EE453 electrical engineering writing question

1. Consider the impulse response h[n] = 0.25δ[n − 1] + 0.5δ[n] + 0.25δ[n + 1]
(a) Determine the magnitude respnose |H(e
jω)|
(b) What is the phase response of this filter? How can you obtain a linear-phase filter from this h[n]?
(c) Obtain a length three linear-phase highpass filter by suitably modifying the coefficients of the
linear-phase version of h[n].
2. Consider a stable, causal IIR transfer function with squared-magnitude response given by
|H(e
jω)|
2 =
9(1.09 + 0.6 cos ω)(1.25 − cos ω))
(1.36 + 1.2 cos ω)(1.16 + 0.8 cos ω)
|H(e
jω)|
2 = H(z)H(z
−1
)|z=e
jω HINT: cos ω 7→ 1
2
(z + z
−1
)
(a) Determine a stable transfer function H(z) such that H(z)H(z
−1
)|z=e
jω satisfies the above squaredmagnitude response
(b) How many stable, distinct transfer functions Hi(z) are there such that:
H1(z)H1(z
−1
)|z=e
jω = H2(z)H2(z
−1
)|z=e
jω = … = Hn(z)Hn(z
−1
)|z=e
jω
(c) Among the different transfer functions Hi(z), identify the minimum-phase, mixed-phase, and maximumphase systems.
(d) Plot the different pole-zero diagrams for each different transfer function, again identifying minimum/maximum/mixed phase
(e) Calculate the all-pass filter which transforms the minimum-phase filter into the maximum-phase
filter