  The recently introduced compressive sensing (CS) framework enables digital
signal acquisition systems to take advantage of signal structures beyond
bandlimitedness. Indeed, the number of CS measurements required for stable
reconstruction is closer to the order of the signal complexity than the Nyquist
rate. To date, the CS theory has focused on real-valued measurements, but in
practice, measurements are mapped to bits from a finite alphabet. Moreover, in
many potential applications the total number of measurement bits is
constrained, which suggests a tradeoff between the number of measurements and
the number of bits per measurement. We study this situation in this paper and
show that there exist two distinct regimes of operation that correspond to
high/low signal-to-noise ratio (SNR). In the measurement compression (MC)
regime, a high SNR favors acquiring fewer measurements with more bits per
measurement; in the quantization compression (QC) regime, a low SNR favors
acquiring more measurements with fewer bits per measurement. A surprise from
our analysis and experiments is that in many practical applications it is
better to operate in the QC regime, even acquiring as few as 1 bit per
measurement.
