Two first-order low-pass filters H 1(j)=1/(1+j/ 1) and H 2(j)=1/(1+j/ 2) are cascaded. Find the resulting second-order undamped natural frequency ωc.

$\omega_1\omega_2/(\omega_1+\omega_2)$

GATE 2020 BM – Question 42

Signals and Systems · Digital Filters · 2 marks · Multiple choice

Two first-order low-pass filters $H_1(j\omega)=1/(1+j\omega/\omega_1)$ and $H_2(j\omega)=1/(1+j\omega/\omega_2)$ are cascaded. Find the resulting second-order undamped natural frequency ωc.

$\sqrt{\omega_1\omega_2}$
$\omega_1+\omega_2$
$\omega_1\omega_2/(\omega_1+\omega_2)$
$\sqrt{\omega_1^2+\omega_2^2}$

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Correct answer: (A) $\sqrt{\omega_1\omega_2}$

Explanation

The cascade denominator in s is s²+(ω₁+ω₂)s+ω₁ω₂, after multiplying by the corner-frequency product. Compare with s²+2ζωns+ωn². Hence ωn²=ω₁ω₂ and ωc=√(ω₁ω₂), A. This natural frequency need not equal the cascade’s overall three-decibel cutoff.

GATE 2020 BM – Question 42

Explanation

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