On Generalized Signal Waveforms for Satellite Navigation (797942), страница 75
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In fact,we only need the Fisher matrix to compute the CRLB, which only depends on the logarithmof the likelihood function. In other words, only in the case of unbiased estimators the Cramér359Cramér Rao Lower BoundRao lower bound is independent of the used estimator. Accordingly, using the unbiasedCramér-Rao lower bound when the estimator is biased can lead to wrong conclusions.Indeed, due to its simplicity, the unbiased Cramér-Rao lower bound is frequently used toassess performance limits. Nevertheless, we must keep in mind that when multipath is presentthe exact form of the estimator´s bias explicitly enters the computation of the bound. As aconsequence, the use of the unbiased Cramér-Rao has to be actually understood as a desperatetry to give a lower bound to a problem which is in reality so complicated and nonlinear thatcomputing the bias is nearly impossible.360Cramér Rao Lower BoundK.1Appendix.
Single-Path Maximum LikelihoodEstimatorThe single-path case (ML-1P) occurs when only the direct signal is present, thus no multipathis considered, and can be modelled as follows:z (t ) = Ae jφ m(t − τ ) + nc (t ) = x(t ) + jy (t )(K.6)where we have assumed that the signal has been Doppler-compensated and striped of the data.Moreover, A denotes the amplitude, φ the phase offset and τ the code delay. For simplicity,we will separate z(t) into its real component x(t) and imaginary component y(t) according to:x(t ) = A cos(φ )m(t − τ ) + nx (t )y (t ) = A sin (φ )m(t − τ ) + n y (t )(K.7)where nx(t) and ny(t) are assumed to be independent real-valued zero-mean Gaussiandistributed variables. The real and imaginary signals are sampled on [0,T] and thus we canalso express our problem in the discrete domain as followsxk = A cos(φ )mk (τ ) + n x ,ky k = A sin(φ )mk (τ ) + n y ,k(K.8)The Maximum Likelihood (ML) estimates of the populational parameters θ = (A,φ,τ) areshown to be obtained by maximizing the probability density function of the observablesx = {x1 , x 2 ,..., x N } and y = {y1 , y2 ,..., y N } conditioned to the real values of the parameters ofthe model:∑ {[ xk − A cos (φ )mk (τ )]2 +[ yk − A sin (φ )mk (τ )]2 }NN⎛ 1 ⎞ − k =1P(x , y / θ ) = ⎜⎟ e⎝ σ 2π ⎠2σ 2(K.9)where σ2 is the noise variance of x(t) and y(t).
In other words, the ML estimates willmaximize the probability to have a set of observables for a given unknown set of multipathparameters.Before calculating the partial derivatives with respect to the parameters of the model, it isworth it to derive some expressions of interest. We define,1R xm (Δτ ) = ∫ x(t )m(t − Δτ )dtT T(K.10)1Rmm (Δτ ) = ∫ m(t )m(t − Δτ )dtT Tand we will assume that the integration time is a multiple of the period of the signal.361Cramér Rao Lower BoundAs we saw above, for the case the estimators are unbiased, the Cramer Rao lower Bound of θ,namely θˆ can be expressed by means of the Fisher information matrix F. In the case of theML-1P problem, the Fisher matrix can be simplified and expressed only as a function of theautocorrelation function of m(t) if we take into account the constraints that result from solvingthe minimization problem. Indeed, the Fisher Hessian matrix simplifies to:⎧⎛ Rmm (0)0⎪⎜′′ (τˆ) + sin φˆ R′ym′ (τˆ)F = 2 E ⎨⎜ 0− Aˆ cos φˆ Rxmσ ⎪⎜0⎩⎝ 0[ ()N()]⎞⎫⎟⎪0⎟⎬ˆA cos φˆ R (τˆ) + sin φˆ R (τˆ) ⎟⎪xmym⎠⎭0[ ()()](K.11)which can be further simplified if we recall the definition of the model in (K.7), and correlatewith the estimated path delay τˆ .
