Low-frequency modulation of sound carry important information for speech and music. The modulation spectrum is commonly obtained by spectral analysis of the sole temporal envelopes of the sub-bands out of a time-frequency analysis. Processing in this domain usually creates undesirable distortions because only the magnitudes are taken into account and the phase data is often neglected. We remedy this problem with the use of a complex wavelet transform as a more appropriate envelope and phase processing tool. Complex wavelets carry both magnitude and phase explicitly with great sparsity and preserve well polynomial trends. Moreover an analytic Hilbert-like transform is possible with complex wavelets implemented as an orthogonal filter bank. By working in an alternative transform domain coined as ldquomodulation subbandsrdquo, this transform shows very promising denoising capabilities and suggests new approaches for joint spectro-temporal analytic processing of slow frequency and phase varying signals.