In usual Fourier transform (FT), the filter is cut-off certain frequency.

This trick is also suitable for wavelet transform (WT). However, there could be some “features” located in high frequency scale (or octave) , a simply cut-off would remove these features.

If the signal to noise level is large, that means the noise has smaller amplitude than that the signal, we can use hard or soft thresholding, which zero any coefficient, which is after the FT or WT,  less then a threshold.

Lets X be the coefficient. The hard thresholding is

Y=\begin{cases} 0, & |X| <\sigma \\ X, & \mbox{else} \end{cases}


The soft thresholding is

Y = \begin{cases} 0, & |X| < \sigma \\ sign(X) f(|X|, \sigma), & \mbox{else} \end{cases}

A popular function

\displaystyle f(x, \sigma) = \frac{x - \sigma}{ X_{max} - \sigma } X_{max}



\displaystyle f(x, \sigma) = x - \sigma