Formulation and Characterization of One-Way Radiometric Tracking with the Iris Radio Using a Chip-Scale Atomic Clock

2.1 Analog Down-Conversion to the Intermediate Frequency

After receiving r(t), the first step is to analog down-convert from f_u to the intermediate frequency (IF) f_IF (approximately 112.5 MHz) by mixing the received signal r(t) with an Iris TCXO signal multiplied up to a specified frequency and then bandpass-filtered (BPF) with a bandwidth B_IF = 14 MHz (the same as the LNA) centered around f_IF. The mixing frequency is defined as follows:

7

where the value for M_IF is selected such that the following holds:

8

and is a known frequency bias. For example, NASA’s CAPSTONE mission has selected M_IF = 2 × 2766/39 for the uplink carrier frequency f_u = 7,204,869,318 Hz to yield . Because the physical oscillator is not operating at exactly its nominal frequency f₀, an additional frequency term will be introduced in the down-conversion beyond that will be accounted for through the dependence on x_I(t), as shown in Equation (3). The down-conversion to IF is performed via the following:

9

where ϕ_IF,0 is the initial phase. By substituting Equation (1) into Equation (9) and reducing, the following is obtained:

10

where the BPF process has eliminated higher-frequency terms that are outside B_IF. here, we define . The resultant noise process ν_s(t) is a narrowband noise that has been shifted down from f_u to f_IF and takes the following form:

11

The mixing process introduces a factor of , and as a result, ν_s(t) has a reduced noise variance of (the component processes ν_i(t) and ν_q(t) still have the same two-sided power spectral density of N₀/2).

2.2 Analog-to-Digital Conversion and Alias Sampling

The next step is to convert the analog IF signal s(t) to a digital signal and alias-sample it to a frequency of 12.5 MHz that prepares the signal for an efficient digital down-conversion to baseband. Digital sampling occurs on the TCXO cycle boundaries or, equivalently, a full cycle of its phase, which can be modeled by using Equation (3) as follows:

12

Solving for t_n yields the resulting difference equation:

13

where is the period of the nominal sample step. Equation (13) reveals that, with a real oscillator, the actual time between samples varies non-uniformly (via the presence of x_I(·)) and is an important artifact to track as we work to construct the range and range rate. If the oscillator is assumed to be perfect (i.e., x_I(t) = 0), then the discrete and continuous arguments reduce to their typical definitions. Solving the difference equation yields the following:

14

where n is the integral number of samples since the start of counting (recall that t₀ is defined to be zero). Note that the two equations in Equation (14) can be used to equivalently express a sampled quantity as a function of n or t_n. Therefore, when it is convenient to identify the index of a function, we will use n, and when we want to identify the associated true time, we will use t_n. Substituting Equation (14) for t in Equation (10) and rearranging yields the following:

15

Now, the specific value for f_IF, as defined in Equation (8), is selected to facilitate the alias-sampling according to the following identity:

16

That is, with T_s = 1/f₀ = 20 ns, we have 112.5/50 = 9/4 = 2 + 0.25, which will alias the signal from the IF 112.5 MHz to 12.5 MHz. Substituting Equation (16) into Equation (15) and rearranging the IF term leads to the following:

17

The following observation now proves useful:

18

Applying this relationship to Equation (17) produces an alias-sampled signal centered at ~ 12.5 MHz (that is, ) that takes the following form:

19

Since nT_s is not readily known, whereas t_n can be estimated via the CSAC 1PPS and its comparison to the TCXO; we replace nT_s with (t_n + x_I(t_n)) (from Equation (14)) to yield, after some rearranging, the following expression for s(n):

20

Similar steps are used to derive an expression for ν_s(n) with the following result:

21

The digital sampling of the noise has changed the autocorrelation of the component noises into a discrete form {ν_i(n), ν_q (n)} with , and their associated two-sided discrete power spectral densities become (Rice, 2009).

2.3 Digital Down-conversion and Lowpass Filtering

As illustrated in Figure 2, s(n) is routed to the DDC where it undergoes a digital down-conversion in frequency to baseband via branching s(n) into the in-phase arm (or I-arm) and quadrature arm (or Q-arm) signals, followed by mixing of the in-phase signal with cos(2πn/4) and quadrature signal with -sin(2πn/4). This yields baseband signals that are now down-sampled by every other value (i.e., n = 2j) and lowpass-filtered (LPF) to obtain the resulting I-arm and Q-arm signals d_i(j) and d_q(j), respectively. These signals can be represented in complex form as follows:

22

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FIGURE 2

Iris DDC

with . The DDC is a traditional multiply–filter–decimate architecture, but has been implemented as a polyphase filter, yielding the multiply–decimate–filter architecture illustrated in Figure 2. The bandwidth of the lowpass filters is 25 MHz centered on 0 Hz and thus does not affect the w(n) noise process that was previously band-limited to B_IF = 14 MHz by the LNA. In sampling every other value, Equation (14) can be rewritten to define a time argument t_i(j) that applies to the I-arm as follows:

23

where now the effective sampling period is doubled to and the associated sampling frequency is f_d = 25 MHz. Similarly, a time argument for the Q-arm is defined as follows:

24

Observe that the absolute sampling time t_q(j) has been shifted by ~ T_s relative to t_i(j).

