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For an all-pass system H(z) = (z⁻¹ - b)/(1 - az⁻¹), where |H(e^-jω)| = 1, for all ω. If Re(a) ≠ 0, Im(a) ≠ 0, then b equals

1. a
2. a*
3. 1/a*
4. 1/a

Correct answer: a*

Solution

In an all-pass filter, the output phase shifts without affecting the amplitude, which requires that the poles and zeros are complex conjugates. Given that the condition |H(e^-jω)| = 1 holds for all frequencies, it follows that the zero at z = b must be the complex conjugate of the pole at z = a, leading to the conclusion that b equals a*.

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