We all know that the Correlation matrix is :
Rxx = E{x.x^H} where E{} denotes expectation and H is the hermitian operator.
In practice, and in most cases, the E{} is replaced by the sample average.
x is an N x 1 column complex vector.
I would like to know how the eigenvalues of Rxx are affected if x were affected by a diagonal matrix C that changes every sample and depends only on a scalar, say 'alfa' i.e.
Rx'x' = E{x'.x'^H} where x' = Cx
or
Rx'x' = 1/N * (C(1)x(1)[C(1)x(1)]^H + ......... + C(N)x(N)[C(N)x(N)]^H )
Dear Ahmad,
you say that C is a diagonal matrix that changes every sample and depends on alpha.
What is the role of alpha here ? Can we write C_i = alpha_i Id ? (Id = matrix identity and i the number of the sample)
or C_i = alpha_i P with P a diagonal matrix that does not depend on the sample ?
Is C a random matrix ?
Note that x must centred to get the covariance matrix.
Thanks
Dear Emma,
C is a diagonal matrix where the (k,k)^th entry is :
C = exp(-j*2*pi* (k-1)alfa). (1)
Note that alfa is changing at each sample update, i.e.
C(1) is a function of alfa(1) as defined in (1)
C(2) is a function of alfa(2)
.
.
C(N) is a function of alfa(N)
And yes alfa is random at each sample update, i.e.
alfa(1) ... alfa(N) are realizations of a uniform random function in [0,alfa_max];
How should i deal with this if my only interest is Rxx ?
Thank you.
Dear Ahmed,
Ususally, x is a function of time t, if C is constant over t, then the expectation operator will ignore it, i.e., E{x'x'^H} = E{Cxx^HC^H} = CE{xx^H}C^H
However, if x is a column matrix then the Covariance matrix is given by
E{x^Hx} in that case, E{x^{H} C^{H}C x} = E{x^H D x},
Given your definition, D = Identity.
Therefore, E{x^Hx}.
Maybe I made a mistake. I 'm sure some expert will correct it sooner or later.
Dear Samrat,
Yes i know, by definition, that's true if C and X were independent
Dear Ahmad,
I thought that C is a diagonal matrix with non-random diagonal. If that is not the case, then, if C is independent of X (which i think is the case for your problem since diagonals of C are functions of alpha and alpha seems to be independent of X), the correlation matrix will be E(CXX^H C^H)=E(E(CXX^HC^H|C))=E(CR_xx C^H)=E(T) where the (i,j)th element of T is t_{ij}=C_ii.C*_jj. r_xx(i,j). So E(t_ij)=E(C_ii C*_jj)r_xx(i,j). From your given information, E(C_ii C_jj*)=E(exp(-2*pi*i(i-j)alpha)) which I think you can find easily.
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics