How Narrow is Narrow Band

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    Ho w narrow is narrowbland?M.Za tman

    Indexing t er m : Narrowband, Cramer-Ruo bound, Super-resolution algorithm s, Array signal processing

    --Abstract: The 'narrowband' assumption is oftenmade in the analysis of array signal processingalgorithms. The author provides a definition forthe notion of narrowband, and the derivation ofan expression which is useful in determining if aparticular scenario qualifies as narrowband. Theexpression derived correctly predicts where thenarrowband assumption fails for some super-resolution algorithms, the Cramer-Rao bound onangle estimation and the signal-to-interferenceplus noise performance of adaptive beamformers.--I Introduction

    The 'narrowband' assumption is often made when ana-lysing the performance of an adaptive array signalprocessing or estimation scheme. A vague definition ofnarrowband given in the literature [l] is that 'the]-e isessentially no decorrelation between signals received onopposite ends of the array'. In this paper, a more pre-cise analytical expression is derived for classifying whena signal environment may be described as narrowband.The expression derived, motivated by the signal model,provides insight into the performance of a number ofsuper-resolution algorithms, the Cramer-Rao lowerbound (CRB) and the signal-to-interference-plus-noiseratio (SINR) performance of adaptive beamformer:;.2 Data modelConsider a uniform linear array (ULA) of N antennaswith an inter-element spacing d, (the results in thispaper are easily extended to an arbitrary array geome-try). The transfer function between the angle of arrival(AOA) from the array broadside 6 and the output ofthe array is represented by the steering vector

    a(8)= [ l , J + , . ,e J ( N - l ) + ] T (1)where represents the transpose operator and27rdf$I=-- sin(8)Cf i s the frequency of the signal incident on the array,and c the propagation velocity. For convenience, thearray is assumed to have half-wavelength spacing at thechosen operating frequency, f o . In array processing, afixed phase is typically used to approximate the timedelay of a signal between elements of an array. Since

    IEE Proceedings online no. 19981670Paper first received 25th March and in revised form 11thAugust 1997The author is with MIT Lincoln Laboratory, 244 Wood St., Lexington,MA 02138, USA

    the inter-element phase, q, epends on both the AOAand frequency, for a linear array, a non-zero-band-width signal appears as an extended angular source,whereas a zero-bandwidth signal appears to come froma discrete angle of arrival. This effect is known as dis-persion and is pictured qualitatively in Fig. 1.

    frequencyFig.1A zero-bandwidth signal appears to come from a single angle of arrival. How-ever, a non-zero-bandwidth signals appear to come from an angular extentcommensurate with its bandwidth. This effect is known as dispersionQualitative picture o dispersion

    For the zero-bandwidth case, the exact covariancematrix of the data received by the array from A4 signalsarriving from AOAs 01, ..., 13, and the (zero mean andspatially white) noise is typically modelled aswhere A = [a(6',), ..., a(@,>], I is an identity matrix, o2is the noise power and " represents the Hermitian(complex) transpose operator. It is also assumed thato2= 1 and that the signals are temporally uncorrelated,thus S is an M x A4 matrix with each signal's SNRalong its leading diagonal and zeros elsewhere. R maybe simply decomposed into the sum of the covariancematrices for each signal RI , ..., R, plus the noise cov-ariance matrix 1. The covariance matrix for the mthsignal may be written as

    R = ASAH+ a21 (3 )

    R, = s ,a(L)a(Q,)H (4)The kith element of R, corresponding to the correla-tion of the mth signal between the kth and ith elementsof the array is

    ( 5 )k l = S , e J 2 . r r m f owhere zk[ is the time delay between the kth and lth ele-ments of the array, and s is the SNR of the mth sig-nal. From eqn. 2 it is easily shown that

    r k l = (1 - k)dsin(Q)/cFor the non-zero-bandwidth case, the covariancematrix may be obtained by integrating eqn. 4 over thedesired range of instantaneous frequency, i.e.

