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The S-transform is becoming popular for time-frequency analysis and data-adaptive filtering thanks to its simplicity. While this transform works well in the continuous domain, its discrete version may fail to achieve accurate results. This paper compares and contrasts this transform with the better known continuous wavelet transform, and defines a relation between both. This connection allows a better understanding of the S-transform, and makes it possible to employ the wavelet reconstruction formula as a new inverse S-transform and to propose several methods to solve some of the main limitations of the discrete S-transform, such as its restriction to linear frequency sampling.