The design of the analog demodulator for receivers with low-resolution analog-to-digital converters (ADC) is investigated. For infinite ADC resolution, demodulation to baseband with M = 2 orthogonal sinusoidal functions (quadrature demodulation) is an optimum design choice. For receive systems which are restricted to ADC with low amplitude resolution we show here that this classical demodulation approach is suboptimal. To this end we analyze the theoretical channel parameter estimation performance based on a simple pessimistic characterization of the Fisher information measure when forming M > 2 analog demodulation channels prior to an ADC with 1-bit amplitude resolution. In order to emphasize that this inside is also true for communication problems, we provide an additional discussion on the behavior of the Shannon information measure under overdemodulation and 1-bit quantization.