    f"+kf o - %R m = / s m ( f .(e, f ) a (Q ,n f (6)

    where 0 is the bandwidth of the signal and s,cr> is thereceived signal power as a function of frequency.Eqn. 6 suggests that, apart from the case where 6' = 0,a non-zero-bandwidth signal always produces a full-

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    rank covariance matrix, since it may be viewed as aninfinite sum of zero-bandwidth signals. However, someof the eigenvalues of such a signal may be small com-pared to the noise level, and will have little effect onthe adaptive beamformer performance. Thus we definean effective rank, which is the number of signal sub-space eigenvalues of the signal-only covariance matrixgreater than 1. In the signal-plus-noise covariancematrix of eqn. 3, the effective rank is the number ofeigenvalues greater than 2 (3dB above the noise floor).The klth element of R, is the integral of eqn. 5 over thedesired bandwidth,so+$

    r k l = / s , ( f )eJzTTk l cif ( 7 )f o - 5For example, if sc is rectangular between b /2 and -bl2 , then the solution to the above integral eqn. 7 is givenby

    r h l = sinc(brk1 smej21Tkl fo ( 8 )(note that we defined s = bs,vo) = Js,v> dJ). Eqn. 7 isalso the inverse Fourier transform of the signals powerspectrum, which is of course the autocorrelation func-tion. This information is used by Compton to deriveeqn. 8 [l]. By noting that eqn. 8 is simply eqn. 5 multi-plied by a term representing signal decorrelationbetween the elements of the array, Compton describesa signal as narrowband whenwhere z is the time delay between the 1st and Nthelements of the array, i.e. there is essentially no decor-relation between signals received at opposite ends ofthe array.The covariance matrix of a number of non-zero-bandwidth signals is calculated by first computing allthe elements of the covariance matrix for each signalaccording to eqn. 8, summing the individual matricesand then adding noise as in eqn. 3. Instead of referringto specific values off0 and b , we will also refer to thefractional bandwidth which we define as

    sinc(brlN) N 1 (9)

    bb f = -f 0

    3 The definition of narrowbandA covariance matrix R may be represented by its eigen-values and eigenvectors such thatwhere E is the matrix of eigenvectors, and A is thediagonal matrix of eigenvalues. It is well known thatthe eigenvectors may be partitioned into two subspaces,a signal subspace and a noise subspace, i.e.where th e subscripts s and y1 refer to the signal andnoise subspaces, respectively. In super-resolution, E, isa minimum-rank orthonormal basis for the signalswhose parameters we wish to estimate. In adaptivebeamforming, the E, is called the interference subspace,since it contains an orthonormal basis for the interfer-ence we wish to suppress [Note I]. For the zero-band-width signal model, it follows from eqn. 3 that the rank

    R = EhEH (11)

    R = E,A,E,H + E,AnE,H (12)

    Note 1: In this paper the radar scenario, where the interference-plus-noisecovariance matrix is estimated from range gates which exclude the targetsignal, is assumed

    of the signal subspace is the same as the number of sig-nals present, i.e. there is a rank-one representation ofeach signal present.For a sufficiently wide bandwidth, the effective rankof the signal subspace is larger than the number of sig-nals present, and the zero-bandwidth signal model nolonger applies. Thus we will define the notion of a nar-rowband signal as follows:If the bandwidth of a signal is such that the secondeigenvalue of the signals noise-free covariance matrixis larger than the noise level in the signal-plus-noisecovariance matrix, then that signal may not bedescribed as narrowband.Simulations may be used to show that as a signalsbandwidth is increased, eigenvalues pop up from thenoise floor one at a time. Hence, an expression for the2 largest eigenvalues is sufficient for the purpose of thispaper. Methods have been proposed for determiningthe eigenvalues of a covariance matrix [2, 31; the deri-vation used here is a special case of the general resultof [2]. For the problem posed here, consider the covari-ance matrix of two uncorrelated zero bandwidth signalswith powers ,ul and p2:R = p1 a1 a? + pu2aza? (13)

