A central use case for the Internet of Things (IoT) is the adoption of sensors to monitor physical processes, such as the environment and industrial manufacturing processes, where they provide data for predictive maintenance, anomaly detection, or similar. The sensor devices are typically resource-constrained in terms of computation and power, and need to rely on cloud or edge computing for data processing. However, the capacity of the wireless link and their power constraints limit the amount of data that can be transmitted to the cloud. While this is not problematic for the monitoring of slowly varying processes such as temperature, it is more problematic for complex signals such as those captured by vibration and acoustic sensors. In this paper, we consider the specific problem of remote anomaly detection based on signals that fall into the latter category over wireless channels with resource-constrained sensors. We study the impact of source coding on the detection accuracy with both an anomaly detector based on Principal Component Analysis (PCA) and one based on an autoencoder. We show that the coded transmission is beneficial when the signal-to-noise ratio (SNR) of the channel is low, while uncoded transmission performs best in the high SNR regime.