Bandpass filter

A bandpass filter passes signal frequencies in a band around some center frequency and blocks everything else. The standard second-order bandpass is built by cascading a highpass and a lowpass: signals must pass through the highpass (which blocks low frequencies) and then the lowpass (which blocks high frequencies). What survives is the band in between.

With cutoffs ${\omega }_{ca}$ (high-pass cutoff, lower frequency) and ${\omega }_{cb}$ (low-pass cutoff, higher frequency), the transfer function is

$H(s)=\frac{{\omega }_{cb}s}{s^2+({\omega }_{ca}+{\omega }_{cb})s+{\omega }_{ca}{\omega }_{cb}}.$

Pole-zero structure

• One zero at $s=0$ (the highpass’s zero — kills DC).
• Two poles in the left half-plane. For widely-separated cutoffs ${\omega }_{ca}\ll {\omega }_{cb}$, the poles are approximately at $-{\omega }_{ca}$ and $-{\omega }_{cb}$. For close cutoffs, the poles can become complex-conjugate, giving a sharper, more resonant bandpass with a peaked response.

Behavior

• At DC ($\omega =0$): $H(0)=0$ (zero at origin kills DC).
• At high frequency: $|H(j\omega )|\to 0$ (two poles dominate the falloff).
• In between: $|H(j\omega )|$ peaks at the center frequency ${\omega }_0=\sqrt{{\omega }_{ca}{\omega }_{cb}}$ (the geometric mean of the cutoffs).

The bandwidth is roughly ${\omega }_{cb}-{\omega }_{ca}$, and the quality factor $Q={\omega }_0/\text{bandwidth}$ measures the sharpness of the peak.

Where it’s used

• Radio receivers: pick out one station’s carrier frequency (and the small modulation sideband around it) while rejecting all other stations. The tunable LC bandpass at the front end of an old AM radio is the canonical example.
• Audio equalization: a bandpass with adjustable center frequency boosts or cuts a specific band (called a “parametric EQ” filter).
• Modulation analysis: passband filters in lock-in amplifiers and superheterodyne receivers extract narrowband signals from wideband noise.
• Communication systems: channel-select bandpass filters between RF stages.

High-Q resonance

When the two cutoffs are close (${\omega }_{ca}$ and ${\omega }_{cb}$ nearly equal), the poles become complex-conjugate, and the bandpass becomes resonant — a peak at ${\omega }_0$ much larger than the value at the cutoffs. This is the regime of RLC resonant circuits in radios, where Q can be 100 or more and the filter passes a band only a few kHz wide around a multi-MHz center frequency.

The mathematics is the same; only the parameter regime changes. Low-Q bandpass: gentle peak, wide band. High-Q bandpass: sharp peak, narrow band.