    The eigen-problem is to find values of A and v suchthatRv = AV (14)Each of the signal subspace eigenvectors is a linearcombination of the signals present, so for the two sig-nal case v may be expressed as

    Thus from eqns. 13 and 15 the product Rv can be writ-ten asRv = piPNai +piy$Nai + p2P$*Na2 + p2yNaz

    (16)where aHa = N (N is the number of array elements,equal to the array gain assuming equal power sensors)and aIHa, = NI). The quantity I) is the cosine of theangle between the two vectors in N-dimensional space.From eqns. 14 and 15 we can rearrange eqn. 16 as thefollowing pair of simultaneous equations in the form ofa 2 x 2 eigen-problem

    v = pal + ya2 (15)

    (17)XP = PlPN + pLly?LNAy = P 2 P $ * N + PzyNThus the size-N eigen-problem has been reduced to asize-two eigen-problem with roots atSolving the resulting quadratic in h gives the followingeigenvalues

    (19)

    A, = NPl(1+ 14) (20)For equal power sources, 1 e. p1 = p2,eqn. 19 simplifiestoOf interest for the problem addressed here is thesmaller eigenvalue (the - of the 2 in eqns. 19 and 20)Now that an expression for the second eigenvalue ofthe covariance matrix has been obtained, the wideband

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    problem needs to be formulated in such a way thateqn. 20 may be applied.By the definition earlier, a signal ceases to be nar-rowband when its effective rank increases from one totwo. Therefore, the rank-two approximation of a non-zero bandwidth by two discrete uncorrelated equalpower sources is an adequate model for the purposes ofthis paper. The discrete sources need to be arranged sothat the mean and variance of their instantaneous fre-quency spectrum correspond to the mean and variancefor the non-zero-bandwidth signal they are meant tomodel. This is easily accomplished for an arbitrary sig-nal pass-band. For the example of a rectangular powerspectrum with bandwidth b mentioned earlier, the vari-ance of the spectrum is given by

    2010ma i-a,0)a,2 -10.--20

    -30

    A two frequency representation of the signal wouldconsist of two delta functions spaced a frequency Keither side of the centre frequency. The varianclz ofsuch a representation is K ~ . or the rank-two represen-tation of the finite-bandwidth signal, the variance ofthe model and reality should be the same, thus

    -.

    u o x x x x x /////.

    //. ///

    /./

    I&and the spacing of the two delta functions is given byb2K =-

    Given both the two signal model of a non-zero-band-width signal, and the expression for the eigenvalues ofa two-signal covariance matrix, both parts need to befused. For an arbitrary array, IqA may be computedfrom the inner product of the two steering vectors a(@7 K) and a(@, f - K), where J is the mean of thereceived signals power spectral density, i.e.

    (24)For several regular array configurations eqn. 24 has aprecise analytical form. For the ULA example used inthis paper, the dependence of the inter-element phaseon both frequency and AOA means the model ofeqn. 23 corresponds to having two signals arrivingfrom bearings equivalent to

    The two signals are spaced in sin(6) bysin(0)b fE=---- 8For the ULA example, the value of /q(n eqn. 20 is aDirichlet function of E

    For the purposes of this paper, the smaller of the twoeigenvalues of eqn. 20 IS of interest. Given s (the SNRof the signal) and from eqn. 27 (or eqn. 24 for anarbitrary array), substituting the two signal model intoeqn. 20 results inI E E Pvoc -Radar, Sonar Navig , Vol 145. N o 2, April 1998

