Let S1 be a causal and stable LTI system with impulse response h1[n] and frequency response H1(ejω). The input x[n] and output y[n] for S1 are related by the difference equation
(a) If an LTI system S2 has a frequency response given by H2(ejω) = H1(−ejω), 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(ejω) has the property that H3(ejω)H1(ejω) = 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 noncausal LTI system whose frequency response is H4(ejω) and whose input x[n] and output y[n] are related by the difference equation:
y[n] + α1y[n − 1] + α2y[n − 2] = β0 x[n],
where α1, α2, 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(ejω)| = |H1(ejω)|.
(d) Let S5 be an FIR filter whose impulse response is h5[n] and whose frequency response, H5(ejω), has the property that H5(ejω) = |A(ejω)|2 for some DTFT A(ejω) (i.e., S5 is a zero-phase filter). Determine h5[n] such that h5[n] ∗ h1[n] is the impulse response of a noncausal FIR filter.
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