Modulated Unit Norm Tight Frames for Compressed Sensing Peng Zhang1, Lu Gan2, Sumei Sun3 and Cong Ling1 1 Department of Electrical and Electronic Engineering, Imperial College London 2 College of Engineering, Design and Physics Sciences Brunel University, United Kingdom 3 Institute for Infocomm Research, A*STAR, Singapore

Outline Background Proposed system Basics of compressed sensing; Structured random operators for compressed sensing; Proposed system Performance bounds; Connection with existing systems; Applications in convolutional compressed sensing Compressive imaging; Sparse channel estimation in OFDM; Conclusions

Principles of Compressed Sensing (CS) Sampling: linear, non-adaptive random projections [Candes-Romberg-Tao-2006]: y = x x: N×1 signal vector; y: M×1 sampling vector (M << N); : M×N measurement matrix; Sparsity: x has a sparse representation under a certain transform  (DCT, wavelet); f = x can be well approximated with only K (K < M) non-zero coefficients; Reconstruction: nonlinear optimization; l1 optimization; Iterative-based methods: OMP, subspace pursuit (SP) etc.; Now, let us a take a closer look at In the most general case, recovery of x from y is ill-posed as it is under-determined.

Principles of Compressed Sensing (CS) Restricted isometry property (RIP) [Candes-Romberg-Tao-2006]: An M×N matrix A= is said to satisfy the RIP with Parameters (K,) if where  represents the set of all length-N vectors with K non-zero coefficients. Now, let us a take a closer look at In the most general case, recovery of x from y is ill-posed as it is under-determined.

Fully random sampling operators : full random matrix with independent sub-Gaussian elements i,j follows the Gaussian or Bernoulli distributions; Optimal bound: Universal: applicable for any  Limitations: High computational cost in matrix multiplication; Huge buffer requirement; Difficult or even impossible to implement;

A wish-list for the sampling operator 

Randomly subsampled orthonormal system [Candes et al. -2006] S: Randomly sampling operator (choose M samples uniform at random): Q: NN Bounded unitary matrix with Example of Q: DCT, DFT, Walsh-Hadamard matrices 7

Structurally random matrices Do and Gan et al. 2008 [ICASSP], 2012 [TSP] Random Sampling (choose M out of N) Fast transform FFT, WHT, DCT Random Sign Flipping ±1  = S F D Fast implementation Universal Random sampling could be difficult! 8

Random Convolution (filtering) J. Tropp, J. Romberg, H. Rauhut, F. Krahmer et al. c x y R R : Deterministic sampling operator; c: random sequence; Lacks universality =I (Identity matrix) 

Outline Background Proposed system Examples of potential applications Basics of compressed sensing; Structured random operators for compressed sensing; Proposed system Performance bounds; Connection with existing systems; Examples of potential applications Compressive imaging; Sparse channel estimation in OFDM; Conclusions

Unit-norm tight frame An M×N matrix U corresponds to a unit norm tight frame if Each column vector has a unit norm: The rows of form an orthonormal family;

Examples of unit norm tight frame Partial FFT or WHT: Partial summation operator; Cascading of identity or Fourier matrices; U=RF or U=RW U= U=[I I … I] or U=[F F … F]

Bounded orthonormal matrix Proposed system The product A= can be written as Bounded orthonormal matrix Random Sign Flipping Unit norm tight frame ±1 A= U D B Many existing systems can be characterized by the above systems: random convolution, random demodulation, compressive multiplexing, random probing… 13

Performance bound Performance bound Main tools in the proof Suprema of chaos processes [Krahmer et al.- 2014] Variations and extensions The diagonal matrix D could be constructed from any sub-Gaussian variables; B could be a near-orthogonal (rectangular) matrix;

Example: Random demodulation Random demodulation [Tropp et al.-2010]

Random demodulation Sampling operator: =UD, U= Sparsifying transform : FFT matrix with column permutation Sparsifying transform Previous work [Tropp et al. 2010]: M≥ O(K log6N) Our bound: M ≥ O(K log2K log2N)

Outline Background Proposed system Basics of compressed sensing; Structured random operators for compressed sensing; Proposed system Performance bounds; Connection with existing systems; Applications in convolutional compressed sensing Compressive imaging; Sparse channel estimation in OFDM; Conclusions

Coded aperture imaging—(existing system) Romberg et al.-2008, Marcia et al.-2009 Low resolution detector array x Random mask D Lens F Lens FH Begin with Focus on our proposed After that, will be presented, followed by a brief conclusion =I Works poorly for natural images

Coded aperture imaging—(existing system) Spatially sparse, =I Fails for this one (spectrally sparse)

Proposed system Sampling Operator: =RFHDF : diagonal matrix made from the Golay sequence; Mask based on  Implementation: Double phase encoding [Rivenson et al. 2010]

Golay sequence Let a=[a0, a1, ..., aN − 1]T (an {1, -1}) and define If a is a Golay sequence, then for all z on the unit circle

Proposed system Sampling Operator: =RFHDF Sparsifying transform : : diagonal matrix made from the Golay sequence; U=RFH B=F  Sparsifying transform : Haar Wavelet, Fourier, (Block) DCT (popular for natural images) 

Simulation results: Compressive imaging Experimental setup Test images: 256256 Lena and Hall; Reconstruction: re-weighted BPDN [Carrillo et al.-2013] Sampling ratio M/N=0.25

Simulation Results: Convolutional CS (b) Reconstructed images of Lena. (a) Results from conventional compressive coded aperture imaging, SNR=11.02 dB; (b) Our proposed algorithm, SNR=29.56 dB;

Simulation Results (a) (b) Reconstructed images of Hall. (a) Results from conventional compressive coded aperture imaging, SNR=9.21 dB; (b) Our proposed system, SNR=24.62 dB;

Outline Background Proposed system Basics of compressed sensing; Structured random operators for compressed sensing; Proposed system Performance bounds; Connection with existing systems; Applications in convolutional compressed sensing Compressive imaging; Sparse channel estimation in OFDM; Conclusions

Sparse channel estimation for OFDM system Existing solutions: y Low rate ADC Meng et al.-2012: R: deterministic sampler; i: Variable with random phase High Peak-to-Average Power Ratio (PAPR) after the IDFT Li et al.-2014: R random sampler; i: Golay sequence; Difficult to implement random sampling

Proposed system : Golay sequence  Low PAPR p(t): random signal Random demodulation : Golay sequence  Low PAPR p(t): random signal Overall Sampling Operator: =UDFFH B=FFH U=

Experimental setup (OFDM) N=1024, M=64; Channel model ATTC (Advanced Television Technology Center) and the Grande Alliance DTV laboratory ensemble E model. Channel impulse response Input signal to noise ratio (SNR): 0dB to 30dB Reconstruction: subspace pursuit (Dai et al.-2009)

Simulation results for low rate OFDM channel estimation

Conclusions Proposed framework A==UDB U: unit norm tight frame, D: random diagonal matrix, B: bounded orthogonal matrix; M ≥ O(K log2K log2N) Improved performance bound for existing system Random demodulation, compressive multiplexing, random probing etc. Novel compressive sensing framework Compressive imaging; Sparse channel estimation for OFDM systems;