A Flexible Photonic Method for Angle-of-Arrival and Frequency Measurements

Abstract

A microwave photonic approach for measuring the angle of arrival (AOA) and frequency is proposed and experimentally demonstrated. The AOA-dependent phase difference and frequency of two received signals were mapped to intensity information through subtractive and differential operations, which were achieved by a delayed superposition structure with phase inversion. By measuring the output signal powers, both the phase difference and frequency of the two signals could be determined. The theoretical analysis results are given in detail. In this proof-of-concept experiment, the system had a phase difference measurement range of 340 degrees, with a maximum error of 2.9 degrees. The frequency measurement covered 1–10 GHz, with a maximum error of 2.2%. The proposed approach offers a straightforward method for measuring the AOA and frequency under the same configuration, which provides new insight into AOA- and frequency-measurement techniques.

1. Introduction

Microwave photonics technology has been extensively studied in recent years due to its great potential in many applications, such as signal processing [1,2], sensing [3,4], signal generation [5], and measurement [6]. Among these applications, the measurement of microwave signal parameters is important in various fields of instrumentation, radar systems, electronic warfare, and wireless communications [7,8,9,10]. In particular, the angle-of-arrival (AOA) measurement is particularly crucial and has consequently attracted considerable attention [11,12]. Conventionally, the AOA is usually measured and processed by digital signal processing technology, but it faces challenges due to bandwidth limitations. Fortunately, microwave photonics offers significant advantages for measuring the AOA, including a wide bandwidth, ultra-low transmission loss, and immunity to electromagnetic interference [13,14,15,16,17].

To date, numerous methods for measuring the AOA based on microwave photonic systems have been reported. The AOA can be determined by measuring different parameters, including the output direct current (DC) voltage [18], the ratio of the two DC voltages [19], or the optical power [20,21,22]. However, these methods are limited to an AOA measurement range of only 90 degrees. In order to broaden the measurement range of the AOA, researchers have addressed the phase ambiguity problem by using optical delay technology [23]. Nevertheless, these schemes have poor system stability. Another photonic solution for eliminating phase ambiguity involves constructing two mapping curves [24,25] or introducing a periodic phase jump [26]. Although these techniques address the phase ambiguity, they still encounter a common issue mentioned in the previous schemes. Specifically, these AOA measurement techniques assume a known input signal frequency to set the antenna spacing for accurate AOA estimation. Obviously, this requires a separate frequency measurement system. In this case, the cost and complexity of the system are increased, and the practicality is reduced. Therefore, it is necessary to design a system for measuring both the AOA and frequency. In [27], the AOA and frequency were converted into the relative amplitude and time interval of the microwave pulses. However, this technique showed an approximate phase measurement error of 6 degrees. Based on temporal pulse shaping, the AOA and frequency can also be measured [28]. While this system achieved a low measurement error, the requirement of high-dispersion devices brings a challenge. Alternatively, a combination of stimulated Brillouin scattering and frequency-to-time mapping [29], or the Talbot effect with pulse interference [30], can be employed to achieve a simultaneous measurement of AOA and frequency. However, the frequency measurement range of these methods lack sufficient flexibility. Furthermore, various approaches have also been investigated, including optical sampling [31,32], an acousto-optic modulator-based optical frequency shift loop [33], and optical sideband sweeping with envelope detection [34]. But these approaches often require the use of expensive equipment, such as a mode-locked laser or dual-parallel Mach–Zehnder modulator, which increases the overall system cost to a certain degree. The development of a cost-effective structure with tunability for AOA and frequency measurement is necessary.

In this paper, we propose a photonic-assisted approach for measuring the AOA and frequency based on a delayed superposition structure with phase inversion, which is both theoretically analyzed and experimentally demonstrated. In our method, the phase difference and frequency information of the microwave signals are converted into intensity information of the output signal through subtractive and differential operations. The AOA and frequency are subsequently determined by measuring the final output powers. It is worth noting that an unambiguous phase difference measurement can be clarified by observing the power variation tendency after increasing the delay. The proof-of-concept experiment showed that the phase difference measurement range was 10 to 350 degrees, with a maximum error of 2.9 degrees at 1 GHz, 3 GHz, 6 GHz, and 9 GHz. The frequency measurement range was 1–10 GHz, with a maximum error of 2.2%. Compared with the previous methods for measuring the AOA and frequency, the proposed configuration only relies on two Mach–Zehnder modulators (MZMs) and a few passive components, which exhibits the advantages of a simple structure, wide tunability, and good measurement accuracy.

