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Most state-of-the-art filtering methods for speech enhancement require an estimate of the noise statistics, but the noise statistics are difficult to estimate in practice when speech is present. Thus, nonstationary noise will have a detrimental impact on the performance of most speech enhancement filters. The impact of such noise can be reduced by using the signal statistics rather than the noise statistics in the filter design. For example, this is possible by assuming a harmonic model for the desired signal; while this model fits well for voiced speech, it will not be appropriate for unvoiced speech. That is, signal-dependent methods based on the signal statistics will introduce undesired distortion for some parts of speech compared to signal-independent methods based on the noise statistics. Since both the signal-independent and signal-dependent approaches to speech enhancement have advantages, it is relevant to combine them to reduce the impact of their individual disadvantages. In this paper, we give theoretical insights into the relationship between these different approaches, and these reveal a close relationship between the two approaches. This justifies joint use of such filtering methods which can be beneficial from a practical point of view. Our experimental results confirm that both signal-independent and signal-dependent approaches have advantages and that they are closely-related. Moreover, as a part of our experiments, we illustrate the practical usefulness of combining signal-independent and signal-dependent enhancement methods by applying such methods jointly on real-life speech.