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Due to the continuously increasing computational power of modern data receivers it is possible to move more and more processing from the analog to the digital domain. This paper presents a compressed sensing approach to relaxing the analog filtering requirements prior to the ADCs in a direct conversion receiver. In the presented solution, the filtered down-converted radio signals are randomly sampled with an average sampling frequency lower than its Nyquist rate, and then reconstructed in a DSP system. To enable compressed sensing, this approach exploits the frequency domain sparsity of the down-converted radio signals. As shown in an experiment presented in the article, when the proposed method is used, it is possible to relax the requirements for the quadrature down-converter filters. A random sampling device and an additional digital signal processing module is the price to pay for these relaxed filter requirements.