Spatial Array Processing

Transcription

1 Spatial Array Processing Signal and Image Processing Seminar Murat Torlak Telecommunications & Information Sys Eng The University of Texas at Austin,

2 Introduction A sensor array is a group of sensors located at spatially separated points Sensor array processing focuses on data collected at the sensors to carry out a given estimation task Application Areas Radar Sonar Seismic exploration Anti-jamming communications YES! Wireless communications 2

3 Problem Statement s (t) s 2 (t) θ θ2 x (t) x 2(t) x (t) x 4 (t) x 5(t) x 6 (t) Find Number of sources 2 Their direction-of-arrivals (DOAs) Signal Waveforms

4 Assumptions Isotropic and nondispersive medium Uniform propagation in all directions Far-Field Radius of propogation >> size of array Plane wave propogation Zero mean white noise and signal, uncorrelated No coupling and perfect calibration 4

5 Antenna Array θ Source X X 2 X X 4 X5 Array Response Vector Far-Field Assumption - Delay Narrowband =) Assumption Phase Shift a() = [;e j2f c4 sin =c ;::: ;e j2fc44 sin =c ] T Single Source Case =) x(t) 2 64 x (t) x 2 (t) x M (t) 75 = 2 64 s (t) s (t, ) s (t, (M, ) ) e,j2f c e,j2f c(m,) 75 s (t) = a( )s (t) where = 4 sin =c 5

6 General Model By superposition, for d signals, x(t) = a( )s (t) ++a( d )s d (t) = dx k= a( k )s k (t) Noise x(t) = dx k= a( k )s k (t) +n(t) = AS(t) +n(t) where A = [a( );::: ;a( d )] and S(t) = [s (t);::: ;s d (t)] T : 6

7 Low-Resolution Approach:Beamforming Basic Idea xi(t) = dx k= = e (i,)(j2f c4 sin k =c) s k (t) = dx k= s k (t)e jw k (i,) where w k = 24 sin( k )=c and i = ;::: ;M Use DFT (or FFT) to find the frequencies fw k g F = [F(w ) F(w M )] = e jw e jw 2 e jw M e j(m,)w e j(m,)w 2 e j(m,)w M 7 5 Look for the peaks in jf(xi(t))j = jf x(t)j 2 To smooth out noise B(wi) = N NX t= jf x(t)j 2 7

8 Beamforming Algorithm Algorithm P Estimate R x = N N t= x(t)x (t) 2 Calculate B(w i ) = F (w i )R x F(w i ) Find peaks of B(w i ) for all possible w i s 4 Calculate k, i = ;::: ;d Advantage - Simple and easy to understand Disadvantage - Low resolution 8

9 Number of Sources Detection of number of signals for d < M, x(t) = As(t) +n(t) Rx = Efx(t)x (t)g=aefs(t)s (t)ga + Efn(t)n (t)g = A {z} Md Rs {z} dd A +n 2 I {z} dm {z } Rs {z } 2 n I where 2 n is the noise power No noise and rank of R s is d Eigenvalues of Rx = ARsA will be f ;::: ; d ;0;::: ;0g: Real positive eigenvalues because Rx is real, Hermition-symmetric rank d Check the rank of R x or its nonzero eigenvalues to detect the number of signals Noise eigenvalues are shifted by n 2 f + 2 n ;::: ; d + 2 n ;2 n ;::: ;2 ng: where > :::> d and >> 0 Detect the number of principal (distinct) eigenvalues 9

10 MUSIC Subspace decomposition by performing eigenvalue decomposition Rx = ARsA + 2 n I = MX k= k e k e k where e k is the eigenvector of the k eigenvalue spanfag = spanfe ;::: ;e d g =spanfe s g Check which a() spanfe s g or P A a() or P? A a(), where P A is a projection matrix Search for all possible such that jp? A a()j2 = 0 or M () = P A a() = After EVD of R x P? A = I, E se s = E ne n where the noise eigenvector matrix E n = [e d+ ;::: ;e M ] 0

11 Root-MUSIC For a true, e j2f c4 sin =c is a root of P (z) = MX k=d+ [;z;::: ;z M, ] T e k e k [;z, ;::: ;z,(m,) ]: After eigenvalue decomposition, - Obtain fe k g d k= -Formp(z) - Obtain 2M, 2 roots by rooting p(z) -Pickdroots lying on the unit circle - Solve for f k g

12 Estimation of Signal Parameters via Rotationally Invariant Techniques (ESPRIT) Decompose a uniform linear array of M sensors into two subarrays with M, sensors Note the shift invariance property a (2) () = 2 64 e jw e j2w e j(m,)w e jw = 4 e j(m,)w 75 ejw = a () e jw General form relating subarray () to subarray (2) A (2) = A () 26 4 ejw 75 = A(): e jw d contains sufficient information of f k g 2

