In this paper we consider the impact of quality of wireless sensor links on the overall detection performance of a large sensor network system. Independent and identical sensors gather observations regarding the presence or the absence of a phenomenon of interest and then transmit their binary decisions to a fusion center over parallel, non-interfering but slow Rayleigh fading, wireless links. We derive asymptotic error exponents of the probability of false alarm and the probability of miss at the fusion center for the following cases: (a) BPSK modulation and (i) maximal ratio combining (ii) equal gain combining (iii) decision fusion (b) BFSK modulation and (i) square law combining (ii) decision fusion. In the case of BPSK, the EGC performs the best for low and moderate SNR, with the DF achieving the next best performance. The DF scheme performs the best for large SNR values, whereas the MRC performs the best for very low SNR values. Similar relative performance results were obtained by others for the case of a finite number of sensors. In the case of BFSK, square law combining outperforms DF, except for large SNR values. Finally, we show how the false alarm and the detection probabilities of the decision of a sensor, as seen at the fusion center, are altered by changes to the threshold of the matched filter receiver.