Because it is quadratic in the (X_{p}) and (gleft( z right))'s, minimizing (xi) is a straightforward linear least squares problem.
Therefore, the mapping Q1 satisfies Equation (4) and so it is quadratic by Theorem 2.2.
By the same reasoning as the proof of Theorem 2.1, it is quadratic.
However, this recombination requires O M 2 · d multiplications, i.e., it is quadratic in M, which is undesirable.
A mapping f : X → Y satisfies the functional equation (1.4) if and only if it is quadratic.
Moreover, one and the same input gives rise to a different number of these kinds of waves, and it is quadratic nonlinearity that determines it.
Moreover, the corresponding computational complexity is shown to be independent of the constellation size, but it is quadratic in the number of transmit antennas.
The mapping Q is quadratic because it satisfies equation (1.2) as follows: ∥ D ˜ 1 Q ( x, y ) ∥ β = lim n → ∞ 1 | k | 2 n β ∥ D ˜ 1 f ( k n x, k n y ) ∥ β ≤ lim n → ∞ 1 | k | 2 n β φ ( k n x, k n y ) = 0. for all x, y ∈ X ; therefore, by Lemma 2.2, it is quadratic.
The mapping Q is quadratic because as follows it satisfies in Equation (1.2): D ̃ 1 Q ( x, y ) β = lim n → ∞ 1 k 2 n D ̃ 1 f ( k n x, k n y ) β = lim n → ∞ 1 k 2 n β D ̃ 1 f ( k n x, k n y ) β ≤ lim n → ∞ 1 k 2 n β φ ( k n x, k n y ) = 0, for all x,y ∈ X, therefore by Lemma 2.2, it is quadratic.
