Hardware Architecture for Direction-of-Arrival (DOA) Estimation Algorithms
- 1. Hardware Architecture for Direction-of-Arrival Estimation Algorithms (IN792)
- 2. ▷ Direction of Arrival (DOA) Estimation refers to the process of determining the direction from which a received signal was transmitted. This is typically achieved using an array of sensors or antennas that capture the signal's wavefronts. 2 DOA Estimation Primer (Definition)
- 3. ▷ Radar: Air traffic control and defense systems. ▷ Sonar: Underwater exploration, navigation and detection ▷ Wireless Communications: Increase spatial diversity ▷ Audio Processing: Hearing aids and home assistants. 3 DOA Estimation Primer (Applications)
- 4. 4 DOA Estimation (Single Source Model) ▷ A far-field source emits a narrowband signal s(t) with center wavelength λ. ▷ The signals received at the M-sensor uniform linear array (ULA) can be written in the form of a vector as: Array Steering Vector Noise Vector Received Signal Vector
- 5. 5 DOA Estimation (Multiple Source Model) ▷ The single source model can also be extended to multiple source model wherein (P<M) sources emit signals onto the array. The corresponding receivA far-field source emits a narrowband signal s(t) with center wavelength λ. The corresponding received signal matrix is given as: Received Signal Matrix Array Manifold Matrix
- 7. 7 MUSIC Algorithm (Basic Principles) ▷ Separation of subspaces- ○ The MUSIC algorithm separates the received signal's subspace via eigenvalue decomposition (EVD) into two components: the signal subspace and the noise subspace. ○ Signal Subspace: Contains the directions of arrival (DOAs) of the incoming signals. ○ Noise Subspace: Orthogonal to the signal subspace and contains no information about the signal. ▷ Orthogonality Property: ○ If 𝑎(𝜃) is the steering vector for an angle 𝜃, it is orthogonal to the noise subspace vectors, resulting in the MUSIC pseudo-spectrum peaks at the DOAs.
- 8. 8 MUSIC Algorithm (Math. Framework) ▷ The covariance matrix of the received signal can be expressed as: ▷ The eigenvalue decomposition of the covariance matrix yields M eigenvalues and eigenvectors, P of which correspond to the signal subspace while the rest (M-P) correspond to the noise subspace.
- 9. 9 MUSIC Algorithm (Math. Framework) ▷ The array manifold matrix A spans the same subspace as the signal eigenvector matrix Us . Hence we have: ▷ Based on the above observation, the MUSIC spectrum can be expressed as: ▷ The DOAs are directly estimated from the P peaks of the MUSIC spectrum (wherein the denominator almost tends to zero).
- 10. 10 MUSIC Algorithm (Hardware Approach) ▷ Existing hardware research aims to reduce computational complexity and delay. ▷ Scopes of improvement: ○ Conversion of covariance matrix into a real matrix for easier calculation of eigenvalues and eigenvectors. ○ Improvement of CORDIC rotation (angle related computation) or parallelize using Jacobi algorithms (eigenvalue computation) for better speed. ○ Eliminate EVD and directly obtain noise-subspace from the sub-matrix of the covariance matrix. ○ Coarse and fine search stages in the final spectral search stage.
- 11. 11 Core Literature (Paper 1) ▷ Implemented on a 6 element Uniform Circular Array (UCA). ▷ Runs at a clock speed of 100MHz. ▷ Estimates DOA in 200 clock cycles (2 µs). ▷ FPGA used: ZedBoard Zynq
- 12. 12 Core Literature (Paper 1) Block diagram of the proposed optimized MUSIC Algorithm
- 13. 13 Key Components (Paper 1) ▷ Virtual Array Reduction Block: ○ Objective- ■ Reduces the number of antenna signals processed ■ Lowers the computational load ■ Improves FPGA resource utilization. ○ Mechanism- ■ Compares signals in pairs to identify the strongest three signals. ■ Outputs the selected signals with their corresponding indices.
- 14. 14 Key Components (Paper 1) Virtual Array Reduction Block
- 15. 15 Key Components (Paper 1) ▷ Covariance Matrix Calculator: ○ Objective- ■ Computes the reduced covariance matrix (3*3) in place of original (6*6) ○ Mechanism- ■ Compute variance using: ■ Compute covariance using:
- 16. 16 Key Components (Paper 1) Covariance Matrix Calculator
- 17. 17 Key Components (Paper 1) ▷ Normalization Block: ○ Objective- ■ Normalizes the covariance matrix to manage resource usage without compromising accuracy. ○ Mechanism- ■ Scales down the values to fit within the available precision, reducing the required number of bits.
- 18. 18 Key Components (Paper 1) Covariance Matrix Normalization Block
- 19. 19 Key Components (Paper 1) ▷ Eigenvalue Decomposition Block: ○ Objective- ■ Decomposes the covariance matrix into its eigenvalues and eigenvectors. ○ Optimizations- ■ Uses an algebraic method for simplification. ■ Largest eigenvalue corresponds to the signal subspace; smaller eigenvalues correspond to the noise subspace. ■ Eigenvector computation based on the row-reduced echelon form.
