Multidimensional GNSS Meta-Signals

2.1 Conjugation and Hyperbolic Modulus

Given the presence of several imaginary and hyperbolic units, it is possible to define several multicomplex conjugation operations (Alpay et al., 2014). In the following, only the *-conjugation (Alpay et al., 2014) is considered. More specifically, the *-conjugate of z_M is given by the following:

zM∗=kMTa2M−1−jMTb2M−15

This conjugate is obtained by negating the sign of all of its imaginary units. The hyperbolic real and imaginary parts of a multicomplex number are obtained using the *-conjugation (Borio, 2024a):

ℜk{zM}=12[zM+zM∗]=kMTa2M−16

and:

Ik{zM}=12j0[zM−zM∗]=1j0jMTb2M−1=kMTDMb2M−17

Here, j₀ is the basic imaginary unit available in all multicomplex spaces including the complex numbers, and D_M is a diagonal matrix whose diagonal elements all equal one or minus one. D_M can be built recursively as follows (Borio, 2024a):

DM=[DM−102M−2×2M−202M−2×2M−2−DM−1]8

with D1=1.02M−2×2M−2 is a square matrix of size 2M−2×2M−2 with all zeros. It is important to note the presence of the index "k" in Equations (6) and (7). This index has been added to distinguish ℜk{⋅} from the real-part operator, ℜ{⋅}, which isolates the real coefficient multiplying the real unit, 1 in a multicomplex number. To avoid confusion, the operators in Equations (6) and (7) are denoted as hyperbolic real and hyperbolic imaginary parts of a multicomplex number. Using Equations (6) and (7), it is possible to write z_M as follows:

zM=kMTa2M−1+j0kMTDMb2M−1=kMT(a2M−1+j0DMb2M−1)=kMTz2M−19

where z2M−1 is a vector of 2M−1 complex coefficients.

The *-conjugation can also be applied to compute the hyperbolic modulus of a multicomplex number:

|zM|k=zM⋅zM∗=kMTr2M−110

where r2M−1 is a vector of real coefficients. Details on the computation of r2M−1 have been reported in prior work (Borio, 2024a). It is important to note that the hyperbolic modulus and the hyperbolic real and imaginary parts belong to DM, the 2M−1-dimensional hyperbolic space defined by kM (Fjelstad & Gal, 1998). In multicomplex numbers, DM plays a role similar to that of real numbers for complex numbers.

2.2 Orthogonal Decomposition

A multicomplex number can be expressed in terms of orthogonal components (Price, 1991):

zM=∑i=02M−1−1zO,ieM,i11

where zO,i are complex coefficients and eM,i are a set of idempotent orthogonal units. The eM,i units are characterized by the following properties:

• orthogonality: eM,ieM,j=0 for i≠j

• idempotency: eM,ieM,i=eM,i

For a multicomplex space of order M, there are 2M−1 idempotent units that can be arranged in the following vector:

eM=[eM,0,eM,1,…,eM,2M−1−1]T12

Using Equation (12), it is possible to write Equation (11) as follows:

zM=eMTz2M−1O13

where:

z2M−1O=[zO,0,zO,1,…,zO,2M−1−1]T14

is a vector with the orthogonal components of zM. The vectors in Equations (12) and (2) are related through the following transformation (Borio, 2024a):

eM=12M−1(DMH2M−1)TkM15

where H2M−1 is a square Hadamard matrix of size 2M−1 (Yarlagadda & Hershey, 1997). Details on the recursive construction adopted in this context for building H2M−1 have been provided in previous work (Borio, 2024a). Equation (15) provides a simple way to compute the orthogonal components of zM from z2M−1:

z2M−1O=(DMH2M−1)Tz2M−116

Orthogonal decomposition plays a fundamental role in the implementation of the different operations on multicomplex numbers because these numbers can be expressed in terms of orthogonal components. The orthogonality and idempotency properties of the eM,i units allow one to operate component-wise, thus reducing the computational and implementation complexity of different operations. For instance, the product of two multicomplex numbers, zM and yM, can be expressed as follows:

wM=zM⋅yM=eMTz2M−1OeMTy2M−1O=enTw2M−1O17

where the components of the vector wMO are obtained as the product of the corresponding elements in z2M−1O and y2M−1O. Similarly, the hyperbolic modulus of a multicomplex number can be expressed as follows:

|zM|k=eMTz―2M−1O=∑i=02M−1−1|zO,i|eM,i18

where z―2M−1O is a vector with the moduli of the orthogonal components of zM. In this case, operations are also performed component-wise.

