ABSTRACT
As discussed in Chapter 2, conventional approximation techniques (Butterworth, Chebyshev, Elliptic, Bessel, etc.) lead to a normalized transfer function denoted low-pass prototype (LPP). The LPP is characterized by a passband frequency VP¼ 1.0 rad=s, a maximum passband ripple AP (or Amax), a minimum stopband attenuation As (orAmin), and a stopband frequencyVs.Ap andAs are usually specified in decibels. Tolerance bounds (also called box constraints) for the magnitude response of an LPP are illustrated in Figure 3.1a. The ratio Vs=Vp is called the selectivity factor and it has a value Vs for an LPP filter. The passband and stopband edge frequencies are defined as the maximum frequency with the maximum passband attenuationAp and the minimum frequency with the minimum stopband attenuation As, respectively. The passband ripple and the minimum passband attenuation are expressed by
AP ¼ 20 log KH vPð Þ
, As ¼ 20 log KH vsð Þ
(3:1) where K is the maximum value of the magnitude response in the passband (usually unity). Figure 3.1b shows the magnitude response of a Chebyshev LPP transfer function with specifications Ap¼ 2 dB, As¼ 45 dB, and Vs¼ 1.6. Transformation of transfer function. Low-pass, high-pass, bandpass, and band-reject transfer functions (denoted in what follows LP, HP, BP, and BR, respectively) can be derived from an LPP transfer function through a transformation of the complex frequency variable. For convenience, the transfer function of the LPP is expressed in terms of the complex frequency variable s, where s¼ uþ jV while the transfer functions obtained through the frequency transformation (low-pass, high-pass, bandpass, or band-reject) are expressed in terms of the transformed complex frequency variable p¼sþ jv.
