Symmetric Overbounding of Correlated Errors

Transcription

1 Smmetric Oerbounding of Correlated Errors Jason Rife, Tufts Uniersit Demoz Gebre-Egziabher, Uniersit of Minnesota ABSTRACT In aiation naigation sstems such as the Ground-Based and Space-Based Augmentation Sstems (GBAS and SBAS) for GPS, it is critical for users to compute a conseratie naigation-error bound during precision approach and landing. This paper describes a method for oerbounding outputs of linear sstems (i.e., conseratiel describing their error distributions). Whereas earlier oerbounding methods appl onl to sets of independent samples, the methods deeloped in this paper handle samples drawn from correlated time series, so long as those samples hae a probabilit distribution which a linear mapping can make sphericall smmetric. Conseratie error bounds constructed with these methods are useful in modeling the worst-case performance of filters under a wide range of enironmental conditions. An example illustrates how smmetric bounding could enable aircraft to appl user-customized filtering to differential GPS corrections in GBAS or SBAS. INTRODUCTION Ground-Based and Space-Based Augmentation Sstems (GBAS and SBAS) proide GPS users with high qualit differential corrections that improe naigation accurac. Depending on the configuration of the augmentation sstem, indiidual or networked reference stations make differential correction measurements and broadcast these to users through terrestrial VHF transmitters (in GBAS) or through a satellite link (in SBAS). In addition to differential corrections, GBAS and SBAS messages also contain integrit information: both in the form of alert messages, which warn of possible satellite failures, and in the form of error parameters, which enable construction of conseratie naigation-error bounds. Aiation users performing precision approach and landing operations rel on this integrit information to ensure the meet the safet requirements imposed b ciil aiation authorities. In practice, users must define confidence bounds which bracket all but the rarest errors, those occurring with a risk of less than -7 (or -9 for full automated landing). The user computes this confidence bound on-the-fl and compares the bound to a maximum allowable error for which the aircraft can still land within its touchdown box. As long as its confidence bound does not

2 exceed this limit, the aircraft can proceed safel with landing haing alidated that its naigation integrit risk meets the federall approed standard. This operational test for integrit relies on conseratism in the user confidence bounds. To certif naigation hardware, manufacturers must demonstrate that their sstems report a user error bound at least as large, or larger, than the actual alue needed to protect the integrit risk requirement. Because the form of actual error distributions ma ar in time or depend on enironmental conditions, it is common to replace actual error distributions with conseratie approximations, called oerbounds. An oerbound proides a conseratie guarantee because its accumulated tail probabilit alwas exceeds the tail probabilit of the actual distribution. This paper introduces methods to derie oerbounds for the output of linear sstems in airborne and other safet-critical applications. Linear sstems, such as filters and estimators, commonl operate on signals corrupted b time-correlated measurement noise. These noise inputs ma result from high-order and time-aring processes. Nonetheless, it ma be desirable to describe the noise process with a lower-order, stationar model (such as a first-order Marko model) that is a conseratie approximation of the actual noise process. The bounding strateg deeloped here, called smmetric oerbounding, can replace time-aring and higher-order processes with stationar, lower-order approximations. Consequentl, smmetric oerbounding enables the description of noise inputs in a compact manner appropriate for communication oer band-limited channels and for computation of real-time bounds on time-aring filters and estimators. The first portion of this paper deelops a quantitatie basis for smmetric oerbounding. Subsequentl, the paper demonstrates the capabilities of smmetric oerbounding b exploring the example of a ariant GBAS architecture which would permit users to define their own linear filters for smoothing differential GBAS corrections. SYMMETRIC OVERBOUNDING This section deelops a theor for oerbounding the output of linear sstems applied to signal inputs corrupted b correlated noise. Preious Research Preious research has established a set of guidelines for oerbounding the error distribution that results from a sum (or a linear combination) of random ariables. These operations are common in GPS, since user positioning errors equal the weighted sum of range measurement errors for each isible GPS satellite.

3 Currentl, the certification of GPS augmentation sstems relies heail on Cumulatie Distribution Function (CDF) oerbounding. This technique constructs an oerbound for a linear combination of random ariables using indiidual oerbounds for each input ariable []. Although the original ersion of CDF oerbounding applies onl to independent, smmetric distributions with a single peak and zero-mean, other research has relaxed these restrictions. Paired-CDF oerbounding and excess-mass oerbounding, for instance, appl to arbitrar independent distributions, een those that are asmmetric, non-zero mean or hae multiple peaks []-[3]. Because it is conenient to approximate actual distributions with Gaussians, other research has inestigated the use of Gaussian bounds to represent independent distributions with heaierthan-gaussian tails [4]-[6]. Additional work has addressed the topic of deeloping Gaussian oerbounds from actual datasets [7],[8]. This paper extends earlier work b generalizing oerbounding to appl to linear sstems (of random signals) and not just to linear combinations (of random ariables). Because random signals are frequentl autocorrelated in time or cross-correlated with each other, these new bounding methods relax the independence constraint assumed in preious oerbounding work. CDF Oerbounding The basic notion of most oerbounding techniques is to define an approximation that conseratiel represents the Probabilit Distribution Function (PDF) for the output of an operation on one or more random ariables. This approximation is conseratie if its integrated tail probabilit eerwhere exceeds the integrated tail probabilit of the actual output distribution. This definition can be quantitatiel expressed as follows, where the oerbar indicates an approximation and where the tilde indicates a dumm ariable used for integration. x x ( ) ( ) ( ) ( ) x x p xdx p xdx, x< x p xdx p x dx, x> x med med () These relationships appl to the right and left distribution tails on either side of the distribution median, x med. Throughout this paper, the cured inequalit smbol will impl the two oerbound conditions gien b (). That is, the following notation indicates that the approximate distribution p oerbounds the actual distribution p. p(x) p(x) () 3