2. Principle

The schematic diagram of the AOA is illustrated by Figure 1. The microwave signals are received by two antennas, with the angle-of-arrival denoted as $\theta$. Due to the difference in the received distance at the two antennas, a relative time delay ${\tau }_1$ is introduced, which can be expressed as

$$\begin{array}{c}\hfill {\tau }_1={\displaystyle \frac{dsin\theta }{c}}\end{array}$$

(1)

where c is the velocity of an electromagnetic wave in a vacuum, and d is the distance between antenna a and antenna b. Typically, to avoid measurement ambiguity [30], d is set to ${\displaystyle \frac{\lambda }{2}}$, where $\lambda$ represents the wavelength of the received microwave signal. The phase difference caused by the time delay between the received signals can be expressed as

$$\phi =2\pi f_m{\tau }_1=2\pi f_m{\displaystyle \frac{\frac{\lambda }{2}sin\theta }{c}}=\pi sin\theta$$

(2)

where $f_m$ represents the frequency of the received signal. Equation (2) indicates that the key to measuring AOA lies in the measurement of the phase difference.

Figure 2 illustrates the proposed AOA measurement scheme that can also be extended to the frequency measurement of microwave signals. After the antennas, the received signal 1 and received signal 2 are independently intensity modulated and represented as two cosine functions with different phases. In case 1, the signal in the lower branch is delayed and overlaps with the signal in the upper branch to perform a subtractive operation. In case 2, another certain delay is applied to the lower branch signal before overlapping, facilitating a differential operation. Through these operations, the phase difference and frequency information of the microwave signals are converted into intensity information in the output signal. By measuring the output powers corresponding to these two operations, the AOA and frequency can be determined. In this diagram, the two received signals, with a phase difference of $\phi$, can be expressed as

$$\left\{\begin{array}{c}{V}_{m}cos\left(2\pi f_{m}t\right)\hfill \\ V_{m}cos(2\pi f_{m}t+\phi )\hfill \end{array}\right.$$

(3)

where $V_m$ denotes the amplitude of the microwave signal.

The continuous wave (CW) emitted by a laser diode (LD) is equally split by a 3 dB coupler and sent into two MZMs. The received signals from two antennas are applied to each of these MZMs as drive signals. To facilitate subsequent subtractive and differential operations, MZM1 is biased at the positive quadrature bias point (+QB), while MZM2 is biased at the negative quadrature bias point (−QB). As we know, a continuous wave can be expressed as

$$E_{\mathrm{in}}\left(t\right)=E_{0}exp\left(j{\omega }_{c}t\right)$$

(4)

where $E_0$ and $w_c$ are the amplitude and angular frequency of the optical carrier. The modulated optical fields of the MZM1 and MZM2 under the condition of small-signal modulation are expressed as

$$\left\{\begin{array}{c}{E}_{\mathrm{out}1}\left(t\right)\approx -{\displaystyle \frac{1}{2}}{E}_{0}exp\left(j{\omega }_{c}t\right)\left[J_0\left(m\right)+2J_1\left(m\right)cos\left(2\pi f_{m}t\right)\right]\hfill \\ E_{\mathrm{out}2}\left(t\right)\approx {\displaystyle \frac{1}{2}}{E}_{0}exp\left(j{\omega }_{c}t\right)\left[J_0\left(m\right)-2J_1\left(m\right)cos(2\pi f_{m}t+\phi )\right]\hfill \end{array}\right.$$

(5)

where $m={\displaystyle \frac{\pi V_m}{2V_{\pi }}}$, and $V_{\pi }$ is the half-wave voltage of the MZM.