When interpreting the time argument results, it is important to note that any quantity that appears as a function of t_i(j) or t_q(j) implies that it is evaluated according to its original sampling by t_n but selected from a list according to its j-th argument. For example, the light time delay τ(t_n) yields a sequence {τ(t₀), τ(t₁), τ(t₂), τ(t₃), τ(t₄), τ(t₅),⋯} that for j = 0,1,2⋯ and τ(t_i(j)) results in a subset of the original sequence with values {τ(t₀), τ(t₂), τ(t₄),⋯} and, similarly, for τ(t_q(j)) yields a subset with values {τ(t₁), τ(t₃), τ(t₅),⋯}. This implies that d_i(j) and d_q(j) have sampled slightly different values for {τ(·), x_I(·), m(·)} and the associated component noise terms. Furthermore, as with the previous sampling that yields a sequence of values indexed by n or values that are identified by their true time dependence by t_n, the time argument for a quantity using j, 2j, or t_i(j) and t_q(j) or 2j + 1 is selected to clarify the relevant functional dependence of that quantity.

Let us begin by examining the I-arm signal d_i(j), which is obtained via the following operation:

25

Using cos(A + πj)cos πj = cos A and sin(A + πj)cos πj = sin A yields the following:

26

where we have defined the total phase ϕ_T since the start of the pass as follows:

27

The in-phase noise ν_d,i(j) takes the following form:

28

where, for notational convenience, we have defined the phase modulating the noise as follows:

29

Because of the dependence on , ν_d,i(j) is still a narrowband noise process. Similarly, the Q-arm signal d_q(j) is obtained as follows:

30

where cos(A + πj + π/2)sin(πj + π/2) = –sin A and sin(A + πj + π/2) sin(πj + π/2) = cos A have been employed, and the quadrature phase noise nu_d, q(j) is expressed as:

31

Careful examination of Equations (26) and (30) reveals that the signal has been converted to baseband with the total phase ϕ_T consisting of constants {Δϕ₀, N}, slowly varying terms such as the light time delay τ(t) and the clock phase offset x_I(t), or the frequency bias , which is much lower than the 25-MHz sampling frequency. In fact, the highest-frequency term is the ranging/data signal m(t – τ(t)); however, this signal is also below 12.5 MHz (the Nyquist limit of the sampling frequency). We will see shortly how this term is separable from the phase terms in the CTL.

2.4 Carrier Tracking and Measuring the Total Count Phase

At this point, the Iris radio performs closed-loop phase tracking of the complex signal d(j) from the DDC to generate estimates of the total count phase that will be used to form the one-way range rate measurements. Phase tracking is accomplished in the CTL, where the output of the CTL’s numerically controlled oscillator (NCO) produces the phase estimate. In addition to its use in phase tracking, the CTL isolates the ranging signal in the imaginary part of the tracking error signal and routes the signal to the STL for range demodulation to form range measurements. A block diagram of the CTL is shown in Figure 21 in Appendix A. The appendix also provides a detailed analysis of the CTL’s processing and resulting phase error variance.

The output of the NCO’s accumulation of phase, as given in Equation (93) in Appendix A, ensures that the phase is a total count that does not wrap frequently (until the 64-bit word length of the sampling register is reached) and enables sampling at a much lower rate. The register for the NCO’s phase accumulation is triggered for access by the Iris firmware via a 1PPS signal and reported out in telemetry on its 1-s boundaries. A continuous total phase is critical for forming range rate (or Doppler) measurements because the phase is sampled on a selected count time T (typically greater than 1 s) and then differenced to compute a total phase change over T. If the phase wrapped on 2π boundaries, it would not be possible to form range rate measurements with sufficient precision for navigation. The 1PPS-sampled total count phase is represented as follows:

32

where m is defined as the index associated with consecutive 1PPS boundaries (that is, t_m – t_m–1 ≅ 1s), the term Δϕ(t_m) is the tracking loop error introduced by the CTL, which we will assume is negligible in later analysis, and is the phase noise with an associated variance . A detailed development of the expression for the variance is developed in Appendix A, with a final expression given in Equation (110). For convenience, we repeat the expression here:

33

where B_CTL is the CTL loop bandwidth, P_c is the carrier power as defined in Equation (83), R_S is the ranging signal-to-noise ratio defined in Equation (108), the coefficients {h₂, h₀, h₋₁, h₋₂} are the Iris TCXO’s component noise spectral density strengths, which take the values given in Equation (106), and .

The Iris radio collects telemetry at a 1 Hz sampling rate with the relevant telemetry fields identified in Table 1 that are needed for forming the total count phase and, to be discussed later, the fields listed in Table 3 for range. The total count phase is sampled on CSAC 1PPS boundaries (discussed in the next section) and recorded in telemetry, as defined in Table 1 for the total count carrier phase field. Before discussing the Iris range demodulation and formation of a range measurement, we turn to Iris timekeeping with the CSAC 1PPS and the formation of one-way range rate measurements.

2.5 Timekeeping on Iris with a CSAC 1PPS

Recall that the true time t_m is unknown; however, because the CSAC 1PPS input is available for comparison with the Iris TCXO, we can use the CSAC’s improved stability to reduce effects from TCXO drift and improve one-way radiometric data accuracy. The clock readings of the Iris radio derived from the TCXO C_I(t) and from the CSAC 1PPS C_C(t) can be represented as follows:

34

where x_I(t) and x_C(t) are the respective clock phases, ε_I(t) denotes the error in reading the Iris time, and ε_C(t) is the error in reading the CSAC time. The Iris time query errors ε_I(t) are at the sub-nanosecond level. For example, the absolute sampling jitter, obtained by integrating the TCXO phase noise profile in Equation (106), yields approximately 1.6 ps. Other clock query errors are also present; however, these turn out to be negligible when examining the use of the CSAC 1PPS to read the TCXO time. Indeed, the dominant error in using C_C(t) arises from the fact that the CSAC is asynchronous with the TCXO. That is, the CSAC 1PPS edge that triggers reading the TCXO will randomly fall somewhere within a TCXO sample interval (with a duration T_s = 20 ns) and is representative of a quantization error. Other clock reading errors are present with the CSAC (for instance, the CSAC specification states that the 1PPS rising edge has a duration of <10 ns, representing another source of error); however, the quantization effect is the largest. Advanced models exist for characterizing quantization error, but a simple approach suffices in which this error is modeled as a zero-mean Gaussian white noise process with variance (Lipshitz et al., 1992). The following applies for the CSAC 1PPS query noise ε_C(t):