4 Oerbounding in this sense is sometimes called Cumulatie Distribution Function (CDF) oerbounding, because the conditions of () appl to the cumulatie (integrated) probabilit in each tail of a PDF. The earliest form of CDF oerbounding was introduced in []. This form of CDF oerbounding applies onl to independent random ariables. B contrast, the current paper deelops a distinct form of CDF bounding that applies to correlated random ariables with linearl mapped smmetric distributions. To distinguish between the two methodologies, this paper will refer to the earlier theor of [] as independent-distribution oerbounding and to the new form of CDF bounding introduced here as smmetric oerbounding. Before introducing smmetric oerbounding, it is useful to reiew the particulars of independent-distribution oerbounding. Independent-Distribution Oerbounding The independent-distribution oerbounding theorem established in [] states that linear combination operations presere CDF oerbounding. This theor applies to linear combinations of the form, which is the sum of two independent random ariables x a and x b scaled b the deterministic coefficients α a and α b. =α axa +α bxb (3) Reference [] proes that the output distribution for is oerbounded b the conolution of the oerbounds for x a and x b, so long as both the oerbounds and the actual error distributions are independent, smmetric, single-peaked and zero-mean. In smbolic form, this result ma be written as ( ) ( ) p p (4) when the approximation p ( ) is computed (after appling the multiplication rule described in the appendix) b the following equation in which the closed circle ( ) indicates the conolution operator, p( ) = pa p b, (5) αa αb αa αb gien that ( ) ( ) ( ) ( ) p x p x and p x p x. (6) a a a a b b b b 4

5 Thus, under appropriate conditions, the PDF of the output of a linear combination ma be computed conseratiel b using conseratie approximations of the input PDFs. The smmetr, single-peak, and zero-mean conditions are not generall significant when the noise is approximatel zero-mean Gaussian; howeer, the requirement that the input random ariables be independent places a significant constraint on man applications. Smmetric Oerbounding This section deelops a new CDF oerbounding theorem that applies to linear combinations of random ariables that are correlated. This method, called smmetric oerbounding, requires that a linear transformation G exist that maps the ector of correlated inputs x to a space z for which the joint PDF is sphericall smmetric. Smbolicall, smmetric oerbounding places the following requirement on the joint PDF p(x). ( ) = ( ) = = ( ) p x-x p r, r z-z z-z G x-x (7) Here the mean of x is x and of z is z. These mean alues ma be suppressed b substituting a new set of random ariables for the originals ( xnew = xold -x ). Thus, without loss of generalit, distribution means will be dropped in the following analsis. Gien that the joint PDF for a set of random ariables exhibits the smmetr implied b (7), it is possible to oerbound the distribution of a linear combination of those ariables x b an appropriate scaling of the distribution for an indiidual element x i. p (x ) p (x ) ξ p ( ξ) p () (8) xi i xi i xi To proe this result and to obtain a alue for the scaling factor ξ, consider a ector of random ariables x linearl combined to form an output using the deterministic weights contained in the ector α. T = α x (9) The probabilit distribution for this combination can be expressed in terms of a linear mapping. T T T ( ) = ( ) = ( ) p p α Gz p α G e z. () Here a unit pointing ector has been introduced equal to 5

6 e T = T α G T α G. () It is furthermore useful to define a scalar coordinate in the z-space along the line defined b the unit pointing ector e. z = T ez. () Appling the multiplication rule, described in the appendix, to () gies the following. p α G p p z α G α G ( ) = = ( ) T z T T z (3) This distribution for the output can be related to an one of the random inputs x i using the z-space as an intermediar. The i-th random input x i is related to the smmetrized ector z through the the i-th row of the smmetrization matrix G: x = Gz. (4) i i B analog to (3), the distribution for x i can be related to the z-space through the following equation. p x = p z. (5) G ( ) ( ) xi i zx x i Here the ariable z is defined to be the scalar coordinate projected along the axis of the ector G i. That is i z x = G z Gi. (6) Because the z-space probabilit is sphericall smmetric, howeer, the marginal probabilit is identical when integrated onto an axis. Thus, spherical smmetr implies that ( ) ( ) p z = p z (7) zx x z when z x = z. Using this smmetr relation, the probabilit distribution for gien b (3) can be related to the distribution for x gien b (5). 6

7 G i G i p( ) = p T x i T α G α G (8) Equation (8) indicates that the PDF for a linear combination is a scaled ersion of the PDF for an one of its inputs. This result is true so long as the joint distribution function for those random inputs is itself a linear mapping of a sphericall smmetric distribution. The scaling rule described b (8) can be used as the basis for oerbounding an set of random ariables x which can be linearl mapped into a sphericall smmetric space. If an of the elements of x can be oerbounded, such that ( ) ( ) p x p x, (9) xi i xi i then the linear combination is, as a consequence of (8), oerbounded b the following approximation, G i G i p ( ) = p x p ( ) T i T. () α G α G This completes the proof of the smmetric oerbounding theorem, originall proposed as (8), and specifies the alue of the scaling factor preiousl referred to as ξ. Comparison of Bounding Methods It is instructie to compare the smmetric oerbounding method with the original independent-distribution oerbounding theorem. The linear map G that was introduced to decorrelate the random-ariable inputs for smmetric oerbounding can also be introduced to decorrelate the random-ariable inputs for independent-distribution bounding. Specificall, this linear transformation G must map correlated random ariables to a space in which their joint PDF is the product of seeral marginal PDFs. ( ) ( ) j p p z, x x = z j z = G x () j If such a linear map exists, then the independent-distribution oerbound of (5) can be generalized to appl not onl to independent random ariables but also to correlated ones. In the case of the linear combination described b (9), the 7

