The problem of distributed estimation of a non-random parameter in a wireless sensor network (WSN) with, in general, correlated observations over a bandlimited channel is addressed. Each distributed node performs amplify-and-forward (AF) processing on its noise corrupted local observations and sends the locally processed data to the fusion center over a wireless channel sharing a common bandwidth. The bandwidth constraint is taken into account by assigning a direct sequence code division multiple access (DS-CDMA) signaling waveform for each node. The communication with orthogonal, equi-correlated and perfect correlated signaling waveforms is considered. We find the best linear unbiased estimator (BLUE) based on the observations at the fusion center. Assuming perfect synchronization in sensor transmissions, we first analyze the asymptotic mean squared error (MSE) performance with correlated observations assuming equal node power and identical channel gains. Next the optimal power allocation scheme is derived to minimize the total power consumption of the network meeting a required MSE. We show that the optimal power scheduling scheme with perfect correlated signaling waveforms has a significant performance over that of using orthogonal signaling. The impact of synchronization errors on MSE performance is also analyzed and it is shown that with small synchronization errors the use of perfect correlated signaling waveforms performs well when compared with orthogonal channels.