4 Blind Source Separation: A Generalized Modal Identification Tool for Civil Structures 41 Fig. 4.1 SWPT binary tree indexing of j;v ψ 0,0 j=1 j=2 ψ1,1 ψ1,0 ψ3,7 ψ3,6 ψ3,5 ψ3,4 ψ2,3 ψ2,2 ψ2,1 ψ2,0 ψ3,0 ψ3,2 ψ3,3 ψ3,1 j=3 Under such conditions, the plot of x1 versus x2 yields the direction vector of partial mixing matrix coefficients. Generally, sparsity is difficult to obtain directly from raw vibration data. However, it is possible in the frequency or in the time-frequency domain. This is the basic idea of sparse transformation. Recently, the author presented a methodology to achieve sparse transformations using Stationary Wavelet Packet Transform (SWPT) [8]. WPT is an extension of the WT and can be implemented by a generalization of the pyramidal algorithm [33, 34]. A wavelet packet is a triple-index function j;v k .t/,where j, k andv can be interpreted as scale, shift and frequency parameter [35, 36], and a SWPT basis is defined as [34]: j;v k .t/ D 1 2j v t k 2j (4.4) where, 1;0.t/ D .t/ and 1;1.t/ D .t/ represent the scaling (father) and wavelet (mother) functions, respectively. For a given scale level j, a binary tree as shown in Fig. 4.1 can be formed using the basis functions, j;v k .t/, whose nodes are indicated by the scale level, j and the frequency parameter, v D0;1;2;::::;.2j 1/. The wavelet packet coefficients at each node .j;v/ are written as: wj;v k .y/ D y.t/ ˇ ˇ ˇ ˇ 2 j = 2 v t k 2j D 1 2 j = 2 Z 1 1 y.t/ v t k 2j dt (4.5) The wavelet packet component signal yj;v.t/ at each node can be expressed as a linear combination of wavelet packet basis functions j;v k .t/: yj;v.t/ DX k wj;v k j;v k .t/: (4.6) For the jth level of decomposition, the original signal y.t/ can be expressed as a summation of all the wavelet packet component signals, yj;v [8]: y.t/ D 2j 1X vD0 yj;v D 2j 1X vD0 Xk wj;v k j;v k .t/ (4.7) Thus, the WPT is a generalization of SWT in that each frequency band of the wavelet spectrum is further sub-divided into finer frequency bands, repeatedly. 4.3 Details of the Algorithm Consider a linear, classically damped, and lumped-mass ns degrees-of-freedom (DOF) structural system, subjected to a broad-band random input force, F.t/: MRx.t/ CCPx.t/ CKx.t/ DF.t/ (4.8)
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