8 generalized form of the independent distribution oerbound is a sequence of coolutions between each of the N independent distributions in the mapped-ariables space z. p( ) = p T z... p T T z () N T α G, α G, α G,N α G,N The oerbound () is analogous to (), where the linear map conerts the input ariables x to a set of output ariables z that is independent, in the former case, and that is sphericall smmetric, in the latter. Here it should be noted, in describing the generalized independent-distribution theorem, that the notation G,j indicates a column ector whereas, in the description of smmetric oerbounding, G i represents a row ector. The oerbounds produced b the smmetric and independent-distribution assumptions are distinct. The independent distribution theorem maps ariables to a space in which the are uncorrelated and independent (in contrast with smmetric bounding for which the mapped ariables are merel uncorrelated). Also, the independent-distribution method allows flexibilit in that each random ariable ma possess a unique PDF (in contrast with smmetric bounding for which all random ariables share PDFs of a common form). Most notabl, howeer, the independent-distribution approach incurs significant computational expense. Compared to the smmetric bound of (), for which the output is simpl a stretched ersion of the input, the independent-distribution bound of () conoles together the PDFs for N inputs. The computational load for repeated conolutions becomes especiall significant when the number of input ariables becomes er large. For moderate numbers of ariables, efficient computation mitigates this computational load [9]. As N approaches infinit, howeer, repeated conolutions are infeasible unless the independent distributions are Gaussian (in which case the independent-distribution and smmetric oerbounds are equialent). Thus, the smmetric oerbounding approach offers a practical solution to oerbounding linear combinations with man correlated inputs, as long as those inputs are a linear mapping of a set of sphericall smmetric random ariables. Application to Linear Sstems In this section, smmetric oerbounding will be applied to bound the outputs of linear sstems, such as filters and estimators. The inputs and outputs of linear sstems are random signals rather than random ariables. These signals represent sets of samples acquired discretel or continuousl through time. Because an essentiall infinite number of samples are processed, the low computational load of smmetric oerbounding is critical. 8

9 Oerbounding applies to linear sstems because their outputs can be iewed as linear combinations of input samples weighted b sstem impulse functions. This connection is most clearl isible for Single-Input, Single Output (SISO) sstems. The dnamics of an linear, time-aring SISO sstem are described b a function f(δt,t) which describes the sstem s response to an impulsie input at time t. The noise output (t) equals the inner product of the input noise (t) with the reflected impulse response f. For a sstem discretel sampled at an interal of T seconds, this inner product is the following. k m= (( ) ) (kt) = f k m T, kt (mt) (3) = f(,kt)(kt) + f(t,kt)(kt T) +... Clearl, the output is a linear combination of weighted input samples. Likewise in the continuous time case, when the sample interal T shrinks to zero, the output is still a linear combination of input samples. (t) = f (t t, t)(t)dt (4) No distinction need be drawn between the behaior of continuous and discretel sampled sstems. Thus it is useful to introduce a generic conolution operator ( ) that takes the form of (3) for discretel sampled sstems and of (4) for continuous time sstems. For compactness, the arguments Δt and t will be suppressed unless the are specificall required for a deriation: = f. (5) An equialent Multiple Input, Multiple Output (MIMO) expression also exists. In the MIMO sstem, the inputs and outputs are ectors of signals. The matrix F contains the impulse functions relating each input in the ector to an output in the ector. = F (6) Here, the matrix conolution operator acts in an analogous manner to matrix multiplication, with conolutions substituted for element-wise products. 9

10 f,(t) f, (t) (t) F = f,(t) f,(t) (t) f,(t) (t) + f, (t) (t) + = f,(t) (t) + f,(t) (t) + (7) Because, b the analog of (3), the aboe equation represents a form of linear combination, it is possible to appl smmetric oerbounding. Howeer, in order to appl smmetric oerbounding, correlated elements of the input noise ector must first be mapped to an uncorrelated, sphericall smmetric form. It is possible to substitute a linear process G and a white noise input w for the correlated noise ector, as shown in Figure, if the linear process conerts the white noise input into a correlated input with statistics matching those of. = G w (8) Gien precise knowledge of the noise process G, the following smmetric oerbound can be obtained. Gi(t) G i(t) p ( (t) ) j j = p i Fj G (t) F j G (t) p j ( j(t) ) (9) Original Sstem F Equialent Sstem w G F Bounding Sstem w G F Figure. Oerbounding Correlated Noise

