We consider the problem of pilot design for channel estimation in multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems in the presence of virtual carriers. The design criterion is the average mean-square error (MSE) of the least square (LS) estimates of the channel frequency response over the data carriers. To maximize bandwidth efficiency, the number of pilots is set to its minimum which is equal to the number of unknown channel coefficients. Both the phase shift orthogonal and disjoint pilot schemes are investigated. For a given pilot placement, a closed-form expression for the pilot power distribution for the disjoint scheme is provided. It is found that the optimal power distribution allocates less power to the pilots that are close to the virtual carriers. Pilot placement is obtained using an exhaustive search. The latter is too complex in systems with large number of subcarriers. A suboptimum pilot placement design for the disjoint pilot scheme is presented. It is shown that, unlike fully loaded systems where the two pilot design schemes have similar performance, in the presence of virtual carriers the optimal disjoint scheme not only has lower complexity but also yields more accurate channel estimation and bit error rate performance.
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