The corresponding envelope intensities are calculated as

$$\left\{\begin{array}{cc}\hfill I_1\left(t\right)& ={\displaystyle \frac{1}{4}}{E}_0^2\left[J_0{\left(m\right)}^2+4J_0\left(m\right)J_1\left(m\right)cos\left(2\pi f_{m}t\right)\right.\hfill \\ \hfill & +2J_1{\left(m\right)}^2+2J_1{\left(m\right)}^{2}cos(2\times 2\pi f_{m}t)\hfill \\ \hfill I_2\left(t\right)& ={\displaystyle \frac{1}{4}}{E}_0^2\left[J_0{\left(m\right)}^2-4J_0\left(m\right)J_1\left(m\right)cos(2\pi f_{m}t+\phi )\right.\hfill \\ & +2J_1{\left(m\right)}^2+2J_1{\left(m\right)}^{2}cos(2\times 2\pi f_{m}t+2\phi )\hfill \end{array}\right.$$

(6)

Equation (6) is simplified to

$$\left\{\begin{array}{c}{I}_1\left(t\right)=DC+Acos\left(2\pi f_{m}t\right)+Bcos(2\times 2\pi f_{m}t)\hfill \\ I_2\left(t\right)=DC-Acos(2\pi f_{m}t+\phi )+Bcos(2\times 2\pi f_{m}t+2\phi )\hfill \end{array}\right.$$

(7)

where $DC$ is a direct current component, $A=J_0\left(m\right)J_1\left(m\right)$, and $B={\displaystyle \frac{1}{2}}{J}_1{\left(m\right)}^2$. Since the values of A and B greatly depend on the modulation index, under the small-signal modulation condition, the value of B is always small and close to zero. Therefore, Equation (7) can be approximated as

$$\left\{\begin{array}{c}{I}_1\left(t\right)=DC+Acos\left(2\pi f_{m}t\right)\hfill \\ I_2\left(t\right)=DC-Acos(2\pi f_{m}t+\phi )\hfill \end{array}\right.$$

(8)

The modulated signals in the upper and lower branches contain only the first-order frequency component. When the lower branch signal is delayed by an optical delay line (ODL), the envelope intensity can be written as

$$I_3\left(t\right)=DC-Acos[2\pi f_m(t+\tau )+\phi ]$$

(9)

Then, the optical envelopes on two branches are combined by a polarization beam combiner (PBC), where the total envelope intensity is expressed as

$$\begin{array}{c}\hfill I_{\mathrm{out}}\left(t\right)=I_1\left(t\right)+I_3\left(t\right)=DC+Acos\left(2\pi f_{m}t\right)-Acos[2\pi f_m(t+\tau )+\phi ]\end{array}$$

(10)

The corresponding photocurrent can be expressed as

$$\begin{array}{c}\hfill i_{\mathrm{out}}\left(t\right)\propto DC+Acos\left(2\pi f_{m}t\right)-Acos\left[2\pi f_m(t+\tau )+\phi \right]\end{array}$$

(11)

2.1. AOA Measurement

When measuring the AOA, the delay difference between the upper and lower branches can be adjusted to satisfy $\tau =0$, and Equation (11) is simplified to

$$i_{\mathrm{out}}\left(t\right)\propto DC+c_{1}sin\left({\displaystyle \frac{\phi }{2}}\right)sin\left(2\pi f_{m}t+{\displaystyle \frac{\phi }{2}}\right)$$

(12)

where $c_1=2A$, which is a constant independent of the phase difference $\phi$.