    noting that each of the two signals has a power of0 . 5 ~ ~ .wing to the formulation of eqn. 13 the value ofA2 has been calculated without the inclusion of noise. Ifnoise is included, then the value of A2 will, assumingthe model of eqn. 3 , be the result of eqn. 28 plus 1. Fo ra signal to be classified as narrowband, i.e. for a sig-nals effective rank to be 1, A2 must be smaller than 1,(equivalent to A2 < 2 or < 3dB above the noise level ifthe noise is included), i.e.From inspection of eqns. 26-29 the value of A2 is afunction of the fractional bandwidth, AOA, power(SNR), the array gain and the dimensions of the array.Comptons result is neither a function of power (SNR)nor the array gain and has no precise threshold. Thuswe claim our result is a more precise definition of nar-rowband.A similar result to eqn. 28 for the case of a rotatingarray was recently derived by Hayward [4], utilisingLees method for determining the eigenvalues [ 3 ] . Forthe case of a linear array, the apparent spreading of asignals AOA due to bandwidth effects is analogous tospreading due to array rotation [5] .

    30 I

    I-10 I0 0.5 1 1.5 2 2.5 3bandwidth,MHzFig.2 Predicted and simulated values of th e second eigenvalue as a func-tion of bandwidth fo r 20-element array with AO A of 30andSNR of SOdBat a centre requency of SOOMHz_ _ ~ ~redicted

    __ predicted + noise* simulation

    In order to demonstrate the accuracy of eqn. 29 forthe second eigenvalue, the results of some simulationsare shown in Figs. 2 and 3. A 20-element half-wave-

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    length spaced ULA was simulated at a centre frequencyof 500MHz with a signal present from 30". In Figs. 2and 3, either the SNR or bandwidth is fixed while theother parameter is varied. The predicted and measuredvalue of the second eigenvalue is plotted as a functionof the changing parameter. It is clear from Figs. 2 and3 that the prediction of the second eigenvalue is accu-rate. In the example scenario the narrowband criterionis met under the following two conditions: with anSNR of 50dB, for bandwidths less than 0.135MHz(0.027% fractional bandwidth), and with a bandwidthof lMHz (0.2% fractional bandwidth), for SNRs ofless than 32dB. Tiny fractional bandwidths and lowSNRs are a requirement for the narrowband criterionto be met.4and root-MUSICThe non-zero-bandwidth propertiesof ESPRITHaving established a criterion for narrowband, we firstinvestigate how it applies to some super-resolutionalgorithms. The results presented here are asymptotic,in that they are for the known (infinite sample) covari-ance case and are intended to give insight into the per-formance of the algorithms. It is assumed that thereader is already familiar with the essence of thesealgorithms, the details of which may be found in [6-91.Since we are assuming a known rather than estimatedcovariance matrix, the results for the various imple-mentations of ESPRIT and MUSIC are the same. Inthe implementation of the algorithms we define the sig-nal subspace using the effective rank criterion describedearlier. Before showing simulation results, we will firstattempt to predict the performance of these algorithmsas the zero-bandwidth model begins to break down.Consider a single signal incident on the array froman angle other than array broadside. According toeqn. 29, as the bandwidth is increased, at some pointthe effective rank of the signal subspace will increasefrom one to two. Both the MUSIC and ESPRIT algo-rithms attempt to find poles on the unit circle whichare related to the AOAs of any signals present. In thecase of non-zero-bandwidth signals, these poles attemptto form a low-rank approximation of the true environ-ment. For the case of a rank-two approximation thetwo poles will be separated in the AOA as described byeqns. 23-26. This is pictured qualitatively in Fig. 4.

