A Pilot Included Column Mean Vanishing Matrix

In this paper, we study a special matrix used in OFDM technology including the pilot vector. This is based on the property of ’column mean vanishing’ and orthogonal columns. We study the spectral decomposition. Using this, we suggest a new method of generating such matrices. Numerical examples are included.


Introduction
Recently, there have arisen a large necessity of developing a new technology in wireless communications.An OFDM or its generalization is a big trend (O.Edfors, et al., 1998;Frederiksen, F. B. & Prasad, R., 2002;Myungsup, K. & Kwak, D. Y.).In this paper, we review the algorithm developed in (Myungsup, K. & Kwak, D. Y.) and study some properties of the OFDM matrix.Based on this we propose a simple method to generate the matrix.In the resulting matrix, we see the pilot column has only two nonzero entries which correspond to zero rows, so that the pilot does not interfere with other data.
Definition 1.1.We say a matrix A has a column mean vanishing (CMV) property if the sum of each column is zero.

Generation of CMV Matrix Having Orthonormal Columns
Let L = n ≫ N. Recall the scheme introduced in (Myungsup, K. & Kwak, D. Y.): Algorithm Orth-pilot 1.Given a N × (N − 2) initial matrix K p with orthonormal columns.

Multiply by L × N matrix
4. Subtract the first row from all the rows, the result is Φ • F −1 (PK p ). Step (2).Permute and Pad Zeros

Perform FFT to get
Starting from an N × M initial matrix, we construct an L × M matrix as follows: Move the last m + 1 rows of K p to the first m + 1 rows of K p .Next fill it with pad with L − M zero rows (called zero padding).This process can be expressed as PK p where Steps ( 3) and ( 4) : IFFT Followed by Subtraction of the First Row Let us use the notation K = (k i j ) and Hence the matrix after step (4) is Here Ǩ * 1 is the matrix all of whose rows are the vector ǩ1 .
Lemma 2.1.The sum of all entries of each column of the matrix K is zero.
Theorem 2.1.The matrix G obtained in step ( 7) satisfies CMV property: Then by Lemma 2.1, we have Hence by (4) we see

Simplification of the Algorithm
In this paper we simplify the algorithm above.We apply the algorithm to an initial matrix having CMV property.First consider the case N = 2m + 1 is odd.We will explain with m = 2, general case follows easily from this.Consider the following N × (N − 1) initial matrix.
Lemma 3.1.If the initial matrix K p (K p,e or K p,o ) satisfies CMV property, then steps ( 1)-( 7) is simplified as

Property of Odd Columns
We assume N is odd.The case of even is similar.Let k i and g i and denote the i-th column of the matrix K p and G respectively.
Lemma 4.1.The odd columns of K p are orthogonal to all other columns of K p .As a consequence, for all odd j, the vector T is an eigenvector of K H p K p corresponding to the eigenvalue 1.
Proof.The orthogonality of odd columns of K p comes from that of K * of (5) since during the transformation of K * to K p in ( 5), ( 7), the odd columns did not change essentially(only the orders are permuted).Let Then K p e j = k j and hence the j -th column of KH p Kp satisfies This means that when j is odd, the j-th columns of K p are orthogonal to all other columns of K p .Clearly (12) implies the second assertion of the lemma.
Example 4.1.For N = 5 we see Note that the zeros in the box keep the odd columns of K p orthogonal to other columns.In view of ( 12), K H p K p has two eigenvectors e j , j = 1, 3 corresponding to the eigenvalue 1.
Theorem 4.2.The odd columns of G = K p (K H p K p ) −1/2 are the same as those of K p .
Proof.From the spectral decomposition of K p = UΣV H we have that of K H p K p : where by ( 12) Λ and V have the following form: Note that for j odd Ve j = e j and for each odd j, This is the same as j-th column of K p (normalization does not change even columns).