2.3 Polar Representation

Multicomplex numbers can also be expressed in polar form as the product of the hyperbolic modulus and a multicomplex exponential (Borio, 2024a):

zM=knTr2M−1exp⁡{jnTΦ2M−1}19

where r2M−1 and Φ2M−1 are real vectors of size 2M−1 containing modulus and phase information, respectively. r2M−1 is the hyperbolic modulus defined in Section 2.1, and Φ2M−1 can be computed as follows:

Φ2M−1=12M−1H2M−1Φ2M−1O20

Φ2M−1O is a vector with the phase of the orthogonal components of zM:

Φ2M−1O=[∠zO,0∠zO,1⋮∠zO,2M−1−1]21

These properties are used in the following to derive multicomplex baseband GNSS acquisition and tracking algorithms.

3 MULTICOMPLEX SIGNAL REPRESENTATION

A set of RF signals can be effectively represented using multicomplex numbers (Borio, 2024a). Consider 2M−1 signals from different RFs:

zM,RF(t)=∑i=02M−1−1xi,RF(t)=∑i=02M−1−1ℜ{xi(t)exp⁡{j02πfX,it}}22

where xi,RF(t) denotes the i-th RF signal, obtained by modulating a baseband signal, xi(t), around fX,i. Using multicomplex numbers, it is possible to express zM,RF(t) as follows (Borio, 2024a):

zM,RF(t)=ℜ{zM(t)exp⁡{2πjMTfMt}}23

where zM(t) is a multicomplex signal obtained by combining the different baseband signals, xi(t), and fM is a vector of frequencies obtained as the Hadamard transform of the original RFs:

fM=12M−1H2M−1fX=12M−1H2M−1[fX,0fX,1⋮fX,2M−1−1]24

Equation (24) is analogous to Equation (20), where frequency is considered instead of phase. Note that for M=1, i.e., the case of a single signal, the classical single-frequency modulation formula is obtained (Proakis & Salehi, 2001). For M=2, the dual-frequency modulation formula involving bicomplex numbers is found (Borio, 2023a). The multicomplex baseband signal zM(t) is obtained as follows:

zM(t)=2M−1∑i=02M−1−1xi(t)eM,i25

where eM,i are the idempotent orthogonal units introduced in Section 2. Thus, the complex baseband signals, xi(t), are the orthogonal components of the multicomplex signal, zM(t).

Equation (23) provides a different way of representing the modulation process of multiple RF signals. This process was illustrated by Borio (2024b) for the Galileo quad-frequency case. A similar interpretation can be given when considering four of the five OS signals offered by BDS. Because four components are considered, four different meta-signals can be constructed by excluding one component from the BDS OS landscape in each case. In the following, the B1C, B1I, B2a, and B2b components are considered. Although the B3I signal could have been considered, it would have led to a meta-signal with characteristics similar to that of the Galileo case detailed by Borio (2024b). A case with different characteristics is preferred to show the generality of the derived approach. The B1C, B1I, B2a, and B2b components can be represented as a single baseband meta-signal with tricomplex numbers, C3. In a traditional setting, one can imagine having four baseband signals independently up-converted to four different RFs; however, the modulation formula in Equation (23) implies a different process. Through the Hadamard transform in Equation (24), four new frequencies are obtained:

• a common carrier frequency, computed as the average of the different RFs

• 2M−1−1 subcarrier frequencies

For M=2, the bicomplex case analyzed by Borio (2023a) is obtained with a single subcarrier frequency. For the case with the four BDS components mentioned above, three subcarriers are found. Thus, the traditional independent modulation process is adjusted as shown in Figure 1. The four BDS baseband signals are represented as a single tricomplex signal, which is initially modulated around a common RF equal to 1380.027 MHz. The first subcarrier is used to split the signals around two intermediate RFs. The second subcarrier brings the different signals around four RFs. Finally, the last subcarrier is used to bring the signals to the final

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FIGURE 1

Schematic representation of the multicomplex modulation process obtained by considering four BDS signal components broadcast on different RFs, where a single meta-signal is up-converted to four RFs using a common carrier and three subcarriers; BOC: binary offset carrier; BPSK: binary phase shift keying

RFs. It is important to note that the frequencies in fM are common to all four BDS signals.

