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Direction Finding Part 2: Array Signal Processing, Errors, Location Calculation

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Title: Direction Finding Part 2: Array Signal Processing, Errors, Location Calculation


1
Direction FindingPart 2 Array Signal
Processing, Errors, Location Calculation
INA Academy Workshop Spectrum Management
Series Workshop 3 "Measurements and Techniques"
  • Prof. Venceslav Kafedziski
  • University "Ss Cyril and Methodius"
  • Skopje, Republic of Macedonia

2
Outline
  • Correlative interferometer.
  • Antenna array processing techniques.
  • Classical beamforming.
  • High resolution methods Capon's beamformer.
  • High resolution methods Subspace methods
    (MUSIC).
  • Display of bearings.
  • Classification of bearings.
  • Error sources.
  • Location calculation method of triangulation and
    single site location.

3
Array processing
  • Classical DF methods usually use several antennas
    or antenna arrays to measure phase differences.
  • Modern DF methods make use of all the information
    received on different elements of the antenna
    array.
  • Array signal processing finds applications in
    mobile communications, radar, sonar, seismic
    exploration.
  • Next generations of wireless systems will use
    antenna arrays - smart antennas. The goal is to
    enhance the desired signal, and null or reduce
    interference.
  • Direction finding in multipath conditions also
    uses smart antennas.

4
Antenna array basics
  • angle f is the azimuth and q90o-elevation angle.
  • a(f,q)1 a1(f,q) a2(f,q) ... aM-1(f,q) is
    called steering vector.
  • a(f,q) describes the phases of the signal at each
    antenna element relative to the phases of the
    signal at the reference element (element 0).

k2p/l is the wave number
5
Correlative interferometer
  • The correlative interferometer is based on a
    comparison of the measured phase differences
    between the antenna elements of the DF antenna
    system with those obtained for the same antenna
    system at all possible directions of incidence.
  • The comparison is made by calculating the
    correlation of the two data sets (or the scalar
    product of two vectors obtained by multiplying
    the coordinates element by element and summing
    the result).
  • Using different comparison data sets for
    different wave directions, the bearing is
    obtained from the data set for which the
    correlation is at a maximum.

6
Correlative interferometer example
  • 5-element antenna array used
  • each column of the data matrix corresponds to
    azimuth direction f and forms a comparison
    vector.
  • the upper data vector contains the measured phase
    differences
  • each column of the reference matrix is correlated
    with the measurement vector
  • The angle associated with the comparison vector
    resulting in maximum K(f) is the bearing.

7
Antenna array processing
  • The development of Digital Signal Processing has
    enabled the use of new approaches for direction
    finding.
  • The requirement for a simple and
    frequency-independent relationship between the
    signals obtained on antenna elements and the
    bearing no longer applies complex mathematical
    relationships can be efficiently computed.
  • High-resolution methods allow the separation of
    several waves arriving from different directions
  • Methods of direction finding are based on
    Direction of Arrival (DOA) estimation. DOA can be
    converted to direction relative to the true north.

8
Antenna array processing
  • The outputs of the individual antenna elements
    are taken to a network which contains test
    signal inputs and multiplexers.
  • The signals are then converted to an
    intermediate frequency and digitized.
  • The digital data are down-converted into the
    baseband. The complex samples of the baseband
    signal ui (i1,2, ... M) are filtered and applied
    to the bearing calculation section.

9
Beamforming
  • Conventional methods of DOA are based on the
    concept of beamforming, i.e. steering antenna
    array beams in all possible directions and
    looking for peaks in the output power.
  • If the antenna array signals ui are multiplied by
    complex weighting factors wi and added, a sum
    signal is obtained which depends on the direction
    of wave incidence.
  • With conventional beamforming algorithms the
    phases of the weighting factors are chosen so
    that the weighted element signals are added in
    phase and thus yield a maximum sum signal for a
    given wave direction.

10
Delay-and-sum method
  • The output signal is given by a weighted sum of
    the element inputs
  • The total output power of the conventional
    beamformer is
  • Ruu is the autocorrelation matrix of the input
    data vector uu1 u2 ... uM'.

11
Beamforming block diagram
  • S
  • Since both u and w are complex vector-columns, y
    is a complex scalar.
  • The maximum of yy is searched for.
  • The direction corresponding to maximum yy is
    the bearing.

