Presentation slide of graduate course "Maths for Signal Processing and Communications"

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