Inside Collection (Course): Digital Signal Processing (Ohio State EE700)

Say that the DWT for a particular choice of wavelet yields an efficient representation of a particular signal class. In other words, signals in the class are well-described using a few large transform coefficients.

Now consider unstructured *noise*, which
cannot be eifficiently represented by any transform, including
the DWT. Due to the orthogonality of the DWT, such noise
sequences make, on average, equal contributions to all
transform coefficients. Any given noise sequence is expected
to yield many small-valued transform coefficients.

Together, these two ideas suggest a means of de-noising a signal. Say that we perform a DWT on a signal from our well-matched signal class that has been corrupted by additive noise. We expect that large transform coefficients are composed mostly of signal content, while small transform coefficients should be composed mostly of noise content. Hence, throwing away the transform coefficients whose magnitude is less than some small threshold should improve the signal-to-noise ratio. The de-noising procedure is illustrated in Figure 1.

Now we give an example of denoising a step-like waveform using the Haar DWT. In Figure 2, the top right subplot shows the noisy signal and the top left shows it DWT coefficients. Note the presence of a few large DWT coefficients, expected to contain mostly signal components, as well as the presence of many small-valued coefficients, expected to contain noise. (The bottom left subplot shows the DWT for the original signal before any noise was added, which confirms that all signal energy is contained within a few large coefficients.) If we throw away all DWT coefficients whose magnitude is less than 0.1, we are left with only the large coefficients (shown in the middle left plot) which correspond to the de-noised time-domain signal shown in the middle right plot. The difference between the de-noised signal and the original noiseless signal is shown in the bottom right. Non-zero error results from noise contributions to the large coefficients; there is no way of distinguishing these noise components from signal components.

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