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 [16], 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 ([C.sub.s]), R.

The results of Table 2 show that the method has a higher performance than traditional Toeplitz or

Circulant matrix method.

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,