ce. 5 Filter Bank Multicarrier Modulation Figure 5.2 Magnitude responses of the CMT pulse-shaping filters at various subcarriers and time instants t = 0 and t=T. Let n = , -2, -1,0,1,1 2, ... denote symbol time index, k = 0,1,2, ... denote symbol frequency index, sk[n] denote the (n,k) data symbol in the time-frequency lattice, and phase shift that is added to the carrier of sk[n]. Accordingly, a CMT waveform that is constructed based on the pulse-shape/prototype filter p(t) is expressed as (5.1) where ank(t)== (5.2) Pnx(t)==(p(t-nT)e (5.3) The synthesis of x(t) according to (5.1) has the following interpretations. The terms an.k(t) may be thought of as a set of basis functions that are used to carry the data symbols sk[n]. The data symbols Sk[n] can be extracted from x(t) in a straightforward manner if an.k(t) are a set of orthogonal basis functions. The orthogonality for a pair of functions U1 (t) and U2(t), in general, is defined as (5.4) For the case of interest here, where the data symbols Sk[n] are real-valued, the orthogonality definition (5.4) can be replaced by the more relaxed definition (5.5) where R{.} indicates the real part. The definition (5.5) is referred to as real orthogonality. 5.3 Theory 87 It is not difficult to show that n = m and k = l (5.6) otherwise and hence, for any pair of n and k, (5.7) To develop an in-depth understanding of the CMT signaling, it is instructive to explore a detailed derivation of (5.6). To this end, we begin with the definition (5.8) and note that this can be rearranged as (5.9) When m = n and k=1, after a change of variable t to t + nT, (5.9) reduces to (5.10) where the second equality follows from the fact that p(t) is a real-valued SR-Nyquist pulse and p(t) = p(-t). When k = l, m # n, and m - n = 2r, where r is an integer, (5.11) where the second equality follows since p(t) is a SR-Nyquist pulse, designed for a symbol spacing 2T. On the other hand, when k = l, but m - + 1, (5.12) Next, consider the case where k - l = 1 and m - n = 2r. In that case, one finds that 