# VOFDM (Vector OFDM) advantages and disadvantages

What are the advantages and disadvantages of using VOFDM over OFDM? I know that one advantage is frequency diversity, what are others?

edit : By VOFDM I mean Xia's method as shown here.

• Perfect question for @MaximilianMatthé Commented Mar 1, 2017 at 12:30
• @DanBoschen sadly, paging doesn't work when the person being paged isn't already in a comment thread :) Commented Mar 1, 2017 at 12:48
• Ah!- Thanks Marcus. I am sure the title of the question will catch his eye. Commented Mar 1, 2017 at 12:50
• Gabe, can you please give a reference, which VOFDM you refer to? It seems there are different definitions out there in the internet. Commented Mar 1, 2017 at 13:02
• @Gabe love this situation. If only there was a way to ask the first author of that paper :) Commented Mar 1, 2017 at 13:41

## 1 Answer

Referring to VOFDM as described in this reference, some thoughts on that (I wrote this summary some years ago, due to an issue in my own research). We can understand VOFDM as quite similar to interleaved SC-FDMA, (this means that a single carrier occupies several frequency bins, which are not grouped together but spread around the full bandwidth of the system). So, assuming the system has $N=KM$ OFDM subcarriers (FFT window length N, N frequency bins), the frequency indices occupied by an interleaved SC-FDMA carrier would be $[l, l+K, l+2K, l+(M-1)K]$, where $M$ is the number of frequency bins per SCFDMA carrier and $K$ is the number of different carriers possible in the full bandwidth. $l=0,\dots,K-1$ is the index of the SC-FDMA carrier. Then, after FFT at the receiver, the received signal for the $l$ carrier is given by

$$Y_l=H_ld_l+n_l$$ where $Y_l$ is the received signal at the frequency bins for the $l$ carrier, $d_l$ is the transmit data for the $l$ carrier and $n_l$ is AWGN. $H_l$ is the equivalent channel matrix, which can be considered a circulant matrix.

For the considerations below, VOFDM is actually doing exactly the same thing (compare eq (4) of this reference). So, how can we compare OFDM and VOFDM?

• The big advantage of OFDM is low complexity at the receiver. A single DFT plus one-tap equalization is sufficient for the detection. In contrast, VOFDM requires additional M-point DFTs on each carrier to perform one-tap equalization.
• In OFDM, each frequency bin can be treated independent from the others (it's orthogonal). This allows great implementation speedups in terms of parallel processing. VOFDM can be only parallelized to $K$ "threads".

• The complexity aspect comes even more into play, when we consider MIMO transmission (e.g. spatial multiplexing). With today's hardware we actually have the chance to use a sphere-decoder in real-time on each carrier to get the optimal receiver performance. For VOFDM, the dimension of the sphere-decoding would be M-times larger, which quickly becomes infeasible for today's hardware.

• VOFDM can exploit frequency diversity, since one data symbol is spread around multiple, far apart, frequency bins. However, in order to harvest this diversity, you'd at least need an LMMSE equalizer. However, even better performance can be obtained by employing a sphere-decoder on the equivalent channel $Y_l=H_ld_l+n_l$. But, this requires very high implementation complexity. Also, it needs to be investigated, if additional (smart) soft-output decoding of the (certainly) encoded input data stream can gain the same/similar diversity in the OFDM case.

• In principle VOFDM can achieve better FER performance when considering iterative receivers due to its frequency diversity and concatenation of a channel code and the constellation constraint of the equivalent channel. However, again, to achieve this performance high calculation complexity needs to be accepted.

• regarding robustness against synchronization errors and channel estimation errors, I'd assume both systems are similar. However, Channel estimation itself can be more challenging for VOFDM, since no orthogonal frequency bins exist which can be nicely used for dedicated pilot frequency bins.

So, to summarize: VOFDM is a nice idea and has potential to perform better in terms of FER than OFDM. But, it needs a lot more complexity.

• Thanks a lot. Why does VOFDM requires additional M-point DFTs over OFDM? In the transmitter you calculate M-point DFT K times and in OFDM you calculate N point DFT one time, isn't it so? Commented Mar 1, 2017 at 21:31
• @Gabe: At the transmitter you are right. But, at the receiver, after the M-point DFTs, you still have the relation $Y_l=H_ld_l+n_l$, where $H_l$ is very similar to a circulant matrix. So, to equalize the Rx signal (e.g. "divide by $H_l$") you need to perform two extra K-point DFT for each carrier. In OFDM, you have the single-tap equalization, i.e. $H_l$ is a scalar. Commented Mar 2, 2017 at 7:28
• Not sure I've understood the order of actions at the receiver. After removing the Cyclic Prefix, I can wait for the N=KM symbols to arrive, then I reshape the N-point vector back to its matrix form (D in the paper) and apply M-point DFT for each row (Did I get it right so far?). After that I'd like to equalize the Rx data, but why do I have to use specifically two (or at all) extra K-point DFTs? what do you mean by for each carrier? Commented Mar 3, 2017 at 6:44
• You are right in your comments. After you perform the reshaping and M-point DFT at the receiver, for each carrier (i.e. for each different $l$) the IO relation $Y_l=H_ld_l+n_l$, where $H_l$ is similar to a $K\times K$ circulant matrix. To "divide" by this matrix you can go for brute-force matrix inversion. But, since you know $H_l$ is circulant you can apply $H_l=U\tilde{H}_lU^H$, (see paper from Yabo Li, reference in my answer) where $\tilde{H}_l$ is a diagonal, which is far easier to invert. But, $U$ is like a K-point DFT (with some additional phase-shift). Commented Mar 3, 2017 at 6:50
• Thanks,your explanation clarifies it a lot. Another thing, How exactly do I get the frequency diversity using this method? the paper says that every time-domain signal (a row in the matrix after IFFT operation) goes through upsampling with rate L and transmitted with delay that equals the row number, but actually the signal is transmitted column-wise so I can't see the upsampling process here, can you point it out? Commented Mar 13, 2017 at 14:17