    Fig.4Many super-resolution algorithms form a low-rank approximation to th e trueenvironment, and find the poles corresponding to the low-rank approximation(i) Low-rank approximation of a non-zero-bandwidth signal; (ii) section of cir-cle covered by the signal's bandwidth; (iii) unit circleNon-zero-bandwidth signal occupying a region on the unit circle

    Thus, when the zero-bandwidth model begins to fail,we expect root-MUSIC and ESPRIT to begin towrongly estimate two AOAs for a single signal insteadof one. The estimated AOAs will be given by eqn. 25for the combinations of bandwidth, SNR and AOAwhere a rank-two approximation of the signal is appro-88

    priate. Predicted and simulation results are plotted inFig. 5 for a single signal incident on a 10-element arrayfrom 30", for fractional bandwidths of 0 to 5% with anSNR of 30dB. Based on eqn. 29, the narrowband sig-nal model becomes invalid at a fractional bandwidth ofabout 0.8%. For fractional bandwidths above 0.8%, thepredicted AOAs based on eqn. 26 agree with the simu-lation results for both algorithms.I ..51

    h5'8 0.50i1LILo.-

    0.49 I I . I0 1 2 3 4 5fractional bandwidth, %

    F i .5 Predicted and simulated AOAs for root-MUSIC and ESPRIT asaj8nction of bandwidthAfter the transition to the rank-two signal model both algorithms gave biasedAOA measurements as predicted_ _ _ _ uredicted transition~ predicted AOA0 root-MUSICx ESPRIT

    The erroneous results for root-MUSIC and ESPRITare caused by the mismatch between the rank of thesignal subspace and the number of signals present. It iswell known in the literature that both MUSIC,ESPRIT and many other super-resolution algorithmsare asymptotically unbiased for the zero-bandwidth sig-nal model. We have just seen that this is not the casefor the non-zero-bandwidth signal model.5Cramer-Rao boundAdaptive beamformer performance and theOne common measure of adaptive beamformer per-formance is the signal-to-interference-plus-noise-ratio(SINR) loss. SINR loss is defined as the ratio of theSINR achieved by the adaptive beamformer to thatwhich could be achieved by a matched filter in theabsence of interference. If w is the adaptive weight vec-tor and v the target steering vector, then SINR loss isdescribed by the equation

    In this Section, we will analyse the performance of theoptimal adaptive weight vector, w = R-'v, i.e. theweight vector which maximises the SINR performanceof the adaptive beamformer. As in Section 4, theresults presented here assume a known covarianceFig. 6 shows plots of SINR loss against azimuth fora single 'jammer' incident on a 20-element array from30" with an SNR of 50dB (the same scenario as Fig. 2) .As the bandwidth is increased, the angular region inwhich the jammer denies coverage grows. Here we willdefine a second performance metric, the useablebeamspace fraction (UBSF). The UBSF is theproportion of the beamspace (evaluated either in sin(0)

    or wavenumber rather than ") where the SINR loss isless than 5dB. The 5-dB-loss figure is somewhat

    matrix.

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    arbitrary, but is used because a 5dB loss limits aradar's performance to about 75% of its interfererice-free detection range.0

    -5mU0!Iz -15v)

    g -10-

    -20

    -25 0 0.2 0.4 0.6 0.8 1sin (azimuth)

    Fig.6Note how the width of the null widens with increasing bandwidth~ 0%

    SINR loss against target AO A for a jammer at 30"1Y"5%)

    _ _ _

    100

    98aP 96m= 94U-0

    92

    90

    Fig.7 UBSFof 50dBEqn. 29 predictswidth of 0.027%~ UBSF

    - rank

    0.1 0.2 0.3 0.4 0.5fractional bandwidth,%

    and estimated rank against bandwidth or a jumm er SN Rthat the narrowband assumption fails at a fractional band-

    3

    YI - - - - - -g08mU"v)3 95 I'3 90 - - - - - I 15 I 10

    0 20 40 60 80 100JNR, dBFig.8 UBSF and estimated rank against jamme r SNR fo r a fractionalbandwidth of 0.2%Eqn. 29 predicts that the narrowband assumption fails at a jammer SNR of32 dB

    For the scenario of Fig. 6, Figs. 7 and 8 show plotsof UBSF as a function of the bandwidth (see Fig. 2)and jammer SNR (see Fig. 3). Additionally, plots areshown of the measured estimated rank of the interfer-ence subspace. Clearly, there is a close relationshipbetween the interference rank and the UBSF. It is alsoobvious that eqn. 29 correctly predicts where the zero-bandwidth ceases to closely predict performance.IE E Proc.-Rudar, Sonar Nuvig., Vol. 145, o . 2, April 1998

