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## EE453 electrical engineering writing question

# 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

DETAILED ASSIGNMENT