1. Introduction

The Laser Interferometer Space Antenna (LISA) is a space based mission to detect gravitational waves in the frequency band from 0.1 mHz to 1 Hz. LISA consists of three spacecraft flying in a heliocentric, Earth-trailing orbit, with separations of L = 5 × 10⁹ m. Each spacecraft (SC) contains two proof masses that are shielded from external disturbances. Bidirectional laser links are maintained between all SC in the constellation. The relative optical phase between the incoming and outgoing lasers, referenced to their respective test masses, is used to track the changes in separation between the proof masses, allowing detection of gravitational waves. The change in separation of the proof masses ΔL must be monitored with a displacement sensitivity of $\mathrm{\Delta }L\le 4\times {10}^{-11}\text{m}/\sqrt{\text{Hz}}$ to give a strain sensitivity of $\mathrm{\Delta }L/L\le {10}^{-20}/\sqrt{\text{Hz}}$ . This sensitivity requirement far exceeds the stability of available laser sources and has motivated the development of a set of post-processing algorithms termed Time Delay Interferometry (TDI) [1, 2].

The TDI algorithms will provide the majority of the required laser frequency noise suppression through post-processing of the optical phase measurement data recorded at each SC. These sensitive phase measurements are performed by a phasemeter aboard each SC. The phasemeter is required to achieve a relative phase sensitivity of $1\mu \text{cycle}/\sqrt{\text{Hz}}$ [3], however this measurement will be limited by shot noise with a spectrum of $10\mu \text{cycles}/\sqrt{\text{Hz}}$ . The phasemeter provides the input to both the TDI algorithms and ranging system.

The phase data is logged at 3.33 Hz relative to each SC's local clock, however the data streams from all SC must be time synchronised to within ≈ 3 ns (1 m) to achieve the required frequency noise suppression [3]. Without a mechanism for timing synchronisation, a time domain technique called fractional-delay filtering is used to resample the phase data using sub-sample interpolation to synthesise a new set of phase data for all SC relative to a common time base [4]. This technique resamples a SC data stream with respect to a new time reference, accounting for clock differences, propagation delays as required by TDI. Experimental demonstration of one proposed TDI combination on an optical test bed has been shown to provide sufficient frequency noise suppression to meet the LISA requirements with the application of fractional-delay filtering [5]. The interpolation and resampling requires knowledge of the relative time and range between all SC, which is measured using a ranging signal modulated onto each laser link [6].

In LISA, the ranging signal will be generated by Pseudo-Random Noise (PRN) phase modulation of the bidirectional laser links. This is similar to the Global Positioning System (GPS), which uses PRN modulation of a Radio Frequency carrier for extraction of timing and position [7]. The PRN modulation itself will consist of a deterministic sequence of binary pulses or “chips”, which vary between ±1 in a pseudo-random fashion. An example of a PRN code are the Maximum Length (M) sequence and Gold codes associated with GPS. The LISA PRN modulation will use 1% of the total optical power and will spread this power over a wide (MHz) bandwidth, minimising the interference with the LISA science measurement below 1 Hz. The ranging signal acts as a time stamp upon the laser phase, indicating when the light left the SC. Measurement of the PRN arrival time at the far SC will constitute a ‘pseudo-range’ τ = t_Tx(t − L/c) − t_Rx(t), the difference between the receiving SC clock t_Rx and the transmitted time stamp t_Tx after propagating at the speed of light c across the arm of length Lm. This measurement is sensitive to two fundamental effects;

The use of the pseudo-range measurements for alignment of TDI phase data is best understood by considering the propagation of a pulse of laser frequency noise as it passes through the LISA constellation. For a simple case, we can attempt to cancel a pulse of laser frequency noise leaving SC ₁ and arriving at SC ₂ using only combinations of phase measurements.

Figure 1 shows the layout of a single LISA arm where a pulse of phase noise ϕ(t) propagates between SC ₁ and SC ₂. The pulse is recorded by SC ₁ as ϕ ₁(t ₁(t)) = ϕ(t) with local clock time t ₁(t) and appears at SC ₂, with its local clock t ₂(t), after propagation delay L/c = T seconds as ϕ ₂(t ₂(t)) = ϕ(t − T). The pulse of noise will be cancelled by differencing the SC phase measurements at the time the pulse was measured by each SC, giving ϕ ₁(t ₁(t)) − ϕ ₂(t ₂(t) = t ₁(t) − T) = 0. In other words, TDI attempts to align each data sample recorded on SC ₂ with clock t ₂ with an equivalent sample from SC ₁, applying sub-sample interpolation where required, such that they a both samples are measuring equivalent phase noise. Determining this alignment requires knowledge of the propagation delay T plus the relationships between the two clocks and is the responsibility of the ranging system.

 

Fig. 1 Showing a pulse of phase noise sent between SC 1 and 2 with propagation time T. Local SC clock times t ₁(t) and t ₂(t) are relative to global time t.

