## 2 The Principles of Polybinary Transformation  
An n-level polybinary transformation can be completed by a series of delay operations and modulo-two sum operations. The operation of n-level polybinary transformation can be expressed as two steps as follows. The first step is to transform the original input binary message sequence \(\{a_{m}\}\) into another binary sequence \(\{d_{m}\}\). The present binary sequence of \(\{d_{m}\}\) is generated by forming the modulo-two sum of the present input sequence of \(\{a_{m}\}\) and the preceding n-2 of \(\{d_{m}\}\). The above-stated mathematical calculation can be expressed as an equation:  
\[d_{m}=a_{m}\oplus d_{m-1}\oplus d_{m-2}\oplus\cdots\oplus d_{m-n+2} \tag{1}\]  
where \(\oplus\) represents the exclusive-or logic operation (modulo-two addition). The second step is to generate an n-level polybinary sequence \(\{p_{m}\}\), and the ith element of this polybinary sequence \(p_{i}\) is proportional to the algebraic sum of the successive n-2 values of the binary sequence \(\{d_{m}\}\). This process is expressed by  
\[p_{i}=E\sum_{k=0}^{n-2}d_{m-k} \tag{2}\]  
The structure for these operations is illustrated in Figure 1, where \(\{a\}\) is the input binary message sequence, \(\{d\}\) is the transformational binary sequence, \(E\) is an arbitrary constant, \(\{p\}\) is the n-level polybinary sequence at the detector and \(TFigures 2a-c and 3a-c are the probability density figures (PDFs) of QPSK and 16 QAM constellations when performing 1-level polybinary transformation, duobinary transformation and tribinary transformation, respectively. The colors from red to dark blue represent the level of probability density of the constellation points; the red color denotes a high level and the blue color denotes a low level. Besides the aforementioned phenomena, the number of constellation points changes after the polybinary transformation. Figure 2 shows that the number of constellation points changes from 4 points to 9 points and 16 points when the original QPSK signal undergoes duobinary transformation and tribinary transformation, respectively. Figure 3 shows that the number of constellation points changes from 16 points to 49 points and 100 points for the original 16 QAM signal after duobinary transformation and tribinary transformation, respectively. Figure 1: Structure of n-level polybinary signal by delay operations and modulo-two sum operations. Figure 2: The PDF of n-level polybinary transformation for QPSK signal: (**a**) 1-level polybinary transformation of original signal; (**b**) duobinary transformation; (**c**) tribinary transformation. Note that the above processes for the bit message sequence and the symbol message sequence are similar. Therefore, the polybinary transformation can be achieved either at the transmitter side or receiver side in a practical system. Compared with the polybinary transformation at the transmitter side, the polybinary transformation at the receiver side can be combined with DSP algorithms to efficiently equalize the impairments. Therefore, we will look into the advantages and benefits of polybinary transformation at the receiver side in this paper. The Relationship among Time-Packing Factor, n-Level Polybinary Transformation and Performance of Decoding Algorithms  
Using the FTN technique in a system means that higher SE is obtained with a smaller time-packing factor. However, an extremely small time-packing factor induces more serious ISI in the received signal sequence, hence increasing the burden of the decoding algorithm and even making the algorithm unworkable. Obtained through numerical simulation, Figure 4a shows the PDF diagram of the original QPSK signal when the time-packing factor equals 0.50. In this case, the SE doubles but the ISI is severely induced, with the result that we cannot clearly distinguish the constellation points. Therefore, we utilize the n-level polybinary transformation for the original QPSK signal with the time-packing factor of 0.50.