Random demodulator in matrix form
The random demodulator (in this definition) compresses the signal by taking samples that are each linear combinations of W/R Nyquist rate samples. So it does not take W/R samples at a time; it takes one sample at a time, each of which contain information corresponding to several (W/R) Nyquist rate samples. It is then up to a reconstruction algorithm to subsequently "disentangle" these into the original signal (approximately) at Nyquist rate.

The random demodulator works by "smearing" the content of the input signal all over the spectrum (by multiplication with the spreading sequence). Provided that the input signal is sufficiently sparse, we can now reconstruct it from just a small portion of the resulting spectrum (selected by the low-pass filter).