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Since the ranging time stamp is encoded onto the carrier phase, it experiences the same propagation delay T and is recorded with the same clock as the phase data. Correspondingly, the pseudo-range measured at SC ₂ is sufficient to identify a) when the sample left SC ₁ and b) when it was recorded at SC ₂. This implicitly compensates for all physical delays (‘T’) and clock deviations. For each phase sample, the pseudo-range represents the delay/advance that must be applied to align the samples between SC using the fractional-delay filtering approach and represents an equivalent physical range estimate R = cT, for speed of light c. The accuracy of the phase data reconstruction and subsequent frequency noise cancellation is limited by the pseudo-range estimate and is the focus of this paper.

As mentioned previously, another source of physical delay is the analog electronic chain that forms the front end of the phasemeter [8]. This delay is indistinguishable from all other physical delays as measured by the optical signals. It will be correctly (and automatically) measured if the ranging signal and the science signal experience a common delay in passing through the analog chain. A PRN ranging signal centered on the heterodyne frequency will therefore produce the correct total physical delay needed for TDI implementation.

Detection of the PRN ranging signal will use a Delay Locked Loop algorithm comparable to those used by GPS receivers [7]. The receiver generates a local copy of the transmitted code, referred to as the ‘demodulation code’, then attempts to time align it with the incoming ranging signal using a Delay Locked Loop (DLL) to track any timing changes. A nominal chip period of 1 μs is proposed to minimise any differential group delay relative to the MHz frequency heterodyne beat notes. To ensure that the DLL is capable of tracking any timing changes, any averaging performed within the DLL must be significantly faster than the expected timing fluctuations. The digital nature of the PRN modulation also lends itself well to the transmission of data between SC and a 20 kbps data communication requirement has also placed upon the ranging system [10]. Initial simulation work has shown the viability of this scheme for use in the LISA environment [9].

In this paper, we present a performance prediction and experimental demonstration of an inter-spacecraft ranging algorithm for LISA under expected system conditions. Section 2 introduces the demodulation of the received ranging signal including phase measurement effects and relevant noise sources. Closed loop pseudo-range estimation is presented in Section 3, where we show that the pseudo-range estimate will be limited by cross-correlation between incoming and outgoing PRN codes to ≈ 0.19 m absolute arm length knowledge over a 0.5 Hz signal bandwidth, meeting the 1 m rms LISA requirement. The presented algorithms were integrated into a prototype LISA phasemeter and Section 4 shows the measured ranging performance on a optical test bed, verifying the performance model. A mechanism for optical data transmission is described in Section 5.

2. System Description

In this section we present a model for the demodulation of an incident ranging signal with a local demodulation code. This work builds the previous analysis of the MF-DLL algorithm [9], extending the model to incorporate several additional effects by including the dispersion effects of the phasemeter, cross-correlation interference and data modulation.

We consider a simplified LISA configuration, concentrating upon the reception of a ranging signal sent from spacecraft SC ₂ and received at SC ₁ as shown in Fig. 2. PRN ranging codes c ₁ and c ₂ are phase modulated onto the respective SC lasers via an Electro-Optic Modulator (EOM). The laser from the far SC ₂, with phase x ₂(t − T) delayed by the propagation delay T, is interfered with the local laser, with phase x ₁, at point A ₁ on SC ₁. The heterodyne beat note, which encodes the difference between the laser phases, is measured at the Detector (D ₁) and tracked by the phasemeter. The phasemeter output is passed to the DLL for demodulation of the ranging signal. The outgoing ranging signal is modulated onto the outgoing laser before the interference occurs at A ₁.

 

Fig. 2 A simplified LISA configuration with spacecraft SC ₁ and SC ₂. The SC ₁ and SC ₂ lasers, with phases x ₁ and x ₂, are interfered on SC ₁ at beamsplitter A ₁ and their phase difference is measured by a phasemeter through photodetector (D ₁). A ranging output is produced by the phasemeter and is passed to the MF-DLL for demodulation.

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The LISA lasers will be sufficiently stable that the phase fluctuations are dominated by the ranging modulations c ₁ and c ₂ ie. x ₂ = βc ₂ and x ₁ = βc ₁, with modulation depth β = 0.14rad for 1% of the carrier power. Additionally, over the MHz scale PRN frequencies the LISA phase measurement noise will be dominated by carrier shot noise, appearing as an additive phase noise source ν_PM with a white spectrum with a value of $10\mu \text{cycles}/\sqrt{\text{Hz}}$ [3]. At these frequencies, the laser frequency noise is significantly lower than shot noise and will not be considered here.

The difference between spacecraft clocks couples into the phase measurement once the heterodyne beat note is measured with the SC ₁ clock t ₁. The pseudo-range incorporates the propagation delay T between SC and clock differences and can be viewed as a time varying delay τ ₂ applied to the SC ₂ laser phase signal. Fortunately, any pseudo-range variations are expected to be slow relative to each pulse period and can be considered static over an averaging period (∼ 50μs). Any long term phase fluctuations will be tracked by the DLL algorithm introduced in Section 3. Setting the global time relative to the SC ₁ time t ≡ t ₁ gives the shot noise limited phase estimate at the detector as Eq. (1).

