The sampling rate of analog-to-digital converters is severely limited by underlying technological constraints. Recently, Tropp et al. proposed a new architecture, called a random demodulator (RD), that attempts to circumvent this limitation by sampling sparse, bandlimited signals at a rate much lower than the Nyquist rate. An integral part of this architecture is a random bi-polar modulating waveform (MW) that changes polarity at the Nyquist rate of the input signal. Technological constraints also limit how fast such a waveform can change polarity, so we propose an extension of the RD that uses a run-length limited MW which changes polarity at a slower rate. We call this extension a constrained random demodulator (CRD) and establish that it enjoys theoretical guarantees similar to the RD and that these guarantees are directly related to the power spectrum of the MW. Further, we put forth the notion of knowledge-enhanced CRD in the paper. Specifically, we show through simulations that matching the distribution of spectral energy of the input signal with the power spectrum of the MW results in the CRD performing better than the RD of Tropp et al.