**3. Wavelet background**

In this chapter, we apply wavelet decomposition using multiple wavelet mothers, like Daubechies. The discrete wavelet transform (DWT) uses a set of basic functions to perform a decomposition over a *x*(*n*) signal in two resultant signals: detailed and approximated signals. The first one is the scaling function, called the basic dilation function. The second one is the main wavelet function. This decomposition is defined by the equation used in [39, 40] and represented as follows:

$$\mathcal{X}(\mathfrak{n}) = \sum\_{j=1}^{J} \sum\_{k \in \mathbb{Z}} d\_{\mathbb{Z}^j}(k) \, \mathfrak{w}\_{J,k}^\*(\mathfrak{n}) + \sum\_{k \in \mathbb{Z}} a\_{\mathbb{Z}^j}(k) \, \mathfrak{d}\_{J,k}(\mathfrak{n}) \tag{1}$$

where (1)*j* is the scale that represents the dilation index and *k* represents the index in time. *J* is the decomposition level and ∗ denotes complex conjugation. The wavelet and scaling functions are defined as

$$\Phi\_{l,k}(n) \quad = \ 2^{-j/2}\phi(2^{-j}n - k) \tag{2}$$

$$
\Psi\_{l,k}(n) = 2^{-j/2} \Psi\{2^{-j}n - k\}\tag{3}
$$

In ϕ*J*,*k*(*n*) and ψ*J*,*k*(*n*), *j* allows the scaling and the wavelet function the dilation or compression. *k* controls the translation in time. The functions ϕ*J*,*k*(*n*) and ψ*J*,*k*(*n*) have the essential properties of low-pass and band-pass Fourier transform, respectively.

The approximation obtained with *a*20(*n*) at scale *j* = 0 is equivalent to the original signal *x*(*n*). The signal *a*2*<sup>j</sup>* (*n*) at lower resolutions represents smoothed *a*2*<sup>j</sup>*−1(*k*). The detailed signals *d*2*<sup>j</sup>* (*n*) are given by the difference between approximate signals *a*2*<sup>j</sup>* (*n*) and *a*2*<sup>j</sup>*−1(*k*). The approximate signals *a*2*<sup>j</sup>* (*n*) and the detailed signals *d*2*<sup>j</sup>* (*n*) are replaced by the following equations:

$$a\_{\mathcal{D}}(n) = \sum\_{k} h\left(k - 2^{j}n\right) a\_{\mathcal{D}^{j-1}}(k) \tag{4}$$

$$d\_{\mathcal{Z}}(n) = \sum\_{k} \mathbf{g}\left(k - \mathcal{Z}^{j}n\right) a\_{\mathcal{Z}^{j-1}}(k) \tag{5}$$

**7**

**Figure 2.**

**Figure 1.**

*Using Wavelets for Gait and Arm Swing Analysis DOI: http://dx.doi.org/10.5772/intechopen.84962*

for gait tracking. The clinical space settings are shown in **Figure 1**. The acceptable capture area was restricted to a distance of 1.5–3.5 m from the camera, which was

*Signals obtained from Kinect. The first image shows the spine base movement, the second shows the movement related* 

*to the left and right ankles, and the third shows the movement related to the left and right wrist.*

able to record at least one full gait cycle during each walking test.

*Graphic interface from eMotion Capture software and acceptable capture area.*

where *h* and *g* represent the coefficients of the discrete low-pass and high-pass filters associated with the scaling function and the wavelet function, respectively. Given that each level of wavelet decomposition generates coefficients of length less than the original signal, it is important to clarify that for the use of the approximation and detail coefficients, it was necessary to perform an interpolation process to adjust the size of the coefficients according to the size of the original signal.