35

Using the CSAC 1PPS rising edges as a trigger, samples are taken of the CSAC time C_C(t), Iris time C_I(t), total count phase (for forming range rate measurements), and code phase Ψ(q) (to be defined shortly and used for forming range measurements). In its raw form, the CSAC time C_C(t) is measured as integer seconds since the time at which the radio was turned on (i.e., the 1PPS count field in Table 1)², and the Iris time C_I(t) has two parts: integer seconds and subseconds corresponding to cycle counts that can be converted into a fraction of a second by dividing by f₀ (i.e., the Iris time [seconds] and Iris time [subseconds] fields from Table 1, respectively). These data can be used to determine the Iris clock phase offset x_I(t_m), where m is now the count number associated with the consecutive 1PPS boundaries (that is, m – (m – 1) = 1, and is ~ 1 s in duration). Using Equation (34), we obtain the following result:

36

where the error ε_I(t) has been neglected because ε_C(t_m) ≫ ε_I(t).

3.1 Method 1: Determining the One-Way Range Rate Using Direct Calibration

Fundamentally, the range rate is obtained by differencing two total count phase values that are separated by a specified integration time; currently, these phase values are time tagged by the CSAC 1PPS and, as shown in Equation (32), has a direct sensitivity to the TCXO clock phase error. In this first approach, the impact of the TCXO is reduced by directly calibrating for the TCXO clock phase by substituting Equation (34) and Equation (36) into Equation (32), reducing, and rearranging, which leads to the following:

37

Recall that Δϕ(t_m) ≈ 0; thus, this term has been ignored. Moreover, in Equation (37), both C_C(t_m) and C_I(t_m) are recorded telemetry values. These terms can be brought to the left-hand side and used to define a calibrated total count phase estimate :

38

where the right-hand side now contains the light time delay τ(t_m) (i.e., the central quantity of interest), a clock phase that is dependent on CSAC stability (rather than the TCXO with worse stability), the CSAC 1PPS clock query noise ε_C(t_m) (dominated by quantization error), the phase ambiguity N_ϕ, and other phase error effects . It is worth noting that, while ε_C(t_m) is large in strength as compared with x_I(t) at 1 s, the effect of ε_C(t_m) diminishes over longer time scales relative to x_I(t) (which grows unbounded and is much larger than x_C(t)). This fact is the central motivation for calibrating the phase with the CSAC time comparison data.

We are now in a position to form an average one-way range rate measurement by differencing two calibrated phases that are separated in time by using the following definition:

39

where for any function f(t) and T = p seconds is the number of “seconds” between telemetry values recorded at time C_C(t_m) and the values recorded p samples earlier at C_C(t_m–p). We emphasize that T is simply a defined value of integral seconds and acts as a scale factor to ensure that Ṙ(t_m, t_m–p) has the correct units; it is not a real elapsed time. For instance, we could have just as easily defined a difference range measurement using that contains the same information as Ṙ(t_m, t_m–p) without dividing by a time duration. Conveniently, the values of p and, hence, T equal the recorded interval of the CSAC 1PPS even though true time does not equal T seconds (c.f., Equation (34)). By substituting Equation (38) into Equation (39), reducing, and rearranging, we obtain the following:

40

where the ambiguity and initial phase offsets have been eliminated and all terms inside the square brackets are non-dimensional. The first bracket represents the quantities that are amenable to further estimation in the orbit determination filter (i.e., spacecraft position and velocity and CSAC clock parameters), whereas the second bracket contain elements that typically comprise the filter’s measurement noise. We make the following observations about the terms in order of their appearance:

• 1) is the average range rate over time T (or average Doppler shift if multiplied by f_u instead of c) and contains the astrodynamics of interest plus other delay effects that modify the signal path length and introduce additional errors, such as atmospheric delays or solar plasma effects.

• 2) represents the average fractional frequency (in range rate) of the CSAC clock phase offset over time T. Note that the Iris TCXO clock phase offset has been eliminated from Equation (40) and has been replaced with the more stable CSAC clock offset. However, because the CSAC still drifts, orbit estimation should also include the components of the clock phase x_C(t_m) as part of the estimation algorithm design. Moreover, because only a time difference appears in Equation (40), the estimation process will be relatively insensitive to the initial clock phase offset. However, because the CSAC time C_C(t_m) is used to tag the measurement, some observability is present in the time tag. Although the filter will be estimating the CSAC components, it is instructive to assess the effect of the CSAC on the range rate for the short time scales typically used with range rate count times, T < 600 s. On these time scales, the CSAC error is dominated by white frequency noise, and its associated range rate uncertainty can be approximated via the following:

  41

  (A more precise expression for the Allan deviation is provided later in Equation (51) when Method 2 is discussed for computing the range rate.) From the CSAC specifications for the Microchip SA-45S, it can be determined that , which yields . Relative to the other errors, this error decreases more slowly because it is proportional to vs 1/T.

• 3) represents the range rate error in reading the CSAC 1PPS leading edge time and is dominated by the quantization error. The uncertainty associated with this error takes the following form:

  42

  which, for the Iris sample period T_s, yields (mm/s). For this test, and likely during operations, this is the largest error source dictating the precision of the one-way range rate derived using this method. In the results, this theoretical uncertainty will be compared with the measured results.

