Designing the Product P0(z) (Advanced Signal Processing Toolkit)

The auxiliary function P₀(z) denotes the product of G₀(z) and H₀(z), as shown in the following equation:

P₀(z) = G₀(z) + H₀(z)

You usually use one of the following three types of filters for P₀(z).

• Maximally-flat
• General equiripple halfband
• Positive equiripple halfband

The maximally-flat filter is defined by the following equation:

P₀(z) = (1 + z^–1)^2pQ(z)

In the Wavelet Design Express VI, the Zero pairs at  (P0) control on the Configure Wavelet Design dialog box specifies the value of the parameter p, which determines the number of zeros placed at π on the unit circle. The more the zeros at , the smoother the corresponding wavelet. The value of p also affects the transition band of the frequency response. A large value of p results in a narrow transition band. In the time domain, a narrower transition band implies more oscillations in the corresponding wavelet. When you specify a value for the parameter p, you can review the frequency response shown on the right-hand side of the Configure Wavelet Design dialog box.

In a general equiripple halfband filter, halfband refers to a filter in which ω_s + ω_p = π, where ω_s denotes the stopband frequency and ω_p denotes the passband frequency, as shown in the following figure.

The positive equiripple halfband filter is a special case of general equiripple halfband filters. The Fourier transform of this type of filter is always nonnegative. Positive equiripple halfband filter is appropriate for orthogonal wavelets because the auxiliary function P₀(z) must be nonnegative.

Use the P₀type control to specify the P₀(z) type. When Wavelet Type is set to Orthogonal, you can set P₀(z) either to Maxflat (default), for a maximally-flat filter, or to Positive Equiripple. When Wavelet Type is set to Biorthogonal, you can set P₀(z) to Maxflat (default), Positive Equiripple, or General Equiripple.

Because all filters, including P₀(z), G₀(z), and H₀(z), are real-valued finite impulse response (FIR) filters, the zeros of these filters are mirror-symmetric about the x-axis in the z-plane. Therefore, for any zero z_i, a corresponding complex conjugate z_i* always exists. If z_i is complex, meaning that if z_i is located off of the x-axis, you always can find a corresponding zero on the other side of the x-axis, as shown in the following figure. As a result, you only need to see the top half of the z-plane to see all of the zeros that are present. After you select z_i, the Wavelet Design Express VI automatically includes the complex conjugate z_i*.

Two parameters are associated with equiripple filters—# of taps and Passband. Use the # of taps control to define the number of coefficients of P₀(z). Because P₀(z) is a type-I FIR filter, the length of P₀(z) must be odd. Use the Passband control to define the normalized passband frequency, ω_p, of P₀(z). The value of ω_p must be less than 0.5. Longer filters improve the sharpness of the transition band and the magnitude of the attenuation in the stopband at the expense of extra computation time for implementation.