According to this, it can be shown that:R xm (τˆ ) = A cos(φ )Rmm (τˆ − τ )R ym (τˆ ) = A sin (φ )Rmm (τˆ − τ )(K.12)If we derivate now two times with respect to τˆ , we have:′′ (τˆ ) = A cos(φ )Rmm′′ (τˆ − τ )R xm′ (τˆ ) = A sin (φ )Rmm′′ (τˆ − τ )R ′ym(K.13)And substituting now these expressions into the Fisher matrix results in⎧⎛ Rmm (0)00⎞⎫⎜⎟⎪⎪′′ (τˆ −τ )− Aˆ A cosφˆ − φ RxmF = 2 E⎨⎜ 00⎟⎬ (K.14)σ ⎪⎜Aˆ A cosφˆ − φ Rxm (τˆ −τ )⎟⎠⎪⎭0⎩⎝ 0N[ ( )][ ( )]As it can be shown, the estimates of A,φ and τ are unbiased and thus weFisher matrix of the ML-1P estimator as follows:000⎛ Rmm (0 )⎞⎛1⎟ N ⎜N ⎜22′′ (0 )′′ (0 )− A R xmF= 2⎜ 00⎟ = 2 ⎜ 0 − A R xmσ ⎜σ2⎜0A R xm (0 )⎟⎠00⎝ 0⎝can express the0 ⎞⎟0 ⎟A 2 ⎟⎠(K.15)where Rmm(0)=1 if we work with the C/N0 defined after the filtering.
Thus, the Cramér-Raolower Bound of the estimates can be correspondingly expressed as:⎛⎞⎜⎟⎛ Aˆ ⎞100⎜⎟⎜ ⎟ σ2 ⎜1⎟0 − 20 ⎟(K.16)CRLB⎜τˆ ⎟ =⎜′′ (0)A R xm⎜⎜ ⎟⎟ N ⎜⎟ˆ1 ⎟⎝φ ⎠⎜⎜ 00⎟A2 ⎠⎝For the particular case of the estimation of the path delay, the Cramer Rao Lower Bound ofthe error will be therefore362Cramér Rao Lower BoundCRLBτˆ = −1⎧∂⎫E ⎨ 2 ln[ p( x / θ )]⎬⎩ ∂τ⎭2=−1′′ (0 )NA Rmm2(K.17)σ2And since N0=BLσ2 with BL=1/2T and 2A2N = 2PN = 2E = C,CRLB = −BLBL=∞CC′′ (0 )(2π )2 ∫− ∞ f 2 Sm ( f )dfRmmN0N0(K.18)which is the well known expression that we can find in the literature. It is interesting to notethat although the phase φ was considered in our derivation for completeness, same resultswould have been obtained for the simplified case with φ=0.The Cramér Rao lower Bound can also be derived if we consider our problem of estimatingthe amplitude and delay of the direct signal as a fitting exercise where only noise is present.Indeed, recalling again the first equation in (K.7), we have:x(t ) = Am(t − τ ) + nx (t )(K.19)If we define now f(A,τ)=Am(t-τ) we can linearize our problem and obtain the estimates andvariance using the minimum least squares approach.
Indeed both approaches result in thesame solution when the noise follows a normal distribution. For simplicity this is theapproach we will adopt next for the case that one multipath signal is present (ML-2P).Once we have calculated the expression of the ML-1P estimator, it is important to note thatthis estimator will not be optimal when multipath is actually present since the CRLB cannotbe used. The alternative would be to use the Minimum Mean-Square Error (MMSE)estimator, optimum in this case, but it presents the problem that its calculation requires anenormous computational power. A solution to this problem is the Multipath MitigationTechnology (MMT) which reduces enormously the computational power needed to solve the2P-ML problem.