• 4) represents the range rate error from the phase noise remaining in the CTL, after it is locked and tracking the Doppler-shifted received phase. This term is the combination of the thermal phase noise related to the received signal strength, the ranging signal, and the oscillator that remains after the CTL is locked and tracking and has the phase variance given in Equation (33). For one-way links, the phase uncertainty is related to the range rate uncertainty as follows:

  43

  As an example, consider a link with a 40-dB margin over the minimum required P_c/N₀ = 20 dB-Hz, thus P_c/N₀ = 60 dB-Hz. For B_CTL = 200 Hz and β = 45°, the resultant range rate uncertainty is (mm/s).

  For P_c/N₀ = 40 dB-Hz (i.e., a 20-dB decrease in link strength), the range rate uncertainty increases to (mm/s). Relative to the other terms discussed above, is not significant; however, for traditional two-way data, this term can be a dominant error term. Moreover, if we examined the Iris one-way range rate without the CSAC 1PPS, this term would be significant.

There are potentially other errors, not delineated above, that should be included in Equation (40) if they are present; examples include multipath errors, phase wind-up errors, and other receiver errors.

3.2 Method 2: Determining the One-Way Range Rate Using Filtered Clock Phase Calibration

As noted in the prior section, quantization effects dominate the range rate errors when Equation (40) is used. It is possible to reduce the direct effects of quantization noise on the range rate calibration by first estimating the phase offset of the Iris TCXO relative to the CSAC by using the CSAC 1PPS/TCXO comparison data. This step requires introducing a clock model that can be used to form an estimation state vector. Let us consider the measurement equation for the clock difference data:

44

This expression can be cast into a state-space form as follows:

45

where the following definitions apply:

46

with a measurement noise strength of . A state-space model for the clock state X(t_m) can be formed by converting the continuous clock model defined in Equation (4) to a discrete model using an approach outlined by Zucca & Tavella (2005). In the present case, the clock model is now an aggregate of the CSAC clock phase x_C(t) and the TCXO clock phase x_I(t), where , , and . The resulting linear time-invariant, three-state discrete clock model takes the following form:

47

48

where and the initial condition X(t₀) is as follows:

49

The noise process G(t_m–p) is the discrete state-space form of ψ_IC(t)/2πf₀ and is distributed normally (i.e., Gaussian) with the following correlated covariance strength Q:

50

where is the noise strength of the oscillator’s white frequency noise component and is the noise strength of the random walk frequency component. These noise strengths can be related to an oscillator’s Allan deviation σ_y(T) at integration time T as follows:

51

Upper bounds for the aging and noise components of the Iris TCXO and CSAC can be obtained from their appropriate Allan deviation specifications (for the CSAC) or measured values (for the TCXO); using Equation (51) yields the values listed in Table 2. Applying these values from Table 2 to the aggregate noise process of the Iris TCXO/CSAC clock difference yields the following values:

52

We now have enough information to form a Kalman filter estimate of the clock state , where . The conventional Kalman filter time update equations take the following form:

53

and the Joseph forms of the measurement update equations are as follows:

54

Using a procedure outlined by Maybeck (1979), it is possible to find the steady uncertainty P^ss of Equation (54) by approximating the Kalman filter with its equivalent continuous form and solving for the steady solution. Doing so yields the following steady-state covariance (the subscript IC notation has been dropped):

55

where T is 1 s for the sampling interval of the 1PPS. Using the values defined above, we can predict the steady-state uncertainty for the TCXO/CSAC clock phase estimate to be , which, when compared with the quantization noise of 5.77 ns, represents a significant improvement in the uncertainty of the clock phase offset over the raw measurement.

We can use the real-time output of the Kalman filter to now form a calibrated range rate estimate. We begin by relating the Iris TCXO clock phase x_I(t_m) to as follows:

56

where is the error in the estimate for x_IC(t_m), which has uncertainty . By substituting Equation (56) and Equation (34) into Equation (32), reducing, and rearranging, we obtain the following:

57

Observe in Equation (57) that has just been computed and C_C(t_m) is a recorded telemetry value; thus, both terms can be brought to the left-hand side to define a calibrated total count phase estimate as follows:

58

As before, we determine the one-way range rate by substituting Equation (58) into Equation (39) to obtain the following:

59

where the terms in the first square bracket match the prior result in Equation (40) and the second square bracket has a slightly different set of errors. The errors with differences from Method 1 are now described.

• 1) represents the range rate error introduced by using the estimate and has a magnitude that can be estimated from the steady-state uncertainties as follows:

  60

  For , we obtain (mm/s). As will be seen with the experimental test results, this estimate is too large by a factor of 5 at T = 1 s but matches the test results on longer time scales.

• 2) represents the range rate error in reading the CSAC 1PPS leading edge time and is dominated by the quantization error; however, the magnitude of this term is greatly reduced relative to the term appearing in Equation (40) because the coefficient now contains only the fractional frequency bias, which is , in the present case. The uncertainty for this term is as follows:

  61

  and takes the value (mm/s). Relative to the prior term, this is not a significant effect and can be neglected.