2.1. Ranging Demodulation

Recovery of the pseudo-range requires tracking of the ranging signal arrival time. This will be accomplished by the Delay Locked Loop described in Section 3, which forms an error signal by demodulating the ranging signal at two offset timing points. The demodulation mechanism itself is performed by correlating the phase estimate against a local copy of the transmitted code ĉ ₂(t, $\widehat{\tau }$ ₂) = c ₂(t − $\widehat{\tau }$ ₂). This local copy, the ‘demodulation’ code, utilises a controllable delay $\widehat{\tau }$ ₂ to estimate the pseudo-range delay τ ₂ that allows the demodulation code to be aligned with the ranging signal.

The unknown timing differences between LISA spacecraft requires that the initial demodulation delay $\widehat{\tau }$ ₂ must be explicitly acquired. Acquisition is accomplished by correlating the phase estimate against the demodulation code while scanning the delay $\widehat{\tau }$ ₂, mapping out the auto-correlation of the ranging code. When the misalignment between the demodulation code and ranging signal Δτ ₂ = τ ₂ − $\widehat{\tau }$ ₂ is within 1 chip period T_P (|Δτ ₂| < T_P), a strong correlation peak will be present. The auto-correlation of a PRN sequence evaluated over a period of 50 chips without noise is shown in Fig. 3a, clearly demonstrating the high correlation at 0 delay. While acquisition will not be discussed here, other sources show acquisition within ±1 chip is achievable [7, 9].

 

Fig. 3 (a) The auto-correlation function of a nominal PRN sequence (solid) measured over 50 chips. The auto-correlation of the Rectangular pulse shape (dashed) is also shown for comparison. (b) describes the overlap between the ranging and demodulation pulses with a small timing misalignment and discounting the effects of shot noise. Region A is the ‘wanted’ correlation, while Region B shows the ‘unwanted’ area introduced by the misalignment.

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After acquisition of the demodulation delay, the correlation between the demodulation code ĉ ₂ and phase estimate ϕ ₁ can be directly evaluated over an N chip interval for a small acquisition misalignment |Δτ ₂| < T _P. By assuming that each code chip is represented by a pulse f(t) of width T_P and auto-correlation R_F(τ), then each chip of the demodulation code will overlap with at most two chips of both the ranging c ₂ and interfering c ₁ codes.

Figure 3b illustrates the demodulation of an ideal ranging signal without shot noise. The first chip is the ‘wanted’ pulse which correlates with demodulation code and the contribution is N chips weighted by area of overlap. In this instance, the overlap area is defined as ${\int }_{{\tau }_2}^{{\tau }_2+T_P}f\left(t-{\tau }_2\right)f(t-{\widehat{\tau }}_2)\text{d}t={\int }_0^{T_P}f\left(t\right)f\left(t-\mathrm{\Delta }{\tau }_2\right)\text{d}t$ , which is the definition the pulse autocorrelation function R_f(.) from Eq. (2) evaluated at the delay Δτ ₂. This dependence is also seen in Fig. 3a, where the M sequence auto-correlation follows that of the Rectangular pulse for small offsets. The second ‘unwanted’ correlation results from coupling in an adjacent chip. For sample n, this contribution is given as the code auto-correlation evaluated at a ±1 chip offset C _ĉ2,c2 [n, ±1], which is then weighted by the overlap area R_f((Δτ ₂ + T_P)mod T_P).

Similarly, the two pulses introduced by the interfering c ₁ code each contribute a cross-correlation term C _ĉ2,c1 [n,m_i] and C _ĉ2,c1 [n,m_i + 1], with weightings R_f($\widehat{\tau }$ ₂ mod T_P) and R_f(($\widehat{\tau }$ ₂ + 1) mod T_P) respectively. Since the interfering code is generated locally, only the changing demodulation delay will affect the alignment with the demodulation code and the cross-correlation will be calculated with an offset m_i = ⌊$\widehat{\tau }$ ₂/T_P⌋, where ⌊x⌋ is the integer part of x. This determines that the normalised demodulation over N chips (τ_Corr = NT_P), will be given by Eq. (3).

When the misalignment is small (|Δτ ₂| ≪ T_P), the output is dominated by the strong, coherent signal from the correlation with the ranging signal, with low level cross-correlation noise from the interfering code. As the misalignment increases, the coherent amplitude will reduce down to the noise levels from the other correlation components. To minimise the interference, the cross-correlation between codes c ₁ and c ₂ should be minimised for all possible delays, however before this effect can be quantified we must discuss the filtering effects introduced by the Phasemeter.

2.2. Phase Measurement Effects and Pulse Shaping

Ideally the phasemeter output would provide a shot noise limited readout of the relative laser phase. In practise, phase measurement algorithms are affected by finite bandwidth and amplitude quantisation, which limit the phase estimate. To track the heterodyne phase, the LISA phasemeter implements a Phase Locked Loop (PLL) with a sophisticated cascade of filters and control systems [11]. Summarised by Fig. 4, the PLL acquires and tracks the phase of the detected heterodyne beat note ϕ ₁ through controller C_PLL. The detector phase signal is mixed with a Local Oscillator (LO) and low pass filtered (LPF) to form an error signal for feedback control of the LO phase. Such a control system locks the LO to be in quadrature with the signal.

