Recently a sampling theorem for a certain class of signals with finite rate of innovation (which includes for example stream of Diracs) has been developed. In essence, such non band-limited signals can be sampled at or above the rate of innovation. In the present paper, we consider the case of such signals when noise is present. Clearly, the finite rate of innovation property is lost, but if the signal-to-noise ratio (SNR) is sufficient, several methods are possible to reconstruct the signal while sampling well below the Nyquist rate. We thus explore the trade-offs between SNR, sampling rate, computational complexity and reconstruction quality. Applications of such methods can be found in acquisition and processing of signals in high bandwidth communications, like ultra wide band communication.