4.1 Symbol Timing Recovery for Range

Iris utilizes its STL for both two-way regenerative ranging and one-way uplink ranging to recover an estimate of the symbol timing and to generate the soft symbols that are sent to the ranging correlator (the index k is the time index of the STL after the integrate-and-dump [I&D] step that operates at the chip rate f_R). A block diagram of the STL is shown in Figure 22, with details of its operation and estimates of the symbol timing variance provided in Appendix B: Iris symbol timing variance. With the STL locked, Iris records the PN chip accumulator (PCA) telemetry defined in Table 3 (i.e., this is the output of the phase accumulator in Figure 22), where the fractional chip phase can be estimated as follows:

65

The estimate has the associated variance given by Equation (129) in Appendix B. For convenience, we repeat the expression for the variance here:

66

where B_STL is the STL loop bandwidth, P_R is the ranging power as defined in Equation (83), and R_S is the ranging signal-to-noise ratio defined in Equation (108).

4.2 Forming the Range Measurement

The light time delay τ(t) of the received range code m(t – τ(t)) can be defined as follows:

67

where t_E is the epoch associated with the ETT and is uniquely associated with the current chip Ψ(q) that defines m(t – τ(t)) as defined in Equation (62). Because the PN sequence has a period of approximately 0.5 s, there will be an ambiguity associated with the measured delay; as a result, the light time can be decomposed further as follows:

68

where N_r is an integer ambiguity associated with the “wavelength” of the range code and Δτ(t) is a fractional portion of the range code. Because the range code is a repeating sequence, the following property holds:

69

where the range signal Φ_R(t – Δτ(t)) is the actual signal that Iris has locked onto and is tracking.

After obtaining the fractional phase of the received chips , the STL routes the soft chips to the component code correlators for acquisition; details of this process have been described by Angkasa et al. (2019) and Hamkins et al. (2016). Once Ψ(k) has been acquired and locked, the value of the index q in Equation (62) is determined (i.e., the location q of the chip Ψ(k) as an element of the sequence Ψ(k – q)Ψ(k + 1 – q)…Ψ(k)…Ψ(k + L – q)) by correlating on the component codes to determine the individual delays from local reference component codes. These individual delays are then combined via the Chinese remainder theorem (Hamkins et al., 2016) to determine a value for q, which is reported in telemetry as the PN chip index from Table 3. As the CSAC 1PPS samples the recorded value of q and the fractional phase of the chip , the following associations can be made from the telemetry:

70

The next important step in determining a value for the light time delay τ(t) is to relate the received code to the ETT as recorded in the telemetry (i.e., the ETT and ETT epoch fields in Table 3). In this first test of uplink one-way ranging with the DSN and CAPSTONE, the ETT will consist of two elements. The first element is a precursor command sent at the beginning of a ranging pass that identifies the UTC epoch on a second boundary. The train of PN sequences begins transmission at the UTC epoch on its second boundary to within ±1 μs. For our analysis, we will assume that the ground transmitter frequency reference has perfect stability; therefore, the transmitted period of each chip T_r can be treated as exact.

Now, each transmitted PN sequence has a reserved field of 32 bits that overwrite the first 32 chips of the sequence; however, bit 31 encodes the PN epoch bit that would have appeared in the zeroth position. These bits encode the integer counter position of the PN sequence that has been an element of the sequence train since transmission started. Hence, the first sequence would have 0, the second 1, and so on. With this, the ETT integer value can be combined with the epoch in the precursor command to determine the time that any selected PN sequence has been transmitted (note that this approach eliminates the range ambiguity N_r as an unknown). Because the start of transmission has an error of ±1 μs (or 300 m), only one PN epoch should be used to initialize the ETT; otherwise, the time of transmission of each sequence would be effectively white noise with an uncertainty of 1 μs, eliminating its utility for orbit determination. Using only one epoch to initialize the ETT, the following can be applied to determine a value for t_E using the recorded telemetry:

71

where is the computed t_E, is the epoch identified in the precursor command, b_E is an unknown (bias) error in the realized , ETT is the integer obtained from the telemetry and determined by associating the ETT epoch to the PN epoch that identifies the specific PN sequence associated with q, and is the fractional chip phase determined using the PCA (see Table 1 for descriptions of these fields). As noted in Appendix B, the STL is operating with the T_d = 2T_s sampling period, but it is possible to recover the full resolution of the T_s sampling by noting the following:

72

Finally, recall that we defined the time t as an elapsed time past some epoch t₀. We now re-introduce the initial epoch t₀ corresponding to the true epoch of when the radio clock began counting. Via the operational spacecraft clock comparison process, an estimate of can be found and is expressed as follows:

73

where x₀ is the initial clock phase offset identified in Equation (4) and represents a random bias error with initialization of the spacecraft clock for epoch . We are now able to form the uplink one-way range measurements. First, we derive a range measurement using the Iris TCXO as the reference:

74

Second, using the superior stability of CSAC as the reference, we find the following:

75

Note that x₀ has been absorbed into the respective clock expressions, C_I(t_m) or C_C(t_m). The expressions for R_I(t_m) and R_C(t_m) are grouped similarly to their range rate counterparts, where the terms involving the light time delay τ(t_m), x_C(t_m), and b_E will contribute to the estimation state defined for the orbit determination filter while the noise terms are uncompensated, additive errors. The associated range error variances for the noise terms are given as follows:

1. cε_I(t_m) is the error associated with sampling the TCXO. Using the associated jitter calculation, the error has an uncertainty of cσ_I, = c(1.6 ps) = 0.0048 m.

2. cε_C(t_m) is the error associated with using the CSAC 1PPS for the sample TCXO time; this term is dominated by the quantization error as identified in Equation (35) and has an uncertainty of cσ_C = 1.73 m.

