On Generalized Signal Waveforms for Satellite Navigation (797942), страница 76
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In fact, for threepaths, one direct signal and two multipath signals, the linearized problem to solve is3x (t ) ≈ ∑ Ai m(t − τ i ) − τ i Ai m′(t − τ i ) + nx (t )(K.29)i =1And the variance of the estimates is shown to be:′ (Δτ 1 )′ (Δτ 2 ) ⎞Rmm (Δτ 1 )A2 RmmRmm (Δτ 2 )A3 Rmm10⎛ A1 ⎞⎛⎜ ⎟⎟⎜2′′ (0 )′ (Δτ 1 ) − A1 A2 Rmm′′ (Δτ 1 ) − A1 Rmm′ (Δτ 2 ) − A1 A3 Rmm′′ (Δτ 2 ) ⎟0− A1 Rmm− A1Rmm⎜ τ1 ⎟⎜⎜ A ⎟ 1 ⎜ R (Δτ )′ (Δτ 1 )′ (Δτ 3 ) ⎟Rmm (Δτ 3 )10− A1Rmm− A3 Rmm1⎟Var ⎜ 2 ⎟ = ⎜ mm′ (Δτ 1 ) − A1 A2 Rmm′′ (Δτ 1 )′′ (0)′ (Δτ 3 ) − A2 A3 Rmm′′ (Δτ 3 )⎟0− A22 Rmm− A2 Rmm⎜ τ 2 ⎟ N ⎜ A2 Rmm⎜ ⎟⎟⎜ R (Δτ )′ (Δτ 2 )′ (Δτ 3 )Rmm (Δτ 3 )10− A1Rmm− A2 Rmm2⎟⎜ mm⎜ A3 ⎟2⎟⎜ A R′ (Δτ ) − A A R′′ (Δτ ) − A R′ (Δτ ) − A A R′′ (Δτ )⎜τ ⎟′′()AR00−21 3 mm23 mm32 3 mm33 mm⎝ 3⎠⎠⎝ 3 mm(K.30)As we did in the case there is only one multipath signal present, we could calculate thedeterminant of the matrix to be inverted above, in order to study the stability of the varianceof the estimates when the multipath delays approach to zero.
Although the exact expressiondoes not look very friendly, it can be shown that when Δτ1 and Δτ2 tend to zero, thedeterminant also approaches to zero and thus the standard deviation also increases to infinitylike in the non constraint ML-2P case. Once again, we can see that the use of biasedestimators seems to be a better solution to some problems.367Cramér Rao Lower Bound368Antisymmetric SequencesLAppendix. Antisymmetric SequencesMBOC is the result of multiplexing BOC(1,1) and BOC(6,1).
In the long process of findingan alternative to BOC(1,1), experts of the STF of the EC realized that some specific BCSsequences seemed to be of special interest for satellite navigation. We have shown in chapter5 that CBCS presented the problem of having a tracking bias when correlated with a pureBOC(1,1) receiver, what lead to further investigations on alternative BCS sequences withparticular symmetry properties. These receive the name of antisymmetric sequences and havebeen studied in [A.R.
Pratt et al., 2006]. In the next lines the most important characteristicsare underlined.Let the binary sequence S={si} of length n define a reversed sequence S ∗ such that:S ∗ = {s n −1−i } for i ∈ 0..( n - 1)(L 1)It is important to note that this sequence S can be any generic BCS signal in principle. Asequence is defined to be symmetric if S ∗ = S , so that:s i = s n −1−i for all i ∈ 0..(n - 1)(L.2)As we can see, such a definition allows solutions for both n even or odd. Furthermore, abinary sequence is defined as antisymmetric if S ∗ = − S , that is:s i = − s n −1−i for all i ∈ 0..(n - 1)(L.3)As we can see, antisymmetric sequences may only be composed with even values of n, sincethere cannot be a central element, x(n-1)/2 which is self inverse.Another important characteristic of any generic BCS sequence is the balance.
As shown in[A.R. Pratt et al., 2006], the balance of a symmetric binary sequence can be determinedthrough a consideration of the sum of the elements. Let S={si} be a symmetric binarysequence of length n, then the sum Σ( S ) of the element values is:n −1n−22i =0i =0Σ( S ) = ∑ si = 2∑ si for n even(L.4)Moreover, assuming that the sequence is binary and n even, it can also be shown thatn−22Σ( S ) = 2∑ si = n mod 4(L.5)i =0Equally, when n is odd, the sum of the elements of S adopts the following form:n −32Σ( S ) = s n −1 + 2∑ s i = n + s n −1 − 1 mod 42i =0(L.6)2As we can recognize, all antisymmetric binary sequences with even n are balanced because oftheir particular construction. Moreover as a consequence of (L.5), the sequence balance, thatmeans Σ(S ) = 0 , can only be achieved for symmetric binary sequences whose lengths are369Antisymmetric Sequencesmultiples of 4.