13 ESPRIT spanfe s g = spanfag and E s = AT - T is a d d nonsingular unitary matrix - T comes from a Grahm-Schmit orthogonalization of Ab in R x = E s s E s + E ne n A H R s A + 2 n I E (2) s = A (2) T and E () s = A () T Es(2) = A (2) T = A () T = Es()T, T Multiply both sides by the pseudo inverse of E () s E ()# s Es(2) = (E () E () ), E () E () T, T = T, T where # means the pseudo-inverse A # = (A sh A), A sh Eigenvalues of T, T are those of

14 Superresolution Algorithms Calculate R x = N P N k= x(k)x (k) 2 Perform eigenvalue decomposition Based on the distribution of f k g, determine d 4 Use your favorite diraction-of-arrival estimation algorithm: (a) MUSIC: Find the peaks of M () for from 0 to 80 - Find f^ k g d k= M () corresponding the d peaks of (b) Root-MUSIC: Root the polynomial p(z) - Pick the d roots that are closest to the unit circle fr k g d k= and ^ k = sin, r kc 2fc (c) ESPRIT: Find the eigenvalues of E ()# s E (2) s, f k g - ^ k = sin, kc 2fc4 4

15 Signal Waveform Estimation Given A, recover s(t) from x(t) Deterministic Method No noise case: find w k such that w k? a( i );i6=k; w k 6? a( k ) A # can do the job A # x(t) = A # As(t) = s(t) With noise, n(t) A # x(t) = s(t) + A # n(t) Disadvantage =) increased noise 5

16 Stocastic Approach Find w k to minimize min Efjw k x(t)j 2 g = a (k)wk= Use the Langrange method min w a k R kw k (k)wk= min Efjw a ( k )w k x(t)j 2 g, min w k = ;w k R kw k + 2(a ( k )w k, ) k Differentiating it, we obtain Rxw k = a( k );orw k = R, x a( k) Since a ( k )w k = a ( k )R, x a( k) =, Then = a ( k )R, x a( k) Capon s Beamformer w k = R, x a( k)=(a ( k )R, x a( k)) 6

17 Subspace Framework for Sinusoid Detection x(t) = dp k= k e ( k+j!k)t Let us select a window of M, ie, x(t) = [x(t);::: ;x(t,m + )] T Then x(t) = = = dx k= dx k= x(t) x(t, ) x(t, M +) 2 64 = 7 5 e,( k +j! k ) dx k= e ( k +j! k )(,M+) k e ( k +j! k )t k e ( k +j! k )(t,) k e ( k +j! k )(t,m+) 75 {z } a( k ) a( k )s k (t) = As(t); k e ( k +j! k )t {z } s k (t) 7 5 where M is the window size, d the number of sinusoids, and k = e k +j! k 7

18 Subspace Framework for Sinusoid Detection Therefore, the subspace methods can be applied to find f k + j! k g Recall x(t) = dx k= k e ( k+j!k)t Then finding f k g is a simple least squares problem 8

19 Wireless Communications co-channel interference Multipaths Direct Path Cellular Telephony Office Building Residential Area Outdoors Personal Communications Services (PCS) To Networks Direct Path Multipath Wireless LAN Direct Path Increasing Demand for Wireless Services Unique Problems compared to Wired communications 9

20 Problems in Wireless Communications Scarce Radio Spectrum and Co-channel Interference Multipath Multipath Multipath Direct Path Base Station Time Desired Signal Reflected Signal Coverage/Range 20

21 Smart Antenna Systems Employ more than one antenna element and exploit the spatial dimension in signal processing to improve some system operating parameter(s): - Capacity, Quality, Coverage, and Cost User One User Two Multiple RF Module Advanced Signal Processing Algorithms Conventional Communication Module 2

22 Experimental Validation of Smart Uplink Algorithm Comparison of constellation before (upper) and after smart uplink processing (middle and lower) imaginary axis imaginary axis imaginary axis real axis Antenna Output real axis Equalized Signal real axis Equalized Signal 2 22

23 Selective Transmission Using DOAs Beamforming results for two sources separated by 20 Power Spectrum Power Spectrum Frequency [Hz], User # Frequency [Hz], User #2 x 0 4 x 0 4 2

24 Selective Transmission Using DOAs Beamforming results for two sources separated by Power Spectrum Power Spectrum Frequency [Hz], User # Frequency [Hz], User #2 x 0 4 x

25 Future Directions Adapt the theoretical methods to fit the particular demands in specific applications Smart Antennas Synthetic aperture radar Underwater acoustic imaging Chemical sensor arrays Bridge the gap between theoretical methods and real-time applications 25