Analysis of Compressive Sensing in Radar Holger Rauhut Lehrstuhl C f¨ ur Mathematik (Analysis) RWTH Aachen Compressed Sensing and Its Applications TU Berlin December 8, 2015 1 / 44
Overview Several radar setups with compressive sensing approaches ◮ Range-Doppler resolution via compressive sensing ◮ Sparse MIMO Radar ◮ Antenna arrays with randomly positioned antennas 2 / 44
Time-Frequency Structured Random Matrices Resolution of Range-Doppler in Radar 3 / 44
Resolution of Range-Doppler Received signal is superposition of delayed and modulated (Doppler shifted) versions of sent signal. Task: Determine delays (corresponding to distances; range) and Doppler shifts (corresponding to radial speed) from subsampled receive signal! 4 / 44
Gabor Systems in Finite Dimensions Translation and Modulation on C m ( T k g ) j = g ( j − k ) ( M ℓ g ) j = e 2 π i ℓ j / m g j . and mod m Time-frequency shifts π ( λ ) = M ℓ T k , λ = ( k , ℓ ) ∈ { 0 , . . . , m − 1 } 2 . For g ∈ C m define Gabor synthesis matrix ( ω = e 2 π i / m ) Ψ g = ( π ( λ ) g ) λ ∈{ 0 ,..., m − 1 } 2 g 0 g m − 1 g 1 g 0 g 1 g 1 · · · · · · · · · ω m − 1 g 2 g 1 g 0 g 2 ω g 1 ω g 2 · · · · · · · · · ω 2 g 2 ω 2 g 3 ω 2( m − 1) g 3 g 2 g 1 g 3 · · · · · · · · · ω 3 g 3 ω 3 g 4 ω 3( m − 1) g 4 = . g 3 g 2 g 4 · · · · · · · · · . . . . . . ... ... . . . . . . . . . . . . ω ( m − 1) 2 g 0 ω m − 1 g m − 1 ω m − 1 g 0 g m − 1 g m − 2 g 0 · · · · · · · · · Use of Ψ g ∈ C m × m 2 as measurement matrix in compressive sensing
Gabor Systems in Finite Dimensions Translation and Modulation on C m ( T k g ) j = g ( j − k ) ( M ℓ g ) j = e 2 π i ℓ j / m g j . and mod m Time-frequency shifts π ( λ ) = M ℓ T k , λ = ( k , ℓ ) ∈ { 0 , . . . , m − 1 } 2 . For g ∈ C m define Gabor synthesis matrix ( ω = e 2 π i / m ) Ψ g = ( π ( λ ) g ) λ ∈{ 0 ,..., m − 1 } 2 g 0 g m − 1 g 1 g 0 g 1 g 1 · · · · · · · · · ω m − 1 g 2 g 1 g 0 g 2 ω g 1 ω g 2 · · · · · · · · · ω 2 g 2 ω 2 g 3 ω 2( m − 1) g 3 g 2 g 1 g 3 · · · · · · · · · ω 3 g 3 ω 3 g 4 ω 3( m − 1) g 4 = . g 3 g 2 g 4 · · · · · · · · · . . . . . . ... ... . . . . . . . . . . . . ω ( m − 1) 2 g 0 ω m − 1 g m − 1 ω m − 1 g 0 g m − 1 g m − 2 g 0 · · · · · · · · · Use of Ψ g ∈ C m × m 2 as measurement matrix in compressive sensing 5 / 44
Radar model (Herman, Strohmer 2008) Emitted signal: g ∈ C m . Objects scatters g and radar device receives the contribution x λ π ( λ ) g = x k ,ℓ M ℓ T k g . T k corresponds to delay, i.e., distance of object M ℓ corresponds to Doppler shift, i.e., speed of the object x k ,ℓ reflectivity of object Received signal is superposition of contribution of all scatteres: � y = x λ π ( λ ) g = Ψ g x . λ ∈ Λ Usually few scatterers so that x ∈ C m 2 can be assumed sparse. 6 / 44
Radar model (Herman, Strohmer 2008) Emitted signal: g ∈ C m . Objects scatters g and radar device receives the contribution x λ π ( λ ) g = x k ,ℓ M ℓ T k g . T k corresponds to delay, i.e., distance of object M ℓ corresponds to Doppler shift, i.e., speed of the object x k ,ℓ reflectivity of object Received signal is superposition of contribution of all scatteres: � y = x λ π ( λ ) g = Ψ g x . λ ∈ Λ Usually few scatterers so that x ∈ C m 2 can be assumed sparse. We will choose g as random vector below. 6 / 44
Reconstruction via compressive sensing Reconstruction of x from y = Ax via ℓ 1 -minimization min � z � 1 subject to Az = y min � z � 1 subject to � Az − y � 2 ≤ η 7 / 44
Reconstruction via compressive sensing Reconstruction of x from y = Ax via ℓ 1 -minimization min � z � 1 subject to Az = y min � z � 1 subject to � Az − y � 2 ≤ η Alternatives: Matching Pursuits Iterative hard thresholding (pursuit) Iteratively reweighted least squares ... 7 / 44
Uniform vs. nonuniform recovery Often recovery results are for random matrices A ∈ R m × N ; choose generator g ∈ C m for Ψ g at random ◮ Uniform recovery With high probability on A every sparse vector is recovered; P ( ∀ s -sparse x , recovery of x is successful using A ) ≥ 1 − ε. 8 / 44
