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Mathematical Foundations for Signal Analysis in Wireless and Mobile Systems

An overview of linear algebra and calculating the discrete Fourier transform for signal analysis in wireless and mobile systems.

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Mathematical Foundations for Signal Analysis in Wireless and Mobile Systems

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  1. Wireless and Mobile Systems for the IoT CMSC 818W : Fall 2019 • Lecture 1.3: Mathematical Foundations • for Signal Analysis (Part 2) Nirupam Roy M-W 2:00-3:15pm CSI 1122

  2. An (incomplete) overview of Linear Algebra 2. Calculating Discrete Fourier Transform (DFT)

  3. An (incomplete) overview of Linear Algebra 2. Calculating Discrete Fourier Transform (DFT)

  4. Properties of a linear function f(x)

  5. System of linear equation

  6. Row view Solution, x=1, y=2

  7. Column view

  8. Column view

  9. Column view What is the solution space or the set of all candidate solutions? Ans: All linear combinations of the columns of the matrix A(m x n)

  10. Column space

  11. Column space Solution space, spanned by the columns of the matrix A ( b vector should lie in this space )

  12. What if the vectors are aligned? What is column rank of the matrix? A column is a linear combination of other columns i.e., the columns are not independent

  13. Approximate solution

  14. Approximate solution Column space of the matrix A

  15. Approximate solution

  16. Approximate solution

  17. Approximate solution Error e is orthogonal to the plane for optimal solution Projection of the vector b on the solution plane Two vectors a & b are orthogonal when their dot product (aTb) is zero.

  18. Approximate solution

  19. Approximate solution Projection matrix

  20. An (incomplete) overview of Linear Algebra 2. Calculating Discrete Fourier Transform (DFT)

  21. Discrete Fourier Transform … Sample number X = [ ] x[0], x[1], x[2], x[3], x[4], … x[N-1] N-1 2 3 4 1 6 5 0

  22. Discrete Fourier Transform … Sample number X = [ ] x[0], x[1], x[2], x[3], x[4], … x[N-1] N-1 2 3 4 1 6 5 0

  23. Discrete Fourier Transform … Sample number N-1 2 3 4 1 6 5 0

  24. Discrete Fourier Transform … Sample number = [ ] , … Freq f1 N-1 2 3 4 1 6 5 0

  25. Discrete Fourier Transform … Sample number N-1 2 3 4 1 6 5 0

  26. Discrete Fourier Transform … Sample number = [ ] , … Freq f2 N-1 2 3 4 1 6 5 0

  27. Discrete Fourier Transform … Sample number What is the MINIMUM number of cycles observable in this model? N-1 2 3 4 1 6 5 0

  28. Discrete Fourier Transform … Sample number N-1 2 3 4 1 6 5 0

  29. Discrete Fourier Transform … Sample number = [ ] … , Freq f0 N-1 2 3 4 1 6 5 0

  30. Discrete Fourier Transform … Sample number What is the MAXIMUM number of cycles observable in this model? N-1 2 3 4 1 6 5 0

  31. Discrete Fourier Transform … Sample number What happens when the ball rotates exactly N times? N-1 2 3 4 1 6 5 0 We can uniquely observe up to (N-1) number of cycles.

  32. Discrete Fourier Transform Freq f0 n=0 n=1 n=2 … … n=N-1

  33. Discrete Fourier Transform Number of cycles … … Freq f0 Freq f1 Freqfm Freq fN-1 n=0 n=1 Sample number n=2 … … n=N-1

  34. Discrete Fourier Transform Number of cycles … … Freq f0 Freq f1 Freqfm Freq fN-1 n=0 n=1 Sample number … … n=2 … … … … … n=N-1

  35. Discrete Fourier Transform + + + Z1 Zm Z0 ZN-1 … … … … … …

  36. Discrete Fourier Transform + + + Z1 Zm Z0 ZN-1 … … … … … …

  37. Discrete Fourier Transform + + + Z1 Zm Z0 ZN-1 … … … … … … N-1 5 0 6 1 4 3 2 = … Sample number

  38. Discrete Fourier Transform Frequency domain coefficients + + + Z1 Zm Z0 ZN-1 … … … … … … x[0] x[1] = X = Time domain samples x[2] … x[N-1]

  39. Discrete Fourier Transform F … … fN-1 fm f1 f0 = DFT matrix,

  40. Discrete Fourier Transform F … … fN-1 fm f1 f0 = DFT matrix, Z Z0 = Fourier coefficients, Z1 … Zm … ZN-1

  41. Discrete Fourier Transform F … … fN-1 fm f1 f0 = DFT matrix, Z Z0 x[0] X = Time series, = Fourier coefficients, Z1 x[1] … x[2] Zm … … ZN-1 x[N-1]

  42. Discrete Fourier Transform Z F X = X F-1 Z = * X F Z =

  43. Discrete Fourier Transform Z F X = X F-1 Z = * X F Z = * f0 * f1 … * fm … * fN-1

  44. Discrete Fourier Transform Z F X x[0] = x[1] X F-1 Z x[2] = … * X F Z x[N-1] = * Z0 f0 Z1 * … f1 Zm = … … * fm DFT IDFT ZN-1 … * fN-1

  45. Discrete Fourier Transform z(m) = DFT: x(n) = IDFT:

  46. Preserving energy in time and frequency domain

  47. Preserving energy in time and frequency domain Is IDFT( DFT(x) ) = x ?

  48. Preserving energy in time and frequency domain Is IDFT( DFT(x) ) = x ? * * X ? N.X X X F F F F = = F … … fN-1 fm f1 f0 = DFT matrix,

  49. Discrete Fourier Transform z(m) = DFT: x(n) = IDFT:

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