 

Fig. 4 Showing a simplified phasemeter Quadrature (Q) with demodulation including mixer, Local Oscillator (LO), Low Pass Filters (LPF) and Controller C_PLL. Signal frequencies greater than the Controller bandwidth f_u appear in the Quadrature (Q) output, while frequencies lower than f_u are suppressed by the controller and imposed onto the LO phase

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The phasemeter cannot track the fast phase fluctuations of the MHz scale PRN used by LISA. The PLL mixer output (Q) will pass the high frequency PRN modulation signal as phase noise, which the PLL will attempt to suppress. At frequencies well above the control bandwidth the phasemeter passes phase fluctuations through to Q with unity response, however any ranging signal frequency components near or below the phasemeter unity gain frequency will be suppressed and phase shifted. As a consequence, the ranging signal will be filtered by the phasemeter transfer function.

The transfer function between the beat note phase at the photodetector output ϕ ₁ and the phasemeter Q channel output ϕ_Q can be approximated as H_PM(s) with Laplace variable s. This transfer function assumes the controller C_PLL is dominated by a 1/f slope around the open loop unity gain frequency f_u ≈ 100 kHz. The closed loop transfer function exhibits a high pass behaviour, with a corner frequency f_u.

For a detector output ϕ ₁, then the phasemeter Q output ϕ_Q(t) will be given by Eq. (4), where the high frequency shot noise is also passed to the output.

This measurement includes the high pass filtering effect through convolution (★) with the phasemeter impulse response h_PM. Figure 5a shows the filtering effect upon a Rectangular pulse f(t) of width T_P = 1 μs, with an unfiltered pulse shown for comparison. The visible decay induced by the filtering causes a measurable delay in the pulse centroid, which will couple into the range estimate.

 

Fig. 5 Plot (a) shows the phasemeter pulse responses for a 1 μs chip with Rectangular and Manchester pulse shapes. Plot (b) shows the pulse correlation functions for Manchester and Rectangular pulses with and without phasemeter filtering.

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To mitigate this effect, the receiver would traditionally match the demodulation code to the filtered signal or include a compensating filter to recover the original pulse shape [14]. These are resource intensive operations. An alternative and complementary solution is to use a Manchester pulse shape, which moves the PRN energy to higher frequencies where the phasemeter transfer function is unity. Manchester encoding of the PRN sequence sends a bipolar pulse of 1, −1 for each code chip and has a pulse shape f_M(t) defined by Eq. (5). This pulse shape doubles the signal bandwidth by guaranteeing a transition between 1 and −1 occurs every chip. Additionally, the code is now balanced for each chip, with zero DC component and reduced low frequency information as since each chip has 0 DC power. The Manchester pulse also passes through the phasemeter with minimal decay as shown by Fig. 5a.

Manchester coding was initially proposed for the ranging signal to limit low frequency interference with the LISA phase measurement [12] [13], however the use of Manchester coding has the added benefit of reduced dispersion through the phasemeter. Figure 5b compares the filtered and unfiltered pulse correlation for Rectangular and Manchester pulses against an ideal demodulation code. The Manchester pulse suffers only a small 0.13dB ‘on-time’ signal loss with minimal decay compared to the significant tail evident in the Rectangular pulse correlation, which is verified by the time domain trace in Fig. 5a. The introduced phase delay can be seen in the asymmetric shape of the correlation functions, which will influence the formation of the closed loop timing error signal to be introduced in Section 3.

The analysis to this point has assumed that only two ranging pulses can correlate with one demodulation pulse, so immediately the Rectangular pulse would break this condition due to the large decay. Figure 5b demonstrates that the decay contribution outside (0, T_P) for the Manchester pulse shape is small (≪ 1%) compared to other terms, allowing us to simply replace the pulse function with the Manchester pulse f_M(t) and auto-correlation R_M(τ). The pulse correlation after the phasemeter is now R_M(Δτ) ★ h_PM = R_PM(Δτ) and the demodulation correlation can be written as;

2.3. Interference Contribution

To evaluate the system performance, we can make an assumption that the correlation between uncorrelated or unaligned PRN codes appears as additive white noise [7, 15]. Whilst the repetitive nature of the PRN sequences guarantees that these correlations will be periodic, the independent LISA clocks will disrupt this periodicity. This allows us to approximate the ‘unwanted’ and ‘interfering’ correlations as white noise sources.

The unwanted auto-correlation appears as a noise source C _ĉ2,c2 [n, ±1] = ν _ĉ2,c2 [±1], with variance ${\sigma }_{{\widehat{c}}_2,c_2}^2=N$ . A similar assumption may be made for the interference cross-correlation terms, where the interfering contribution is reduced to two independent noise sources C _ĉ2,c1 [n,m_i] = ν _ĉ2,c1 [m_i] and C _ĉ2,c1 [m_i + 1] = ν _ĉ2,c1 [m_i + 1], with variance ${\sigma }_{{\widehat{c}}_2,c_1}^2=N$ . This allows a different interpretation of A _ϕQ,ĉ2 [n] as shown in Eq. (6) below.

