Let S₁ be a causal and stable LTI system with impulse response h₁[n] and frequency response H₁(e^jω). The input x[n] and output y[n] for S₁ are related by the difference equation 

y[n] – y[n – 1] + ¼ y[n – 2] = x[n].

(a) If an LTI system S₂ has a frequency response given by H₂(e^jω) = H₁(−e^jω), would you characterize S₂ as being a lowpass filter, a bandpass filter, or a highpass filter? Justify your answer.

(b) Let S₃ be a causal LTI system whose frequency response H₃(e^jω) has the property that 

H₃(e^jω) H₁(e^jω) = 1.

 Is S₃ a minimum-phase filter? Could S₃ be classified as one of the four types of FIR filters with generalized linear phase? Justify your answers.

(c) Let S₄ be a stable and non-causal LTI system whose frequency response is H₄(e^jω) and whose input x[n] and output y[n] are related by the difference equation: 

y[n] + a₁y[n – 1] + a₂y[n – 2] = β₀ x[n],

where α₁,  ₂, and β₀ are all real and nonzero constants. Specify a value for α₁, a value for α₂, and a value for β₀ such that |H₄(e ^jω)| = |H₁(e^jω)|.

(d) Let S₅ be an FIR filter whose impulse response is h₅[n] and whose frequency response, H⁵(e^jω) has the property that H₅(e^jω) = |A(e^jω)|² for some DTFT A(e^jω) (i.e., S₅ is a zero-phase filter). Find h₅[n] such that h₅[n] * h₁[n] is the impulse response of a non causal FIR filter.