We analyze it more in detail in the next chapter.363Cramér Rao Lower Bound364Cramér Rao Lower BoundK.2Appendix. Two-Path Maximum LikelihoodEstimatorFor the case that one multipath signal is present, we can express the Doppler-compensatedbaseband signal model (ML-2P) as followsz (t ) = A1 e jφ1 m (t − τ 1 ) + A2 e jφ 2 m (t − τ 2 ) + n c (t ) = x (t ) + jy (t )(K.20)As done in the case of the ML-1P problem above, we separate the real and imaginarycomponents as followsx(t ) = A1 cos(φ1 )m(t − τ 1 ) + A2 cos(φ2 )m(t − τ 2 ) + nx (t )(K.21)y (t ) = A1 sin (φ1 )m(t − τ 1 ) + A2 sin (φ2 )m(t − τ 2 ) + ny (t )or after sampling, the equivalent discrete problem is shown to be:xk = A1 cos(φ1 )mk (τ 1 ) + A2 cos(φ2 )mk (τ 2 ) + nx, k(K.22)yk = A1 sin(φ1 )mk (τ 1 ) + A2 sin(φ2 )mk (τ 2 ) + ny , kNow, in order to apply the Least Mean Squares (LMS) approach, we linearize the multipathmodel equations defined in (K.21) as follows:x(t ) ≈ A1m(t − τ 1 ) − τ 1 A1m′(t − τ 1 ) + A2 m(t − τ 2 ) − τ 2 A2m′(t − τ 2 ) + nx (t )(K.23)According to this, the discrete matrix problem to solve will be in this case⎛ x1 ⎞ ⎛ m(t1 − τ 1 ) − A1m′(t1 − τ 1 ) m(t1 − τ 2 ) − A2 m′(t1 − τ 2 ) ⎞⎛ A1 ⎞ ⎛ n1 ⎞⎜ ⎟ ⎜⎟⎜ ⎟ ⎜ ⎟⎜ x2 ⎟ ⎜ m(t 2 − τ 1 ) − A1m′(t 2 − τ 1 ) m(t 2 − τ 2 ) − A2 m′(t 2 − τ 2 ) ⎟⎜ τ 1 ⎟ ⎜ n2 ⎟⎜ ...
⎟ = ⎜⎟⎜ A ⎟ + ⎜ ... ⎟............⎜ ⎟ ⎜⎟⎜ 2 ⎟ ⎜ ⎟⎜ x ⎟ ⎜ m(t − τ ) − A m′(t − τ ) m(t − τ ) − A m′(t − τ )⎟⎜ τ ⎟ ⎜ n ⎟NNNN111222 ⎠⎝ 2 ⎠⎝ N⎠ ⎝⎝ N⎠(K.24)what we can express by means of matrices as followsX = M ′β + N(K.25)Since the variance of the LMS estimator of M is shown to be σ 2 Mˆ = (M ′M ) σ 2 , we need to−1calculate the matrix M’M which, after some math, is shown to simplify to′ (Δτ ) ⎞10Rmm (Δτ )A2 Rmm⎛⎜⎟2′′ (0 )′ (Δτ ) − A1 A2 Rmm′′ (Δτ )⎟0− A1 Rmm− A1 Rmm⎜M ′M = ⎜⎟′ (Δτ )− A1 Rmm10R (Δτ )⎜ mm⎟⎜ A R′ (Δτ ) − A A R ′′ (Δτ )′′ (0 ) ⎟⎠0− A22 Rmm1 2 mm⎝ 2 mm(K.26)365Cramér Rao Lower Boundwhere Δτ=τ2-τ1.
As we can see, M’M is a function of the multipath delay and thus we will beable to represent the standard deviation (equivalent to the root mean square in this case sinceall the estimates are unbiased) as a function of the delay of the multipath signal.Now that we have calculated the matrix M’M, the variance of the estimates is shown to be⎛ Aˆ1 ⎞⎜ ⎟⎜ τˆ ⎟−1(K.27)Var ⎜ 1 ⎟ = (M ′M ) σ 2ˆA⎜ 2⎟⎜ τˆ ⎟⎝ 2⎠The result coincides perfectly with that we would have obtained if we would have maximizedthe probability density function since the noise is Gaussian distributed.The expression of the inverse matrix of F looks a little bit complicated but to study thecharacteristics of the ML estimates we just have to look at the determinant, since after theinversion this is in the denominator.
Thus, evaluating the determinant when Δτ tends to zerowe can study the behaviour of the bound. This can be expressed as follows:{[][]}′′ (0) − Rmm′′ (Δτ )Rmm(Δτ )] + R2 mm(Δτ ) −1 R′′2 mm(Δτ ) − R′′2 mm(0)M ′M = A12 A22 R′4 mm(Δτ ) + 2R′2 mm(Δτ )[Rmm(K.28)It is interesting to note that although all the estimates are unbiased, what is a good property inprinciple, when the separation of the direct and delayed multipath paths Δτ approaches tozero, the determinant of M’M makes the variance of the error tend to infinity.This is a very important characteristic of any unconstraint ML estimate in presence ofmultipath. In fact, unbiased estimators are preferred in principle but their quality degradesconsiderably for small path separations since the standard deviation increases when Δτ tendsto zero.366Cramér Rao Lower BoundK.3Appendix.Multiple-PathLikelihood EstimatorMaximumOnce we have solved the case of one multipath signal present (ML with 2 paths), it isstraightforward to generalize the results to any number of multipath signals.
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