Rearranging the terms in the above equation we get zm(1+m)=-(1+m)+cm (15) which. with a bit of algebra can be written as m(1-c+z)+1=0 Equation(16)is a polynomial equation in m whose coefficients are polynomials in z. We will refer to such polynomials, often derived from canonical equations as Stieltjes(transform) polynomials for the remainder of this paper. As discussed, to obtain the density using (5) we need to first solve(16) for m in terms of z. Since, from (16), we have a second degree polynomial in m it is indeed possible to analytically solve for its roots an obtain the density. Figure 1: Level density for BN=(1/N)XnXn with c= 2 This level density, sometimes referred to in the literature as the Marcenko-Pastur distribution, is b dF(a) b√(x-b-)(b+=x (17) where b+=(1+va and I(- b+ is the indicator function that is equal to 1 for b-<z<b+ and 0 elsewhere Figure 1 compares the histogram of the eigenvalues of 1000 realizations of the matrix B IX*X wit N=100 and n= 200 with the solid line indicating the theoretical density given by(17)for c=n/N=2 ance matrices that appear often in array processing applications em that is motivated by the sample covari- We now consider a modification to the marcenko-Pastur thee 3.1 The Sample Covariance Matrix In the previous section we used the Marcenko-Pastur theorem to examine the density of a class of random matrices BN= LX*TnXn. Suppose we defined the N x n matrix, YN=X:Tn/ 2, then BN may be written Recall that Tn was assumed to be diagonal and non-negative definite so Tl/ can be constructed uniquely up o the sign. If YN were to be interpreted as a matrix of observations, then BN written as(18)is reminiscent� � � Rearranging the terms in the above equation we get zm(1 + m) = −(1 + m) + cm (15) which, with a bit of algebra, can be written as m z 2 + m(1 − c + z) + 1 = 0. (16) Equation (16) is a polynomial equation in m whose coefficients are polynomials in z. We will refer to such polynomials, often derived from canonical equations as Stieltjes (transform) polynomials for the remainder of this paper. As discussed, to obtain the density using (5) we need to first solve (16) for m in terms of z. Since, from (16), we have a second degree polynomial in m it is indeed possible to analytically solve for its roots and obtain the density. dFB/dx 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 −1 0 1 2 3 4 5 6 7 x Figure 1: Level density for BN = (1/N)X∗Xn with c = 2. n This level density, sometimes referred to in the literature as the Marˇcenko-Pastur distribution, is given by dF B(x) = max 0, 1 − c δ(x) + (x − b−)(b+ − x) I[ b− ,b+] (17) dx 2πx where b± = (1±√c)2 and I[b ,b+] is the indicator function that is equal to 1 for b− < z < b+ and 0 elsewhere. − 1 Figure 1 compares the histogram of the eigenvalues of 1000 realizations of the matrix BN = N Xn ∗Xn with N = 100 and n = 200 with the solid line indicating the theoretical density given by (17) for c = n/N = 2. We now consider a modification to the Marˇcenko-Pastur theorem that is motivated by the sample covari￾ance matrices that appear often in array processing applications. 3.1 The Sample Covariance Matrix In the previous section we used the Marˇcenko-Pastur theorem to examine the density of a class of random 1 ∗ 1/2 matrices BN = N X∗TnXn. Suppose we defined the N × n matrix, YN = XnTn , then BN may be written n as 1 ∗ BN = YN YN . (18) N 1/2 Recall that Tn was assumed to be diagonal and non-negative definite so Tn can be constructed uniquely up to the sign. If YN were to be interpreted as a matrix of observations, then BN written as (18) is reminiscent