Principle and performance analysis of coherent tracking sensor based on local oscillator beam nutation

Ke Deng; Bing-Zhong Wang; Gu-Hao Zhao; Jian Huang; Peng Zhang; Da-Gang Jiang; Zhou-Shi Yao

doi:10.1364/OE.22.023528

1. Introduction

Free-space optical (FSO) communication has become a major technical trend for satellite communication due to its high data rate. With high receiving sensitivity and immunity to the sun, coherent communication is the first choice for long-distance high-data-rate optical links such as low-earth orbit (LEO) to geosynchronous-earth orbit (GEO) or GEO to optical-ground station (OGS) links. Many current optical communication systems are based on coherent technologies, such as the German in-orbit validation laser communication terminals for the LEO-LEO link between the TerraSAR-X observation satellite and the near-field infrared experiment laser communication terminal (NFIRELCT) (5.625Gbps), the LEO-GEO link between the AlphaSat and the Sentinel 1A and 2A satellites (1.8Gbps) [1–6], and the developing American laser communications relay demonstration (LCRD) (2.88Gbps) [7], and the Japanese next generation optical communication terminals (2.5Gbps) [8].

Coherent terminals tend to apply coherent tracking technologies, and coherent tracking sensors are integrated into the receiver to simplify the optical path and detector assembly. Coherent tracking technology is immune to background light, and the noise equivalent angle (NEA) is smaller than that of incoherent tracking technology due to the nature of the coherence. In coherent systems, the receiving sensitivity is limited by shot noise, and coherent conditions make receiving highly selective. Low-level background radiation such as starlight has a negligible effect on the detecting process. Under high-level background radiation such as sun, when the system is appropriately filtered, the sensitivity degrades, but the signal is not interrupted as it would be with incoherent systems. The integrated coherent tracking sensor uses light for communication to obtain the boresight error. Due to the differences between the communicating bandwidth and the noise equivalent bandwidth (NEB) of the tracking loop, the signal-to-noise ratio (SNR) of tracking sensor is very high. When the SNR is sufficient for communication, the NEA is very small.

At present, there are two primary coherent tracking sensor technologies: coherent tracking sensors based on binary detectors and coherent tracking sensors based on fiber nutation. The binary-detector-based coherent tracking systems are applied on the laser communication terminal (LCT) from the German Aerospace Research Establishment (DLR) [9] and were verified by in-orbit experiments on the TerraSAR-X and NFIRE satellites. The fiber-nutation-based coherent tracking systems were developed by the Massachusetts Institute of Technology [10–13] and adopted by NASA’s lunar laser communications demonstration (LLCD) project [14, 15] and the RUAG Space’s OPTEL laser terminal family [16, 17]. In this paper, we propose a third type of coherent tracking sensor, which utilizes the nutating local-oscillator beam to obtain the boresight error from the received communication light and is integrated in the coherent communication receiver.

The working principle of the binary-detector-based coherent tracking sensor is similar to a quadrant detector (QD). Each of the balanced detectors in the in-phase branch of the optical hybrid output is split into two equal parts by a linear dead zone. The two linear dead zones in the detectors are perpendicular. The boresight error is obtained by measuring and calculating the normalized difference of the interfering light photocurrent from each binary detector [9]. As the number of individual active parts increases, the detector requires large number of amplifiers, adders, and subtractors in the subsequent processing circuitry, increasing the complexity of the circuit. Combining the two signals from two different parts of the binary detector causes a little more noise than a single active area detector, leading to minor receiver sensitivity losses.

The fiber-nutation-based coherent tracking sensor obtains the boresight error by synchronously demodulating the fiber nutation and the signal envelope fluctuation of the mixed light between the received light being coupled into fiber and the local oscillator (LO) [10–13]. Due to the difference between the space light field and the fiber waveguide mode, there is a theoretical minimal coupling loss of about 0.86dB. Additionally, fiber nutation introduces loss, so the total loss of the receiving sensitivity is more than 1dB. Furthermore, the fiber may fracture due to material fatigue, resulting in low reliability when the fiber is used as a nutation device. In addition, the fiber device needs to be radiation hardened when it is used in space, which is a hindrance to the volume and weight.