Interestingly, Eq. (6) predicts that changes in the demodulation delay $\widehat{\tau }$ ₂ will modulate the interference contributions through the pulse auto-correlation. The interference pulse correlation weightings R_f($\widehat{\tau }$ ₂mod T_P) and R_f((τ_{̂ 2} + 1)mod T_P) are each dependent upon the delay, acting as a time varying gain to the correlation noise. On short time scales, performance predictions must be used directly Eq. (6), however on longer averaging periods we can simplify the result.

When the demodulation delay is tracking the long terms changes in the pseudo-range, the interference code fluctuations will average over the Manchester pulse auto-correlation to appear as a single noise source ν _ĉ2,c1 with zero mean and variance ${\sigma }_{{\widehat{c}}_2,c_2}^2=2N{T}_P^2/3$ . Similarly, the unwanted noise term will be given as R_f ((Δτ ₂ + T_P)mod T_P) ν _ĉ2,c2 [±1] = ν _ĉ2,c2 with zero mean and a timing error dependent variance ${\sigma }_{{\widehat{c}}_2,c_2}^2(\mathrm{\Delta }{\tau }_2)=N{R}_f{\left((\mathrm{\Delta }{\tau }_2+T_P)mod{T}_P)\right)}^2{\text{rad}}^2$ .

Accumulating all the correlation noise into a single noise term ν_A, with variance ${\sigma }_A^2(\mathrm{\Delta }{\tau }_2)=2{\beta }^2/3N+N{\left(\beta R((\mathrm{\Delta }{\tau }_2+T_P)mod{T}_P)/{\tau }_{\mathit{\text{Corr}}}\right)}^2{\text{rad}}^2$ , the demodulated correlation A _ϕQ,ĉ2 [n] can be written as a timing error dependent random variable given by Eq. (7).

To place this result within the context of LISA, we can consider Eq. (3) with a chip rate of 1 MHz and a correlation period τ_Corr of N = 50 Chips (50 μs or 20 kHz correlation rate). The modulation depth is β = 0.14 for 1% of the carrier power. For perfect demodulation, the correlation noise terms give a noise power ${\sigma }_A^2(0)=2{\beta }^2/3N{\text{rad}}^2\approx 2.25\times {10}^{-4}{\text{rad}}^2$ . The shot noise contribution of $10\mu \text{cycles}/\sqrt{\text{Hz}}$ gives a noise variance of ${\sigma }_{\mathit{\text{PM}}}^2\approx 8\times {10}^{-5}{\text{rad}}^2$ over a 20 kHz bandwidth. This highlights that the demodulation noise is dominated by the interference introduced by the outgoing ranging code, with a noise power > 10 dB larger than shot noise. For perfect demodulation alignment with Δτ ₂ = 0, the Signal to Interference Noise ratio (SINR) is 3N/2. The noise models in the presented in this section were verified by simulation.

3. Timing Tracking

Recovery of the pseudo-ranging information requires that the delay of the incident ranging signal τ ₂ be determined to within 3 ns (or 1 m range) and is equivalent to demodulating the ranging signal phase to within 10⁻³ chips. Changes in this delay will be driven by two dominant timing effects; the LISA arm length fluctuations and the relative clock differences between spacecraft. The delay locked loop derives a timing error signal from an ‘early’ and a ‘late’ correlation to track the resulting changes in code arrival time. The early / late demodulation code is nominally advanced/ delayed half a chip from the ‘on-time’ timing point, however the exact separation may be reduced for better sensitivity at the expense of acquisition reliability. Any change in arrival time will add power to either the early or late correlation, indicating the direction of the change. The difference between the correlation forms a timing error signal proportional to the timing offset Δτ ₂.

We define the early and late codes as ${\widehat{c}}_2^{\mathit{\text{early}}}(t)={\widehat{c}}_2(t-T_{\text{P}}/2)$ and ${\widehat{c}}_2^{\mathit{\text{late}}}(t)={\widehat{c}}_2(t+T_P/2)$ respectively, with associated correlations $A_{{\varphi }_Q,{\widehat{p}}_2^{\mathit{\text{early}}}}$ and $A_{{\varphi }_Q,{\widehat{p}}_2^{\mathit{\text{late}}}}$ . The timing error signal is dependent upon Δτ ₂ and is given by

The correlations $A_{{\varphi }_Q,{\widehat{p}}_2^{\mathit{\text{early}}}}$ and $A_{{\varphi }_Q,{\widehat{p}}_2^{\mathit{\text{late}}}}$ can each be expanded using Eq. (7) with a timing offset of Δτ ₂ = ±T_P/2 as appropriate. This gives the relation in Eq. (8) for the error signal output as a function of timing error. The phasemeter and interference noise contributions have been grouped into noise sources ${\nu }_{\mathit{\text{PM}}}^{\mathit{\text{error}}}={\nu }_{\mathit{\text{PM}}}^{\mathit{\text{late}}}-{\nu }_{\mathit{\text{PM}}}^{\mathit{\text{early}}}$ and ${\nu }_A^{\mathit{\text{error}}}={\nu }_A^{\mathit{\text{late}}}-{\nu }_A^{\mathit{\text{early}}}$ respectively. The noise terms ${\nu }_A^{\mathit{\text{late}}}$ and ${\nu }_A^{\mathit{\text{early}}}$ each have a delay dependent variance ${\sigma }_A^2(\mathrm{\Delta }{\tau }_2={\text{T}}_{\text{P}}/2)=11{\beta }^2/12N$ .