    Now let us consider the Cramer-Rao bound (CRB)for non-zero-bandwidth signals. We shall use the adap-tive beamforming scenario used above, and look at theCRB for the angle error of a OdB SNR target in thepresence of the 50dB jammer at 30".The unit of choicewill be the beam-split ratio (frequently used in theradar community), which is given by the ratio of thearray's beamwidth to the RMS angle error computedusing the CRB.In order to compute the CRB, let us take the vectorU of the unknown parameters. In our scenario, theunknown parameters are the target's AOA and magni-tude [Note 21. The CRB for the error covariance matrixof an unbiased estimator of the real parameters U = [ u l ,u2, ...I is given by the inverse of the Fisher InformationMatrix (FIM) J . Assuming that the data obeys the zeromean multivariate Gaussian p.d.f., the FIM for the sto-chastic signal model is conveniently computed usingBang's formula [l11 as

    where R, he exact covariance of all the observed data,is a function of U, and Ji j is the ijth element of J . Thedetails of the computation of J and the resulting CRBon the variance of the angle errors for the non-zerobandwidth case are given in the Appendix.

    1 2 ,

    -1 -0.5 0 0.5 1sin (azimuth)Fig.9 C R B against targe t signal A O A with a jamm er at 30"The width of the region denied by the jammer increases with bandwidth. Also,note how at the wider bandwidth the target accuracy degrades off broadsideowing to the effects of dispersion

    ~ 0%1%>5% )_ _ -

    Fig. 9 shows plots of the CRB as a function of thetarget AOA for the radar and interference scenariodescribed above using a OdB SNR target. As the band-width increases, the region around the jammer wheregood angle accuracy is denied increases with band-width. Additionally, at the higher bandwidths the effectof dispersion on the target away from the array broad-side reduces the achievable angle accuracy. In a similarfashion to the SINR-loss case earlier, we can define aCRB UBSF. The CRB UBSF is the proportion of thebeamspace where a beam-split ratio better than 5 canbe achieved. Fig. 10 shows a plot of both the CRBUBSF and interference rank against bandwidth. As inthe adaptive beamforming scenario results reportedabove, there is a close relationship between the CRBUBSF and the interference rank, and eqn. 29 correctlyNote 2: For the radar scenario assumed in [Note I], it can be shown thatth e CRB for the target's parameters is unaffected by the estimation of theinterference-plus-noisecovariance matrix, since the cross terms in theFIM are zero [lo]

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    predicts the region where the zero-bandwidth modelfails to accurately predict performance.100 I 1 318

    0 0.1 0.2 0.3 0.4 0.5fractional bandwidth, %Fig. 10

    __ UBSFCR B UBSF us ufunct ion of bandwidth

    6 ConclusionsIn this paper, an expression was derived which accu-rately defines the notion of narrowband as used inadaptive array signal processing and estimation. Theresulting expression, which is a function of fractionalbandwidth, AOA, SNR array gain and aperture, pro-vides more insight into the validity of the narrowbandassumption than previous definitions. Simulations wereused to verify that the expression correctly predictswhere the narrowband assumption fails in adaptivebeamformer and super-resolution algorithm perform-ance, and in computing the Cramer-Rao bound forAOA estimation.Additionally, it was shown that the root-MUSIC andESPRIT algorithms are biased estimators of the AOAin the presence of non-narrowband signals. A by-prod-uct of the derivation of the narrowband expression pre-dicts the bias.7 AcknowledgmentsThis work was sponsored by the United States Depart-ment of the Navy under Air Force contract F19628-95-C-0002. Opinions, interpretations, conclusions andrecomendations are those of the author, and are notnecessarily endorsed by the United States Air Force.8 References