In this paper, we present a LO-beam nutation-based coherent tracking sensor. In this technology, the boresight error is calculated by synchronously demodulating the LO nutation and the signal envelope fluctuation of the mixed light between received light and the LO. Similarly to binary-detector-based coherent tracking sensors, with surplus power, the LO light field can be spatially reshaped to match to the received light field to improve the coherent efficiency with acceptable increase in collimator’s size and mass, and laser’s power consumption. A beam deflector will be added to the LO optical path in the coherent receiver, and an envelope detector must be added to the receiver circuits. The beam deflection device could be either acousto-optic or electro-optic, providing a substantial increase in reliability compared to the fiber nutation detector. The single element detectors used in this coherent tracking sensor will have smaller active area than that of binary-detector-based tracking sensor, which would be favorable for increasing the communication rate and decreasing the circuit noise.

2. Principle of coherent tracking sensor based on local oscillator beam nutation

Based on the principles of optical interference, when the focus of the received light does not perfectly overlap with the focus of LO beam, the coherence efficiency decreases. Then, the photocurrent from the detector and the voltage signal after the transimpedance preamplifier will both decrease. When the focus of the LO beam moves in circular route on the detector active area and the nutating center coincides with the focus of the received light, the coherent signal envelope is constant. Otherwise, the signal envelope periodically fluctuates. The terminal controller (TECO) determines the boresight error by synchronizing the periodic coherent signal envelope fluctuation and the position of the LO focus. For example, consider the BPSK receiver based on a decision-directed phase-locked loop. Figure 1 presents the principle block diagram of a coherent tracking sensor based on LO beam nutation that is integrated into this BPSK receiver.

Fig. 1 Principle block diagram of coherent tracking sensor based on local oscillator beam nutation, which is integrated in a BPSK receiver.

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The photocurrent generated from a photodiode can be written as

\begin{array}{l} i_{D} (t) \propto \iint_{D} I (\vec{r}, t) d s = \iint_{D} [E_{L O} (\vec{r}, t) + E_{S} (\vec{r}, t)] {[E_{L O} (\vec{r}, t) + E_{S} (\vec{r}, t)]}^{*} d s \\ = P_{L O} (t) + P_{S} (t) + 2 \sqrt{P_{L O} (t) P_{S} (t)} Re {γ (t)} \end{array}

Where $I (\overset{⇀}{r}, t)$ is the intensity distribution in the interference plane, $E_{L O} (\overset{⇀}{r}, t)$ and $E_{S} (\overset{⇀}{r}, t)$ are the complex electrical field amplitude distribution of the LO beam and received light, respectively, P_LO(t) and P_S(t) are the power of the LO beam and received light, respectively. For simplicity, all interference results are calculated along the entrance pupil of the telescope. The incident angle in the following analysis is also along this plane. The value for the inner position of the optical system can be obtained by multiplying the magnification of the optical system.

The complex degree of coherence γ(t) in Eq. (1) is defined as

γ (t) = \frac{\iint_{D} E_{L O} (\vec{r}, t) E_{S}^{*} (\vec{r}, t) d s}{\sqrt{\iint_{D} {| E_{L O} (\vec{r}, t) |}^{2} d s \iint_{D} {| E_{S} (\vec{r}, t) |}^{2} d s}}

From Eq. (1), we see that the real part of the complex degree of coherence affects the amplitude of the receiving signal. The complex degree of coherence cannot be measured directly; however, it can be obtained from the real part measurement in two special cases: (1) the amplitude distribution of the two light fields is uniform or (2) the phase distribution is the same. In the performance analysis of coherent free-space optical communication, we focus only on the real part of the complex degree of coherence. It is defined as the coherence efficiencyη(t):

η (t) = Re {γ (t)} = \frac{\iint_{D} | E_{L O} (\vec{r}, t) | | E_{S} (\vec{r}, t) | \cos [Δ ω t + Δ ϕ (\vec{r}, t)] d s}{\sqrt{\iint_{D} {| E_{L O} (\vec{r}, t) |}^{2} d s \iint_{D} {| E_{S} (\vec{r}, t) |}^{2} d s}}