A second interpretation of Equation (23) can be obtained by explicitly writing the scalar product jMTfM and separating the carrier and subcarrier frequencies:

zM,RF(t)=ℜ{[zM(t)exp⁡{2π∑i=12M−1−1jifit}]exp⁡{j02πf0t}}26

where f0=12M−1∑i=02M−1−1fX,i is the carrier frequency and fi with i>0 are the subcarrier frequencies. The term in square brackets in Equation (26) can be interpreted as a single wideband meta-signal with an overall bandwidth spanning the full frequency range occupied by the original RF signals. Thus, Equation (26) provides a mechanism for constructing GNSS meta-signals with more than two sideband components, which is achieved by introducing 2M−1−1 subcarriers.

Equation (23) also provides an alternative paradigm for the down-conversion and processing of the different GNSS signals shown in Figure 1: instead of using independents tracking loops operating on the different RFs, a single PLL can be used to track a common center frequency together with 2M−1−1 SPLLs tracking the subcarrier components arising from Equation (26).

The RF signals in Equation (22) pass through the communication channel and are recovered by a multifrequency GNSS receiver. This receiver can adopt several synchronous front-ends, one for each RF band. Each front-end down-converts a different xi,RF(t) and recovers its baseband version. The communication channel attenuates the different signal components, delays them, and introduces Doppler shifts and carrier phases. Noise terms are also introduced. In this way, each received baseband component can be modeled as follows (Kaplan & Hegarty, 2017):

xrx,i(t)=αixi(t−τ0)exp⁡{j02πf0,it+j0φ0,i}+ηi(t)27

where αi is the attenuation introduced by the communication channel and τ0 is the delay assumed to be common to all signals. Note that different delays can be present for the different signal components. These different delays can arise from, for instance, different hardware paths on both the satellite and receiver sides. Equation (27) assumes that all of the components involved in the processing are aligned with respect to the same delay, τ0. This condition can be achieved by using a calibration stage before the full meta-signal approach is applied. Inter-frequency biases are usually stable over time and can be estimated during a calibration phase. A possible receiver implementation of such a phase is briefly introduced in Section 5, where a calibration stage is adopted to realign in time the different received signals. f0,i and φ0,i are the Doppler frequency and carrier phase affecting the i-th received signal. ηi(t) is additive white Gaussian noise with independent real and imaginary parts.

The different xrx,i(t) components are sampled and digitized. A common sampling interval, Ts, is assumed. After analog-to-digital conversion, Equation (27) is as follows:

xrx,i[n]=xrx,i(nTs)=αixi(nTs−τ0)exp⁡{j02πf0,inTs+j0φ0,i}+ηi[n]28

where n is the time index. ηi[n] are independent and identically distributed (i.i.d.) zero-mean complex random processes with real and imaginary parts, each with the same variance, ση2. This variance depends on several factors related to the antenna and front-end implementation. A commonly adopted model for ση2 is as follows:

ση2=N0BRx29

where BRx is the front-end one-sided bandwidth and N0 is the power spectral density (PSD) of the input noises. The same noise PSD, N0, and front-end bandwidth, BRx, are assumed for the different RF channels. A good approximation for BRx is half the sampling frequency, fs=1Ts:

BRx≈fs230

Finally, the received baseband digital multicomplex signal is reconstructed using Equation (25):

zrx,M[n]=2M−1∑i=02M−1−1xrx,i[n]eM,i=AzM(nTs−τ0)exp⁡{2πjMTfM,0nTs+jMTPM,0}+η[n]31

where:

A=∑i=02M−1−1αieM,i32

is a multicomplex constant modeling the attenuation introduced by the communication channel. fM,0 and PM,0 are two vectors obtained from the Doppler frequencies and carrier phases affecting the different received signal components:

fM,0=12M−1H2M−1[f0,0f0,1⋮f0,2M−1−1]PM,0=12M−1H2M−1[φ0,0φ0,1⋮φ0,2M−1−1]33

η[n] is a multicomplex noise process whose orthogonal components are the noise terms ηi[n].

Equation (31) is a multicomplex generalization of the standard model used to describe received single-frequency GNSS signals. The model in Equation (31) is used in the following to design multicomplex acquisition and tracking algorithms. This model suggests that instead of independently estimating the different Doppler frequencies and carrier phases, it is also possible to work in a transformed domain, estimating fM,0 and PM,0.

