A new GLONASS FDMA model

Abstract

We introduce a new formulation of the double-differenced GLONASS FDMA model. It closely resembles that of CDMAbased systems and it guarantees the estimability of the newly defined GLONASS ambiguities. The formulation is made possible because of our defining new concept of integer-estimability and the analytical construction of a special integer matrix canonical decomposition. As a result, an easy-to-compute new design matrix is created that automatically establishes the integer-estimability of the ambiguities. The presented model is generally applicable, and its close resemblance to its CDMA-counterparts implies that available CDMA-based GNSS software is easily modified and that existing methods of integer ambiguity resolution can be directly applied. Also, because of their similar structure, many of the available CDMA results with corresponding insights can now be directly translated to the GLONASS case. We make use of this property to provide insight into the ambiguity resolution capabilities of the model and to analyze the characteristics of the GLONASS ambiguity dilution of precision.

GLONASS ambiguity resolution

“…The new GLONASS FDMA DD model of (Teunissen 2019) is given as in which p ∈ ℝ 2(m−1) and ∈ ℝ 2(m−1) denote the DD pseudorange (code) and phase observables, m is the number of tracked satellites, e = (1, 1) T , ⊗ denotes the Kronecker product, G ∈ ℝ (m−1)× is the relative receiver-satellite geometry matrix, = diag( 1 , 2 ) is the diagonal matrix of wavelengths; L ∈ ℝ (m−1)×(m−1) is a full-rank, easy-to-compute lower-triangular matrix, b ∈ ℝ is the baseline vector ( = 3 in the absence of a Zenith Tropospheric Delay, otherwise = 4 ) and a ∈ ℤ 2(m−1) is the newly defined GLONASS integer ambiguity vector. For background information on…”

“…This close resemblance implies that available CDMA software is easily modified and that existing methods of integer ambiguity resolution can be directly applied. The entries of the lower-triangular matrix L are given as (Teunissen 2019) where a 1(i+1) = a i+1 − a 1 . The integers i and i are given by in which a i = 2848 + i , i ∈ [−7, +6] are the channel numbers, g 1 = a 1 and g i = GCD(a 1 , … , a i ) (1 < i ≤ m) , with GCD denoting the Greatest Common Divisor.…”

“…, with e m−1 being the (m − 1) -vector of ones. In Teunissen (2019), it is shown that the performance of the model is invariant for the choice of differencing matrix.…”

“…In some of these extensions, for instance, in the long-baseline phase-only case, the integer-estimability of the ambiguities will change. In Teunissen (2019), it is shown how to recover in such cases the integer-estimability again. Also note that, although (1) is formulated for the dual-frequency case, its single-frequency variants are easily obtained by replacing e = (1, 1) T by 1 and = diag( 1 , 2 ) by 1 or 2 , respectively.…”

“…In Teunissen (2019), a new formulation of the double-differenced (DD) GLONASS FDMA model was introduced. It closely resembles the CDMA-based systems, and it guarantees the estimability of the newly defined GLONASS ambiguities.…”