2.1 Envelope detector for coherent signal

The far-field light transmitted from long distances is only partially received by the telescope, which we consider to be far-field light sampling by the telescope aperture. The receiving light in range of the sample aperture can be treated as a plane wave, and the field transformed through the telescope optics is treated as a beam-compressed plane wave. The initial output of the LO is Gaussian, which can be reshaped to a plane wave after reverse-Gaussian-profile filtering. Therefore, both the amplitude distributions of the LO and the received light fields are uniform. Using Eq. (3), the coherence efficiency of paths I and Q can be written as

η_{I} (t) = \frac{1}{S} \iint_{D} \cos [Δ ω t + Δ ϕ (\vec{r}, t)] d s = \frac{1}{S} \sqrt{a^{2} (t) + b^{2} (t)} \cos [Δ ω t + a \tan \frac{b (t)}{a (t)}]

η_{Q} (t) = \frac{1}{S} \iint_{D} \cos [Δ ω t + Δ ϕ (\vec{r}, t) - \frac{π}{2}] d s = \frac{1}{S} \sqrt{a^{2} (t) + b^{2} (t)} \sin [Δ ω t + a \tan \frac{b (t)}{a (t)}]

where

\begin{array}{l} a (t) = \iint_{D} \cos [Δ ϕ (\vec{r}, t)] d s \\ b (t) = \iint_{D} \sin [Δ ϕ (\vec{r}, t)] d s \end{array}

Here, S is the area of the integration domain, Δω is the heterodyne frequency of the LO to the received light, and $Δ φ (\overset{⇀}{r}, t)$ is the phase difference between the LO and the received light.

The direct current is cancelled in balanced photodetectors, while the alternating current is transformed to a voltage signal by the transimpedance amplifier. The voltage signals in the I and Q paths can be written as

v_{I} (t) = 2 R r_{L} η_{I} (t) \sqrt{(1 - k) P_{L O} P_{S} (t)} + n_{I} (t)

v_{Q} (t) = 2 R r_{L} η_{Q} (t) \sqrt{k P_{L O} P_{S} (t)} + n_{Q} (t)

Where R is the responsibility of the photodetectors, and r_L is the resistance of the transimpedance. Because the LO output power is constant, the symbol P_LO(t) is modified to P_LO. The additive noises in the I and Q paths are n_I(t) and n_Q(t), respectively, and in coherent systems, the additive noises are primarily formed from shot noise. The power spectral density (PSD) can be written as [18–25]:

S_{N I} (f) = S_{N Q} (f) = 2 e R P_{L O} r_{L}^{2}

We can determine the amplitude envelope of the coherent signal by multiplying the amplification factors k and 1-k by the square of the signal in the I and Q paths and summing the results:

v (t) \approx k v_{I}^{2} (t) + (1 - k) v_{Q}^{2} (t) = s (t) + n (t)

We only consider the zero and first order terms in the Zernike polynomials for the phase difference. Without loss of generality, the phase difference distribution can be written as

Δ ϕ (\vec{r}, t) = Δ ϕ_{0} (t) + x \tan [φ (t)] \approx Δ ϕ_{0} (t) + x φ (t)

Where φ(t) is the angle between the LO wavefront and the received light wavefront, and x is the abscissa. Here, Δϕ₀(t) is the zero order phase difference of the LO and received light. Then, Eq. (6) can be written as

\begin{array}{l} a (t) = S \cos Δ ϕ_{0} (t) \frac{2 J_{1} [\frac{2 π R_{D}}{λ} φ (t)]}{\frac{2 π R_{D}}{λ} φ (t)} \\ b (t) = S \sin Δ ϕ_{0} (t) \frac{2 J_{1} [\frac{2 π R_{D}}{λ} φ (t)]}{\frac{2 π R_{D}}{λ} φ (t)} \end{array}

In order to simplify the expression, we set ψ(t) = 2πR_Dφ(t)/λ, where R_D is the radius of the telescope entrance pupil. In Eq. (12), J₁(∙) is the first order Bessel function. Substituting Eqs. (4)–(8) and Eq. (12) into Eq. (10),the voltage envelope of the signal s(t) becomes

s (t) = 4 R^{2} r_{L}^{2} k (1 - k) P_{L O} P_{S} (t) {2 J_{1} [ψ (t)] / [ψ (t)]}^{2}