4 MULTICOMPLEX CAF AND BASEBAND PROCESSING

Standard single-frequency GNSS acquisition and tracking algorithms have the primary goal of maximizing the CAF between the received samples and locally generated code and carrier replicas. The CAF, in turn, is obtained by maximizing the likelihood associated with the received samples. A bicomplex CAF was obtained by Borio (2023a) by considering a dual-frequency GNSS signal model. A multicomplex CAF is obtained here by starting from the signal model in Equation (31).

4.1 Multicomplex CAF

Assuming a Gaussian model for η[n], the likelihood associated with Equation (31) can be expressed as follows:

p(z∣Ω)=1(2πσ2)2M−1exp⁡{−12σ2|zrx,M[n]−c∼(nTs−τ)exp⁡{2πjMTfdnTs+jMTP}|2}34

where Ω={τ,fd,P} is the set of unknown parameters to be estimated and c∼(⋅) is a multicomplex local replica of the signal codes, mimicking the role of AzM(⋅) in Equation (31). Symbol |⋅| denotes the Euclidean norm of a multicomplex number and should not be confused with the hyperbolic modulus. The Euclidean norm is computed as the square root of the sum of the squares of the components of a multicomplex number. The generation of c∼(⋅) is discussed in Section 4.2 and Section 4.5. For the moment, c∼(⋅) is assumed to be known at the receiver side. When N i.i.d. samples are considered, i.e., when the signal is integrated over NTs samples, the following joint likelihood is found:

p(z∣Ω)=1(2πσ2)N2M−1exp⁡{−12σ2∑n=0N−1|zrx,M[n]−c∼(nTs−τ)exp⁡{2πjMTfdnTs+jMTP}|2}35

where z is a vector containing the N multicomplex samples, zrx,M[n], and Ω={τ,fd,P} is the set of unknown parameters to be estimated. Maximum likelihood estimates, ΩML, are found through the maximization of Equation (35) or, equivalently, through the following minimization process:

ΩML=arg⁡minΩ∑n=0N−1|zrx,M[n]−c∼(nTs−τ)exp⁡{2πjMTfdnTs+jMTP}|236

This is a minimum mean square error problem that can be solved by expanding the squares in Equation (36):

∑n=0N−1|zrx,M[n]−c∼(nTs−τ)exp⁡{2πjMTfdnTs+jMTP}|2=∑n=0N−1|zrx,M[n]|2+∑n=0N−1|c∼(nTs−τ)exp⁡{2πjMTfdnTs+jMTP}|2−2∑n=0N−1ℜ{zrx,M[n]c∼∗(nTs−τ)exp⁡{−2πjMTfdnTs−jMTP}}37

In Equation (37), an expansion of the squared norm of the sum of two multicomplex numbers has been used. A proof of this property is provided in Appendix A. The fact that a phase rotation, i.e., a multiplication by a multicomplex exponential, does not change the Euclidean norm of a multicomplex number is also used to restate the minimization problem in Equation (36) as follows:

ΩML=arg⁡maxΩℜ{∑n=0N−1zrx,M[n]c∼∗(nTs−τ)exp⁡{−2πjMTfdnTs−jMTP}}38

In analogy to standard single-frequency GNSS signal processing, it is possible to define a multicomplex CAF:

Cz(τ,fd)=∑n=0N−1zrx,M[n]c∼∗(nTs−τ)exp⁡{−2πjMTfdnTs}39

We can then restate Equation (38) as follows:

ΩML=arg⁡maxΩℜ{Cz(τ,fd)exp⁡{−jMTP}}40

As briefly discussed in Section 2, multicomplex numbers can be written in polar form; thus, we have the following:

Cz(τ,fd)=|Cz(τ,fd)|kexp⁡{jMTPCAF}41

where |⋅|k is the hyperbolic modulus and PCAF is a vector with the CAF phases. PCAF can be determined through the Hadamard transform of the phases of the orthogonal components of Cz(τ,fd) (Borio, 2024a). Using Equation (41), the optimization problem in Equation (40) can be further simplified as follows:

(τ,fd)ML=arg⁡maxτ,fdℜ{|Cz(τ,fd)|k}PML=PCAF42

Equation (42) is a general multidimensional extension of the single-frequency maximum likelihood estimation process for determining the parameters of the received GNSS signals and can be solved, in analogy to the single-frequency case, with a two-step process. Acquisition is initially used to determine rough estimates of the signal delay and Doppler frequencies, whereas tracking is adopted to evaluate carrier phases, refine code and frequency estimates, and track signal variations. Multicomplex acquisition and tracking algorithms are discussed in the next sections.

