change generated are
$${T_D}={{{l_0}\sqrt {{\varepsilon _r}} } \mathord{\left/ {\vphantom {{{l_0}\sqrt {{\varepsilon _r}} } c}} \right. \kern-0pt} c}$$
6
$$\Delta \varphi =\beta {l_0}$$
7
where \({\varepsilon _r}\) is relative dielectric constant of the medium and \(\beta\) is phase constant of the cable. Comparing Eq. (4) to (7), equations can be formed. The \({l_0}\), which determines the delay and phase change of the RE signal, can be calculated.
Through calculation and simulation, within the constraints of the experimental conditions, it can be found that the composite signal has good performance when \({l_0}\)= 0.683 m, the equivalent radar velocity is \(0.246c\), and frequency of the composite signal is 121.35 MHz.
To make it convenient to connect power splitters and radiating elements, the shortest length of unequal-length cable is 0.20 m, and parameters of unequal-length cables are shown in Table 3.
Table 3
Parameters of unequal-length cables
Cable number
|
Length /m
|
Attenuation /dB
|
Relative phase /rad
|
Delay /ns
|
1
|
0.20
|
0.0735
|
0
|
0.95238
|
2
|
0.88
|
0.3232
|
-3.1093
|
4.1905
|
3
|
1.57
|
0.5767
|
0.1113
|
7.4762
|
4
|
2.25
|
0.8264
|
-2.9980
|
10.714
|
5
|
2.94
|
1.0799
|
0.2226
|
14.000
|
6
|
3.62
|
1.3296
|
-2.8867
|
17.238
|
7
|
4.30
|
1.5794
|
0.2872
|
20.476
|
8
|
4.99
|
1.8328
|
-2.7754
|
23.762
|
Power splitters and the digital module are connected with 8 m cables, and radiating elements are connected to power splitters with unequal-length cables. With the experiment set up in spatial, time and frequency domain, the time and phase accuracy of RE signals have a significant influence on the experimental result.
In order to test the transmission delay of cables and power splitters, 2 channels of the digital module are set to emit signals, with channel 1 and 2 connected to the oscilloscope and the device to be tested separately. The result is displayed in Table 4. According to the test, the average delay difference between unequal-length cables is 3.258 ns. Compared with 3.252 ns ideal delay difference, the average delay error is less than 0.01 ns (corresponding to 2.1 mm cable length error), and variation range of the delay error is within ± 0.3ns. Thus, the transmission delay error of cables and power splitters has little influence on the composite signal.
Table 4
Transmission delay performance test
Device tested (Channel 0)
|
Delay relative to channel 1 / ns
|
Transmission delay/ ns
|
Delay difference/ ns
|
without cable
|
13.333
|
0
|
/
|
8 m cable
|
24.778
|
11.445
|
/
|
8 m cable, power splitter and 0.20 m cable
|
27.079
|
13.746
|
/
|
8 m cable, power splitter and 0.88 m cable
|
30.225
|
16.892
|
3.146
|
8 m cable, power splitter and 1.57 m cable
|
33.371
|
20.038
|
3.146
|
8 m cable, power splitter and 2.25 m cable
|
36.742
|
23.409
|
3.371
|
8 m cable, power splitter and 2.94 m cable
|
40.000
|
26.667
|
3.258
|
8 m cable, power splitter and 3.62 m cable
|
43.371
|
30.038
|
3.371
|
8 m cable, power splitter and 4.30 m cable
|
46.405
|
33.072
|
3.034
|
8 m cable, power splitter and 4.99 m cable
|
49.888
|
36.555
|
3.483
|
Simulation and analysis
According to the experimental parameters in Table 2 and Table 3, spectrums of the 121.35 MHz composite signals based on the 8-element short array and 64-element long array in the direction along the array are simulated respectively.
Simulation and analysis in the ideal case
The simulated composite signal spectrums based on the 8-element short array and 64-element long array in the ideal case are shown in Fig. 4(a) and Fig. 4(c). Spectrum comparisons of the emission signals, the composite signals and random phase composite signals are presented in Fig. 4(b) and Fig. 4(d), and comparisons indicate that phase and delay of RE signals have significance in the LF signal generation process.
In the short-array experiment, with the short array, few radiating elements and close distance between the receiving antenna and the array leading to increase of the carrier and harmonic component, the simulation results show that the composite signal based on the short array can generate apparent low-frequency component. Meanwhile, PSLR of the composite signal based on the long array is -14.8182 dB, ISLR is -6.9479 dB (83.20%), and the energy utilization ratio of the composite signal to the emission signal is about 79.97%.
Simulation and analysis of error influence
Random errors and system errors exist in short/long-array experiment. Simulation and analysis indicate that ± 0.5 cm spacing error, ± 1 ns delay error and ± 10° phase error of RE signals, and ± 0.05 normalization amplitude error (corresponding to -0.45dB to 0.42dB) have little influence on the composite signal, when the errors above are random errors and distribute uniformly. In the process of the experiment, parameter errors of the prototype can be controlled within the ranges above. Therefore, influence of system errors is simulated and analyzed in this section.
The clock frequency and time resolution of the digital module is 1 GHz and 1 ns, and delay system error may be caused, which is less than 1 ns and exists in the short/long-array experiment. In the long-array experiment, power splitters and radiating elements are connected by cables, and they can lead to the phase system error16,17. Due to the limitation of the experimental environment and power of the RE signals, the receiving antenna is close to the antenna array, which makes the amplitudes of received RE signals differ, and the performance of the composite signal is affected.
The simulated composite signal spectrums based on the short array and long array under the influence of system errors are shown in Fig. 4(e) and Fig. 4(f). Figure 4(e) presents the composite signal spectrum based on the short array when the receiving antenna is 6 m away from the array and 0.1 ns delay system error exists. Figure 4 (f) shows the composite signal spectrum based on the long array when the receiving antenna is 20 m away from the array with 0.1 ns delay system error and 30° phase system error existing. After accumulation, the delay system error can reach 0.7 ns. Under the condition that system errors exist, the simulation results indicate that delay system error has little effect on the composite signal of the short array, while the simultaneous delay error and phase system error lead to the increase of carrier component of the composite signal based on the long array.
The phase of the RE signal is designed according to the parameters when the receiving antenna is set in the direction along the array. When the receiving antenna deviates from the direction along the array, the composite signal frequency is higher than the designed frequency. When the receiving antenna is located in the normal direction of the array, the peak frequency of the composite signal is the same as the carrier frequency of RE signals. In the condition that the receiving antenna is 22 m/24 m away from the near end of the array and 38°/51° deviates from the array direction, and influenced by system errors mentioned in Fig. 4(f), the simulated composite signal spectrums based on the long array are shown in Fig. 4(g) and Fig. 4(h).