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The wavelet coefficients are calculated by passing through a series of filters. DWT can be implemented efficiently by using a multi‐resolution analysis (MRA) of fast wavelet transform invented by Mallat [9].

MRA computationally decomposes into a two‐scale relation with various time and frequency resolutions by DWT, which is composed of a series of low‐pass and high‐pass filters.

An L‐level wavelet decomposition is illustrated in detail. The process starts with inputting of length N=2^L into a low‐pass filter g[k] and a high‐pass filter h[k], then is convolved with g[k] and h[k] for generating two vectors and of length N/2, respectively.

The contents of and are approximation and detail coefficients of DWT at the first level, respectively. Further, is used as an input for obtaining wavelet coefficients and at the second level of resolution with the length of coefficients being N/4. In other words, a recursive relationship exists between the approximation and detail coefficients at successive levels of resolution. Therefore, the general decomposition form of wavelet coefficients of length N at the j level can be expressed as follows:

(2.2)

(2.3)

where

 jscale parameters j = 1, 2, …, L;

 Lnumber of decomposition levels;

 N data length of a discrete signal;

 n translation parameter ;

  nth approximation coefficient at level j;

  nth detail coefficient at level j;

 g[k] DWT low‐pass filters; and

 h[k] DWT high‐pass filters.

Figure 2.16 illustrates an example of three‐level DWT decomposition. A signal with available bandwidth 2,000 Hz is decomposed into the sets and , where {A₁, D₁}, {A₂, D₂}, and {A₃, D₃} represent the approximate (gray (A_i) blocks) and detail (white (D_i) blocks) coefficients in levels 1–3, respectively.

Figure 2.16 Three‐level decomposition tree of the DWT.