- 20. 20 Key Components (Paper 1) Eigenvalue and Eigenvector Computation Block
- 21. 21 Key Components (Paper 1) Optimized EVD Block
- 22. 22 Key Components (Paper 1) ▷ Spectral Peak Search Block: ○ Objective- ■ Finds the direction of arrival (DOA) by searching for peaks in the signal subspace spectrum. ○ Mechanism- ■ Searches within the 3dB intercept points of the maximum receiving antenna. ■ Calculates the pseudo-spectrum to identify peaks corresponding to DOAs.
- 23. 23 Key Components (Paper 1) Spectral Search and Final Angle Calculator Block
- 24. 24 Design Performance (Paper 1)
- 25. 25 Core Literature (Paper 2) ▷ Implemented on a 6,8 element Uniform Linear Array (ULA) and Nested Array (NA). ▷ FPGA Used: Virtex-6, Zynq 7000
- 26. 26 Core Literature (Paper 2) Overall Structure of the proposed MUSIC Algorithm Hardware Acceleration
- 27. 27 Key Components (Paper 2) ▷ Covariance Matrix Computation Block: ○ Objective- ■ Compute the covariance matrix for use in the MUSIC algorithm ○ Mechanism- ■ Separate hardware structures for two designs: Design 1 (general purpose) and Design 2 (optimized for fixed input sizes). ■ Design 1: Utilizes nested modules to calculate matrix multiplication, addressing various input configurations, and stores the real and imaginary parts of the input data separately.
- 28. 28 Key Components (Paper 2) Overall structure for computing Covariance Matrix (CM) Sparse cum linear array compatible design for CM Parallelism enabled design for Uniform linear array CM
- 29. 29 Key Components (Paper 2) ▷ Covariance Matrix Computation Block: ○ Mechanism- ■ Compute the covariance matrix for use in the MUSIC algorithm ■ Design 2: Employs parallel computation for fixed-size inputs, calculating only the upper triangular matrix due to its Hermitian property, enhancing efficiency. ■ Step-b: Vectorizes the matrix, calculates the sparse array arrangement, and forms a new vector by removing duplicated virtual arrays. ■ Step-c: Performs spatial smoothing by dividing the virtual array into sub-arrays, obtaining a covariance matrix that meets the full rank condition, crucial for distinguishing signal and noise subspaces.
- 30. 30 Key Components (Paper 2) CM vectorizer (step-b) block for sparse array Spatial smoothing (step-c) block for sparse array
- 31. 31 Key Components (Paper 2) ▷ Covariance Matrix Computation Block: ○ Optimization- ■ Design 1: Flexible configuration to support various input data sizes and structures, using nested module architecture for stepwise multiplication. ■ Design 2: High parallelism, with simultaneous output of valid element values using multiple memory and complex multiplication modules. ■ Step-b: Efficiently removes duplicate virtual arrays and forms a new vector for further processing. ■ Step-c: Applies spatial smoothing to ensure the covariance matrix of the virtual array maintains full rank, enabling accurate signal subspace identification.
- 32. 32 Key Components (Paper 2) ▷ Complex Jacobi Decomposition Block: ○ Objective- ■ Perform eigenvalue decomposition of the Hermitian matrix to determine eigenvalues and eigenvectors. ○ Mechanism- ■ Utilizes the Jacobi algorithm for iterative construction of plane rotation matrices to diagonalize the Hermitian matrix. ■ Hardware implementation updates the eigenvalue and eigenvector matrices in parallel, improving computation speed.
- 33. 33 Key Components (Paper 2) Complex Jacobi Algorithm Hardware architecture for complex Jacobi algorithm
- 34. 34 Key Components (Paper 2) ▷ Complex Jacobi Decomposition Block: ○ Optimization- ■ Focuses on the upper triangular matrix to find the largest off-diagonal element, reducing computational complexity. ■ Simplified trigonometric calculations derive sine and cosine from tan(2θ), minimizing the need for complex hardware functions. ■ Parallel updating of matrices to expedite iterations and improve overall processing time.
- 35. 35 Key Components (Paper 2) Unitary Matrix computation block in Jacobi architecture cs module in Unitary Matrix computation structure
- 36. 36 Key Components (Paper 2) ▷ Peak Search Block: ○ Objective- ■ Identify the direction of arrival (DOA) by searching for peaks in the pseudo-spectral function. ○ Mechanism- ■ Distinguishes signal and noise subspaces by comparing eigenvalues of the covariance matrix. ■ Calculates the pseudo-spectral function using pre-stored direction vector values to expedite processing. ■ Employs a three-stage search process to accurately pinpoint source angles by identifying troughs instead of peaks, simplifying calculations.
- 37. 37 Key Components (Paper 2) Hardware architecture for distinguishing noise subspace Hardware architecture for computing pseudo-spectrum
- 38. 38 Key Components (Paper 2) ▷ Complex Jacobi Decomposition Block: ○ Optimization- ■ Stores pre-calculated direction vector values in ROM to reduce computational overhead. ■ Changes peak search to trough search, eliminating high-delay division operations. ■ Parallel computation in matrix multiplications during peak search for faster and more efficient processing.
- 39. 39 Design Performance (Paper 2)
- 40. 40 Design Performance (Paper 2)
- 41. 41 Design Performance (Paper 2)