12
Beamforming regular arrays
  • If the antenna array has a regular geometry, the
    weighting factors can be calculated analytically.
  • For a Uniform Linear Array (ULA) with antenna
    spacing D the weights equal to
  • (assuming that qp/2) can steer the antenna beam
    to any desired direction f0.

13
Beamforming - analysis
  • Consider a signal s(k) impinging on the array at
    an angle f0. The power at the beamformer output
    can be expressed as
  • The output power is maximized when wa(f0). The
    receiver antenna has the highest gain in the
    direction f0, when wa(f0). This is because
    wa(f0) aligns the phases of the signal
    components arriving from f0 at the antenna
    elements, causing them to add constructively.

14
Beamforming - analysis
  • In the classical beamforming approach for DOA
    estimation, the beam is scanned over the angular
    region of interest in discrete steps by forming
    weights wa(f) for different f and the output
    power is measured
  • If we have an estimate of the input
    autocorrelation matrix and know the steering
    vectors a(f) for all f's of interest (through
    calibration or analytical computation), it is
    possible to estimate the output power as a
    function of the DOA f, called spatial spectrum.
    The DOA can be estimated by locating peaks in the
    spatial spectrum.

15
Beamforming spatial resolution
  • The spatial resolution depends on the mainlobe
    width.
  • Figure gives an example for ULA and weights equal
    to 1.
  • Delay-and-sum method has a low resolution.
  • The resolution can be increased by increasing M,
    but this increases complexity and storage.

16
Super resolution DF methods
  • If in the frequency channel of interest unwanted
    waves are received in addition to the wanted
    wave, conventional beamforming may lead to
    bearing errors.
  • If the power of the interference wave component
    is smaller than that of the wanted wave
    component, the direction finder can be designed
    to minimize the bearing errors by choosing a
    sufficiently wide antenna aperture.
  • If the interference wave component is greater or
    equal to the wanted wave component, the
    interference waves have to be also determined in
    order to be able to eliminate them. This means
    that the secondary maxima in the spatial spectrum
    have to be evaluated too.

17
Super resolution DF methods
  • The limits in determining secondary maxima are
    reached if
  • the ratio between primary maximum and
    secondary maxima becomes too small, or
  • the angle difference between wanted wave
    and interference wave is smaller than the width
    of the main lobe
  • By optimizing the weighting factors, the level of
    the secondary maxima can be reduced but at the
    same time the width of the primary maximum is
    increased. The aim of the super-resolution (SR)
    DF method is to resolve this problem.

18
Capon's method
  • Capon's minimum variance method attempts to
    overcome the poor resolution problems associated
    with classical beamforming.
  • The technique uses some of the degrees of freedom
    to form a beam in the desired look direction,
    while simultaneously using the remaining degrees
    of freedom to form nulls in the direction of
    interfering signals.
  • Minimizes the contribution of the undesired
    interferences by minimizing the output power
    while maintaining the gain along the look
    direction to be constant (unity)
  • minwEy(k)2minwwHRuuw subject to
    wHa(f0)1

19
Capon's method
  • Capon's method is also called a minimum variance
    method, since it minimizes the variance (average
    power) of the output signal while passing the
    signal arriving in the look direction without
    distortion.
  • The output power of the array, as a function of
    DOA is given by Capon's spatial spectrum
  • By computing and plotting the spectrum over the
    whole range of f, the DOA's can be estimated by
    locating the peaks in the spectrum.

20
Capon's method
  • Capon's method has better resolution than the
    delay-and-sum method.
  • The resolution strongly depends on the
    signal-to-noise ratio.
  • Capon's method fails if signals that are
    correlated with the Signal-of-Interest are
    present. The correlated components may be
    combined destructively in the process of
    minimizing the output power.
  • Capon's method requires the computation of a
    matrix inverse, which can be computationally
    expensive for large antenna arrays.

21
Subspace methods
  • The subspace methods are aimed at eliminating the
    effect of noise. This can be done by splitting up
    the M-dimensional space spanned by the antenna
    element outputs into a signal subspace and a
    noise subspace.
  • These methods exploit the structure of a more
    accurate data model for the case of arrays of
    arbitrary form.
  • MUSIC algorithm is a high resolution MUltiple
    SIgnal Classification technique based on
    exploiting the eigenstructure of the input
    covariance matrix. Provides information about the
    number of incident signals, DOA of each signal,
    strengths and cross correlations between incident
    signals, noise power, etc.