Uniform vs. nonuniform recovery Often recovery results are for random matrices A ∈ R m × N ; choose generator g ∈ C m for Ψ g at random ◮ Uniform recovery With high probability on A every sparse vector is recovered; P ( ∀ s -sparse x , recovery of x is successful using A ) ≥ 1 − ε. Recovery conditions on A ◮ Null space property ◮ Restricted isometry property 8 / 44
Uniform vs. nonuniform recovery Often recovery results are for random matrices A ∈ R m × N ; choose generator g ∈ C m for Ψ g at random ◮ Uniform recovery With high probability on A every sparse vector is recovered; P ( ∀ s -sparse x , recovery of x is successful using A ) ≥ 1 − ε. Recovery conditions on A ◮ Null space property ◮ Restricted isometry property ◮ Nonuniform recovery A fixed sparse vector is recovered with high probability using A ∈ R m × N ; ∀ s -sparse x : P (recovery of x is successful using A ) ≥ 1 − ε. 8 / 44
Uniform vs. nonuniform recovery Often recovery results are for random matrices A ∈ R m × N ; choose generator g ∈ C m for Ψ g at random ◮ Uniform recovery With high probability on A every sparse vector is recovered; P ( ∀ s -sparse x , recovery of x is successful using A ) ≥ 1 − ε. Recovery conditions on A ◮ Null space property ◮ Restricted isometry property ◮ Nonuniform recovery A fixed sparse vector is recovered with high probability using A ∈ R m × N ; ∀ s -sparse x : P (recovery of x is successful using A ) ≥ 1 − ε. Recovery conditions on A ◮ Tangent cone (descent cone) of norm at x intersects ker A trivially. ◮ Dual certificates 8 / 44
Restricted isometry property (RIP) Definition The restricted isometry constant δ s of a matrix A ∈ C m × N is defined as the smallest δ s such that (1 − δ s ) � x � 2 2 ≤ � A x � 2 2 ≤ (1 + δ s ) � x � 2 2 for all s -sparse x ∈ C N . 9 / 44
Stable and robust recovery Theorem (Cand` es, Romberg, Tao ’04 – Cai, Zhang ’13) √ Let A ∈ C m × N with δ 2 s < 1 / 2 ≈ 0 . 7071 . Let x ∈ C N , and assume that noisy data are observed, y = A x + e with � e � 2 ≤ τ . Let x # by a solution of z � z � 1 � A z − y � 2 ≤ τ. min such that Then � x − x # � 2 ≤ C σ s ( x ) 1 √ s + D τ, � x − x # � 1 ≤ C σ s ( x ) 1 + D √ s τ for constants C , D > 0 , that depend only on δ 2 s . Here σ s ( x ) 1 = z : � z � 0 ≤ s � x − z � 1 . inf Implies exact recovery in the s -sparse and noiseless case. 10 / 44
Dual certificate Theorem (Fuchs 2004, Tropp 2005) For A ∈ C m × N , x ∈ C N with support S is the unique solution of min � z � 1 subject to A z = A x if A S is injective and there exists a dual vector h ∈ C m such that ( A ∗ h ) j = sgn( x j ) , | ( A ∗ h ) ℓ | < 1 , j ∈ S , ℓ ∈ S . Corollary Let a 1 , . . . , a N be the columns of A ∈ C m × N . For x ∈ C N with support S, if the matrix A S is injective and if |� A † S a ℓ , sgn( x S ) �| < 1 for all ℓ ∈ S , then the vector x is the unique ℓ 1 -minimizer with y = A x . Here, A † S is Moore-Penrose pseudo inverse. 11 / 44
Dual certificate Theorem (Fuchs 2004, Tropp 2005) For A ∈ C m × N , x ∈ C N with support S is the unique solution of min � z � 1 subject to A z = A x if A S is injective and there exists a dual vector h ∈ C m such that ( A ∗ h ) j = sgn( x j ) , | ( A ∗ h ) ℓ | < 1 , j ∈ S , ℓ ∈ S . Corollary Let a 1 , . . . , a N be the columns of A ∈ C m × N . For x ∈ C N with support S, if the matrix A S is injective and if |� A † S a ℓ , sgn( x S ) �| < 1 for all ℓ ∈ S , then the vector x is the unique ℓ 1 -minimizer with y = A x . Here, A † S is Moore-Penrose pseudo inverse. One ingredient: Check that � A ∗ S A S − I � 2 → 2 ≤ δ < 1. 11 / 44
Stability and robustness via dual certificate Theorem Let x ∈ C N and A ∈ C m × N with ℓ 2 -normalized columns. Denote by S ⊂ [ N ] the indices of the s largest absolute entries of x . Assume that (i) there is a dual certificate u = A ∗ h ∈ C N with h ∈ C m s.t. � h � 2 ≤ 3 √ s . � u T c � ∞ ≤ 1 u T = sgn( x ) T , 2 , (ii) � A ∗ T A T − I � 2 → 2 ≤ 1 2 . Given noisy measurements y = A x + e ∈ C m with � e � 2 ≤ τ , the x ∈ C N of noise-constrained ℓ 1 -minimization satisfies solution ˆ x � 2 ≤ 52 √ s τ + 16 σ s ( x ) 1 . � x − ˆ 12 / 44
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