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

1. Direction FindingPart 2: Array Signal Processing, Errors, Location Calculation Prof. Venceslav Kafedziski University "Ss Cyril and Methodius" Skopje, Republic of Macedonia

2. 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. 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. 4 Antenna array basics angle f is the azimuth and q=90o-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).

5. 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. 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. 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. 8 Antenna array processing

9. 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. 10 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 u=[u1 u2 ... uM]'. Delay-and-sum method

11. 11 Beamforming � block diagram S

12. 12 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 q=p/2) can steer the antenna beam to any desired direction f0. Beamforming � regular arrays

13. 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 w=a(f0). The receiver antenna has the highest gain in the direction f0, when w=a(f0). This is because w=a(f0) aligns the phases of the signal components arriving from f0 at the antenna elements, causing them to add constructively.

14. 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 w=a(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. 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. 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. 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. 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): minwE[|y(k)|2]=minwwHRuuw subject to wHa(f0)=1

19. 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. 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. 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. 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 A=[a(f1) a(f2) ... a(fD)] s=[s1, ... , sD]' is the signal vector, and n=[n1, ... ,nM] is a noise vector with components of variance sn2.

23. 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 Ruu=E[uuH]=ARssAH+sn2I 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. 24 MUSIC The M-D smallest eigenvalues of Ruu are equal to sn2, and the eigenvectors qi, i=D+1, ... ,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, i=D+1, ... ,M, the DOAs f1,f2, ... ,fD, can be determined.

25. 25 MUSIC To form the noise subspace, we form a matrix Vn containing the noise eigenvectors qi, i=D+1, ... ,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. 26 Examples of Capon and MUSIC

27. 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. 28 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. Multichannel direction finders

29. 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. 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 n<1, Class B for 1<n<6.5, Class C for 6.5<n<25, class D for n>25.

31. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 SNR[dB] for bearing fluctuation of 1�.

43. 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. 44 Location calculation steps

45. 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. 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. 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. 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. 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. 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. 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. 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.