Applications of Random Matrix Theory in Wireless Underwater Communication

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1 Applications of Random Matrix Theory in Wireless Underwater Communication Why Signal Processing and Wireless Communication Need Random Matrix Theory Atulya Yellepeddi May 13, Eigenvalues of Random Matrices, Spring Final Project Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 1 / 10

2 Signal Processing and the Law of Large Numbers Want to estimate a population size m from n observations Methods and Analysis based on statistics of population as n Usually works if n m Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 2 / 10

3 Signal Processing and the Law of Large Numbers Want to estimate a population size m from n observations Methods and Analysis based on statistics of population as n Usually works if n m Modern signal processing- cases when n m Parameter of a time-varying system (eg., underwater communication) Population size grows as observations grow (eg., social networks) Huge population size (eg., huge array beamformers) Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 2 / 10

4 Signal Processing and the Law of Large Numbers Want to estimate a population size m from n observations Methods and Analysis based on statistics of population as n Usually works if n m Modern signal processing- cases when n m Parameter of a time-varying system (eg., underwater communication) Population size grows as observations grow (eg., social networks) Huge population size (eg., huge array beamformers) Random Matrix Theory: excellent at making predictions in such scenarios Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 2 / 10

5 Why Random Matrix Theory? Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 3 / 10

6 Why Random Matrix Theory? For Mathematicians, 30 So with reasonable population sizes m 30 or more, we can make highly accurate predictions which work for small number of observations. Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 3 / 10

7 Why Random Matrix Theory? For Mathematicians, 30 So with reasonable population sizes m 30 or more, we can make highly accurate predictions which work for small number of observations. Almost Anything is IID Gaussian Results for IID Gaussian ensembles carry over, in practice, to all sorts of ensembles (with some caveats!) if they are reasonably like Gaussian, and more or less independent. Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 3 / 10

8 Least Squares Channel Estimation- the Problem Transmit u(n)... m 1 vectors, independent from time to time E[u(n)] = 0 E[u(n)u (m)] = Rδ(n m) Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 4 / 10

9 Least Squares Channel Estimation- the Problem Transmit u(n)... m 1 vectors, independent from time to time E[u(n)] = 0 E[u(n)u (m)] = Rδ(n m)... across channel w 0... outputs: d(n) = w 0u(n) + v(n) v(n) is noise of power σv 2 Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 4 / 10

10 Least Squares Channel Estimation- the Problem Transmit u(n)... m 1 vectors, independent from time to time E[u(n)] = 0 E[u(n)u (m)] = Rδ(n m)... across channel w 0... outputs: d(n) = w 0u(n) + v(n) v(n) is noise of power σv 2... what s w 0? Don t know R Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 4 / 10

11 Least Squares Channel Estimation- the Problem Transmit u(n)... m 1 vectors, independent from time to time E[u(n)] = 0 E[u(n)u (m)] = Rδ(n m)... across channel w 0... outputs: d(n) = w 0u(n) + v(n) v(n) is noise of power σv 2... what s w 0? Don t know R Least Squares Solution = n i=1 u(i)d (i) {}}{ ŵ(n) = R 1 (n) z(n) (1) }{{} = n i=1 u(i)u (i)+δi Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 4 / 10

12 Least Squares Channel Estimation-the Challenge Idea is as n, the Sample Covariance Matrix R(n) n R. Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 5 / 10

13 Least Squares Channel Estimation-the Challenge Idea is as n, the Sample Covariance Matrix R(n) n R. Performance: the channel estimation error ɛ(n) = w 0 ŵ(n) Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 5 / 10

14 Least Squares Channel Estimation-the Challenge Idea is as n, the Sample Covariance Matrix R(n) n R. Performance: the channel estimation error ɛ(n) = w 0 ŵ(n) If n (practically, n m) is assumed: E[ ɛ(n) 2 2] 1 n σ2 vtr{r 1 } (2) Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 5 / 10

15 Least Squares Channel Estimation-the Challenge Idea is as n, the Sample Covariance Matrix R(n) n R. Performance: the channel estimation error ɛ(n) = w 0 ŵ(n) If n (practically, n m) is assumed: E[ ɛ(n) 2 2] 1 n σ2 vtr{r 1 } (2) Bad predictions for n m Occurs in applications like underwater acoustic communication, where channel varies quickly Or when taking observations is expensive Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 5 / 10

16 Least Squares Channel Estimation-the Challenge Idea is as n, the Sample Covariance Matrix R(n) n R. Performance: the channel estimation error ɛ(n) = w 0 ŵ(n) If n (practically, n m) is assumed: E[ ɛ(n) 2 2] 1 n σ2 vtr{r 1 } (2) Bad predictions for n m Occurs in applications like underwater acoustic communication, where channel varies quickly Or when taking observations is expensive Random Matrix Theory allows better predictions Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 5 / 10

17 The Random Matrix Theory Results Let M k (m, n) = 1 m E [ Tr ( R k (n) )]. Then... Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 6 / 10

18 The Random Matrix Theory Results Let M k (m, n) = 1 m E [ Tr ( R k (n) )]. Then... Channel Estimation Error: E [ ɛ(n) 2 2] = m ( σ 2 v M 1 (m, n) + δ 2 M 2 (m, n) δσ 2 vm 2 (m, n) ) (3) Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 6 / 10

19 Moments: Independent Input Observations Let A = 1 n XX T 1 /2. 2 key ideas to compute the moments: Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 7 / 10

20 Moments: Independent Input Observations Let A = 1 n XX T 1 /2. 2 key ideas to compute the moments: Assume m, n are large but c = m is a constant. Then, n eigenvalue density of R(n) can be approximated (Marcenko Pastur law): µ R(n) (x) µ Φ (x) = 1 ( ) x δ n µ A n (4) Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 7 / 10

21 Moments: Independent Input Observations Let A = 1 n XX T 1 /2. 2 key ideas to compute the moments: Assume m, n are large but c = m is a constant. Then, n eigenvalue density of R(n) can be approximated (Marcenko Pastur law): µ R(n) (x) µ Φ (x) = 1 ( ) x δ n µ A n Use the following to compute moments: 1 lim m m E[Tr ( B k) ] = t k µ B (t) dt (5) (4) Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 7 / 10

22 Moments: Independent Input Observations Let A = 1 n XX T 1 /2. 2 key ideas to compute the moments: Assume m, n are large but c = m is a constant. Then, n eigenvalue density of R(n) can be approximated (Marcenko Pastur law): µ R(n) (x) µ Φ (x) = 1 ( ) x δ n µ A n Use the following to compute moments: 1 lim m m E[Tr ( B k) ] = t k µ B (t) dt (5) (4) Compute Moments Using: M k (m, n) t k µ Φ (t) dt (6) Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 7 / 10

23 Predictions Made- Gaussian Input Gaussian input, Channel Length=30, SNR=40 db 10 simulations 0 theory n>>m 10 db number of observations Figure : Channel Estimation MSE vs Number of Observations Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 8 / 10

24 Predictions Made- Gaussian Input Uniform +1/ 1 input, Channel Length=30, SNR=5 db 5 simulations theory n>>m 0 db number of observations Figure : Channel Estimation MSE vs Number of Observations Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 9 / 10

25 Conclusions RMT makes nice predictions about signal processing systems running with a small number of observations Leads to identifying phenomena that were previously unknown Simple tools, but widely applicable More sophisticated tools available... how to use? Atulya Yellepeddi RMT Appl. to Underwater Wireless Comm Course Project 10 / 10