Radar Signal Demixing via Convex Optimization

Transcription

1 Radar Signal Dmixing via Convx Optimization Gongguo Tang Dpartmnt of Elctrical Enginring Colorado School of Mins DSP 2017 Joint work with Youy Xi, Shuang Li, and Michal B. Wakin (Colorado School of Mins)

2 Background - Airborn Radar Systm Pulsd Dopplr Radar rsiding on an airborn platform: (a)

3 Background - Airborn Radar Systm Systm Stup: Equipd with uniformly spacd linar array antnna (ULA) with N lmnts Transmits a cohrnt burst of M pulss at a constant puls rptition frquncy (PRF) f r = 1/T r. For ach (puls rptation intrval) PRI, L sampls ar collctd to covr th rang intrval

4 Background - Airborn Radar Systm Th rcivd data from on (cohrnt procssing intrval) CPI consists of LMN basband sampls Th blu slic: a spac-tim snapshot for all pulss at fixd rang gat (a) CPI datacub (b) A spac-tim snapshot

5 Background - Airborn Radar Systm Targt signal: Th rcivd targt signal is corrlatd both spatially and tmporally du to th lmnts intrvals and Dopplr ffct. Undr narrowband assumption, a singl targt s contribution is modld as X mn = c T jn2πf TS jm2πf TD. Spatial frquncy: f T S frquncy: f T D = 2v T λ 0 f r = d λ 0 cos(θ T ) sin(φ T ); Normalizd Dopplr

6 T D Background - Airborn Radar Systm Targt signal: Th rcivd targt signal is corrlatd both spatially and tmporally du to th lmnts intrvals and Dopplr ffct. Undr narrowband assumption, a singl targt s contribution is modld as X mn = c T jn2πf TS jm2πf TD. Spatial frquncy: f T S frquncy: f T D = 2v T λ 0 f r = d λ 0 cos(θ T ) sin(φ T ); Normalizd Dopplr TS j 2 f ( N1) 1 TS j 2f ( N1) j 2f ( M 1) TS j 2 f 1 T T j 2f S j 2f D 11 1 T j 2 f D 1 j 2 f T D ( M 1)

7 TS TD TS TD Background - Airborn Radar Systm Targt signal: Th rcivd targt signal is corrlatd both spatially and tmporally du to th lmnts intrvals and Dopplr ffct. Undr narrowband assumption, a singl targt s contribution is modld as X mn = c T jn2πf TS jm2πf TD. Spatial frquncy: f T S frquncy: f T D = 2v T λ 0 f r = d λ 0 cos(θ T ) sin(φ T ); Normalizd Dopplr TS j 2 f ( TS j 2 f N1) 11 1 j 1 1 T T 2f S j 2f D T j f D 2 1 j 2f ( N1) j 2f ( M 1) D j 2 f T ( M 1) T j 2 f S TS j 2 f ( T j f D 2 TD j 2 f 1 N1) T T j 2f S j 2f D 1 ( M 1) j 2f ( N1) j 2f ( M 1) a M ( f whr a ( f M a ( f N TD TD TS ) a ( f N ) 1, ) 1, TS T j 2f D T j 2f S ),,,, j T j 2f D ( M 1) T T 2f ( N1) T S

8 J S J S Background - Airborn Radar Systm Jamming signal: Th barrag nois jamming, whos signal is spatially corrlatd but tmporally uncorrlatd, is considrd. A jammr s contribution is modld as X mn = h m jn2πf J S 11 J J S j 2 f ( N1) j 2 f ( N1) j 2 f S 1 1 j 2 f J S 1 j 2 f J S j 2 f J S ( N1) 11 1 J S j 2 f 1 11 j 2 f ( N 1) 1 1 h a ( f whr a ( f ) T h 1, 1,, 1 N N J S J S ) 1, J j 2f S,, J j 2f ( N1) T S

9 Background - Airborn Radar Systm Cluttr signal: Th plant s surfac is th major sourc of th cluttr signal. Th rcivd cluttr signal has th sam structur of th targt with f C D = 2vaTr d f C S := βf C S. Brnnan s rul: cluttr signal livs in a low-dimnsional subspac of dimnsion approximatly N + (M 1)β (a) Th Earth surfac (b) Th spctrum

10 Background - STAP STAP: Spac-tim adaptiv procssing (STAP) wights th collctd data adaptivly in ordr to nhanc th targts signal whil attnuating th intrfrncs (jamming, cluttr) and nois. H z w x z x w

11 Background - STAP STAP: STAP can b viwd as solving an optimization problm to maximiz th signal-to-intrfrnc-plus-nois ratio (SINR). maximiz w w H x T 2 w H I n 2 w - th STAP wights vctor. x T - th targt signal vctor. I n - th intrfrncs and nois signal vctor.

12 Background - STAP STAP: STAP can b viwd as solving an optimization problm to maximiz th signal-to-intrfrnc-plus-nois ratio (SINR). maximiz w w H x T 2 w H I n 2 w - th STAP wights vctor. x T - th targt signal vctor. I n - th intrfrncs and nois signal vctor. According to th Cauchy Schwarz inquality, w H x T 2 w H I = < I 1 n x T, In H w > 2 n 2 In H w 2 Th optimum SINR achivs whn I 1 n x T 2 In H w 2 = I 1 In H w 2 n x T 2. I H n w = I 1 n x T w = (I ni H n ) 1 x T.

