# 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

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