22
MUSIC
  • If there are D signals incident on the array, the
    received input data vector at an M-element array
    can be expressed as a linear combination of the D
    incident waveforms and noise
  • where A is the matrix of steering vectors
  • Aa(f1) a(f2) ... a(fD)
  • ss1, ... , sD' is the signal vector, and
    nn1, ... ,nM is a noise vector with components
    of variance sn2.

23
MUSIC
  • The received vectors and the steering vectors can
    be visualized as vectors in an M-dimensional
    vector space.
  • The input covariance matrix is
  • RuuEuuHARssAHsn2I
  • where Rss is the signal correlation matrix.
  • The eigenvectors of the covariance matrix Ruu
    belong to either of the two orthogonal subspaces,
    the principal eigen subspace (signal subspace)
    and the non-principal eigen subspace (noise
    subspace).
  • The dimension of the signal subspace is D, while
    the dimension of the noise subspace is M-D.

24
MUSIC
  • The M-D smallest eigenvalues of Ruu are equal to
    sn2, and the eigenvectors qi, iD1, ... ,M,
    corresponding to these eigenvalues span the noise
    subspace.
  • The D steering vectors that make up A lie in the
    signal subspace and are hence orthogonal to the
    noise subspace.
  • By searching through all possible array steering
    vectors to find those which are orthogonal to the
    space spanned by the noise eigenvectors qi,
    iD1, ... ,M, the DOAs f1,f2, ... ,fD, can be
    determined.

25
MUSIC
  • To form the noise subspace, we form a matrix Vn
    containing the noise eigenvectors qi, iD1, ...
    ,M.
  • Then aH(f)VnVnHa(f)0 for f corresponding to the
    DOA of a multipath component.
  • The DOAs of the multiple incident signals can be
    estimated by locating the peaks of a MUSIC
    spatial spectrum
  • The resolution of MUSIC is very high even in low
    SNR.
  • The algorithm fails if impinging signals are
    highly correlated.

26
Examples of Capon and MUSIC
Comparison with classical beamforming (a) Capon
beamformer with SNR100. (b) Capon beamformer
with SNR10. (c) MUSIC algorithm with SNR10.
(a)
(b)
(c)
27
Display of bearings
  • The display of the DF results is of great
    importance as an interface to the operator.
  • Distinction is to be made whether the display is
    the DF result of a single channel or of a
    multichannel direction finder.
  • In a single-channel display, the following
    parameters are usually indicated numeric DF
    value, azimuth in polar coordinates, elevation as
    bar graph or polar diagram (combined with azimuth
    display), DF quality, level, histogram of DF
    values, DF values versus time (waterfall).

28
Multichannel direction finders
  • Multichannel direction finders are implemented
    with the aid of digital filter banks (FFT and
    polyphase filters).
  • These direction finders allow quasi-simultaneous
    direction finding in a frequency range from some
    100 kHz up to a few MHz. Scan mode is
    additionally provided to cover larger frequency
    ranges.
  • Usually the following display modes are provided
    DF values versus frequency, DF values versus
    frequency and time (e.g. by using different
    colours for the DF values), level versus
    frequency (power spectrum), level versus time and
    frequency (using different colours for level
    values), histograms.

29
DF below 30 MHz
  • Propagation between transmitter and receiver may
    involve different modes, including a ground wave,
    and single or multiple reflections on E or F
    layers of ionosphere.
  • Since the horizontal stratification of the
    ionosphere and the ion-density are not stable,
    the reflections are irregular.
  • DF operating below 30 MHz are susceptible to
    errors induced by reflections of transmissions
    from the ionosphere.
  • The error in the bearing due to a given angle of
    inclination relative to the reflective surface
    increases with the angle of elevation of the ray
    received by the direction finder. Critical are
    situations with multiple reflections.

30
Classification of HF bearings
  • Reccomendation ITU-R SM.1269 classifies HF
    bearings into four classes A, B, C and D.
  • Classes A, B, C are defined as having probability
    less than 5 that bearing errors exceed 2, 5, and
    10 degrees respectively, and class D is for
    larger errors.
  • The errors in bearings are due to equipment,
    site of the DF, propagation, and operator. The
    total error variance n is equal to the sum of the
    error variances due to equipment, site of the DF,
    propagation, and operator.
  • Classes in terms of error variance Class A for
    nlt1, Class B for 1ltnlt6.5, Class C for 6.5ltnlt25,
    class D for ngt25.