11 This equation defines an oerbound for the j-th element of the output noise ector in terms of the oerbound for the i-th element of the input noise ector. Here G i is the i-th row of the noise process matrix G and F j the j-th row of the linear sstem impulse matrix F. The linear sstem oerbound (9) is analogous to the linear combination oerbound () except that the linear sstem s impulse matrix F replaces the linear combination s weighting ector α. As in (), the numerator represents the stretching associated with decorrelating the noise samples while the denominator represents the stretching associated with linearl combining those decorrelated noise samples. To ensure that the -norm of (9) is well defined, this oerbound is restricted to noise signals and impulse response functions in the L signal space. The -norm of an N-element signal ector has the following form: i N G (t) = G ( Δt,t)dΔt. (3) k= i,k BOUNDING THE NOISE MAPPING FUNCTION The preious section deried an equation to oerbound the outputs of a linear sstem. This oerbound, (9), relies strongl on precise knowledge of the mapping matrix G, which shapes white noise to hae statistical properties that match those of the correlated input noise. This section examines the noise mapping more closel, quantifing the relationship between G and the correlation of the input noise signals. Subsequentl, this section addresses the replacement of the actual map G with an approximation G (as shown in the bottom diagram of Figure ). This replacement is necessar because G is a function of the input noise signals. In fact, it is not possible to determine G precisel because of statistical estimation error. Determination of G is especiall challenging for nonstationar noise processes. B using a conseratie, reduced-order approximation G, howeer, it is possible to ensure bounding een when the estimate of the actual noise map G is uncertain. Statistical Estimation of the Mapping Function In order to appl the linear sstem oerbound of (9), it is necessar to relate the noise map G to the correlation functions for the input noise signals. For this purpose, it is useful to define a correlation matrix Q, in which each element is a correlation function belonging to the L signal space. For a ector of random signals inputs, the corresponding correlation matrix is

12 denoted b a leading superscript: Q. The diagonal elements of Q are the autocorrelation functions of the elements of. Similarl, the off-diagonal elements of Q are the cross-correlation functions between the elements of the signal ector. T ( ) Q= E (3) Here the open circle notation ( ) indicates a matrix correlation operation, analogous to the matrix conolution operation of (7). The impulse-function matrix G in the oerbound (9) is directl related to this correlation matrix Q. In the statistical analsis of linear sstems, it is well established that an autocorrelation function can be obtained b correlating the sstem s impulse response with itself. A cross-correlation function can be obtained b correlating the impulse response functions of two different sstems. Similarl, the elements of the correlation matrix Q can be constructed from the impulse response elements of G b substituting (8) into (3). ( ) E( ( ) ( )) Q= E = G w w G (3) T T T This relationship is further simplified b noting that the white noise signals w are not cross-correlated and that the are autocorrelated onl at a zero time-shift (Δt = ). The power in all the white-noise signals is assumed, without loss of generalit, to hae a constant alue c. ( i wj) E w c δ (t) i= j = (33) i j Ealuating (3) using the white-noise properties of (33) it is possible to express the correlation matrix Q as the square of the impulse response matrix G. ( ) Q= c G G (34) T In practice, it is more conenient to estimate the correlation functions for a set of statistical data, rather than the impulse response function. For instance, if the noise process is slowl time aring, a statistical estimate of ˆQ is obtained b correlating the signals of the noise ector and aeraging those correlation functions oer M + adjacent time samples.

13 M i,j ( ) M i ( ) j m= M ˆQ Δ t,t = k m (k n m) (35) Δ t = nt, t = kt If the elements of the signal ector are the same (i = j), this equation estimates their autocorrelation as a function of the time difference (nt) relatie to the current time (kt). If the elements of the signal ector are not the same, the equation estimates their cross-correlation function. Based on the elements of the correlation matrix, estimated through (35), and their relationship to the noise map, gien b (34), it is possible to create an approximate oerbound of the following form (where the hat superscripts indicate statistical approximations). Gˆ i(t) Gˆ i(t) ˆp ( (t) ) j j = p i Hˆ ˆ j(t) Hj(t) (36) Here H represents the total-sstem impulse response, which maps the white noise inputs through both the linear sstem F and the noise map G. = H w (37) H = F G (38) The method for computing the approximate oerbound, (36), will be discussed in more detail in the following section. The statisticall estimated correlation matrix ˆQ does not necessaril guarantee conseratie oerbounding. Accordingl, a subsequent section discusses how to coert (36) from an approximate form to a guaranteed oerbound. Approximate Oerbounding Using Statistical Correlation After obtaining autocorrelation and cross-correlation estimates for each of the input signals in the ector an approximate oerbound can be constructed. In order to obtain the approximation (36), it is necessar to relate the -norms in the numerator and denominator of the equation to the correlation function estimates of (35). In establishing this relationship, it is useful to note that the alue of the correlation matrix (3), ealuated at a zero time-shift (Δt = ), has the following form. 3

14 N i,j ( ) i,k j,k k= Q, t = c G G dδt (39) B comparing equation (39) to the -norm definition (3), it is eident that the diagonal elements of the correlation matrix are related to the -norms of each row of the the impulse function matrix G as follows. ( ) G i = c Q i,i,t (4) Using the estimated coariance of (35) in this equation results in a -norm estimate (denoted b a hat): ( ) G ˆ = c Qˆ,t. (4) i i,i This statistical estimate approximates the numerator of the oerbound (36). Computing the denominator requires further analsis. The -norm of the j-th row of the total sstem impulse response matrix H can be related to the output ariable correlation matrix b ( ) H = c Q,t, (4) j j,j where the elements of the matrix Q are the autocorrelation and cross-correlation functions of the output signal. ( ) Q= c H H. (43) T The premise of linear sstem oerbounding is that onl the input noise signal for the linear sstem F need be described. As a consequence, it should be possible to oerbound the output noise of the sstem without actuall measuring the statistics of the output signals. It is thus necessar to express the output correlation matrix Q in terms of the input correlation matrix Q. Expanding the elements of Q gies the following. Q = H H T i,j i j ( Fi,l Gl,m ) ( Gk,m Fj,k ) = k l m (44) Using the following identit ( f g) ( f g) = ( f f) ( g g), (45) 4

