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

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