31
Classification of HF bearings
  • Series of N measurements can be conducted, from
    which the mean and the variance can be computed.
    If am is the mean, the variance is computed
    according to
  • To decrease variance, n groups of measurements
    are performed on the same transmitter. Each group
    will have a different mean value. The variance of
    the mean is going to decrease

32
DF above 30 MHz
  • A basic system consists of two or more
    remote-controlled direction-finding stations and
    a manned monitoring station.
  • System can have two or more mobile stations
    equipped with a direction finder and monitoring
    equipment, connected via VHF radio data links.
  • The bearings are classified for accuracy into
    four classes A, B, C and D.
  • Classes A, B, C are defined as having probability
    less than 5 that bearing errors exceed 1, 2, and
    5 degrees respectively, and class D is defined as
    resulting in errors larger than Class C.

33
DOA versus LOB
  • It is important to differentiate the direction of
    arrival DOA from geographic line of bearing
    (LOB).
  • The DOA does not relate to a geographical
    direction. The DOA is a measurement that results
    in a relative angle between an emitter and a
    specific DF antenna orientation.
  • The LOB is a measurement that contains a
    combination of the errors introduced in the DOA
    measurement and those contributed by the
    determination of the magnetic heading as adjusted
    with the true north deviation. It is with these
    factors that a line can be plotted on a map.

34
Error sources
  • The DF accuracy is affected by a number of
    influences
  • Wave propagation (usually disturbed by obstacles)
  • Signals radiated by the emitters are modulated,
    limited in time and their carrier frequency is
    often unknown
  • Received field is additionally superimposed by
    noise, co-channel interferers
  • Tolerances and noise in the DF system

35
Multiwave-related errors
  • The simple case of a plane wave occurs seldom in
    practice. In a real environment, other waves have
    usually to be taken into account which result
  • from other emitters in the same frequency channel
    (incoherent interference) or
  • from secondary waves (caused by reflection,
    refraction, diffraction coherent in-channel
    interference)
  • Usually, a large number of waves is involved.

36
Multiwave-related errors
  • The direct wave component with the amplitude 1
    arrives from an angle of 90.
  • There is a secondary reflected wave.
  • If the majority of waves arrives from the
    direction of the emitter, the DF error can be
    reduced by increasing the antenna aperture.

37
Synchronization errors
  • The receive sections of most multireceiver
    direction finders are calibrated for synchronism
    with the aid of a test generator prior to the DF
    operation. The transmission parameters in each
    section are measured in magnitude and phase, and
    the level and phase differences are stored. In
    the DF process the measured values are corrected
    by the stored difference values before the
    bearing is calculated.
  • Special attention is to be given to the frequency
    response of the filters. Digital filters have the
    advantage that they can be implemented with
    absolutely identical transfer characteristics.

38
Synchronization errors
  • Consider a 2-element interferometer. Different
    gain and phase in the receive sections cause DF
    errors that are smaller if the relative antenna
    aperture D/l increases.

39
Modulation errors
  • Usually the carrier signal of the emitter is
    modulated with a complex modulation function
    (complex envelope).
  • The modulation can affect the DF result in
    several aspects
  • Different envelope delay distortion in the DF
    channels
  • With sequential antenna scanning modulation
    function is not sufficiently stationary for the
    duration of the measurement or cannot be
    compensated for by other measures prior to DF
    evaluation
  • Possible decorrelation between the antenna
    elements if the spacing between the elements is
    greater than the coherence length Lk c0/B (B is
    the signal bandwidth).

40
Noise errors
  • Interference caused by noise has a limiting
    effect on the sensitivity of a DF system.
    Sensitivity is the field strength at which the
    bearing fluctuation remains below a certain
    standard deviation.
  • Consider internal noise produced in the system
    (antenna amplifier, DF converter, A/D converter)
  • For a 2-element interferometer the uncorrelated
    noise in the two receive sections causes
    statistically independent phase variations
    according to the signal-to-noise ratio

41
Noise errors
  • Mapping of the phase variation to virtual
    variations of the DF antenna positions yields for
    the bearing error
  • Error is smaller for larger relative antenna
    aperture D/l.