    7

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    COMPTON , R.T.: Adaptive antennas (Prentice Hall, 1988)HUDSON, J .: Adaptive array principles (Peter Peregrinus, Lon-don, 1981)LEE, H.B.: Eigenvalues and eigenvectors of covariance matricesfor signals closely spaced in frequency, IEEE Trans. Signal Proc-ess., 1992, 40, pp. 2518-2535HAYWARD, S.D.: Effects of motion on adaptive arrays, IEEProc. Radav Sonar Navig., 997, 144, pp. 15-20ZATMAN, M. A.: Production of adaptive array troughs by dis-persion synthesis, Electron. Lett., 1995, 31, pp. 2141-2142PAULRAJ, A., ROY, R. , and KAILATH, T.: A subspace rota-tion approach to signal parameter estimation, Proc. IEEE, 1986,pp. 10441045ROY, R., and KAILATH, T.: ESPRIT - estimation of signalparameters via rotational invariance techniques, IEEE Trans .Acoust . Speech Signal Process., 1989, 37 , pp, 984995BARABELL, A.J.: Improving the resolution performance ofeigenstructure-based direction finding algorithms. Proceedings ofICASSP-83, Boston, MA, May 1983, pp. 336-339RAO. B.D. . and HARI. K.V.S.: Performance analvsis of root-MUSIC, IEEE Trans. Acoust. Speech Signal Proceis., 1989, 37 ,pp, 1939-1949PhD dissertation, Yale University, New Haven, CT, 1971

    10 WA RD, J. : Personal communication11 BANGS, W.J .: Array processing with generalized beamformers.

    9 Appendix: Non-zero bandwidth Cramer-Raobound on angle estimationIn this Appendix, we detail the computation of thenon-zero bandwidth CRB for angle estimation used inSection 5 . First, we will describe the computation ofthe elements of the Fisher information matrix (FIM)denoted by the symbol J , and then we will convert theresulting error variances into our unit of choice-beam-split ratio. For the scenario considered, the twounknown parameters are the target signals AOA, 8,and power (SNR) s . However, it will prove convenientto use the inter-element phase(for a half-wavelength space array) rather than 0 itself.In order to compute the FIM using eqn. 31, the partialderivatives to the data covariance matrix R withrespect to (w.r.t.) w and s are required. The covariancematrix of M uncorrelated signals plus noise isassuming a noise variance of 1. From eqn. 33 it is clearthat the partial derivatives of R w.r.t. the ith signal areindependent of the other M - 1 signals present and thenoise. Thus the partial derivatives required here may beobtained from the non-zero-bandwidth covariancematrix of the target signal only. Rewriting eqn. 8, theklth element of the target signal covariance matrix isgiven by

    w = nsin(8) ( 3 2 )

    R = RI +Ra + . . .+Rn/r+ I ( 3 3 )

    e j n w sin (n gw )rk 1 = ssinc n f w ejnw = s

    (.%U)(34)

    (? 1where n = k - 1 (recall that bf is the fractional band-width). The partial differential of eqn. 34 w.r.t. s issimply

    i.e. the klth element of the target signals covariancematrix assuming an SNR of 1. After working throughthe product-quotient form of eqn. 34 , its derivativew.r.t. o sark1 -__ -dW

    n+eJnw sin ( n 2 u )--s 2 (36)

    (&W)From eqn. 35 , the partial derivative of R w.r.t. the tar-get signals power is simply the target signals covari-ance matrix with an SNR of 1. For the partialderivative of R w.r.t. w , eqn. 36 must be computed foreach combination of k and 1. Utilising eqn. 31 , the ele-ments of the FIM are then given by

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    Let us define C = J-I. The variance of w is then givenby the first element of C, denoted cll . The RMS angle An N-element array produces N beams between sin(0)= -1 and sin(@ = 1. Thus the beam-split ratio is givenn

    error in sin(0) space is given by by(39)Lbeam-split ratio =-e

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