15 the output ariable correlation matrix Q, as described b (44), can be rewritten in terms of the input signal correlation matrix Q. ( ) ( ) Q = F F G G i,j i,l j,k l,m k,m k l m ( ) = F F Q k l i,l j,k l,k (46) Using this relationship, equation (4) can be rewritten as an estimate based on input ector statistics. ( ) H ˆ = c Qˆ,t (47) j j,j ( ) Qˆ = F F Qˆ (48) j, j i,l j,k l,k k l Together equations (36), (4), and (47) proide a means to approximate the distribution for the outputs of a linear sstem. In contrast to the guaranteed oerbound of (9), which assumed perfect knowledge of the noise mapping function G, the approximation distribution (36) is not guaranteed to be an oerbound. Uncertaint in Characterizing the Noise Process This section describes wh the approximate distribution of (36) is not guaranteed to oerbound the output distribution for a linear sstem of interest. As an illustratie example, consider the greatl simplified case of a linear combination of two ariables X and Y. For this case, it is assumed that the ariables X and Y are correlated and that their joint PDF is sphericall smmetric. A constant probabilit ellipse (solid cure) represents the nominal joint PDF in Figure. Because statistical estimation is nois, ellipses for the estimated correlation function ma not match the nominal cure. As a result, linear combinations based on estimated joint PDFs ma hae more or less probabilit mass in their tails than the nominal PDF. Consider two estimates of the nominal constant-probabilit cure, one which is more positiel correlated (dotted line) and one which is less positiel correlated (dashed line). Linear combinations inoling the nominal and approximate distributions are shown around the margin of Figure. For cases in which the linear-combination weights hae the same sign, the more correlated approximation (dotted line) produces a conseratie oerbound but the less correlated approximation (dashed line) does not. The conerse is true when the weights hae opposite signs. 5

16 X (m) Nominal (4σ cure) More Positiel Correlated Less Positiel Correlated Z = (-.38)X + (.9)Y.5.5 PDF Z = ()X + ()Y.5 Y (m).5.5 PDF -.5 Z = (.38)X + (.9)Y PDF -.5 Z = (.9)X + (-.38)Y Z = ()X + ()Y Z = (.9)X + (.38)Y Z = (.7)X + (.7)Y.5 PDF.5 PDF Figure. Bounding the Sum of Correlated Variables To guarantee oerbounds for an linear combination (regardless of the signs of the weights α) or for an linear sstem (regardless of the sign of the impulse functions F), it is critical to bracket uncertaint in the estimate of the noise map G. In graphical terms, the more and less positiel correlated error ellipses of Figure (dotted and dashed) bracket the actual error ellipse (solid cure). The following section quantifies this notion b deeloping a method to bracket the best aailable estimate ˆQ of the correlation function matrix to account for estimation uncertaint. Conseratiel Approximating the Noise Process Map In order to ensure a worst-case error bound that coers uncertaint (and nonstationar statistics), it is important to allow tolerance in the estimation of the correlation matrix Q. This tolerance is labeled Q' and represents the difference between the estimated and actual correlation matrices. Q= Qˆ + Q (49) If the uncertain term Q' can be bracketed b a reduced-order model, such as a first-order Marko process, then it is computationall efficient to incorporate the correlation uncertaint into an oerbound, een if that oerbound must be computed in real-time. This reduced-order model, which must upper and lower bound the uncertaint in Q', is labeled Δ. i,j(t) c Q i,j(t) Δ (5) 6

17 The elements of Δ are modeled correlation functions that bound uncertaint in the actual correlations. As long as the Δ matrix conseratiel brackets Q', an oerbound can be guaranteed for the output of the linear sstem. This oerbound takes the following form. G i G i p ( ) j j = p i H j H j p G H i i pi j H j ( ) G (5) Here the -norms that scale the oerbounding distribution are replaced b a conseratie oerbound in the denominator (indicated b an oerbar) and a conseratie underbound in the numerator (indicated b an underbar). It is straightforward to derie from (46), (49) and (5) that ( ) Q = F F Qˆ j, j i,l j,k l,k k l ( ) + F F Δ Q k l i,l j,k l,k j,j (5) and that an upper bound for the numerator of (5) is thus ( ) H = c Q,t H. (53) j j,j j The absolute alue of the second term of (5) ensures that the signs of the elements of the Δ matrix and the impulse functions of the F matrix align to produce a worst-case bound. A lower bound is required for the row-ector norm of G. A form of this lower bound can be obtained b substituting (49) into (4) and appling the bracketing relationship of (5). This time, rather than adding to the correlation matrix estimate to obtain an upper bound, the bracket matrix Δ is subtracted to obtain a lower bound. ˆ ( ) Q (, t ) = max Q (, t ) Δ (, t ), i,j i,j i,j Q (,t ) i,j (54) ( ) G = c Q,t G (55) i i,i i 7