4.2 Multicomplex Local Codes

On the transmission side, a single-frequency GNSS signal is usually expressed as follows (Kaplan & Hegarty, 2017):

xi(t)=di(t)ci(t)43

where ci(t) is a pseudorandom code used to spread the signal on a large frequency band. di(t) is the navigation message, which can be absent if a pure pilot channel is considered. Different ci(t) codes are selected for different satellites from a family of almost orthogonal sequences. In this way, it is possible to broadcast several signals on the same frequency with limited interference. On the receiver side, a digital replica of ci(t) is synthesized and used to estimate the signal parameters with a process similar to that of Equation (42). ci(t) may be obtained as the product of primary and secondary codes (Kaplan & Hegarty, 2017); moreover, several components, for instance, the data and pilot signals, may be present. The latter case is not addressed here; rather, a single component per frequency is considered. Moreover, it is assumed that all components are referenced to the same reference phase.

Similar to the single- and dual-frequency cases (Borio, 2023a; Kaplan & Hegarty, 2017), local codes should mimic the behavior of the incoming signal, zrx,M[m], defined in Equation (31). In this respect, the local code c∼[n]=c∼(nTs) can be built by using the orthogonal decomposition formula in Equation (25) and taking into account the relationship between the different received components:

c∼[n]=A∼∑i=02M−1−1ci[n]eM,i=∑i=02M−1−1α∼ici[n]eM,i44

where ci[n]=ci(nTs) is the local code of the i-th signal component and A∼ is an estimate of the multicomplex amplitude factor, A, defined in Equation (32). The real number, α∼i, corresponds to the coefficients of A. In the following, it is assumed that A∼ is known or fixed by the receiver. For instance, A∼ can be obtained by considering known power relationships between received components (Borio, 2023a; Wang et al., 2017). A∼ can also be determined through a calibration process by accounting for the full reception chain and including the antenna gain at different frequencies. As previously mentioned, a single component per frequency is considered in this paper. When a data signal is considered, the presence of a navigation message, di(t), must be accounted for. This consideration can be addressed in a manner similar to that of the mixed data/pilot case discussed by Borio (2023a) for the bicomplex case by adopting bit-insensitive carrier/subcarrier discriminators. This case is briefly discussed in Section 4.5. Finally, note that in Equation (44), the constant 2M−1 has been omitted. The presence of constants in the locally generated codes does not influence the optimization process discussed in the previous section.

4.3 Multicomplex Acquisition

The first step in GNSS signal processing is the so-called acquisition process (Kaplan & Hegarty, 2017), which aims to provide initial estimates of the signal code delay and Doppler frequencies. In the single-frequency case, the acquisition process implements a search on a finite grid with the aim of maximizing the absolute value of the CAF. In the bicomplex case, acquisition is performed in a similar manner, where the absolute value operator is replaced by the real part of the hyperbolic modulus (Borio, 2023a). This structure is maintained when moving to higher dimensions with multicomplex numbers. The general multicomplex acquisition process is illustrated in Figure 2. The baseband multicomplex signal, zrx,M[n], is multiplied by a delayed replica of the signal codes, c∼∗(nTs−τ), and by the local carrier and subcarriers. In Figure 2, this last multiplication is represented as a single block with a multicomplex exponential. Moreover, a vector of Doppler frequencies is considered. The resulting multicomplex signal is integrated over NTs seconds, and the multicomplex CAF in Equation (39) is found. Finally, the real part of the hyperbolic modulus of the CAF is retained. Different values of τ and fd are tested, and the real part of the hyperbolic modulus of the CAF is maximized. The delay and Doppler frequencies corresponding to the CAF maximum are the estimates obtained by the acquisition process. Note that the dimensionality of the search space considered in acquisition can be reduced by considering the relationships between the different Doppler frequencies: each Doppler component is proportional to the corresponding signal center frequency. For instance, a meta-signal carrier frequency equal to 1380.027 MHz is found for the BDS quad-frequency case depicted in Figure 1 along with three subcarrier components with different frequencies. These frequencies can be used to constrain the Doppler vector as follows:

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FIGURE 2

Schematic representation of the general multicomplex acquisition process, where multiplication by the carrier and subcarrier components is represented as a single multicomplex block

fd=[fd,0fd,1⋮fd,2M−1−1]=fd,0[1f1f0⋮f2M−1−1f0]45

where fi are the components of the fM vector introduced in Equation (24). Using this constraint, the acquisition search space is reduced to two dimensions, as in the single-frequency case.

