Walsh codes are the most common orthogonal codes used in CDMA applications. A set of Walsh codes of length n consists of the n rows of an n * n Walsh matrix. That is, there are n codes, each of length n. The matrix is defined recursively as follows:

W1 = (0) W2n = a Wn Wn Wn Wn b

where n is the dimension of the matrix and the overscore denotes the logical NOT of the bits in the matrix. The Walsh matrix has the property that every row is orthogonal to every other row and to the logical NOT of every other row. Show the Walsh matrices of dimensions 2, 4, and 8.