## 2 System Model of DBT-MC-FTN
### DBT-MC-FTN Transmitter  
DBT-MC-FTN is essentially a multi-carrier scheme. Denote \(x_{m}(d)\) as the \(d\)th complex QPSK symbol modulated on the \(m\)th subcarrier, then the continuous-time transmitted signal of DBT-MC-FTN signaling can be expressed as  
\[g(t)=\sum_{d=0}^{D-1}\sum_{m=0}^{M-1}x_{m}(d)w\left(t-d\beta T_{s}\right)e^{j2 \pi\frac{m\alpha t}{T_{s}}}, \tag{1}\]  
where \(w(t)\) is the continuous-time unit energy pulse with the Nyquist symbol interval \(T_{s}\), \(M\), and \(D\) are, respectively, the numbers of occupied subcarriers and the symbols transmitted on each subcarrier. \(\alpha(\alpha\leq 1)\) and \(\beta(\beta\leq 1)\) denote the frequency packing factor (FPF) and the time packing factor (TPF), respectively. We set \(v=\alpha\cdot\beta\) as the joint time-frequency packing factor. For \(\alpha=1\) and \(\beta=1\), an orthogonal system is obtained, while for \(\alpha<1\) or \(\beta<1\), frequency-domain or time-domain packing occurs between the subcarriers or symbols. Figure 2 illustrates the time-frequency grid comparison between the multicarrier FTN system and the orthogonal systems, from which the relationship between the desired signal and introduced ISI/ICI can be recognized clearly. As a result, the improvement of spectral efficiency is obvious. By sampling the signal in Equation (1) at the sampling period of \(T_{s}/N_{s}\), the discrete signal can be written as  
\[g(q)=\frac{1}{\sqrt{N_{s}}}\sum_{d=0}^{D-1}\sum_{m=0}^{M-1}x_{m}(d)w\Big{(} \Big{(}q-dN_{f}\Big{)}\Big{)}_{Q}e^{j2\pi\frac{m\alpha t}{N_{s}}}, \tag{2}\]  
where \(N_{f}=\beta N_{s}\), \(0\leq q\leq Q-1\), \(\frac{1}{\sqrt{N_{s}}}\) is a normalization scaling factor. \(w(n)\) is a \(N_{s}\)-orthogonal pulse with the length of \(L\), where \(\sum_{n=0}^{L-1}w(n)w^{*}(n-dN_{s})=0\) when \(d\
eq 0\). The operator of \(((\cdot))_{Q}\) represents the Modulo-\(Q\) operation where \(Q=D\times N_{f}\) and \(Q>L\). By applying \(Q\)-point IDFT operation to both sides of Equation (2), the frequency-domain expression of DBT-MC-FTN signal can be derived as  
\[G(i)=\sqrt{D}\sum_{m=0}^{M-1}X_{m}(i-m\alpha\beta D)W(i-m\alpha\beta D), \tag{3}\]  
where \(0\leq i\leq Q-1\), \(X_{m}(i)\) is the \(D\)-point DFT of \(x_{m}(d)\) and \(W(i)\) is the \(Q\)-point DFT of \(w(n)\). Equation (3) can be interpreted as follows. The data symbols transmitted on each subcarrier are first transformed into the frequency domain with \(D\)-point DFT, and then extended circularly to \(Q\) points. Each \(Q\)-point output signal is multiplied by a frequency-domain window whose length is variable according to the TPF \(\beta\). After the summation of the \(M\)\(Q\)-point signals circularly with the shift interval  
Figure 2: Illustration of the time-frequency grid of multicarrier FTN symbols and orthogonal symbols. \(ma\beta D\), \(Q\)-point IDFT, and CP padding, finally, DBT-MC-FTN signaling is generated, as shown in Figure 1a. Note that the two-dimensional time-frequency packing can be realized in a unified manner via the overall packing factor \(v\). The time-domain packing is implemented by adjusting the window length, and the frequency-domain packing is realized by multi-carrier cyclic shift summation with the FPF \(\alpha\). Change the window length and the multi-carrier summation cyclic shift interval at the same time, time-frequency packing can be simultaneously realized. It is worth pointing out that the proposed DBT-MC-FTN scheme can be viewed as the fast-than-Nyquist extension of SC-OFDM (i.e., DFT-S-OFDM). In the case of \(M=1\), \(\alpha=\beta=1\), and the window is a rectangular pulse, obviously, it becomes the traditional SC-OFDM. If the rectangular pulse is used, the proposed scheme without time-frequency packing will transformed into an \(N\times\)SC-OFDM scheme. To take the general cases into consideration, the coefficients of all the \(M\)\(D-\)point DFTs will be packed, sequentially mapped, and superimposed on each subcarrier of the \(Q\)-point IDFT within the DBT-MC-FTN symbol; hence, this is the faster-than-Nyquist extension of SC-OFDM.