The noise n(t) is then

\begin{array}{l} n (t) = 8 R r_{L} \sqrt{k} \sqrt{k (1 - k) P_{L O} P_{S} (t)} J_{1} [[ψ (t)]] / [[ψ (t)]] \cos (Δ ω t + Δ ϕ_{0} (t)) n_{I} (t) + ... \\ + 8 R r_{L} \sqrt{(1 - k)} \sqrt{k (1 - k) P_{L O} P_{S} (t)} J_{1} [[ψ (t)]] / [[ψ (t)]] \sin (Δ ω t + Δ ϕ_{0} (t)) n_{Q} (t) + ... \\ + k n_{I}^{2} (t) + (1 - k) n_{Q}^{2} (t) \\ \approx 4 R r_{L} \sqrt{k (1 - k) P_{L O} P_{S \max}} {\sqrt{k} \cos (Δ ω t + Δ ϕ_{0} (t)) n_{I} (t) + \sqrt{(1 - k)} \sin (Δ ω t + Δ ϕ_{0} (t)) n_{Q} (t)} \end{array}

Where P_Smax is the maximum received power of the communication link, and the noise PSD can be written as

S_{N} (f) \approx 8 R^{2} r_{L}^{2} k (1 - k) P_{L O} P_{S \max} S_{N I} (f) = 16 e R^{3} r_{L}^{4} k (1 - k) P_{L O}^{2} P_{S \max}

2.2 The algorithm for boresight error detection and its equivalent linear model

The LO beam nutates around the boresight of the optical system. As shown in Fig. 2, φ_N is the half-nutation angle, and α(t) is the azimuth boresight error for the received light and angle of arrival (AOA). We suppose that a negligible change occurs in the boresight error during a half-circle nutation of the LO beam and analyze the azimuthal component of the boresight error. The process for determining the elevation is identical.

Fig. 2 Schematic diagram of the intersection angle between wavefronts of LO and received light in nutation process. Point C.T. is the AOA coordinate of the received light from counter terminal.

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When the tracking has been established, the boresight error is small. Therefore, the angle between the LO and the received light wavefront can be approximated as

\begin{array}{l} φ (t) \approx φ_{N} - α (t) \\ φ (t - T / 2) \approx α (t - T / 2) + φ_{N} \end{array}

By substituting Eq. (16) into Eq. (13) and Eq. (10), the voltage envelope of the coherent signal can be approximated to the first-order as

\begin{array}{l} v (t) \approx 4 k (1 - k) R^{2} r_{L}^{2} P_{L O} P_{S} (t) {\frac{2 J_{1} {\frac{2 π R_{D}}{λ} [φ_{N} - α (t)]}}{\frac{2 π R_{D}}{λ} [φ_{N} - α (t)]}}^{2} + n (t) \\ \approx 4 k (1 - k) R^{2} r_{L}^{2} P_{L O} P_{S} (t) {{[\frac{2 J_{1} (ψ_{N})}{ψ_{N}}]}^{2} - S F (φ_{N}) \cdot α (t)} + n (t) \end{array}

Where Ψ_N = 2πR_Dφ_N/λ. In a similar way, we find

v (t - T / 2) \approx 4 k (1 - k) R^{2} r_{L}^{2} P_{L O} P_{S} (t - T / 2) {{[\frac{2 J_{1} (ψ_{N})}{ψ_{N}}]}^{2} + S F (φ_{N}) \cdot α (t - T / 2)} + n (t - T / 2)

where the slope factor SF(φ_N) is

S F (φ_{N}) = {d {[\frac{2 J_{1} (ψ (t))}{ψ (t)}]}^{2} / d φ (t) |}_{φ_{N}} = \frac{16 π R_{D}}{λ} \frac{J_{1} (ψ_{N})}{ψ_{N}} [\frac{J_{0} (ψ_{N})}{ψ_{N}} - \frac{2 J_{1} (ψ_{N})}{ψ_{N}^{2}}]