Figure 6 shows the noiseless error signal ε _ϕQ,c2 against timing error Δτ ₂. The error signal is approximately linear for errors within | Δτ ₂| ≤ T_P/2, however the error signal zero crossing point occurs at an offset from the ideal timing point Δτ ₂ = 0. This is a direct consequence of the delay introduced by the phasemeter and results in an offset of the zero crossing point Δτ ₀ which is dependent upon the phasemeter closed loop decay time 1/α by,

 

Fig. 6 Showing the timing of the early and late demodulation correlations. Comparing the demodulated signals gives the DLL timing error signal ε _ϕQ,c2(Δτ ₂) shown in for the Manchester pulse shape. The error signal zero occurs away from ideal timing point Δτ ₂ = 0 due to phasemeter induced dispersion.

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For α = 2π × 10⁵, the timing offset is Δτ ₀ ≈ 75.2 ns or ≈ 22.6 m with a ∼ 0.82 dB Signal to Noise Ratio (SNR) loss. Importantly, without automatic gain control of the digitized heterodyne beat note, any amplitude deviations will couple into the phasemeter open loop bandwidth f_u though gain scaling of the 1/f slope. Consequently the bandwidth f_u and the error signal zero crossing point will vary with any amplitude variation. As the effective unity gain frequency scales with amplitude noise, this will introduce in an additional source of time varying, timing offset which must be compensated for. Around the target phasemeter operating amplitude A_P, the coupling between amplitude noise ΔA_P and timing offset is Δτ ₀ ≈ 75.2 + 70ΔA_P/A_P ns. Allocating 0.03 ns error to this error source requires knowledge or control of the fractional amplitude fluctuation to within ΔA_P/A_P ≈ 4×10⁻⁴. This can either be accomplished by automatic gain control or calibrated in offline post-processing using the to the amplitude information recorded by the phasemeter at a fidelity of ΔA_P/A_P ≈ 10⁻⁶.

3.1. Closed Loop Timing Analysis

With an error signal defined, we can now consider closing the timing control loop to form a timing estimate. LISA specifications require an absolute timing estimate within ≈ 3 ns(or 1 m) for all sample points. After initialising the demodulation delay through the acquisition system, the delay estimate $\widehat{\tau }$ ₂ will be updated by the error signal output. Any orbit or clock effects are significantly slower than the ranging system will operate [3], hence a second order Integral, Double Integral (II ²) controller is selected, which maximises low frequency tracking while minimising the sensitivity to high frequency noise. The closed loop system is modelled to predict the effect of phase and interference noise propagating through the timing control loop and an associated noise floor is derived for the timing estimate $\widehat{\tau }$ ₂. A diagram of the system under consideration is show in Fig. 7.

 

Fig. 7 Block diagram of the Closed Loop Timing Control system including ADC, Phasemeter with impulse response h_PM, PRN generator with timing estimate $\widehat{\tau }$_i, correlation noise sources ν_PM and ν_A and Controller C. The timing error signal is calculated using the difference between Early and Late demodulations.

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For small timing deviations around the zero crossing point, the error signal is approximately linear and is given by ε _ϕQ,c2 (Δτ ₂) = βN/τ_Corr(R_PM(Δτ ₂ + T_P/2) − R_PM(Δτ ₂ − T_P/2)) ≈ 2βN/τ_Corr Δτ ₂. This linear operation requires the acquisition algorithm to accurately identify the demodulation delay to within |Δτ ₂| < T_P/2 for this analysis to be valid. From Fig. 6, we can see this assumption is only valid when the timing offset from 0 is accounted for. Within the linear region, the error signal from Eq. (8) can be written as;

We define the error signal gain G = ε _ϕQ,c2(Δτ ₂)/Δτ ₂ = 2βN/τ_Corr and the II ² controller C _II² (s) =k_I/s + (k _I²/s)², with Laplace variable s and controller gains k_I = 5 × 10⁻³ and k _I² = 2.5 × 10⁻⁴ selected for a closed loop bandwidth of ≈ 1.4 kHz.

