Editing Orthogonal frequency-division multiplexing (section)

== Vector OFDM (VOFDM) ==
VOFDM was proposed by Xiang-Gen Xia in 2000 (''Proceedings of ICC 2000'', New Orleans, and ''IEEE Trans. on Communications'', Aug. 2001) for single transmit antenna systems. VOFDM replaces each scalar value in the conventional OFDM by a vector value and is a bridge between OFDM and the single carrier frequency domain equalizer (SC-FDE). When the vector size is <math>1 </math>, it is OFDM and when the vector size is at least the channel length and the FFT size is <math>1 </math>, it is SC-FDE.

In VOFDM, assume <math>M </math> is the vector size, and each scalar-valued signal <math>X_n </math> in OFDM is replaced by a vector-valued signal <math>{\bf X}_n </math>of vector size <math>M </math>, <math>0\leq n\leq N-1 </math>. One takes the <math>N </math>-point IFFT of <math>{\bf X}_n, 0 \leq n \leq N - 1 </math>, component-wisely and gets another vector sequence of the same vector size <math>M </math>, <math>{\bf x}_k, 0 \leq k \leq N - 1 </math>. Then, one adds a vector CP of length <math>\Gamma </math> to this vector sequence as

: <math>{\bf x}_0, {\bf x}_1, ..., {\bf x}_{N-1}, {\bf x}_0, {\bf x}_1, ..., {\bf x}_{\Gamma-1} </math>.

This vector sequence is converted to a scalar sequence by sequentializing all the vectors of size <math>M </math>, which is transmitted at a transmit antenna sequentially.

At the receiver, the received scalar sequence is first converted to the vector sequence of vector size <math>M </math>. When the CP length satisfies <math display="inline">\Gamma \geq \left\lceil \frac{L}{M} \right\rceil </math>, then, after the vector CP is removed from the vector sequence and the <math>N </math>-point FFT is implemented component-wisely to the vector sequence of length <math>N </math>, one obtains

: <math>{\bf Y}_n = {\bf H}_n {\bf X}_n + {\bf W}_n,\,\, 0 \leq n \leq N - 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1) </math>

where <math>{\bf W}_n</math> are additive white noise and <math display="inline">{\bf H}_n = {\bf H}\mathord\left(\exp\mathord\left(\frac{2\pi jn}{N}\right)\right) = {\bf H}(z)|_{z=\exp(2\pi j n/N)} </math> and <math>{\bf H}(z)</math> is the following <math>M \times M </math> polyphase matrix of the ISI channel <math display="inline">H(z) = \sum_{k=0}^L h_k z^{-k}</math>:

: <math>\mathbf{H}(z) = \left[ \begin{array}{cccc}
  H_0(z) & z^{-1} H_{M-1}(z) & \cdots & z^{-1} H_1(z)\\
  H_1(z) & H_0(z) & \cdots & z^{-1} H_2(z)\\
  \vdots & \vdots & \vdots & \vdots \\
  H_{M-1}(z) & H_{M-2}(z) & \cdots & H_0(z)
\end{array}\right]</math>,

where <math display="inline">H_m(z) = \sum_l h_{Ml+m}z^{-l} </math> is the <math>m </math>th polyphase component of the channel <math>H(z), 0 \leq m \leq M - 1</math>. From (1), one can see that the original ISI channel is converted to <math>N </math> many vector subchannels of vector size <math>M </math>. There is no ISI across these vector subchannels but there is ISI inside each vector subchannel. In each vector subchannel, at most <math>M </math> many symbols are interfered each other. Clearly, when the vector size <math>M = 1 </math>, the above VOFDM returns to OFDM and when <math>M > L </math> and <math>N = 1 </math>, it becomes the SC-FDE. The vector size <math>M </math> is a parameter that one can choose freely and properly in practice and controls the ISI level. There may be a trade-off between vector size <math>M </math>, demodulation complexity at the receiver, and FFT size, for a given channel bandwidth. Equation (1) is mathematically new for an ISI channel, when the vector size <math>M>1</math>. 

Note that the length of the CP part in the sequential form does not have to be an integer multiple of the vector size, <math>\Gamma M</math>. One can truncate the above vectorized CP to a sequential CP of length not less than the ISI channel length, which will not affect the above demodulation.

Also note that there exist many other different generalizations/forms of OFDM, to see their essential differences, it is critical to see their corresponding received signal equations to demodulate. The above VOFDM is the earliest and the only one that achieves the received signal equation (1) and/or its equivalent form, although it may have different implementations at transmitter vs. different IFFT algorithms. 

It has been shown (Yabo Li et al., ''IEEE Trans. on Signal Processing'', Oct. 2012) that applying the MMSE linear receiver to each vector subchannel (1), it achieves multipath diversity and/or signal space diversity. This is because the vectorized channel matrices in (1) are pseudo-circulant and can be diagonalized by the <math>M</math>-point DFT/IDFT matrix with some diagonal phase shift matrices. Then, the right hand side DFT/IDFT matrix  and the <math>k</math>th diagonal phase shift matrix in the diagonalization can be thought of the precoding to the input information symbol vector <math>{\bf X}_k</math> in the <math>k</math>th sub vector channel, and all the vectorized subchannels become diagonal channels of <math>M</math> discrete frequency components from the <math>MN</math>-point DFT of the original ISI channel. It may collect the multipath diversity and/or signal space diversity similar to the precoding to collect the signal space diversity for single antenna systems to combat wireless fading or the diagonal space-time block coding to collect the spatial diversity for multiple antenna systems. The details are referred to the IEEE TCOM and IEEE TSP papers mentioned above.