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Institute for Communication
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Noise Reduction – Weighting Rules

In this section we will briefly discuss a few weighting rules for audio noise reduction. Examples can be found in the next section.

  • Spectral Subtraction
    One of the first weighting rules proposed for audio noise reduction was the spectral subtraction [Boll-79]. One version of it is based on the subtraction of the short-term power spectral densities of the noise and noisy signal. The magnitude spectrum |Y(f)| of the output signal can then be stated as

    |Y(f)| = ( |X(f)|2 - Rnn(f) )0.5

    whereas the short-time phase of the noisy input signal is used for reconstruction.
    Although the noise level is reduced by the spectral subtraction, a serious disadvantage is that there will remain an unnatural sounding residual noise, called 'musical tones'.
  • Wiener Filter
    The Wiener filter  rule is derived from the optimal filter theory, see e.g. [Vaseghi-96]. It is based on minimizing the mean squared error between the speech S(f) and the estimate Y(f)

    E{ |S(f) - Y(f)|2 } ---> min.,

    and leads to the solution

    G(f) = |S(f)|2 / ( |S(f)|2 + Rnn(f) ).

    The result when using the Wiener rule above also suffers from an unnatural sounding residual noise. However compared to the results of spectral subtraction the occurrence of 'musical tones' can be reduced.
  • Weighting Rules of Ephraim and Malah
    In [Ephraim, Malah-1984] and [Ephraim, Malah-1985] two weighting rules were proposed which were derived by taking the distribution of the magnitude spectra of the speech and the noise signal into account. Here it is assumed that either real and imaginary part of the magnitude spectra are Gaussian distributed. The expressions are relatively complicated and not shown here.
    In comparison to the spectral subtraction and the standard Wiener rule, the weighting rules proposed by Ephraim and Malah reduce the occurrence of 'musical tones' in the processed signal once more.
  • Weighting Rules based on Super-Gaussian distributions
    The assumption that either real and imaginary part of the magnitude spectra of speech and noise are Gaussian distributed is not true in reality, especially not for speech signals. A better approximation is obtained by using a Super-Gaussian distribution (e.g., Laplacian or Gamma) for the speech coefficients. For this several MMSE and MAP estimators can be found in literature, e.g., in [Martin-02] and [Lotter, Vary-04].
  • Psychoacoustical Weighting Rules
    When dealing with audio signals intended for a human listener, the properties of the human auditory system should be taken into account. An interesting effect is the one of auditory masking. This means that a weak signal is inaudible if a stronger signal is present at the same or a nearby frequency. One says that the weak signal is masked by the stronger one. The effect of auditory masking can be advantageously used for noise reduction as well. For example in [Gustafsson et al.-02] the aim is to preserve a certain amount of background noise in the signal which is masked by the speech signal. The result is that 'musical tones' as well as other artifacts are almost eliminated and that the distortion of the speech remains low.