The closed loop transfer function can be calculated from Fig. 6 as:

The interference noise contribution is ${\sigma }_{A,\mathit{\text{error}}}^2=2{\sigma }_A^2\left(\mathrm{\Delta }{\tau }_2=T_P/2\right)=\frac{11{\beta }^2}{6N}{\text{rad}}^2$ which converts to a PRN phase noise spectrum of $\approx 268\mu \text{cycles}/\sqrt{\text{Hz}}$ . The equivalent interference noise timing error will be $\approx 1\hspace{0.17em}\text{ns}/\sqrt{\text{Hz}}$ with an equivalent ranging error of 0.7 ns or approximately 0.2 m rms over 0.5 Hz bandwidth. Contributions from the phase measurement and shot noise ${\nu }_{\mathit{\text{PM}}}^{\mathit{\text{error}}}$ are at $\approx \sqrt{2}\times 10\mu \text{cycles}/\sqrt{\text{Hz}}$ , which has an equivalent timing noise spectrum of $\approx 5\times {10}^{-2}\text{ns}/\sqrt{\text{Hz}}$ and an error of 3.5 × 10⁻² ns or ≈ 10 mm rms at 0.5 Hz bandwidth.

Once the 75.2 ns delay has been compensated, the interference limited ranging precision will meet the LISA requirement for 1 m absolute range knowledge.

4. Test bed results

The ranging system was integrated with the LISA phase measurement systems and tested on the JPL TDI test bed, a ‘proof-of-concept’ demonstration for the effectiveness of TDI [5]. The realtime ranging between platforms provides the capacity to further verify TDI with the required LISA signals and processing. The PRN ranging was implemented on a National Instruments 7833 FPGA development platform, enabling rapid prototyping and design verification.

The TDI test bed imitates a collapsed LISA architecture with two independent dummy spacecraft nodes separated by static 1.25 m arm length. The basic experimental setup is shown in Fig. 2, with further details available in [5]. LISA-like signalling is used to gather all measurements required by TDI, successfully demonstrating the suppression of laser frequency noise by many orders of magnitude. Importantly, each dummy spacecraft uses an independent clock source to drive all digitisation, phase measurements, digital signal processing and PRN generation. The frequency offset between the two node clocks used in the testbed nodes has been measured in [5] and shows a frequency error of ≈ 1 ppm between stations. Each node generates a unique PRN ranging signal, using a 17 bit ‘Maximum Length’ sequence truncated to 128,000 chips, for modulation onto the laser using an Electro-Optic Modulator. Reliable acquisition of the code can be accomplished in ≈ 12.8s by averaging over 50 chips with a T_P/2 separation between candidate timing points.

The static arm length of the test bed determines that the pseudo-range measurement will be dominated by the difference between node clocks. For a 1 MChip/s chip rate, the 1 ppm difference between clocks amounts to a slip rate of 1 Chip/s. For a single direction pseudo-range measurement, this will appear as a ∼ 300 m/s change in spacecraft separation, much faster than the maximum ∼ 10 m/s velocity predicted for LISA orbits [16]. This ramp between pseudo-range measurements is clearly visible in Fig. 8a, however because the ramp is antisymmetric between stations it will be cancelled in the round trip range measurement.

 

Fig. 8 (a) shows the raw inter-node pseudo-range measurements. The clock difference causes the node timing to drift, appearing as a range ramp of ∼ 300 m/s. This effect is anti-symmetric and is removed in the round trip estimate. (b) shows a comparison of the detrended GHz Clock sideband and A-B pseudo-range. Differencing the clock sideband and pseudo-range allows estimation of the residual ranging error. The zoom section demonstrates sub-ns pseudo-range error with a 0.5 Hz signal bandwidth.

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In the LISA baseline design, a 6 GHz clock sideband signal is derived from the spacecraft local clock and modulated onto the laser in addition to the ranging signal. The primary use for this sideband signal is for removal of the clock noise contribution to the phase measurement which scales directly with the heterodyne frequency [17], however the beat note between the GHz sidebands also provides an estimate of the difference between the phase of the clocks they are generated from. Using about 10% of the optical power, the sideband beat note provides a high SNR readout of the clock difference and, due to the static arm lengths in the test bed, can be used to calibrate the performance of the ranging system.

Figure 8b shows this clearly, where the detrended single direction ranging estimate and GHz clock sideband are tracking the same clock deviations. The difference between the two independent measurements, shown in the difference trace, demonstrates that each sample of the ranging estimate, updated at 3.33 Hz, suffers a ≈ 0.65 ns rms error relative to the clock sideband signal when averaged to a 0.5 Hz signal bandwidth. This is equivalent to ≈ 0.19 m rms range error and demonstrates close agreement with the ≈ 0.7 ns or 0.2 m rms error predicted in Section 3.1.

Figure 9 shows the Root Power Spectral Density (RPSD) of the experimental results achieved for the bidirectional ranging estimate (blue) and 6 GHz sideband signals (dark green) which show close agreement at low frequencies. These traces are overlayed onto the white interference (dashed) and shot noise (solid) level predictions from Section 3, with interference noise present at $\approx 1\hspace{0.17em}\text{ns}/\sqrt{\text{Hz}}$ and a shot noise contribution of $\approx 5\times {10}^{-2}\text{ns}/\sqrt{\text{Hz}}$ . The difference between the clock sideband signal and the pseudo-range shows an estimate of the residual ranging noise spectrum (light green). The ranging estimate spectrum closely matches the predicted white interference level, indicating the ranging measurement is limited by interference noise. This is further verified by the single code trace, which represents a single direction ranging measurement (red) with the interfering code deactivated. This trace give a better estimate of the true ranging noise floor and appears to be limited by the known testbed displacement noise limit of $\approx 20\mu \text{cycles}/\sqrt{\text{Hz}}$ .