18 A Procedure for Oerbounding Linear Sstems The final form of the smmetric oerbound for the output of a linear sstem is gien b equation (5). Because this equation embeds seeral significant assumptions, it is useful to define a procedure that alidates those assumptions prior to constructing the oerbound. This procedure consists of fie steps.. Inspect Spherical Smmetr Assumption. Define Oerbound for an One Input Variable 3. Ealuate Statistical Correlation Functions 4. Estimate and Bracket all Correlation Functions 5. Compute Output Oerbound Using (5) Each of the steps of this procedure will be applied in the example of the following section. BOUNDING A GPS DATASET To demonstrate the principle of smmetric oerbounding for a linear sstem, this section considers a GPS application. Specificall, smmetric oerbounding is applied to demonstrate how aircraft could be permitted to define their own customized smoothing filters for the differential correction measurements broadcast b GBAS or SBAS. In a conentional augmentation sstem, a set of ground stations generates differential correction measurements based on the known location for each reference antenna. The augmentation sstem filters the raw differential correction measurements and transmits the smoothed corrections to users. In standard GBAS sstems, this filter is linear time-inariant with a -s time constant []. For arious reasons, airborne users might benefit from appling longer or shorter smoothing times. Multiple smoothing rates, for example, could be applied to detect certain hazardous conditions such as the occurrence of an anomalous ionospheric storm []. Current GBAS and SBAS sstems appl filtering to raw differential corrections at the ground station, rather than in the user aionics, for two primar reasons. First, differential corrections can be transmitted with higher precision if smoothing reduces the maximum and minimum alues that must be transmitted. Second, the fact that the smoothing process is well defined permits the output noise to be characterized a priori based on accumulated statistics. If sufficient precision is 8

19 aailable to transmit raw corrections, then smmetric oerbounding can alleiate this second consideration b proiding a model of the raw differential corrections that enables output bounding, using (5), een if the form of the user s measurement filter is not known to the ground station. This section applies oerbounds to a representatie dataset to illustrate the potential for a modified GBAS with user-defined filtering of differential corrections. The example deelops an input-noise bound for GPS measurements drawn from the National Geodetic Sure s Continuall Operation Reference Station (NGS-CORS) database []. The noise model deried from this dataset is applied to approximate the output of a set of low-pass smoothing filters with different time constants. The predicted output of the measurement filters, based on oerbounding, is compared to the actual output when the filters are applied to the GPS data. The comparison alidates the oerbounds b demonstrating that the oerbound tails alwas contain equal or greater probabilit than the tails of the actual output error distributions. Data Set The data analzed in this section were acquired from a Wide Area Augmentation Sstem (WAAS) reference station at the Oakland, California airport. This station is designated as ZOA in the NGS-CORS database. The measurement samples analzed describe a 45 minute data record starting at a UTC hour of :: on 8 August 6. Samples were aailable at s interals. Differential corrections can be generated for the ZOA data b subtracting the pseudorange measurements from the known range, based on sure, of the station relatie to each GPS satellite. These differential corrections would be applied b nearb users to subtract out common-mode errors, such as ionosphere and troposphere delas that are highl correlated between users and the reference station. The measurement error in the differential correction signal consists of noise that is not common-mode between the reference station and user. For the purposes of this paper, it is assumed that common mode errors change graduall oer time. In fact, this assumption is quite reasonable under nominal conditions. As such, the uncorrelated measurement error is estimated as the residual after subtracting off a fourth-order polnomial fit from the ZOA range measurement data. The noise model deeloped in the remainder of this section describes these residual pseudoranges. 9

20 User-Defined Pseudorange Smoothing In GBAS and SBAS, differential corrections are filtered with a digital low-pass filter that combines information about the underling carrier signal with the code-phase ranging measurements [3],[4]. Often called a carrier-smoothing or Hatch filter, this class of filter uses low-noise carrier measurements of relatie position to complement noisier code measurements of absolute position. The Hatch filter thus compensates for platform motion using relatie ranging measurements proided b the carrier phase and enables long duration filtering of nois, absolute range measurements proided b the code phase. Under most conditions, the code-phase noise dominates that of the carrier-phase in the Hatch filter s output error. Therefore it is representatie to model onl the noise associated with the code-phase pseudorange residuals. For this example, the codephase noise will be modeled using the ranging residuals measured at ZOA. Considering onl code-phase errors, the following low-pass filter results. T T (k) (k) = τ τ (k ) (56) For standard GBAS and SBAS sstems the sample interal T is.5 s and the time constant τ is s. As the data in this example were sampled at Hz, howeer, T will be set to s in this analsis. The impulse function for the low-pass filter of (56) is the following. f(k) ( ) T T k = (57) τ τ Because this filter is applied separatel to each satellite ranging signal, this section will not consider cross-correlation among satellites and will therefore decouple the analsis of the smoothing filters for each ranging source. Consequentl, the lowpass filter will be considered as a SISO sstem (5) rather than a MIMO sstem (6). Although this decoupling of ranging signals simplifies the example presented in this paper, cross-correlations between signals would need to be considered in a complete analsis of the naigation-error bound, as the GPS position solution combines the ranging measurements for all satellites. Step : Inspect Spherical Smmetr Assumption Because the number of input ariables is essentiall infinite, it is not possible to demonstrate formall that all inputs of a ector ma be mapped to a set of sphericall smmetric ariables. Spherical smmetr must therefore be accepted as a

