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We propose a novel receiver for ultra-wideband impulse-radio communication in bursty applications like wireless sensor networks. It is based on the principle of compressed sensing, and exploits the sparsity of the transmitted signal to achieve reliable demodulation. Instead of a full-fledged high-rate A/D, a modest number of projections of the received signal are acquired using analog correlators, and a joint decoding of the time of arrival and the data bits is performed from these under-sampled measurements via an efficient quadratic program. The receiver does not use wideband analog delay lines, and is robust to large timing uncertainty, hence the transmitter need not waste power on explicit training headers for timing synchronization. Moreover, the receiver can operate in a regime of heavy inter-symbol interference (ISI), and allows a very high baud rate (close to the Nyquist rate). Its performance is shown to remain close to the maximum likelihood receiver under every scenario of under-sampling, timing uncertainty, ISI, and delay spread.