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Modulation filtering is a class of techniques for filtering slowly-varying modulation envelopes of frequency subbands of a signal, ideally without affecting the subband signal's temporal fine-structure. Coherent modulation filtering is a potentially more effective type of such techniques where, via an explicit product model, subband envelopes are determined from demodulation of the subband signal with a coherently detected subband carrier. In this paper we propose a coherent modulation filtering technique based on detecting the instantaneous frequency of a subband from its time-frequency representation. We devise theory to show that coherent modulation filtering imposes a new bandlimiting constraint on the product of the modulator and carrier as well as the ability to recover arbitrarily chosen envelopes and carriers from their modulation product. We then formally show that a carrier estimate based on the time-varying spectral center-of-gravity satisfies the bandlimiting condition. This bandwidth constraint leads to effective and artifact-free modulation filters, offering new approaches for potential signal modification. However, the spectral center-of-gravity does not, in general, satisfy the condition of arbitrary carrier recovery. Finally, the results from modulation-filtering a speech signal are then used to validate the theory.