3. cv_ε(t_m) is the STL symbol timing error contributing to the range error in Equation (66) and takes the following form:

   76

   To elucidate the performance of the STL at recovering range, it is useful to plot Equation (76) as a function of P_r/N₀ and for various loop bandwidths B_STL, as shown in Figure 3. The test results shown later use an operating range with B_STL = 10 Hz and P_r/N₀ = 60.8 dB-Hz, which yields a range uncertainty of 0.38 m (1σ). However, during flight, the expected B_STL = 200 Hz and P_r/N₀ ∈ (50.8,60.8)dB-Hz yield an expected range uncertainty between 11.4 m for the lower power limit and 1.7 m for the higher limit (1σ).

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FIGURE 3

1σ range error uncertainty from the Iris STL for various loop bandwidths as a function of range signal power to noise (Pr/No)

The ETT bias error cb_E will likely change from pass to pass, and the orbit filter will be required to estimate its value. Additionally, there are uncertain path delays between modulating the signal on the carrier and exiting the feed horn of the transmission antenna. This additional error can vary from a few meters to many tens of meters. A conservative a priori uncertainty for cb_E is cσ_E= 320 m (representing a root sum square [RSS] of the path length error and the modulation time error) and should be reset on each pass. Finally, recall that the initial clock phases x_I,0 and x_C,0 have been resolved via ground to spacecraft commanding to an accuracy of 1 ms (equivalently, 300 km). This, like b_E, is a bias error that will be estimated by the orbit filter but, unlike b_E, will be a constant for the duration that the radio remains on. Methods for improving the ETT bias calibration and initializing the onboard spacecraft clock are a topic of future investigation.

5.1 Method 1 – Direct Phase Calibration Using Equation (38)

We start by examining the first method, which uses direct calibration for TCXO effects; the Allan deviation (as determined using an overlapping Allan deviation [OAD] computation) of the raw total count phase (normalized to time deviations by dividing by f_u) and the calibrated phase computed from Equation (38) (normalized with ) are shown in Figure 6. Also shown is the OAD of the raw total count phase with its deterministic drift removed. With a constant carrier and no Doppler shift, this normalization will result in the OAD of the raw phase, representing the stability of the TCXO, and the calibrated phase, representing the stability of the TCXO/CSAC combination (which includes quantization effects). Note that the raw phase OAD at 1 s is 4 × 10⁻¹¹, whereas the specification for the TCXO at 1 s is < 1 × 10⁻⁹, indicating that the actual TCXO in the Iris radio has a much better stability than specified. After approximately 100 s, the raw phase has a slope that is proportional to τ (the Allan deviation integration time – not to be confused with the light time delay or the range rate integration time), which is representative of the oscillator’s frequency drift. This growth is the motivation for attempting to calibrate using the CSAC 1PPS, which exhibits better stability on longer time scales. However, on time scales under 1,000 s, the OAD shown in Figure 6 for the calibrated phase is dominated by quantization effects (as indicated by a slope proportional to 1/τ). Indeed, the OAD at 1 s is ~ 6 × 10⁻⁹ and, when multiplied by the speed of light, yields a rate of 1,800 mm/s, which is close to the range rate measured at 1 s via Method 1. After 200 s, the calibrated phase crosses over the raw phase OAD curve. This cross-over point represents the required integration time needed for the associated one-way range rate derived using the calibrated phase to begin having improved noise characteristics as compared with the range rate derived from the raw phase.

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FIGURE 6

OADs of the raw total count phase, the same phase with its frequency drift removed, and the calibrated phase for Method 1, all normalized by the carrier frequency

The range rate is computed via Equation (40) for the integration times T ∈ {1,10,30,60,90,120,150,180,300}s and, as expected, results in a precision that is inversely proportional to the integration time, as indicated by Equation (42) (and, to a lesser degree, by Equations (41) and (43)). Recall that the predicted dominant uncertainty arises from the quantization error effect when the CSAC is used in comparison with the TCXO, as given by Equation (42). The short-term noise from the TCXO and the thermal noise are much smaller than the quantization effect. These uncertainties and the associated standard deviation of the measured range rate are shown in Figure 7 on a log–log scale, where the range rate statistics have been computed after the removal of any deterministic bias, rate, and acceleration terms, as determined via a least-squares fit (i.e., “detrending”).

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FIGURE 7

One-way range rate uncertainty as determined using Method 1

The theoretical result is computed using the RSS of Equations (41)–(43), and the experimental result is the standard deviation of the range rate residuals after the removal of deterministic biases, rates, and accelerations.

It is clear from the apparent linear slope of the experimental results that the uncertainty is proportional to 1/T (after approximately 10 s) and is consistent with the theoretical results given by Equations (41)–(43). The most significant deviation between the experimental and theoretical results arises for the fundamental 1-s integration time, where the predicted theoretical range rate uncertainty is 2,451 mm/s and the experimental result is 1,797 mm/s, corresponding to a 27% difference. For all other measured count times, the percent difference is within ±6.5%, with a mean of only −0.005%. This agreement indicates that the theoretical one-way range rate uncertainties given in Equations (41)–(43) provide a reasonable prediction of the actual uncertainties.

Another measure of the range rate precision obtained using Method 1 is the OAD computed using the range rates (now normalized by dividing by the speed of light) for different integration times, as shown in Figure 8. By comparing the curves in Figure 8 with the OAD of the normalized calibrated total count phase in Figure 6, we observe that the OADs for the normalized range rates are close to those of the normalized phase. This result is expected (because the normalized range rate is equivalent to the fractional frequency derived from the normalized phase) and shows that the range rate calculations are correct.