13 Background - Atomic norm Atomic norm Givn a st of paramtrizd atoms A = {a(ω) : ω Ω}, dfin th atomic norm: x A = inf{ c a : x = c aa} a A a A Atomic norm gnralizs th l 1 norm a signal x with small x A mans that it has a spars rprsntation w.r.t. th st of atoms. Atomic st: A T { a M (f T D ) a N (f T S ) : f T D [0, 1), f T S [0, 1) } A J { h a N (f J S ) : f J S [0, 1), h 2 = 1 }

14 Problm Stup Goal: sparat x T, x J and x C from th rcivd signal Targt: x T = Jamming: x J = y = x T + x J + x C K k=1 L l=1 c k a M (f T D k ) a N (f T S k ) d l h l a N (f J S l ) Cluttr: x C = Bz, B = approx span({a M (βf) a N (f)}) Normalizd spatial & tmporal string vctors: a N (f) 1 [ 1, j2πf,, j2πf(n 1)] N a M (f) 1 M [ 1, j2πf,, j2πf(m 1)] Atomic st: { } A T a M (f T D ) a N (f T S ) : f T D [0, 1), f T S [0, 1) { } A J h a N (f J S ) : f J S [0, 1), h 2 = 1

15 Problm Stup Atomic Norm Minimization: Atomic norm: x T AT inf minimiz x T,x J,z x T AT + λ x J AJ subjct to y = x T + x J + Bz { c k : x T = k { x J AJ inf d l : x J = l k l } c k a M (f T D ) a k N (f T S k ) d l h l a N (f J S l ) }

16 Problm Stup Atomic Norm Minimization: minimiz x T,x J,z SDP Formulation: minimiz x T,x J,z T,t,u,W subjct to x T AT + λ x J AJ subjct to y = x T + x J + Bz 1 2 tr(bktop(t)) t + λ 2 tr(top(u)) + λ 2 tr(w) [ [ bktop(t) xt Top(u) XJ x H T y = x T + x J + Bz t ] 0, X H J W ] 0, bktop(t): a block Toplitz matrix gnratd by T x J = vc(x J )

17 Problm Stup Atomic Norm Minimization: Dual Analysis: minimiz x T,x J,z x T AT + λ x J AJ subjct to y = x T + x J + Bz maximiz q q, y R subjct to q A T 1, q A J λ, B H q = 0 Dual atomic norm: q A T q A J sup x AT 1 sup x AJ 1 q, x R = q, x R = sup sup f T D,f T S [0,1) f J S [0,1) q, am (f T D ) a N (f T S), (IM a H N (f J S))q 2,

18 Problm Stup Atomic Norm Minimization: Dual Analysis: minimiz x T,x J,z x T AT + λ x J AJ subjct to y = x T + x J + Bz maximiz q q, y R subjct to q A T 1, q A J λ, B H q = 0 Frquncis (f S, f D ) with q, am (f T D ) a N (f T S ) = 1 ar stimatd targt locations. Frquncis f S with (IM a H N (f J S ))q 2 = λ ar stimatd jammr spatial locations

19 Simulations Paramtrs stup: Rcivd noisy signal y(64 1): antnna array contains 8 uniform lmnts with half-wavlngth spacing. 8 PRI within ach CPI. St radar antnna array to b alignd with airborn vlocity and th ratio of platform spd tims PRI ovr array lmnts spacing to 1, i.., β = 1. 2 Targts frquncis: (f T S, f T D ) = {(0.2, 0.8), (0.4, 0.1), (0.5, 0.5), (0.2, 0.2), (0.7, 0.4), (0.7, 0.7), (0.9, 0.5), (0.7, 0.1), (0.9, 0.9), (0.4, 0.8)}. Jamming frquncis: f J S = {0.4, 0.7}. Complx whit nois with zro man and standard dviation Normalizd dopplr frquncy Spatial frquncy (a) Rcivd signal y Normalizd dopplr frquncy Spatial frquncy (b) Targts-only signal

20 Simulations Paramtrs stup: Rcivd noisy signal y(64 1): antnna array contains 8 uniform lmnts with half-wavlngth spacing. 8 PRI within ach CPI. St radar antnna array to b alignd with airborn vlocity and th ratio of platform spd tims PRI ovr array lmnts spacing to 1, i.., β = 1. 2 Targts frquncis: (f T S, f T D ) = {(0.2, 0.8), (0.4, 0.1), (0.5, 0.5), (0.2, 0.2), (0.7, 0.4), (0.7, 0.7), (0.9, 0.5), (0.7, 0.1), (0.9, 0.9), (0.4, 0.8)}. Jamming frquncis: f J S = {0.4, 0.7}. Complx whit nois with zro man and standard dviation Spatial frquncy Normalizd dopplr frquncy Spatial frquncy (a) Dmixing with STAP (b) Dmixing for jamming with ANM (c) Dmixing for targts with ANM

21 Simulations Discussion: Targts ovrlap with jamming Targts and jamming frquncis can b rcovrd but not th signal magnituds. Targts ovrlap with th cluttr Targts can not b dtctd. Targts ar wll sparatd from jamming and cluttr Prfct targts rcovry Spatial frquncy Normalizd dopplr frquncy Spatial frquncy (a) Dmixing with STAP (b) Dmixing for jamming with ANM (c) Dmixing for targts with ANM

22 Conclusions Formulatd targt localization and jammr/cluttr rjction in spac-tim radar signal procssing as a dmixing problm Solv th dmixing problm using atomic norm minimization to xploit signal structurs Driv an SDP to approximatly solv th atomic norm minimization Th proposd atomic norm minimization outprforms STAP Futur work: dvlopmnt of thortical prformanc guarants

23 Qustions?