21 fundamental assumption. Howeer, it is nonetheless prudent to inspect certain necessar conditions to erif qualitatiel this assumption. A reasonable check for spherical smmetr is to confirm isuall that the autocorrelation and cross-correlation functions for each input signal are well behaed to first order. Specificall, a scatter plot can be generated for each input that compares signal alues between subsequent epochs (autocorrelation check) or between pairs of input signals at the same epoch (crosscorrelation check). Best fit ellipses of constant probabilit can then be added to the scatter plot. If the points in these plots appear uniforml distributed around the circumference of the best-fit elliptical bands, then the isual inspection erifies the smmetr assumption. If the points in the plot clearl skew or cluster and do not uniforml scatter around the circumference of the elliptical bands, then isual inspection contradicts the assumption of spherical smmetr. In this latter case, when the isual inspection ields a negatie result, smmetric oerbounding should not be applied to analze the data set. For the CORS range residuals from ZOA, isual autocorrelation inspection does erif the smmetr assumption for all satellites. Figure 3 illustrates the smmetr inspection test for PRN 6. The axes of the scatter plot represent the alues of the prompt residual ( k ) and the preceding residual ( k- ) errors. The cross-coariance between these two sets of residuals was used to construct an inspection coariance matrix P. Ellipses of constant probabilit are plotted for four radii (r =,,3,4) where the radius parameter is defined using the inspection coariance matrix as follows. [ ] k = k k k r P (58) The densit of the data points around each error ellipse contour appears to be approximatel uniform. Consequentl, the model of spherical smmetr (subject to a linear mapping) appears to be a reasonable representation of these data. Because the spherical smmetr assumption models the obsered joint distribution for the measurement residuals well, the smmetric oerbounding rule of (5) can be applied to this satellite. The same qualitatie test was repeated for each PRN in iew. Because some satellites were subject to time-aring ranging residuals, howeer, the scatter plots for these satellites were generated oer multiple shorter interals of data (oer which time-ariation was minimal). With this caeat for time-ariation, all satellites passed the isual inspection test for spherical smmetr.

22 Data Points Radial Units k (m) k- (m) CDF (from r = ) - Empirical CDF Oerbound k (m) k (m) Figure 3. Range Residuals for PRN 6 Step : Define Oerbound for an One Input Variable The residual range errors for the dataset are well modeled b Gaussian distributions. Figure 3 shows an empirical CDF for each tail of the range residuals for PRN 6. The empirical CDF (solid line) is compared to a Gaussian CDF (dashed line) for which the standard deiation, sigma, has been inflated 3% relatie to the sigma of the data. The inflated Gaussian CDF is clearl an oerbound to the empirical data in the sense of the oerbound definition (). For the purposes of this example, the far tails of the actual error distribution are assumed also to be bounded b the inflated Gaussian CDF (dashed line). To alidate this far tail bounding assumption would require the analsis of significantl more data collected oer multiple das. Similar Gaussian models bound all satellites in iew during the 45 minute data window. Howeer, seeral of the satellites exhibit significant nonstationarit. The Gaussian oerbound was thus ealuated oer onl short duration windows to reduce the seerit of time ariation. Standard deiations of the data, Figure 4 for each PRN j. ˆσ j, (compiled for sequential 3 s windows) are illustrated in

23 PRN 4 PRN 9 Residual Error Standard Deiation (m) - PRN 4 PRN PRN 7 PRN 6 PRN 8 PRN 8 - PRN Data Window Figure 4. Standard Deiation of Range Residuals Step 3: Ealuate Statistical Correlation Functions Constructing a smmetric oerbound requires both that the distributions of the input ariables are oerbounded (Steps and ) and that the correlation functions for those ariables are also appropriatel bounded. For the present example which analzes low-pass filters (acting independentl on each ranging signal), onl the autocorrelation functions for each satellite need be considered. Because of nonstationarit in the input noise process, statistical autocorrelation functions were computed for each satellite oer relatiel-short, sequential windows of 3 s in duration. For this purpose, the correlation estimate equation (35) was used, with M set to 3. Although the short length of the correlation window was necessar to capture nonstationar effects (such as phase shifts in oscillator signals and time-ariations in the noise power), a wider window would otherwise hae been desirable to mitigate statistical sampling errors. As an alternatie means of reducing sampling errors, these windowed autocorrelation functions were normalized to a peak alue of one and aeraged. Windows were not considered in the first and last minutes of the data set, to proide margins for large-correlation time shifts, Δt. Excluding these points, fie sequential data windows were defined for each satellite aboe the horizon at the start of the test (but onl four windows were defined for PRN 4, which rose later). Statisticall estimated correlation functions, aeraged oer all windows for each indiidual satellite, are plotted in Figure 5. 3

24 Step 4: Estimate and Bracket all Correlation Functions After erifing spherical smmetr (Step ), identifing an oerbound for the input distribution (Step ), and computing the autocorrelation functions for each satellite (Step 3), the critical step in deeloping the smmetric oerbound (5) is the replacement of the autocorrelation function with a conseratie, reduced-order model. This low-order model must also decompose the autocorrelation function into an estimate and a bracket on the deiation of the actual autocorrelation from that estimate, as described b (49). The estimated autocorrelation need not be an unbiased estimate; in most cases, it is adantageous to define an upper and a lower bracket around the actual autocorrelation functions, as plotted in Figure 5 (dashed lines), and to set the estimate ˆQ equal to be the midpoint of the upper and lower bounds. For simplicit in this example, the estimated correlation function ˆQ is modeled as the correlation function for white noise: a delta function with a zero alue for all time shifts awa from Δt =. The bracket Δ around the estimate ˆQ is constructed from a composite of first-order Marko processes. The composite Marko model sums two first-order processes, as shown in Figure 6. In the figure, these processes are illustrated as a pair of mapping functions ( A g and B g ) each applied to a distinct Gaussian white noise input ( A w and B w). The composite model, described b the following pair of impulse functions, is substituted for the input-noise signal for each satellite j. A B g(k) =β ( β ) σˆ ρ k A A j g(k) =β ( β ) σˆ ρ k B B j (59) The parameters used to define this model were the inerse time-constants (β A = /5 s - and β B = /3 s - ) and the power fraction (ρ = /36) for the two Marko processes. These parameters were determined, b trial and error, to bound the autocorrelation functions empiricall computed from the sampled data. The correlation functions for this composite Marko process are illustrated b the dashed cures in Figure 5, intended to represent upper and lower bounds on the empirical data. The bracketing autocorrelation matrix Δ thus maps two white noise signals w into a single correlated input. At a time shift of zero (Δt = ), the bracket is set to zero because the signal is perfectl autocorrelated,. ΔΔ ( t,t) = Δ t = A A B B ( ) g g+ g g Δt (6) 4