Similarly, when the tracking has been established, the varing range of the received power is very small. And the sampling frequency 2/T is much higher than the pointing error frequency, where T is the period of nutation. Therefore, P_S(t) ≈P_S(t – T/2), and we can write

\begin{array}{l} v (t) - v (t - T / 2) \\ \approx - 4 k (1 - k) R^{2} r_{L}^{2} P_{L O} P_{S} (t) S F (φ_{N}) [2 α (t) - \frac{T}{2} \frac{d α (t)}{d t}] + n (t) - n (t - T / 2) \end{array}

\begin{array}{l} v (t) + v (t - T / 2) \\ \approx 4 k (1 - k) R^{2} r_{L}^{2} P_{L O} P_{S} (t) {2 {\frac{2 J_{1} [ψ_{N}]}{[ψ_{N}]}}^{2} - S F (φ_{N}) \frac{T}{2} \frac{d α (t)}{d t}} \end{array}

Here, $(T / 2) [d α (t) / d t]$ is the small first-order quantity of α(t – T/2), and so

S F (φ_{N}) \frac{T}{2} \frac{d α (t)}{d t} < < S F (φ_{N}) α (t)

Furthermore, because SF(φ_N)α(t) is the small first-order quantity of ${2 J_{1} {(2 π R / λ) [φ_{N} - α (t)]} / {(2 π R / λ) [φ_{N} - α (t)]}}^{2}$ , we find

S F (φ_{N}) α (t) < < {\frac{2 J_{1} [ψ_{N}]}{[ψ_{N}]}}^{2}

So that

S F (φ_{N}) \frac{T}{2} \frac{d α (t)}{d t} < < 2 {\frac{2 J_{1} [ψ_{N}]}{[ψ_{N}]}}^{2}

Therefore, in Eq. (21), $S F (φ_{N}) (T / 2) [d α (t) / d t]$ is negligible.

The optical-axis error can be calculated as follows:

e_{α} (t) = \frac{v (t) - v (t - T / 2)}{v (t) + v (t - T / 2)} \approx \frac{- S F (φ_{N})}{{\frac{2 J_{1} [ψ_{N}]}{[ψ_{N}]}}^{2}} [α (t) - \frac{T}{4} \frac{d α (t)}{d t}] + n^{'} (t)

Where n’(t) is

n^{'} (t) \approx [n (t) - n (t - T / 2)] / [8 k (1 - k) R^{2} r^{2} P_{L O} P_{S \max}]

The PSD of n’(t) can be written as

S_{N^{'}} (f) \approx e / [2 k (1 - k) R P_{S \max}]

Finally, the linear transfer function of the LO-beam nutation-based coherent tracking sensor can be approximated as

G_{S} (s) = \frac{E_{α} (s)}{A (s)} = \frac{- S F (φ_{N})}{{\frac{2 J_{1} [ψ_{N}]}{[ψ_{N}]}}^{2}} (- \frac{T}{4} s + 1)

where A(s) is the Laplace transform of the optical axis error α(t).

Suppose that the nutation period of LO beam is T = 30 μs. Then, the nutation frequency is approximately 33.3kHz. The normalized frequency response characteristics of the LO-beam nutation-based coherent tracking sensor are shown in Fig. 3.

Fig. 3 Normalized frequency response characteristics of coherent tracking sensor based on LO beam nutation.

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Figure 3 shows that when the angular frequency of boresight error falls below 10⁴rad/s, there is little deviation of the magnitude gain or the phase delay. Because the vibration spectrum primarily falls in this range, we can approximate the sensor normalized gain by 1. Then, the phase lag is less than 10°. Therefore, the frequency response characteristic in this range is appropriate for engineering applications. When the angular frequency of boresight error is higher than 10⁴rad/s, the frequency response characteristics change significantly. The sensor gain increases, while the phase lag quickly increases to −90°. This demonstrates that the amplitude-frequency characteristics of the subsequent devices should be cut off after 10⁴rad/s, and a large phase lag should be avoided.

3. Impact of nutation angle on system performance

3.1 Impact of nutation angle on tracking error

The noise generated from the detectors and electric circuits will enter the tracking loop, as shown in Fig. 4. The tracking error caused by these noises is the noise equivalent angle (NEA).

Fig. 4 Linear model of a tracking loop.

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Here, Θ_α(f) is the azimuthal component of AOA, and G_FPA(f) is the transfer function of fine point assembly (FPA).

