Traditional compression involves sampling a signal at the Nyquist rate, then reducing the signal to its essential
components via some transformation. By taking advantage of any sparsity inherent in the signal, compressed
sensing attempts to reduce the necessary sampling rate by combining these two steps. Currently, compressive
sampling operators are based on random draws of Bernoulli or Gaussian distributed random processes. While
this ensures that the conditions necessary for noise-free signal reconstruction (incoherence and RIP) are fulfilled,
such operators can have poor SNR performance in their measurements. SNR degradation can lead to poor reconstruction despite using operators with good incoherence and RIP. Due to the effects of incoherence-related signal
loss, SNR will degrade by M/K compared to the SNR of the fully sampled signal (where M is the dimensionality
of the measurement operator and K is the dimensionality of the representation space).
We model an RF compressive receiver where the sampling operator acts on noise as well as signal. The signal
is modeled as a bandlimited pulse parameterized by random complex amplitude and time of arrival. Hence, the
received signal is random with known prior distribution. This allows us to represent the signal via Karhunen-Loeve expansion and so investigate the SNR loss in terms of a random vector that exists in the deterministic KL basis. We are then able to show the SNR trade-off that exists between sampling operators based on random
matrices and operators matched to the K-dimensional basis.