Last decade has witnessed a major research interest on wearable fall detection systems. Sampling rate in these detectors strongly affects the power consumption and required complexity of the employed wearables. This study investigates the effect of the sampling frequency on the efficacy of the detection process. For this purpose, we train a convolutional neural network to directly discriminate falls from conventional activities based on the raw acceleration signals captured by a transportable sensor. Then, we analyze the changes in the performance of this classifier when the sampling rate is progressively reduced. In contrast with previous studies, the detector is tested against a wide set of public repositories of benchmarking traces. The quality metrics achieved for the different frequencies and the analysis of the spectrum of the signals reveal that a sampling rate of 20 Hz can be enough to maximize the effectiveness of a fall detector.