Performance Analysis of Analog Network Coding with Imperfect Channel Estimation in a Frequency-Selective Fading Channel

Broadcast nature of the wireless channel enables wireless communications to make use of network coding at the physical layer (PNC) to improve the network capacity. Recently, narrowband and later broadband analog network coding (ANC) were introduced as a simpler implementation of PNC. The ANC schemes require two time slots while in PNC three time slots are required for bi-directional communication between two nodes and hence ANC is more spectrum efficient. The coherent detection and self-information removal in ANC require accurate channel state information (CSI). {In this paper, we theoretically analyze the bit error rate (BER) performance with imperfect knowledge of CSI for broadband ANC using orthogonal frequency division multiplexing (OFDM), where the channel estimation error is modeled as a zero-mean complex Gaussian random variable. We investigate the BER performance for three cases: (i) the effect of imperfect self-information removal due to channel estimation (CE) error with fading tracking errors, (ii) the effect of imperfect self-information removal due to CE error without fading tracking errors}, and (iii) the ideal CE case. We discuss how, and by how much, our results obtained by theoretical analysis can be used for design of broadband ANC system with the imperfect knowledge of CSI. Our results show that imperfect channel estimation due to the noise effect has less impact on self-information removal than the imperfect channel estimation due to fading tracking errors. The tracking against fading is an important problem for accurate self-information removal as well as coherent detection and thus, the effect of channel time-selectivity is also theoretically studied. The achievable BER performance gains due to the polynomial time-domain channel interpolation are investigated using the derived close-form BER expressions and it was shown that the broadband ANC schemes with practical CE in a time- and frequency-selective channel should include a more sophisticated channel interpolation techniques since the impact of Doppler shift has prevalent effect on the achievable BER performance.