Similar to the bicomplex case, the multicomplex cost function maximized in acquisition, ℜ{|Cz(τ,fd)|k}, can be expressed in terms of the CAFs of the sideband components, xrx,i[n]. Using the orthogonal decompositions in Equations (31) and (44), the multicomplex CAF can be written as follows:

Cz(τ,fd)=2M−1∑n=0N−1[∑i=02M−1−1xrx,i[n]eM,i∑j=02M−1−1α∼jcj∗(nTs−τ)eM,j]exp⁡{−2πjMTfdnTs}46

Moreover, it is possible to show the following:

exp⁡{−2πjMTfdnTs}=∑i=02M−1−1exp⁡{−j02πfh,inTs}eM,i47

where fh,i are the elements of the frequency vector obtained as the inverse Hadamard transform of fd:

fh=[fh,0fh,1⋮fh,2M−1−1]=H2M−1Tfd48

In this way, Equation (46) takes the following form:

Cz(τ,fd)=2M−1∑n=0N−1∑i=02M−1−1α∼ixrx,i[n]ci∗(nTs−τ)exp⁡{−j02πfh,inTs}eM,i=2M−1∑i=02M−1−1α∼i∑n=0N−1xrx,i[n]ci∗(nTs−τ)exp⁡{−j02πfh,inTs}eM,i=2M−1∑i=02M−1−1α∼iCx,i(τ,fh,i)eM,i=2M−1A∼∑i=02M−1−1Cx,i(τ,fh,i)eM,i49

where Cx,i(τ,fh,i)=∑n=0N−1xrx,i[n]ci∗(nTs−τ)exp⁡{−j02πfh,inTs} are the standard single-frequency CAFs computed for the different signal components, xrx,i[n]. Equation (49) provides an orthogonal decomposition formula for the multicomplex CAF. Using Equation (49), it is finally possible to compute ℜ{|Cz(τ,fd)|k} in terms of single-frequency components:

R{|Cz(τ,fd)|k}=R{|2M−1∑i=02M−1−1α∼iCx,i(τ,fh,i)eM,i|k}=2M−1R{∑i=02M−1−1α∼i|Cx,i(τ,fh,i)|eM,i}=2M−1∑i=02M−1−1α∼i|Cx,i(τ,fh,i)|R{eM,i}=∑i=02M−1−1α∼i|Cx,i(τ,fh,i)|50

Similar to the bicomplex CAF, the cost function maximized during the acquisition process is a weighted sum of the moduli of the single-frequency CAFs. As reported by Borio (2023a) for the dual-frequency case, this is a form of a weighted linear detector, which was originally studied by Marcum (1960) in the context of radar detection. Note that, as for the bicomplex case, Equation (50) is independent from the data bits that could be present in the different single-frequency components. This independence is due to the fact that modulus operations in Equation (50) remove the dependency on the data bits.

In this section, the theoretical foundations for multicomplex acquisition were outlined. However, it is important to remark that several factors may render the computation of Equation (50) impractical. The primary factor is that Equation (50) implies that all of the different signal components must be integrated on a common time interval. This step is necessary to implement the summation in Equation (39), which is at the basis of the CAF computation. This step may not be feasible in acquisition without the implementation of specific approaches accounting for different code durations. For instance, in the BDS quad-frequency case considered in Figure 1, the BeiDou B1C signal features a 10−ms primary code duration. All of the other signals, i.e., the B1I, B2a, and B2b components, are characterized by a primary code duration equal to 1 ms. While these last three signals can be easily integrated on a common time interval equal to 1 ms, the B1C component creates issues. In principle, it is possible to use partial integrations on the B1C component; however, in acquisition, it is not known which of the ten portions of the primary code should be used. This issue does not arise in tracking, where an initial estimate of the code delay is available and the ambiguity on the B1C code is already solved. Several approaches can be developed for solving this issue in acquisition. For instance, the ten possible code delays for the B1C component could be tested. Extended integrations could also be used on the other components (Borio, 2011). Such efforts, however, will increase the computational load associated with the acquisition process. The actual implementation of such approaches is outside the scope of this paper.