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FIGURE 8

OAD of the range rate computed using Method 1 for selected integration times

It is useful to examine the results for integration times of T = 1 s, which is the fundamental data collection interval, and T = 60 s, which is the typical integration time used in deep space navigation. Results for the 1-s integration time are shown in Figure 9, with the top plot exhibiting the range rate computed from the raw total count phase, the middle plot showing the range rate computed from Equation (40), and, as performed for the results in Figure 7, the bottom plot showing the calibrated range rate with the bias, rate, and acceleration effects removed or “detrended” as determined by a least-squares fit. The range rate in the top plot illustrates the effect of a drifting TCXO with nearly a 50-Hz reduction in the frequency bias over 8 h. Additionally, the combined effect of and the TCXO frequency deviation can be seen, with a value near 57 kHz. The range rate in the middle plot represents the data that would be used for orbit and clock determination, where the clock is now effectively the CSAC rather than the TCXO. The detrended curve removes the deterministic effects of the CSAC and can be used to derive the range rate noise characteristics and the associated standard deviation shown in Figure 7. Notably, we observe the trimodal distribution of the range rate and its similarity to the second difference in Figure 4.

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FIGURE 9

Range rate results for Method 1 with T =1 s; range rate from the uncalibrated phase (top), calibrated range rate (middle), and calibrated range with bias, rate, and acceleration effects removed (i.e., “detrended”) (bottom)

The range rate values from the bottom, detrended plot are shown as a histogram in Figure 10. The bulk of the data lies at the center, and the estimated kernel density (dashed black line) suggests conformity to a uniform-density distribution at the center (a consequence of being dominated by quantization error). As the integration time increases, the sampling begins to approach a normal distribution, as will be seen for the next integration time of T = 60 s.

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FIGURE 10

Histogram of the calibrated, detrended range rate for Method 1 with T =1 s

Turning to T = 60 s, the computed values for the three cases (raw, calibrated, and calibrated/detrended) of the range rate are shown in Figure 11, and an associated histogram of the calibrated/detrended data is shown in Figure 12. In the middle and bottom plots in Figure 11, the range rate data appear unimodal with a more normal distribution. This distribution is confirmed in Figure 12, where the estimated kernel density and fitted normal distribution are well matched, indicating that the distribution is indeed normal. Using this approach yields a one-way range rate uncertainty of 40.5 mm/s (1σ), which is 400 times larger than the typical two-way range rate of 0.1 mm/s provided by standard DSN services to a typical deep space transponder. This result motivates examination of the second method.

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FIGURE 11

Range rate results for Method 1 with T =60 s; range rate from the uncalibrated phase (top), calibrated range rate (middle), and calibrated range with bias, rate, and acceleration effects removed (i.e., “detrended”) (bottom)

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FIGURE 12

Histogram of the calibrated, detrended range rate for Method 1 with T =60 s

5.2 Method 2 – Filtered Phase Calibration Using Equation (58)

In this method, a clock phase estimate is obtained via a Kalman filter and used to calibrate the total count phase as in Equation (58). We begin by examining the properties of ; in this case, the prefit residuals are the clock difference data ΔC(t_m), and the postfit residuals are computed from . These residuals are shown in Figure 13, with the prefit residuals at the top and the postfit residuals at the bottom.

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FIGURE 13

Prefit clock residuals (top) and postfit residuals (bottom)

Recall from the differenced clock data in Figure 4 that, at roughly 81,000 s, there is an upward shift of the clock differences, which results from the relative clock drift between the TCXO and CSAC combined with the quantization effect of the clock measurements. This shift is the source of the transition seen in the postfit residuals during the same period (with the x-axis now shifted by 60,675 s and now occurring near 21,000 s) and reflects the Kalman filter reacting to the increasing magnitude of the clock differences. Later, we will see that this transition impacts the noise levels on the calibrated range rate.

The mean of the postfit residuals is 0.42 ns, and the associated standard deviation is 5.61 ns, resulting in a root mean square of 5.62 ns. We can compare this result to the theoretical postfit residual uncertainty derived from the Kalman filter, which is . The actual steady-state solution uncertainty from the filter is 1.34 ns (compare to the prediction in Equation (55) of 1.38 ns) and ; together, these values yield a theoretical steady-state postfit residual uncertainty of 5.92 ns, which compares favorably with the measured value of 5.62 ns. The OADs of the prefit residuals, postfit residuals, actual filter clock solution, and normalized raw total count phase are shown in Figure 14.

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FIGURE 14

OAD of the clock filter prefit residuals (blue), postfit residuals (orange), clock phase solution (green), and normalized raw total count phase (red)

The clock phase estimate (green curve) is the Kalman filter estimate of the TCXO/CSAC difference. The OAD curve of the raw total count phase normalized by (red curve) is a representation of the actual TCXO phase x_I(t_m) AD in this test configuration; therefore, when the green curve is closer to the red curve, this indicates a better filter estimate. Clearly, accurately captures the drift characteristics of x_I(t_m) after approximately 100 s (that is, the CSAC drift contribution is not significant for x_IC(t_m) or ). Prior to that time, the solution approaches the actual TCXO phase x_I(t_m), but the effects of quantization errors (i.e., the measurement noise in the filter) and CSAC short-term noise prevent full recovery. The largest difference occurs at 1 s, with an OAD for of 2.5 × 10⁻¹⁰ while the OAD for x_I(t_m) is 4 × 10⁻¹¹; however, at 100 s, the difference becomes insignificant. Even with these differences, can be used as a reasonable proxy for x_I(t_m) to reduce the range rate errors introduced by the TCXO drift and the effects of quantization noise, as compared with Method 1. As a final comment, the OAD of the postfit residuals is approximately white phase noise because its slope is close to 1/T, indicating that the Kalman filter is producing an optimal estimate for x_IC(t_m).