42
Noise errors
  • The variations caused by noise can be reduced by
    averaging. The variance improvement through
    averaging over K values is
  • Figure shows the two effects together antenna
    aperture and averaging. Parameter is SNRdB for
    bearing fluctuation of 1.

43
Location calculation
  • The location calculation is directly dependent on
    the quality of bearings of the various direction
    finding stations. Bearings should be analyzed at
    both DF stations and monitoring stations.
  • At DF stations classifying the bearings in case
    of the presence of interference, eliminating
    aberrant shootings, calculating the mean value
    and the variance of shootings.
  • At the monitoring station determining the
    bearings to be used for the location calculation,
    calculating the position of the probable
    transmission point, calculating the uncertainty
    ellipse.

44
Location calculation steps
  • Location calculation steps are given in the
    following figure.
  • The computer program "TRIANGULATION", that
    generally follows these steps, is available
    through ITU.

45
Determining reliable bearings
  • If there exist fast links between Monitoring
    Centre and DF stations, the Calculation Centre is
    provided, for each direction finder, with a
    number of elementary bearings. In that case,
    simple processing only is used at DF stations.
  • For slow links, making reliable bearings should
    be performed at the DF station. Making a reliable
    result is achieved by averaging the elementary
    bearings regarding the signal of interest for the
    Monitoring Centre.

46
Eliminate non-convergent bearings
  • On each shooting, the area covered by the own
    angular error of each of the shootings is
    constructed.
  • When areas partly overlap, shootings can be
    associated.
  • The association including the largest number of
    shootings is kept as the best one.
  • Shootings that are not parts of the best
    association are discarded.

47
Location calculation
  • The optimum point is searched applying the least
    squares method.
  • If P is any point and d1, d2, d3, ... the
    angular variations to be applied to each bearing
    to intersect P and n1, n2, n3, ... the
    variances of various bearings, define
  • The optimum point is the point minimizing Sp.
  • The uncertainty ellipse is the area centered
    around the optimum point. This calculation is
    performed from typical deviations from the
    various bearings.

48
SSL for HF emissions
  • The single site location (SSL) system allows
    determining the geographical position of a
    transmitter with a single radio direction-finder.
    Processing system on data supplied by the radio
    direction-finding (azimuth, elevation) associated
    with ionospheric predictions, allows estimating
    the transmitter distance with regard to the radio
    direction-finder, ranging up to 2500 km.
  • SSL radio direction finders simultaneously
    deliver the azimuth and elevation angle of the
    signal received by the antenna array. Propagation
    is by ground wave and ionospheric wave through
    multiple paths corresponding to different
    elevation angles.

49
SSL
  • The basis of SSL is the so-called Classical
    Method of Range Estimation. Assumes that the
    actual HF propagation may be modeled by assuming
    that reflection takes place from a simple
    horizontal mirror at the appropriate height.

50
SSL
  • There are two contra-polarized waves (circular or
    elliptical) - an ordinary (O) and an
    extraordinary (X) wave.
  • Location range calculation process calculating
    elevation histograms, filtering elevation
    diagrams, determining elevation packets,
    determining four basic distances for LH and RH
    polarization of O and X waves, and determining
    the final distance.
  • Limitations of SSL multi-hop propagation leads
    to calculated distances which are shorter than
    the real distances, and reflection may take place
    from layers of different heights.

51
Conclusions
  • Modern Direction Finders use digital signal
    processing.
  • Modern array signal processing methods for DF are
    used in conditions when many signals are present
    and achieve excellent resolution in the presence
    of noise.
  • To estimate bearing accuracy and calculated
    location accuracy it is important to estimate
    bearing errors.
  • Therefore, it is important to understand the
    error sources and the classification of bearing
    errors.
  • There is an increased need for monitoring
    personnel to understand principles of digital
    signal processing, which significantly improves
    and facilitates spectrum monitoring.

52
Literature
  1. Smart Antennas for Wireless Communications IS-95
    and Third Generation CDMA Applications, J.C.
    Liberti and T. S. Rappaport, Prentice Hall, 1999.
  2. Adaptive Filter Theory, S. Haykin, Prentice Hall,
    1991.
  3. Multidimensional Digital Signal Processing, D. E.
    Dudgeon and R. M. Merserau, Prentice Hall, 1984.
  4. International Telecommunication Union Spectrum
    Monitoring Handbook, ITU, 2002.
  5. Introduction into Theory of Direction Finding,
    Rohde Schwarz.
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