Random Matrix Theory (PhD Students)


Lecturer:
Prof. Dr.-Ing. R. Müller

Lecture:
Time and Place: Mo 9:15 - 10:00, N 5.17

Description:

Random Matrix Theory course for PhD students of LNT.

Homework:

  1. Derive the decorrelator (try with real signals first).
  2. Derive the MMSE detector
  3. Show the equivalence of the SINR of the LMMSE detector using the matrix inversion lemma.
  4. Derive the SINR of the LMMSE detector
  5. Find examples of convergence in a) distribution, b) in probability, c) in mean, d) almost surely
  6. Calculate the limiting distributions for matrix elements with doubled variance.
  7. Resolve the inconsistency of pages 50 and 51.
  8. Find a random matrix whose eigenvalues are asymptotically uniformly distributed in an ellipse (which is not a circle).
  9. Calculate the Stieltjes transform of the binary (-1,+1) distribution.
  10. Derive the fixed-point equation for the multiuser efficiency out of the theorem for sums of random matrices.
  11. Find the limit distribution of Girko's law for a random matrix with a checkerboard-like variance profile.
  12. Compare the eigenvalue distribution of i.i.d. CDMA against i.i.d. CDMA with antenna arrays for finite matrix size in Matlab. Use channel gains with unit amplitude and random phase! What do you observe?
  13. Compare the singular value distribution of the product of two steering matrices for a uniform linear array with the singular value distribution of an i.i.d. random matrix of the same size in Matlab. Observe the influence of the number of scatterers!
  14. Calculate the first 8 moments of the asymptotic eigenvalue distribution of the a) semicircle law, b) quarter circle law and compare them to the Catalan numbers.
  15. How many stages of an optimally weighted LPIC are needed to approach the performance of the LMMSE detector up to 0.1 dB for users with equal powers, i.i.d. random spreading and load 1/2?
  16. Find the distribution of the sum of two binary (0,1) free random variables.
  17. How can you create two free binary random variables out of two free Haar distributed random variables?
  18. Find the distribution of the product of two binary (0,1) free random variables.
  19. Find the distribution of the product of three, four, five, etc. binary (0,2) free random variables (eventually in terms of Stieltjes transforms).
  20. Find the distribution of the product of two binary (+1,-1) free random variables.
  21. Proof that there exists no eigenvalue distribution such that R(w)=-w.
  22. Let S(z) be the S-transform of the asymptotic eigenvalue distribution of the random matrix X and c be a real scalar. Find the S-transform of cX.
  23. Proof that the S-transform of an asymptotic eigenvalue distribution cannot be S(z)=1+z.
  24. Let {(A,B,C),(D,E),(F,G)} form a free family. Simplify the expressions Tr(AD^2FD), Tr(ADB^2EA+A^2EB^2D).
  25. Given a Gaussian asymptotically large random matrix, how can you create two free random matrices without the help of a random number generator?
  26. Derive the Boltzmann distribution by means of Lagrange multipliers.
  27. Calculate the multiuser efficiency of a LMMSE detector with equal power users with random signatures in the large-system limit. Assume that the total number of users is equal to the spreading factor, but only half of the users are active, and that the receiver is not aware that some users are inactive.
  28. Re-derive the 1st example of replica calculations for a random matrix with i.i.d. entries and a detector with perfect knowledge of the channel and the data distribution.
  29. Proof that the identity matrix I is free from any other matrix using the definition of freeness. Intuitively explain why that is!

NOTE:

This course will be carried out in English language.

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