 

Fig. 9 Showing the spectrum of the closed loop timing estimate (blue) achieved by the test bed compared against the GHz clock signal (dark green). The predicted interference (dashed) and shot noise (solid) levels are displayed for comparison. Measurement of a single code (red), with interfering code deactivated, shows the limit achievable without interference. The difference (light green) trace shows the residual ranging noise after subtraction of the clock drift.

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The interesting feature of the RPSD is the clear cross-over frequency between the clock signal and the interference noise. This indicates that the optimal averaging period for the ranging signal will be ≈ 0.2 Hz, which produces a ≈ 0.38 ns rms error. Changing the averaging bandwidth away from this optimal value will either allow more white interference noise to couple into the measurement or suppress the clock signal, increasing estimation error in both cases. Importantly, this cross-over frequency is clock specific and does not indicate the optimal filter bandwidth for LISA. As such, we have selected a 0.5 Hz bandwidth as the nominal figure of merit as it is close to the corner frequency for the single code measurement (without interference).

The presence of the noise spikes seen at around 1 Hz (and 0.8 Hz) in the ranging psd corresponds to the ≈ 1 Chip/s slip rate. The spike occurs as a consequence of the Manchester coding scheme. In the early and late signal calculation, the intentionally misaligned demodulation of the Manchester code produces a strong residual tone at the incoming chip frequency (≈ 1 MHz ± 1 Hz). After decimation inside the DLL, this tone is is aliased down give an oscillation in the error signal at the 1 Hz chip slip rate. This effect can be mitigated by improved DLL filtering or filtering the ranging output prior to TDI.

5. Data Modulation and Transmission

The binary nature of PRN modulation naturally permits the transmission of data upon the ranging signal without significantly degrading performance. A single data bit may be encoded into the sign of the ranging signal, with each bit represented by one correlation period (50 PRN code chips). To transmit a ‘1’ bit the transmitter intentionally inverts its outgoing ranging signal over the 50 chip interval. This inversion is detected at the receiver as a negative correlation. For a ‘0’ bit, the signal is transmitted uninverted and a positive correlation is detected.

To assist with data packet synchronisation, a frame can be defined using the repetition length of the PRN code as shown in Fig. 10. For this reason, the PRN sequence was truncated to 128,000 code chips (128 ms) for data alignment. Within this frame, we are able to transmit 2,560 bits at 50 chips per bit, exactly meeting the LISA requirement of 20 kbps. The repetition length of the code is a well defined synchronisation event, allowing the frame ‘start bit’ to be readily identified and the data packets correctly reconstructed. An alternate arrangement would involve transmission of a pilot bit pattern, which would erode the data bandwidth available for communications.

 

Fig. 10 Showing the implemented data protocol for the optical communications system. Each bit is represented by 50 code chips, which are formed into a data packet. The data packet boundary is identified by the repetition length of the PRN code (2560 bits or 128,000 chips) meeting the data rate requirement of 20 kbps.

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Whilst in principle it is possible to select a significantly shorter code (1024 chips) which feature reduced cross-correlation interference when averaged over one code length, the introduction of data transmission with these optimised codes has been shown to degrade the ranging performance [9]. This degradation results from the unknown data applied to the ranging code which destroys the optimised cross-correlation against the interfering code. The use of a significantly longer code sequence acknowledges that this degradation will occur and allows for a longer averaging period of the random cross-correlation fluctuations caused by data transmission. A secondary advantage of the longer code is the reduced ambiguity range introduced by the code repetition.

Since the bit period is equal to the code correlation length, the timing error signal must account for any common sign inversion of the early and late signals resulting from the data. This can be readily accomplished by considering the magnitude of the early and late signals by redefining the error signal as ${\epsilon }_{{\varphi }_Q,c_2}(\mathrm{\Delta }{\tau }_2)=\Vert A_{{\varphi }_Q,{\widehat{p}}_2^{\mathit{\text{late}}}}\Vert -\Vert A_{{\varphi }_Q,{\widehat{p}}_2^{\mathit{\text{early}}}}\Vert$ . Within the linear range of the error signal (‖Δτ ₂‖ ≪ T_P/2), all previous analysis remains applicable.

6. Conclusions

In this paper we have documented the design, analysis and experimental demonstration of a ranging and optical communication sub-system for LISA. Analytic models for the received signal and noise source were presented, showing that the dominant system noise source results from cross-correlation interference between the incoming and outgoing ranging codes. The system noise sources were shown to lead to a ranging estimate noise floor at approximately $0.3\hspace{0.17em}\text{m}/\sqrt{\text{Hz}}$ and a raw ranging output with ≈ 0.19 m rms error per 3.33 Hz sample with a 0.5 Hz signal bandwidth. This level will successfully meet the LISA requirement of 1 m. Integration of the ranging system in prototype LISA hardware and operation within a LISA-like optical test bed verified the predicted performance levels and enables the use of realistic PRN signalling for future TDI experiments.