4.4 Multicomplex Tracking

After initial estimates of τ and fd have been determined, tracking can start. This iterative process maximizes the CAF and estimates the signal phases, implemented via tracking loops, each operating on a different parameter. In the single-frequency case, two coupled loops are used: a DLL for the delay and a PLL for the carrier phase. In the bicomplex case (Borio, 2023a), an additional loop, the SPLL, is adopted to track the subcarrier component. When the general multicomplex case is considered, the set of transmitted RF GNSS signals is represented as the product of a multicomplex baseband component, a carrier bringing the signals to RF, and 2M−1−1 subcarriers. This model was introduced in Equation (26) and suggests a tracking architecture based on the following:

• one DLL to jointly track the delay affecting the signals received from the different frequencies and modeled as a single multicomplex component,

• one PLL to track the common meta-signal carrier with the nominal center frequency obtained as the average of the center frequencies of the different single-frequency components, and

• 2M−1−1 SPLLs to track the different subcarriers. For M=3, i.e., the quadfrequency case obtained by considering tricomplex numbers, three SPLLs are needed.

This general architecture is depicted in Figure 3: each loop consists of elements commonly found in the tracking loop literature (Kaplan & Hegarty, 2017; Van Dierendonck, 1997). The incoming signal, zrx,M[n], is multiplied by the local carrier, subcarriers, and code components and integrated over N samples. The process is repeated for three different code delays selected around the last delay estimate obtained during the last loop update. In this way, three correlators, the early, prompt, and late correlators, are found. These correlators are used by the code discriminator to compute an estimate of the residual delay error that is filtered through the loop filter. Finally, the DLL filter output is used to drive the local code generation performed by the code numerically controlled oscillator (NCO). The prompt correlator is also used to compute carrier and subcarrier errors by the carrier and subcarrier discriminators. These errors are filtered and used to drive the local carrier and subcarrier generation implemented by the NCOs. While these operations are commonly applied in single-frequency receivers, it is important to remark that they are implemented here using multicomplex operations. For instance, all of the addition and multiplication operations required for computing the correlators are multicomplex. In the next section, a discussion is provided on how the architecture depicted in Figure 3 can be implemented in terms of standard components.

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FIGURE 3

Schematic representation of the general multicomplex tracking architecture, with the inclusion of multicomplex functional blocks

To complete the description of multicomplex tracking, it is necessary to derive the different code, carrier, and subcarrier discriminators. These terms are found from the likelihood function in Equation (42). More specifically, the code discriminator can be obtained by modifying the single-frequency noncoherent early-minus-late envelope discriminator (Kaplan & Hegarty, 2017, p. 468) as follows:

Dc=ℜ{|E|k}−ℜ{|L|k}ℜ{|E|k}+ℜ{|L|k}51

where Dc denotes the code discriminator output and E and L are the multicomplex early and late correlator outputs. In the next section, Equation (51) will be expressed in terms of single-frequency components. Note that Equation (51) has been obtained by replacing the modulus operator appearing in the single-frequency case with the real part of the hyperbolic modulus. The code discriminator in Equation (51) has the same functional expression as the code discriminator found for bicomplex numbers. Carrier and subcarrier discriminators are found by considering the polar representation of the prompt correlator:

P=|P|kexp⁡{jMTPp}=|P|kexp⁡{∑i=02M−1−1jiφp,i}52

where P is the multicomplex prompt correlator. The vector with its phases is given as follows:

Pp=[φp,0φp,1⋮φp,2M−1−1]53

Thus, the carrier and subcarrier discriminators must compute Pp from P, which can be achieved by using the orthogonal decomposition of P and a Hadamard transform. This process will be more fully discussed in the next section. φp,0 is the carrier discriminator output, whereas the other components of Pp are used by the subcarrier loops.