25 PRN: 4 PRN: PRN: 9 PRN: - - PRN: 7 PRN: PRN: 6 PRN: 7 - PRN: Time (s) Aeraged Autocorrelation Marko Brackets Figure 5. Autocorrelation Functions for Residual Error Figure 5 shows that the bracketing model bounds all of the aeraged autocorrelation functions, except that of PRN 8. The autocorrelation function for PRN 8 is a special case, as sinusoidal oscillations are obsered in the satellite-aeraged correlations. These oscillations indicate strong multipath errors, which occur when GPS signals reflect from nearb surfaces and corrupt GPS receier tracking of the direct signal. 5

26 Original Sstem f Bounding Sstem w A g + + f w B g Figure 6. Oerbounding Correlated Noise Instead of extending the brackets to bound these multipath oscillations of PRN 8, an alternate solution is to define the estimate Q ˆ to incorporate the multipath process. Handling multipath in this manner presumes that the phenomenon ma in fact be modeled as a random process. Such a model is appropriate because, with the exception of the strongest multipath waeforms that result from reflections off large planar surfaces (e.g. aircraft surfaces, buildings, or the ground), the occurrence of multipath is difficult to predict and model. Een when the occurrence of multipath can be predicted, the phase and amplitude of the periodic errors are generall not known, except in that the displa moderate correlation from da to da oer periods of one to two weeks. In this sense, multipath errors can be treated as the result of a noise process responding to a random input signal. The behaior of the PRN 8 multipath noise is consistent with that of a second-order random process drien b white noise. The oscillation maintains a relatiel consistent period, of 5 s in duration, but changes phase and amplitude infrequentl. Breaking the autocorrelation windows into short records helps to reeal this periodic structure, as correlation effectiel strips awa phase information so that the windowed autocorrelations all align. Modeling the multipath noise process as a lightl-damped, second order impulse function results in an autocorrelation function that is cosinusoidal. This cosine is summed with a delta-function (white-noise) autocorrelation estimate for the remaining thermal noise and diffuse multipath. The combined estimated autocorrelation is the following. 6

27 ( ( ) ) ˆ 5 j a cos Δ t + ( a) δ( Δt) σ PRN 8 ˆQ = σδδ ˆ j ( t) PRN 8 (6) For PRN 8, the amplitude a of the cosine term can be determined b manual scaling. Embedding the multipath ringing into the estimate of ˆQ reduces the residual error which the Δ matrix must bound. The upper plot of Figure 7 shows the difference between the empiricall computed autocorrelation function and the estimated autocorrelation for PRN 8 gien b (6). This remainder is well bounded b the bracket function Δ. Thus, following multipath remoal for PRN 8, the proposed Δ model brackets the aeraged correlation functions for all the satellites obsered in the dataset (lower plot of Figure 7). Aeraged Autocorrelation for PRN 8 after Multipath Remoal Aeraged Autocorrelation for all PRNs after Multipath Remoal on PRN s Figure 7. Impact of Multipath Decomposition Step 5: Compute Output Oerbound The final remaining step is to compute the oerbound of the output noise for one or more user filters. For this example, the user filter is assumed to take the form of the model first-order filter of (56) with a user-defined time constant τ. Because the form of the input oerbound (from Step ) was Gaussian, the output oerbound is also Gaussian. Howeer, the output distribution oerbound is stretched according to (5). The degree of stretching depends on the -norm bounds g and h. According to (55), the alue of this bound for g is determined b ealuating a combination of the estimated and bracketing autocorrelation functions at a time-shift of zero (Δt = ). The bracket matrix Δ has a zero alue when Δt =, as 7

28 specified b (6). The alue of the estimated correlation function ˆQ at a zero time shift is alwas equal to the ariance ˆσ j of the oerbounding Gaussian for an PRN j. Thus, for the SISO filter considered, g = cσ ˆ j. The upper bound on the total-sstem impulse function h can be found using (5). Because the filter impulse function (57) is eerwhere positie, the absolute alue signs of (5) are not required. The resulting bound is ( ) ( ˆ ) Q= f f Q+Δ. (6) Here f is gien b (57), ˆQ b (6) and Δ b (6). Because the upper bound on the -norm of h is ( ) =, (63) h c Q,t the constant c cancels out of the stretching factor used to establish the smmetric oerbound for the output. g g p( ) = p h h p ( ) (64) The output oerbound thus has a Gaussian distribution that is inflated 3% (in Step ) and b an additional scaling factor (in Step 5) equal to the ratio of the -norms bounds of g and h. Discussion This section compares the output oerbound, deried as (64), to the actual distributions that result from analzing the statistics of the filter output signal. Because the data are Gaussian, these results are most easil compared through analsis of the standard deiation parameter, sigma. When two Gaussians are compared, the distribution with the larger sigma oerbounds the distribution with the smaller sigma, according to the oerbound definition (). B considering first-order filters of a range of different time constants, it is possible to alidate the smmetric bound b showing that the computed oerbound sigmas do in fact alwas exceed the sigmas that results when filters are applied to the actual data. Indeed, the oerbounding method produces a conseratie approximation of the output error for all nine satellites in the dataset. Figure 8 illustrates this result for three representatie satellites: PRNs 4, 7, and 8. 8