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This paper focuses on finding the best basis for the synthesis of audio signals and the tracking of slow or fast changing instantaneous frequencies. We propose a penalty based basis selection scheme which allows the random sampling strategy of compressed sensing to reduce the sample size for sparse signals. A greedy heuristic scheme is used for signal reconstruction based on an iterative refinement. Numerical experiments on synthetic and real sounds show the effectiveness of our scheme by achieving high accuracy of tracking instantaneous frequencies with a small number of selected basis.