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In this paper, the compressed sensing (CS) methodology is applied to the harmonic part of sinusoidally-modeled audio signals. As this part of the model is sparse by definition in the frequency domain, we investigate how CS can be used to encode this signal at low bitrates, instead of encoding the sinusoidal parameters (amplitude, frequency, phase) as current state-of-the-art methods do. We extend our previous work by considering an improved system model, by comparing our model to other schemes, and exploring the effect of incorrectly reconstructed frames. We show that encouraging results can be obtained by our approach, although inferior at this point compared to state-of-the-art. Good performance is obtained using 24 bits per sinusoid as indicated by our listening tests.