After a DC block, the power of the output signal can be written as

$$P=c_1^2{sin}^2\left({\displaystyle \frac{\phi }{2}}\right)$$

(13)

Equation (13) shows that the output power depends on the phase difference, allowing for the phase difference to be calculated from the measured power. As shown in Figure 3, the normalized power variation with a phase difference follows a sine-squared relationship, resulting in a symmetrical power distribution in this curve. This symmetry brings an uncertainty, which means the phase difference could fall within either the 0°–180° range or the 180°–360° range at the same power level. However, the slopes of the power variation are different in these two ranges. Within the range of 0° to 180°, the power increases as the phase difference increases, while within the range of 180° to 360°, the power decreases as the phase difference increases. Therefore, by observing the power variation tendency, one can distinguish the phase differences under the same power value and eliminate ambiguity in the measurement. According to Equation (13), the curve becomes flat near phase differences of 0°, 180°, and 360°, where the power–phase difference sensitivity is low and small power changes are difficult to detect, leading to larger errors. In contrast, the curve is steepest near 90° and 270°, where the sensitivity is the highest. As a result, the same phase difference change causes different power variations depending on the region, which may amplify or suppress measurement errors. To reduce this nonlinearity, the system should preferably operate in high-sensitivity regions.

2.2. Frequency Measurement

According to reference [35], a differential operation on a signal $Acos\left(2\pi f_{m}t\right)$ is carried out in two steps. First, the signal undergoes a phase inversion, resulting in $-Acos\left(2\pi f_{m}t\right)$, and then the two signals are overlapped with a relative time delay ${\tau }^{\prime }$. Therefore, as shown in Equation (11), when the ODL is set to $2\pi f_m\tau +\phi =-2\pi f_m{\tau }^{\prime }$, the system can perform the differential operation. Additionally, for a differentiator, the time delay ${\tau }^{\prime }$ is often set to tens of picoseconds [36]. Fortunately, with an accuracy better than 1 picosecond, the ODL can be easily adjusted to perform the differential operation.

Consequently, Equation (11) can be rewritten as

$$i_{\mathrm{out}}\left(t\right)\propto DC+Acos\left(2\pi f_{m}t\right)-Acos\left[2\pi f_m(t-{\tau }^{\prime })\right]$$

(14)

Specifically, Equation (14) is simplified to

$$\begin{array}{cc}\hfill i_{\mathrm{out}}\left(t\right)& \propto DC+{\tau }^{\prime }\times {\displaystyle \frac{d}{dt}}\left[Acos\left(2\pi f_{m}t\right)\right]\hfill \\ \hfill & =DC+c_2{f}_{m}sin\left(2\pi f_{m}t\right)\hfill \end{array}$$

(15)

where $c_2=-2A\pi {\tau }^{\prime }$, which is a constant independent of the frequency $f_m$.

It can be seen in Equation (15) that after a differential operation, the frequency of the signal is mapped to the amplitude. When the DC is removed by a DC block, the power of the output electrical signal is expressed as

$$P_{\mathrm{out}}=c_2^2{f}_m^2$$

(16)

According to Equation (16), the frequency of the microwave signal can be calculated by measuring the output power.

3. Experimental Results and Discussion

To demonstrate the feasibility of the proposed scheme, a proof-of-concept experiment was conducted based on the setup shown in Figure 4. An LD emitted a CW with the center wavelength of 1550 nm, which was split into the upper and lower branches via a 3 dB coupler. The optical field in the upper branch passed through a polarization controller (PC1) and entered MZM1 biased at the positive quadrature bias point. The optical field in the lower branch passed directly through another polarization controller (PC2) into MZM2, which was biased at the negative quadrature bias point. Here, by changing the polarization state of the CW by aligning PC1 and PC2, two MZMs could obtain the best modulation effect. Two RF signals were generated by an external microwave source to emulate signals with an unknown frequency and phase difference from two antennas. After the MZMs, the modulated signal in the lower branch was delayed by a tunable ODL and overlapped orthogonally with the modulated signal in the upper branch through a PBC, which effectively eliminated the random interference or beating noise between the two optical fields. Meanwhile, the polarization states of two signals could be independently adjusted by PC3 and PC4, which ensured an equal power distribution between the upper and lower branches of the PBC. After the PBC, the optical signal was converted into an electrical signal by a PD and then passed through a DC block to eliminate the DC component. The power of the final output RF signal was measured using a microwave power meter.

