In the 1975, James Mazo found out the fact that the binary sinc-pulse can be transmitted faster than the Nyquist theorem states without increase of bit error rate, despite of ISI. Later in 2005, the idea was extended to frequency domain e.g. reduce subcarrier spacing. The transmit signal of FTN can be expressed as $$ x(t) = \sum_{k=-\infty}^{\infty}\sum_{n=0}^{N_{\textrm{sub}}-1} s_{k,n} g\left(t-k\Delta_{T}T\right)e^{j2\pi\frac{\Delta_{F}}{T}nt}, $$ where

• $\Delta_{T} \le 1$ is the time compression factor, which means the pulses are transmitted faster by the factor $1/\Delta_{T}$
• $\Delta_{F} \le 1$ is the frequency compression factor, which means the spectral efficiency is increased by the factor $1/\Delta_{F}$