The PSD of the NEA can be determined through a linear system analysis:

S_{N E A} (f) = S_{N^{'}} (f) {| G_{L} (f) / G_{S} (f) |}^{2}

Where G_L(f) is the closed-loop transfer function of the tracking loop in the frequency domain:

G_{L} (f) = G_{S} (f) G_{F P A} (f) / [1 + G_{S} (f) G_{F P A} (f)]

Here, S_N’(f) is a constant, and the NEA can be written as

N E A = \sqrt{\int_{0}^{\infty} S_{N E A} (f) d f} = \sqrt{S_{N^{'}} (f) B_{n}}

Where B_n is the noise equivalent bandwidth.

B_{n} = \int_{0}^{\infty} {| G_{L} (f) / G_{S} (f) |}^{2} d f

We show an example of the relationship between the half-nutation angle and NEA in Fig. 5. The primary calculation parameters are defined as follows: the receiving optical power is −55dBm, the aperture of the telescope is 150mm, the wavelength of the signal and LO beam is 1μm, and the power splitting ratio of the I path to the Q path is 99:1. The frequency characteristics of a supposed G_L(f) are shown in Fig. 6. The −3dB bandwidth of FPA in this loop is 1500Hz. The FPA is controlled by a digital PID controller, and the update rate of the loop is 20kHz. The parameters of the PID are automatically tuned by the software.

Fig. 5 The relationship between half-nutation angle and NEA.

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Fig. 6 The frequency response characteristic of the tracking loop.

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From Fig. 5, when the half-nutation angle is larger than 0.5μrad, we find that the NEA is less than 0.02μrad. Therefore, the NEA can be ignored in a physical system. As the nutation angle increases, the coherence efficiency sharply decreases. Thus, by satisfying the tracking precision, the nutation angle should be minimized. In next section, we analyze the impact of the nutation angle on communication receiver sensitivity.

3.2 The impact of nutation angle on communication receiver sensitivity

The SNR of the homodyne BPSK coherent receiver without phase locking error can be written as

S N R = \frac{{| v_{I} |}^{2}}{{| n_{I} |}^{2}} = \frac{4 R^{2} r_{L}^{2} (1 - k) P_{L O} P_{S} (t) {2 J_{1} [ψ (t)] / [ψ (t)]}^{2}}{S_{N I} B}

Where B is the single sideband noise equivalent bandwidth of the communication signal. Therefore, the sensitivity degradation caused by nutation is $10 \log {{2 J_{1} [ψ (t)] / ψ (t)}^{2}}$ . We show the relationship between the half-nutation angle and receiving sensitivity degradation is shown in Fig. 7.

Fig. 7 The relationship between the half-nutation angle and the receiving sensitivity degradation.

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Figure 7 shows that the receiving sensitivity quickly degrades as the nutation angle is increased. Combined with our earlier analysis, we find that the best half-nutation angle range is 0.5–1μrad. In this range, the receiving sensitivity degradation is approximately 0.2–0.6dB, and the NEA can be ignored.

4. Conclusion

We present a coherent tracking sensor based on LO beam nutation in the coherent optical communication receiver. The results of the theoretical inference and simulation studies show that when the half nutation angle lies between 0.5 and 1μrad, the NEA is less than 0.02μrad, and the degradation of the receiving sensitivity ranges from 0.2 to 0.6dB. The proposed LO nutation-tracking system is a new approach with comparable performance to existing systems and with potential implementation advantages, depending upon the application. The LO-beam nutation-based coherent tracking sensor has a simple structure, which is easy to realize. The sensor is integrated into a typical coherent receiver by adding a beam deflector to the LO optical path in the coherent receiver, and an envelope detector to the receiver circuits. The LO-beam nutation-based coherent tracking sensor provides a new choice for coherent free-space optical communication system designs.

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Principle and performance analysis of coherent tracking sensor based on local oscillator beam nutation

Abstract

1. Introduction

2. Principle of coherent tracking sensor based on local oscillator beam nutation

2.1 Envelope detector for coherent signal

2.2 The algorithm for boresight error detection and its equivalent linear model

3. Impact of nutation angle on system performance

3.1 Impact of nutation angle on tracking error

3.2 The impact of nutation angle on communication receiver sensitivity

4. Conclusion

References and links

Cited By

Figures (7)

Equations (33)

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