3.1. AOA Measurement

At the beginning, the measurement of the AOA was investigated. The AOA-dependent phase difference was adjusted from 10° to 350° in increments of 10°. Figure 5a illustrates the experimental results of the phase difference measurement at 1 GHz. The results show a good agreement between the experimental data and the theoretical predictions, and the trend of power variation aligned with Figure 3. Consequently, the AOA could be simply calculated from the phase difference by using Equation (2). Figure 5b illustrates the corresponding measurement errors. Evidently, the phase difference was successfully measured from 10° to 350°, with a maximum error of 1.5° at 1 GHz.

To verify the tunability of the system, the input frequencies were set to 3 GHz, 6 GHz, and 9 GHz, respectively. The results of the phase difference measurement and the corresponding measurement errors are presented in Figure 6. At an input frequency of 3 GHz, the maximum error in the phase difference measurement was 2.9°. Additionally, at input frequencies of 6 GHz and 9 GHz, the maximum errors in the phase difference measurements were 2.6° and 2.8°, respectively. The AOA measurements were successfully realized at different frequencies. The results demonstrate that the phase difference measurements from 10° to 350° were successfully achieved at different frequencies, with a maximum error of 2.9°. Although two MZMs were used in our scheme, they were the simplest commercial modulators, and their bias points could be independently adjusted. Therefore, compared with similar works [21,23,24,27,33], which used a mode-locked laser or DP-MZMs, our method achieved competitive measurement accuracy while offering flexibility and tunability.

3.2. Frequency Measurement

We then evaluated the frequency measurement capability of the system. According to Equation (14), during the frequency measurement process, the ODL needed to be initially adjusted to minimize the output RF power. Next, the ODL was further adjusted to introduce a small delay ${\tau }^{\prime }$. At this point, a differential operation was performed on the microwave signal. The microwave signal frequency was set from 1 to 10 GHz, with intervals of 1 GHz. By recording the output power value after the differential operation, we calculated the frequency of the microwave signal and its corresponding errors, which are presented in Figure 7. The frequency measurement range from 1 to 10 GHz was successfully achieved, with a maximum error of 2.2%.

Based on the experimental demonstration above, the principle of the approach was successfully verified, and the proposed system could accurately measure both the AOA and frequency. However, due to the use of a fiber-based structure, along with polarization-sensitive components, such as MZMs and the PBC, the system is susceptible to environmental perturbations, which cause power variations, leading to experimental errors. These errors may be reduced by isolating the system from the environment or developing a waveguide structure. Moreover, the nonlinearities of MZMs also affect the measurement accuracy. For example, for a 1 GHz signal, if the signal’s amplitude A increases to 1.01 A due to the nonlinear effect, the resulting phase difference error is approximately 0.65° according to Equation (13), and a frequency error of about 1%, as calculated by Equation (16), is introduced. By carefully adjusting the MZM bias point and modulation voltage, the signal remains within the modulator’s linear operating region, and these errors can be mitigated. In addition, the accuracy of delay adjustment has an impact on the overall system performance. Considering the employed ODL precision of 0.05 ps, a delay mismatch of 0.1 ps was assumed, which may result in a phase difference error of approximately 0.036° and a frequency error of around 0.71%, as estimated from Equation (16). This can be improved by using a high-precision optical delay line or a closed-loop control system that dynamically compensates for delay errors based on real-time feedback.

4. Conclusions

We propose a novel photonic approach for the measurement of the AOA and frequency. The method applies subtractive and differential operations to microwave signals, converting the phase difference and frequency into intensity information that can be extracted through a power measurement. The experimental results show that the maximum phase difference measurement error was 2.9 degrees (0.85% of the full phase difference measurement range) across a measurement range of 340 degrees. Furthermore, by adjusting the optical delay line, the system could also measure the frequency. The results indicate a maximum error of 2.2% across the frequency range of 1–10 GHz. Overall, the system accurately measured the phase differences, eliminated the phase difference ambiguity, and achieved a wide measurement range. Additionally, it could measure the frequencies without increasing the system’s cost. Characterized by a simple and cost-effective structure, along with ease of tuning, it is a promising candidate for application in modern radar systems.