This is a necessary condition for the existence of some zero sum sequences,but not sufficient.We further define the properties of a product sequence Z={zi} constructed from 2 binarysequences, X and Y. Z will also have length n and element values {+1,-1}. In principle, X andY are restricted to be either symmetric or antisymmetric sequences. Z is derived by formingthe inner product from the two seed sequences, X and Y, as follows:z i = {xi .
y i } for i ∈ 0..(n - 1)(L.7)The sequence, Z={zi}, may be balanced or unbalanced, as described above, or may besymmetric or antisymmetric. If, for example, the sequences X and Y are both symmetric sowill also be Z. If both X and Y are antisymmetric, Z results to be a symmetric sequence. Thisis shown next:xi = − x n −1−iy i = − y n −1−i(L.8)z i = xi y i = (−1) 2 x n −1−i y n −1−i = z n −1−iIn addition if one of X or Y is antisymmetric, and the other is symmetric, Z will beantisymmetric. In fact, the class of sequences X, Y which can form sequences Z withsymmetric properties is wider than the class with seed sequences which are symmetric orantisymmetric.As shown in [A.R.
Pratt et al., 2006], the circular crosscorrelation between two binarysequences, X and Y, of length n, is defined as:n −1Rxc, y (r ) = ∑ xi yk =i =0n −1− r∑xi =0iyi + r +n −1∑xi=n−riyi + r − n(L.9)k = (i + r ) mod nwhere k = (i + r ) mod n . This operation corresponds to the formation of another type ofderived sequence, Z={zi}, where the family members are determined through circular shifts ofthe original sequence members. In fact,z r i = xi y k where k = i + r mod n(L.10)( )As we can see, the cross-correlation Rxc, y (r ) is then the balance, Σ Z r of the sequence Z r .In the following lines some examples of antisymmetric sequences are given for variouslengths n. For length n = 4 , it can be shown that there are two distinct antisymmetricsequences:x1 = {+1,+1,−1,−1}x2 = {+1,−1,+,1 − 1}(L.11)For n = 6 , there are 3 distinct sequences, while for n = 8 we can find 6.
In the same manner,for n = 10 we have 10 and for n = 12 there are 20 such sequences. These were tabulated inchapter 4.7.2 with their cross-correlation values at zero offset.370Interference ModelMAppendix. Interference ModelIn this Appendix we present the most important expressions that are necessary to assess thedegradation that a signal causes on another signal in the shared band. As we have shown inchapter 4.7.8, the methodology is based on the idea of measuring the degradation of a desiredsignal in terms of the reduction of its effective C/N0.
This figure was also analyzed inAppendix H. This can be caused by either the background noise where all the non GNSSsignals are included or by another interfering signal of the same or similar nature. To thisgroup belong all the types of interference from other GNSS signals. We can furtherdistinguish the different GNSS interference sources following this classification:•••IIntra: This type of interference is commonly known as intra-system interference and isdue to the signals coming from satellites that belong to the same system as the desiredsignal.IInterop: This type of interference corresponds to the equivalent noise introduced in thereceiver by an interfering signal coming from a satellite of a different constellation butwith the same signal structure as that of the desired signal.IInter: This type of interference comes from signals with a different signal structure nomatter whether the signal belong to the same system or a different one.If we put now all the GNSS sources of interference together, we have as interfering power:I Total = I Intra + I Inter + I Interop(M.1)Moreover, we define the equivalent noise power density of each of the interfering signals as:NXIX =∑Cj =1jκ js(M.2)βr∫β G ( f + f )df2d−where•••••••dop sr2X defines the type of interference according to the definitions above,NX represents the number of satellites,Cj is the received power of satellite j,βr is the receiver bandwidth,Gd is the power spectral density of the desired signal s,f dop s is the doppler frequency offset of the desired signal s,κ js is the spectral separation coefficient between signal j and the desired signal s.
Aswe saw in chapter 5, this is defined as follows:κ js = ∫βr2−βr() ()G ij f + f dop j Gs f + f dop s df(M.3)2371Interference Modelwhere,• G ij ( f ) is the power spectral density of the interfering signal j•f dop j is the doppler frequency offset of the desired signal j.The GNSS receiver of our model is supposed to stay stationary at the earth’s surface so thatonly the motion of the satellites will be responsible for the Doppler offsets observed at userlevel. Moreover, we know that while for GPS the absolute Doppler frequency offset does notexceed 4.4 kHz, the range is narrower for Galileo, being the maximum value of 3.3 kHz in theE1/L1 band as shown in Figure M.1 next [S.
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