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The output of the fantastic reconstruction two-channelfind more filter bank is [34]Y(z)=12[Go(z)G1(z)+Ho(z)H1(z)]X(z)+12[Go(?z)G1(z)+Ho(?z)H1(z)]X(?z),(sixteen)the place INH6 X(z), G(z), and H(z) will be the z-transform of your input signal, evaluation filters and synthesis filters respectively. For ideal reconstruction: first of all, the alias phrase X(?z) have to be zero, henceGo(?z)G1(z)+Ho(?z)H1(z)=0.(17)This may be achieved by lettingG1(z)=Ho(?z),??H1(z)=?Go(?z).(18)Second, the distortion phrase have to be continual or even a pure delay time; which is,Go(z)G1(z)+Ho(z)H1(z)=2z?l,(19)in which l denotes atech support time delay.Equation (18) may also be written asHo(z)=G1(?z),??H1(z)=?Go(?z).(20)Substituting this in (19)Go(z)G1(z)?G1(?z)Go(?z)=2z?l,(21)Po(z)?Po(?z)=2z?l,(22)in which Po(z) denotes the products of two low-pass filters Go(z) and G1(z).

Equation (21) indicates that all odd terms of solution of two low-pass filters need to be zero for purchase l, in which l has to be odd and in many cases purchase terms are arbitrary [35]. Hence it may possibly be written asPo(n)={0n??odd,??n��l2n=larbitraryn??even.(23)Hence it can be concluded that the design process of two-channel filter bank for a new wavelet is reduced into two steps as follows:design of filter Po(z) which satisfies (23),factorize Po(z) into Go(z) and G1(z). In this work, the filter Po(z) is designed based on the characteristics of fPCG signals. The common requirements of this design are linear phase, minimum phase, and orthogonality of the filter. A fourth-order low-pass Butterworth filter is chosen because the transition width requirement is not stringent for the given cut-off frequency.

This will also help in reducing the computational complexity. The Butterworth filter satisfies the conditions for perfect reconstruction. It has linear frequency response in the pass band as compared to Chebyshev Type I/Type II and elliptic filters. The filter Po(z) is factorized into Go(z) and G1(z), and then the coefficients of Ho(z) and H1(z) are derived using (20) [32, 36]. Figure 6 shows the impulse response of the four filters computed for construction of filter bank of the new wavelet ��fetal.��Figure 6Impulse response for the reconstruction and decomposition filters of fetal.The wavelet and scaling functions are then derived from the coefficients of these filters using (6) and (8), respectively. Figure 7 shows the wavelet and scaling functions of the ��fetal�� wavelet.

Figure 7Wavelet and scaling function of ��fetal�� wavelet.With these wavelet and scaling functions, the wavelet and scaling coefficients for multiresolution analysis are obtained. The developed wavelet ��fetal�� is now ready to use. All the discrete analysis functions, including dwt, idwt, and wavedec. can operate on the new wavelet. Similarly, all the continuous analysis functions, including cwt, wscalogram, and the corresponding GUI tools, can also operate on the new wavelet.