4.5 Implementation in Terms of Standard Components

The multicomplex tracking architecture described above can be implemented using standard tracking elements operating on complex/real numbers. This is possible because of the orthogonal decomposition property of multicomplex numbers: the single-frequency baseband GNSS signals are the orthogonal components of the resulting multicomplex signal (Borio, 2024a). In this section, the implementation of the multicomplex tracking architecture is discussed in terms of standard components. The DLL is analyzed first. Then, the PLL/SPLL implementations are analyzed.

The multicomplex DLL discussed in the previous section can be implemented using standard complex operations according to the schematic representation provided in Figure 4. The multicomplex signal, zrx,M, has been expressed in terms of its orthogonal components, the baseband signals xrx,i[n], with 0≤i≤2M−1−1. Each xrx,i[n] is correlated in a standard manner using a dedicated single-frequency correlation block. These blocks are labeled as “channels” in Figure 4 and implement the functionalities depicted in Figure 5, including the code and carrier NCOs. In Figure 4, only the code NCOs are highlighted. Each correlation channel provides a set of (complex) early, late, and prompt correlators:

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FIGURE 4

Schematic representation of the multicomplex DLL implemented using standard complex operations

Signals from different frequencies are processed independently using standard correlation channels. Correlator outputs from individual single-frequency channels are fed into a single code discriminator. The output is passed to a single loop filter that controls the different code NCOs.

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FIGURE 5

Schematic representation of a standard single-channel correlation block

{Px,i,Ex,i,Lx,i}i=02M−1−154

These standard complex correlators are related to the multicomplex correlators through Equation (49):

P=2M−1A∼∑i=02M−1−1Px,ieM,iE=2M−1A∼∑i=02M−1−1Ex,ieM,iL=2M−1A∼∑i=02M−1−1Lx,ieM,i55

and can be jointly used to compute the discriminator output in Equation (51). By using Equation (55), one can obtain the following:

Dc=∑i=02M−1−1α∼i(|Ex,i|−|Lx,i|)∑i=02M−1−1α∼i(|Ex,i|+|Lx,i|)56

Dc is filtered using a standard loop filter with real coefficients (Kaplan & Hegarty, 2017). The output of the code loop filter is then fed to the channel NCOs that generate the standard local codes required for the correlation process. In this way, the multicomplex DLL is implemented using standard components.

The multicomplex PLL and SPLLs are jointly represented in Figure 6 as a function of standard processing blocks. In this case, the different input signals, xrx,i[n], are represented separately and processed using standard correlation blocks. The different prompt correlator outputs, Px,i, are independently passed through standard carrier discriminators, and their phases, φx,i, are found. Note that the carrier discriminators can be independently adapted to the type of signal at their input. For instance, if xrx,i[n] is a pilot channel, a four-quadrant arctangent discriminator can be used. If xrx,i[n] is a data channel, a Costas arctangent discriminator can be used. In principle, joint data/pilot combining strategies can also be adopted by adapting the single-channel correlation block and the associated discriminator (Borio, 2023b). In this way, the vector with the phases of the orthogonal components of P is found:

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FIGURE 6

Schematic representation of the multicomplex PLL/SPLL implemented using standard complex operations

Signals from different frequencies are processed independently using standard correlation channels. Correlator outputs from individual single-frequency channels are fed into standard carrier discriminators. The different carrier discriminator outputs are passed to a Hadamard carrier/subcarrier multidimensional filter whose outputs drive the different carrier NCOs.

Px=[φx,0φx,1⋮φx,2M−1−1]57

The phase vector Px is related to the carrier and subcarrier phase in Equation (53) through a Hadamard transform:

Pp=12M−1H2M−1Px58

This operation is performed by the Hadamard carrier/subcarrier multidimensional filter depicted in Figure 6. This block receives the different phases, φx,i, as input and implements Equation (58) to obtain the different carrier and subcarrier phase errors, φp,i. These errors are filtered independently using dedicated loop filters. It is noted that narrowband loop filters can be adopted for the subcarrier components characterized by reduced nominal frequencies. Finally, the Hadamard carrier/subcarrier multidimensional filter implements an inverse Hadamard transform that provides Doppler estimates related to the individual single-frequency channels. A schematic representation of the Hadamard carrier/subcarrier multidimensional filter is provided in Figure 7. The outputs of the Hadamard carrier/ subcarrier multidimensional filter are directly connected to the carrier NCOs of the single-frequency correlation blocks closing the different loops.

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FIGURE 7

Schematic representation of the Hadamard carrier/subcarrier multidimensional filter