Using to calibrate the total count phase according to Equation (58), we compute the associated OAD of the raw total count phase and the calibrated phase , with results shown in Figure 15. The calibrated phase results obtained using Method 2 are shown in green, and for comparison, the prior Method 1 results are shown in red. Also shown is the raw total count phase with its deterministic drift removed. Both raw results – regular and drift-removed – are the same as in Figure 6 because they represent the same data.

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FIGURE 15

OADs of the raw total count phase (blue), the same phase with its frequency drift removed (orange), the calibrated phase based on Method 2 (green), and, for comparison, the calibrated phase based on Method 1 (red), all normalized by the identified carrier frequency

A careful examination of the OAD results for in Figure 15 reveals much better performance (i.e., lower Allan deviation values) for Method 2 versus Method 1 on all integration times below 5,000 s, and after 5,000 s, the results are commensurate. Hence, Method 2 produces better phase stability than Method 1. The OAD results from Method 2 for integration times of 1–55 s lie above those generated by the TCXO; for longer integration times, the calibrated phase has lower OAD values than the TCXO and exhibits characteristics representative of the CSAC.

Turning to the range rate calculation of Equation (59) for Method 2 and for integration times of T ∈ {1,10,30,60,90,120,150,180,300} s, we obtain the measured range rate uncertainties shown in Figure 16 as the blue curve, labeled “Method 2 exp.” Similar to our analysis of Method 1, we compare these data with the theoretical result; for Method 2, the theoretical result is Equation (60), shown as the orange dash–dot curve. We also compare these results with the experimental results from Method 1, shown as the green curve.

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FIGURE 16

One-way range rate uncertainty as determined via Method 2

The theoretical result (orange dash–dot curve) is computed from Equation (60), and the experimental result (blue curve) is the standard deviation of the range rate residuals after the removal of deterministic biases, rates, and accelerations. Experimental results for Method 1 (green curve) are shown for comparison.

The theoretical results for Method 2 with a 1-s integration are larger than the experimental results by a factor of 5, but on longer time scales, the theoretical results match the experimental results more closely. After 40 s, the experimental results for Method 2 begin to slightly exceed the theoretical results; it is suspected that this is the effect of the CSAC instability (i.e., the second term in Equation (59)) beginning to affect the range rate uncertainty. In fact, this instability may be dominant relative to the term. For instance, at 100 s, the specified CSAC stability is 3 × 10^–11. This stability can be converted to a range rate by multiplying by the speed of light, which results in an uncertainty of 9 mm/s. The interpolated 100-s experimental range rate uncertainty for Method 2 in Figure 16 is 8–9 mm/s, which presents a very good comparison and suggests that CSAC errors are beginning to dominate. In this figure, the experimental range rate crosses over the theoretical result at T = 30 s, where the uncertainty is ~18 mm/s. Converting to an Allan deviation yields 6 × 10^–11, which is commensurate with the specified CSAC Allan deviation levels shown in Table 2. Therefore, after 30 s, the range rate errors begin to be dominated by CSAC effects. Compared with Method 1 (experimental results shown in green), Method 2 produces smaller range rate uncertainties for all time scales investigated. From this, we can conclude that Method 2 is preferable for deriving more accurate one-way range rate measurements.

In Figure 17, we plot the OADs computed from the range rates based on Method 2 (normalized by the speed of light) for selected integration times. These OAD results can be compared with the calibrated phase OAD results based on Method 2, as shown in Figure 15; as expected, the results match closely, indicating that the one-way range rate calculations are correct.

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FIGURE 17

OAD of the range rate computed from Method 2 for selected integration times

We now show the specific range rate results obtained via Method 2 for integration times of T = 1 and 60 s. To facilitate this comparison, the raw, calibrated, and calibrated plus detrended range rates are shown for both integration times in Figure 18, as well as associated histograms. We start by examining the results for T = 1 s; relative to Method 1, which showed three discrete distributions, Method 2 produces a unimodal distribution with a standard deviation of 101 mm/s, 18 times smaller than the equivalent standard deviation from Method 1. Superimposed on the distribution is the associated normal distribution, and a deviation is observed relative to the estimated kernel density. Indeed, the kernel density has a shape that more closely matches a uniform distribution, suggesting that the quantization effects are still dominant for this integration time. In the results for T = 60 s, the estimated kernel density more closely matches the associated normal distribution, indicating that the statistics of the range rate errors are becoming more characteristic of Gaussian white noise. This is an important result in that the associated orbit determination filters typically treat the measurement noise as Gaussian white noise; therefore, the range rate results for T = 60 s match the orbit determination assumptions better than those for the shorter integration time. This was also true for the range rates obtained via Method 1 at T = 60 s, but the standard deviation for Method 2 is only 11 mm/s (compared with a standard deviation of 41 mm/s for Method 1). As noted previously, the effect of the CSAC (at least as specified³) begins to dominate at T = 30 s. This trend implies that one-way range rates computed using Method 2 begin to have accuracies that are similar to those that would be derived if the Iris radio used the CSAC as its local oscillator rather than the TCXO.

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FIGURE 18

Range rate results for Method 2 with T =1 and 60 s

The top and bottom left images are for T =1 s, and the top and bottom right images are for T =60 s. Top plots show the range rate from the uncalibrated phase (top), calibrated range rate (middle), and calibrated range with bias, rate, and acceleration effects removed (i.e., “detrended”) (bottom) for their respective integration times. Bottom plots show an associated histogram of the calibrated and detrended range rates.

Finally, in the calibrated range rate results for T = 60 s shown in Figure 18, the residuals become noisier at a time point near 22,000 s. This noise is an artifact of the quantization effects seen in the delta time residuals of Figure 13, where the differences switch from the bottom to the top.