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The Constant Q transform has found use in the analysis of musical signals due to its logarithmic frequency resolution. Unfortunately, a considerable drawback of the Constant Q Transform is that there is no inverse transform. Here we show it is possible to obtain a good quality approximate inverse to the Constant Q transform provided that the signal to be inverted has a sparse representation in the Discrete Fourier transform domain. This inverse is obtained through the use of lo and l1 minimisation approaches to project the signal from the constant Q domain back to the Discrete Fourier Transform domain. Once the signal has been projected back to the Discrete Fourier Transform domain, the signal can be recovered by performin and inverse Discrete Fourier Transform.
Fitzgerald, D., Cranitch, M., Cychowski, M.: Towards an inverse constant Q transform. An Audio Engineering Society Convention Paper presented at the 120th. Convention, Paris, France, May 20th. - 23rd., 2006. http://www.audioresearchgroup.com/