Given a pair (B, 1/T ) of a bandwidth and a reference rate, the excess bandwidth β is defined as
β 2BT − 1 and the Nyquist interval F is defined as F f : − 1 2T ≤ f < 1 2T .
Given a pair (B, 1/T ) and a deterministic function p(t) having the continuous-time Fourier transform
(CTFT) P (ξ) ∞ −∞ p(t)e −j2πξt dt, the VFT p(f ) of p(t) is defined as a vector-valued function of f ∈ F that is equivalent to P (ξ). In particular, the kth entry of p(f ) is given by [p(f )] k P f + k−L−1 T for k = 1, 2, · · · , 2L + 1, where L ⌈β/2⌉.
Given a pair (B, 1/T ) and an SOCS random process N(t) with cycle period T having the auto-
correlation function r N (t, s), the matrix-valued PSD R N (f ) of N(t) is defined as a matrix-valued function of f ∈ F , whose (k, l)th entry is given by [R N (f )] k,l R (k−l) N (f +(l − L − 1)/T ) for k, l = 1, 2, · · · , 2L+ 1, where R (k) N (ξ) is the CTFT of r (k)
N (τ ) that is obtained by applying the Fourier series expansion to
r N (t, t − τ ), i.e., r N (t, s) = ∞ k=−∞ r (k) N (t − s)e j2πkt/T .
In using the above definitions, it is assumed that the parameter B is chosen as bandwidth in complex baseband over which the Rx can observe and process a signal and that the parameter 1/T is chosen as the symbol transmission rate of the Tx. It is also assumed that the frequency band over which the Tx can emit non-zero power is identical to the frequency band of the Rx. For a general case where these two frequency bands are different, the notion of virtual legacy Rx's and the orthogonal constraint at the virtual legacy Rx's can be employed as is done in [15] for the transmission of a proper-complex **data** sequence.