In the Wavelet Design Express VI, use the Factorization (Type of G0) control to specify the factorization type.

After you determine P₀(z), the next step is to specify how P₀(z) is factorized into the analysis lowpass filter G₀(z) and the synthesis lowpass filter H₀(z), respectively. The factorizing process is not unique. For a given P₀(z), you have the following four options for creating G₀(z) and H₀(z).

• Arbitrary—No specific constraints are associated with this filter. The following figure shows an example of arbitrary factorization. The blue crosses represent the zeros of G₀(z), and the red circles represent the zeros of H₀(z). Click on the zero you want to select to switch the zero from that of G₀(z) to that of H₀(z) and vice versa.

  

• Minimum Phase—All of the zeros of G₀(z) are contained inside the unit circle, as shown in the following figure. All the zeros of H₀(z) are the reciprocal of the zeros of G₀(z). The Wavelet Design Express VI automatically generates the zeros for H₀(z) and G₀(z). You cannot switch the zeros between G₀(z) and H₀(z). The minimum phase filter possesses minimum phase-lag. When P₀(z) is maximally-flat and G₀(z) is minimum phase, the resulting wavelets are the Daubechies wavelets.

  

• Linear Phase—Any zero and its reciprocal must belong to the same filter, as shown in the following figure. When you switch a zero of G₀(z) to that of H₀(z), the reciprocal of the zero also switches to H₀(z). When you switch a zero of H₀(z) to that of G₀(z), the reciprocal of the zero also switches to G₀(z). This option is available only if the filter is biorthogonal.

  

  In the time domain, a linear phase implies that the coefficients of the filter are symmetric or antisymmetric. Linear phase filters have a constant group delay for all frequencies. This property is required in many signal and image feature-extraction applications, such as peak detection and image edge detection.
• B-Spline—This option is available only if Wavelet Type is Biorthogonal and P0 type is Maxflat. In this case, the analysis lowpass filter G₀(z) and the synthesis lowpass filter H₀(z) are defined by the following equations, respectively: G₀(z) = (1 + z^–1)^kH₀(z) = (1 + z^–1)^2p–kQ)z) where k is specified with the Zeros at π(P0) control, and p is determined by the Zero pairs at π(P0) control. The Wavelet Design Express VI automatically generates the zeros of G₀(z) and H₀(z) based on the settings for k and p. You cannot switch the zeros between G₀(z) and H₀(z). The following figure shows an example of B-Spline factorization: