Asymptotic Equivalence of SC LMMSE-FDE to Continuous-Time LMMSE Equalizer

In this paper, it is shown that single-carrier (SC) linear minimum mean-squared error (LMMSE) frequency domain equalizer (FDE) is asymptotically equivalent to continuous-time (CT) LMMSE equalizer in the limit of a very large block length. The equivalence is established in the sense that the mean-squared error (MSE) of the SC LMMSE-FDE converges to that of the CT LMMSE equalizer. First, the asymptotic MSE of the SC LMMSE-FDE is derived. Then, the necessary and sufficient condition for the asymptotic equivalence is obtained. It is shown that if the received signal is sampled slower than the Nyquist rate then no fixed receive filter followed by a symbol-rate sampler can achieve the MSE of the CT LMMSE equalizer, given an arbitrary channel response. On the contrary, if the received signal is sampled faster than the Nyquist rate then the MSE of the CT LMMSE receiver can be asymptotically achieved by any square-root Nyquist receive filter, of which energy spectral density is flat over transmitted signal's frequency band. Representative numerical results are provided and discussions on convergence rate are also provided.