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This document summarizes key concepts from a tutorial on mathematics for signal processing and communications. It discusses similarity and diagonalization of matrices, spectral decomposition of Hermitian matrices, singular value decomposition (SVD) and how it can be used for image compression. It also provides an example of how SVD can be used to cancel interference in MIMO wireless systems by pre-multiplying the transmitted signal with one SVD matrix and post-multiplying the received signal with another SVD matrix.
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Variable-length codes can be used to encode letters or symbols using codewords of varying lengths. The document reviews mathematical basics relevant to analyzing source coding and data compression, including:
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Tutorial2
- 1. EENGM0014 Mathematics for Signal Processing and Communications Tutorial 2 Soon Yau Cheong University of Bristol 14 Oct 2016 Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications14 Oct 2016 1 / 9
- 2. Tutorial 1 Last week’s tutorial and Matlab 2 Revision on last lecture 3 Example and demonstration 4 C Programming Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications14 Oct 2016 2 / 9
- 3. Similarity Similar Matrix If A and B are square matrices, A is similar to B if there is an invertible matrix X such that A = X B X−1 A and B have same characteristic polynomials hence the same eigenvalues. Diagonalisation If matrix A has linearly independent set of eigenvectors A = X Λ X−1 where Λ is diagonal matrix Diagonalisation Property Ak = XΛk X−1 Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications14 Oct 2016 3 / 9
- 4. Spectral Decomposition Hermitian matrix can be decomposed into A = X Λ XH It has: real eigenvalues eigenvectors corresponding to different eigenvalues are orthorgonal Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications14 Oct 2016 4 / 9
- 5. Spectral Decomposition A = XΛXH = x1 · · · xn λ1 0 ... 0 λn xH 1 ... xH n = λ1x1xH 1 + λ2x2xH 2 + ... + λnxnxH n Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications14 Oct 2016 5 / 9
- 6. Singular Value Decomposition Any M × N matrix A can be decomposed into: A = UΣV H where U (M × M) and V(N × N) are unitary matrices and Σ (M × N) is ’diagonal’ Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications14 Oct 2016 6 / 9
- 7. Image Compression using SVD MATLAB DEMO (run the code to see full resolution images) Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications14 Oct 2016 7 / 9
- 8. MIMO [Andrea Goldsmith, ”Wireless Communication”,2005] Claim: We can cancel interference and restore X by performing SVD on the channel matrix, [U,S,V]=svd(H) and 1. pre-multiplying X with V, ˆX = VX 2. multiply received signal, ˆY with UH, i.e. Y = UH ˆY Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications14 Oct 2016 8 / 9
- 9. Proof say X = x1 x2 · · · xn H Y = UH ˆY = UH H ˆX = UH (UΣV H )(VX) = (UH U)Σ(V H V )X = ΣX = λ1 0 ... 0 λn x1 ... xn = λ1x1 ... λnxn Soon Yau Cheong (University of Bristol) EENGM0014 Mathematics for Signal Processing and Communications14 Oct 2016 9 / 9