Concatenating these vectors yields a matrix, called circulant matrix
, which can be expressed as
Then, transform the convolution based on circulant matrix
in linear system.
In , an orthogonal circulant matrix
transform (OCT) based precoding scheme is proposed, which achieves a relatively flat signal-to-noise ratio (SNR) profile over the signal bandwidth and alleviates significant BER degradation at high frequency carriers.
The circulant matrix
[a] can be diagonalized by [V] as shown in Equation (18), where [[V].sup.H] is the Hermitian matrix of [V].
First, we embed Toeplitz matrix [T.sub.K] into a 2K x 2K circulant matrix
[C.sub.2N] as follows [18, 19]:
where [mathematical expression not reproducible] is an N x N circulant matrix
, with the first columns [mathematical expression not reproducible], and [mathematical expression not reproducible] are circular complex white Gaussian noise with zero mean and covariance matrix [mathematical expression not reproducible] that is allocated to the relay training, that is, [mathematical expression not reproducible] is allocated to the amplified received data, where 0 < [gamma] < 1.
together with the rank-2[n.sub.x] primary circulant matrix
C, that is, the circulant matrix
whose first row has second element one and all other elements zero, allows constructing a banded matrix
The matrix C is chosen as Strang's circulant matrix
The results of Table 2 show that the method has a higher performance than traditional Toeplitz or Circulant matrix
are special Toeplitz matrices as each column of a circulant matrix
is a circulant shift of its preceding column.
One can obtain the same description Q = [D.sup.T]D, where D is also a circulant matrix