We wish to use impulse invariance or the bilinear transformation to design a discrete-time filter that meets specifications of the following form: 

For historical reasons, most of the design formulas, tables, or charts for continuous-time filters are normally specified with a peak gain of unity in the passband; i.e., 

Useful design charts for continuous-time filters specified in this form were given by Rabiner, Kaiser, Herrmann, and Dolan (1974).

(a) To use such tables and charts to design discrete-time systems with a peak gain of (1 + δ₁), it is necessary to convert the discrete-time specifications into specifications of the form of Eq.(P7.3-2). This can be done by dividing the discrete-time specifications by (1 + δ₁). Use this approach to obtain an expression for δ₁ and δ₂ in terms of δ₁ and δ_2. 

(b) In Example 7.2, we designed a discrete-time filter with and maximum passband gain of unity. This filter can be converted to a filter satisfying a set of specifications such as those in Eq. (P7.3-1) by multiplying by a constant of the form (1 + δ₁). Find the required value of δ₁ and the corresponding value of δ₂ for this example, and use Eq. (7.19) to determine the coefficients of the system function of the new filter.

(c) Repeat Part (b) for the filter in Example 7.3.

1 -- 1 < |H(el")]I s 1 + 81. 0 s \wi < wp. |H(ei)l s d2. 1 1 s|H(jN)l < 1. 0 < [2]