Hi guys, Does anyone know how to solve this problem: A zero mean stationary signal having a correlation matrix R is applied to a FIR filter with coefficient vector w. Show that the average power of the filter output is given by w'Rw. Find also an expression for the output power spectrum using the elements of w and R. What would be the output power spectrum if the FIR filter coefficients are stochastic with zero mean and a covariance of rho^2*B, where B is a diagonal matrix? Thanks a million!
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Started by ●March 14, 2007
Reply by ●March 14, 20072007-03-14
sundown wrote:> Hi guys, > Does anyone know how to solve this problem: > > A zero mean stationary signal having a correlation matrix R is applied to > a FIR filter with coefficient vector w. Show that the average power of the > filter output is given by w'Rw. Find also an expression for the output > power spectrum using the elements of w and R. > What would be the output power spectrum if the FIR filter coefficients are > stochastic with zero mean and a covariance of rho^2*B, where B is a > diagonal matrix?Assuming that your instructor know how, the answer is "yes". Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●March 14, 20072007-03-14
On Mar 14, 4:10 pm, "sundown" <pohk0...@ntu.edu.sg> wrote:> Hi guys, > Does anyone know how to solve this problem: > > A zero mean stationary signal having a correlation matrix R is applied to > a FIR filter with coefficient vector w. Show that the average power of the > filter output is given by w'Rw. Find also an expression for the output > power spectrum using the elements of w and R. > What would be the output power spectrum if the FIR filter coefficients are > stochastic with zero mean and a covariance of rho^2*B, where B is a > diagonal matrix? > > Thanks a million!If it's a signal (scalar) then it won't have a correlation matrix - it will be scalar - a variance. You can have a matrix based on regressors of the signal say X - I expect that is what he means. Then say y(k)=W'X where W is a weight vector from an FIR filter. also E[XX']=R find E[yy'} = E[W'XX'W] = W'RW. QED
Reply by ●March 15, 20072007-03-15
>If it's a signal (scalar) then it won't have a correlation matrix - it >will be scalar - a variance. >You can have a matrix based on regressors of the signal say X - I >expect that is what he means. Then > >say y(k)=W'X where W is a weight vector from an FIR filter. > >also >E[XX']=R > >find > >E[yy'} = E[W'XX'W] = W'RW. QED > >Thanks !!
