 such that g(t) = 0, t E [0, M.\T]. Thus, {8k,m(t)} are also finite and causal, with gk.m(t) = 0, t [0, MAT], where the parameter M = M + MS - 1 denotes the number of subsymbol spacing that the pulses {gk,m(t)} span. Based on that, we define the periodic pulse gr(t), where gr(t) = g(t), t [0, T], and T = . As a result, (4.4) Here, ug(t)=1,te[0,T]is: a rectangular window, and the set Mon = {0,Ms 1} defines the set of active subsymbols, where dk,m,b Mon. Therefore, (4.5) where xb(t), which is denoted as the modulation block, is a finite signal of length T. The multicarrier signal is a results of multiplexing the modulated blocks, where Ts is the spacing between the blocks, and To = T - -Ts=(Mo - 1)AT is the overlapping among blocks. 4.1.1.1 Discrete-Time Representation We consider N = MK samples per block, such that T = Ni where Fs is the sampling fre- quency. By sampling (4.4), we get 4 Generalized Frequency Division Multiplexing: Unified Multicarrier Framework For practical implementation, we assume that P = ATFs, Q=NN= = = AfT are integer num- bers. P is the size of subsymbol spacing measured in time samples, and Q is the subcarrier spacing in number of frequency samples, where a frequency sample corresponds to 212 Therefore, the discrete pulse can be expressed as (4.6) Here, <.>N is the modulo-N operator, which is used due to the periodicity of gr(t). The phase term can be considered as part of the data symbols. The frequency domain representation is achieved by means of N-DFT, where (4.7) 4.1.1.2 Relation to Gabor Theory The pulses in (4.6) belong to Gabor time-frequency lattices [17], with gka,mp[n] = gT[<n - a=Q/N, B = P. To uniquely demodulate the data symbols dk,m from a given x[n], Wexler-Raz duality condition [18] must be satisfied. Thus, there exists a pulse h[n] that attains (h,gk/p,m/a) = aBSk,08m,0, where 8k/p,m/a is the dual Gabor lattice, Sij refers to Kronecker delta, and <.> denotes the inner product.2 Furthermore, h is the demodulator prototype pulse such that dk,m = (x, Nka,mp)- It has been rig