The graphs in Figure P7.36 depict four frequency-response magnitude plots of linear-phas...

The graphs in Figure P7.36 depict four frequency-response magnitude plots of linear-phase FIR filters, labelled |Aⁱ_e(e^jω)|, i = 1, 2, 3, 4. One or more of these plots may belong to equiripple linear-phase FIR filters designed by the Parks–McClellan algorithm. The maximum approximation errors in the passband and the stopband, as well as the desired cutoff frequencies of those bands, are also shown in the plots. Please note that the approximation error and filter length specifications may have been chosen differently to ensure that the cutoff frequencies are the same in each design.

(a) What type(s) (I, II, III, IV) of linear-phase FIR filters can |Aⁱ_e(e^jω)| correspond to, for i = 1, 2, 3, 4? Please note that there may be more than one linear-phase FIR filter type corresponding to each |Aⁱ_e(e^jω)|. If you feel this is the case, list all possible choices.

(b) How many alternations does each |Aⁱe(e^jω)| exhibit, for i = 1, 2, 3, 4?

(c) For each i, i = 1, 2, 3, 4, can |Aⁱ_e(e^jω)| belong to an output of the Parks–McClellan algorithm?

(d) If you claimed that a given |Aⁱe(e^jω)| could correspond to an output of the Parks–McClellan algorithm, and that it could be type I, what is the length of the impulse response of |Aⁱe(e^jω)|?