Autoregressive Interference

Autoregressive Interference

Let us assume that the NBI signal is modeled as a pth order AR process, where p << N:

Equation 7.94

where {e_n} is an i.i.d. Gaussian sequence with variance v². Supposing that R_i is positive definite, we first drive a closed-form expression for . Using (7.94), we can write the following:

Equation 7.95

or, in compact form,

Equation 7.96

where A is the matrix appearing on the left-hand side of (7.95), i_N-p=[i_n, i_n-1, ..., i_n-N+p+1]^T, i_p=[i_n-N+p, i_n-N+p-1, ..., i_n-N+1]^T, and e_N-p=[e_n, e_n-1, ..., e_n-N+p+1]^T. Multiplying both sides of (7.96) by their transposes and taking expectations, we obtain

Equation 7.97

that is,

Equation 7.98

where and are, respectively, the N x N and pxp autocorrelation matrices of the interference signal. Since A is nonsingular, then

Equation 7.99

Equation 7.100

Partition the N x N matrix A into the following four blocks:

Equation 7.101

where A₁₁ is of dimension (N - p) x (N - p), and A₁₂ is of dimension (N - p) x p. Substituting (7.101) into (7.100), we can write

Equation 7.102

Now most of the elements of are explicitly given by (7.102), except for the southeast p x p block. But notice that is a Toeplitz matrix, and the inverse of a nonsingular Toeplitz matrix is persymmetric (i.e., it is symmetric about its northeast-southwest diagonal) [158]. Therefore, the elements of the southeast p x p block of can be found in the northwest p x p block, which have already been determined. Hence, with the aid of persymmetry, is completely specified by (7.102). Straightforward calculation of (7.102) then shows that is a bandlimited matrix, with bandwidth 2p+1. Since it is symmetric, we need only to specify the upper p + 1 nonzero diagonals, as follows:

Equation 7.103

where D_k contains the (N - k) elements on the kth upper (lower) diagonal of , k = 0,1,...,p.

Next we consider the output SINR of the linear MMSE detector when the interferer is an AR signal. For the sake of analytical tractability, and to stress the effectiveness of the MMSE detector against the narrowband AR interference (versus the background noise), we consider the output SINR when there is no background noise (i.e., s² 0). Using (7.103), we have

Equation 7.104

Equation 7.105

where in (7.104), we have made the approximation that D_K[i] = D_K[N/2], 0 k p, 0 i N–k–1, since when N >> p, it seen from (7.103) that on each nonzero diagonal most of the elements are the same; and in (7.105) we used the approximation and thus dropped the second term in (7.104). The output SINR is then

Equation 7.106

As will be seen in Section 7.5, this SINR value is the same as an SINR upper bound given by the nonlinear interpolator NBI suppression method in the absence of background noise.