# On the Diversity Order of UW-OFDM

Heidi Steendam^{*}

In unique word (UW) OFDM, the signal is constructed such that the guard interval is part of the FFT interval. Typically, first the data signal is constructed with zeros at the time instants of the guard interval, and later the guard interval is filled with the known samples of the unique word. One of the main questions is: what is the performance of UW-OFDM? Will it have a better performance than CP-OFDM? In order to answer this question, we look at the error rate performance of UW-OFDM. We derive analytical expressions for the bit error rate, from which we can obtain the diversity order. It turns out that when the code generator matrix $\mathbf{G}$, needed to add the redundancy in the signal, is full rank, the standard UW-OFDM system reaches the maximum diversity order. This is in contrast with standard CP-OFDM, where only diversity order one can be reached, unless additional precoding is applied. Further, in the paper, we use the basics of matrix algebra to strip down the structure of the UW-OFDM code generator matrix $\mathbf{G}$. With the help of the basis vectors of the null space of the reduced FFT matrix corresponding to the zero samples in the time domain, we are able to simplify the requirements for optimal receiver design, as e.g. to obtain maximum diversity, maximum coding gain, minimum redundant energy, optimal BLUE or LMMSE data estimation, ... These results can be used in the design of the optimal UW-OFDM signal.

Mathematics Subject Classification: 94A05

Keywords: diversity, construction of UW-OFDM, optimal performance

Minisymposion: Unique Word OFDM