Highpass filter

A highpass filter passes signal frequencies above a cutoff and attenuates lower frequencies. The simplest first-order highpass has transfer function

$H(s)=\frac{s}{s+{\omega }_c},H(j\omega )=\frac{j\omega }{j\omega +{\omega }_c}.$

Behavior

• At DC ($\omega =0$): $H(0)=0$. DC is completely blocked — the filter is AC-coupled.
• As $\omega \to \infty$: $|H(j\omega )|\to 1$. High frequencies pass unattenuated.
• At $\omega ={\omega }_c$: $|H(j{\omega }_c)|=1/\sqrt{2}$ — the half-power cutoff, a $3\text{dB}$ drop, same as for the lowpass.

Pole-zero structure

Same pole as the lowpass at $s=-{\omega }_c$, plus a zero at $s=0$. The zero at the origin is what kills the DC component — the numerator $s$ goes to zero when $s=0$.

The pattern: an extra zero at the origin transforms a lowpass into a highpass with the same cutoff.

LPF + HPF = 1

There’s a clean conceptual identity:

$\text{HPF}(s)+\text{LPF}(s)=\frac{s}{s+{\omega }_c}+\frac{{\omega }_c}{s+{\omega }_c}=\frac{s+{\omega }_c}{s+{\omega }_c}=1.$

The highpass and lowpass at the same cutoff preserve the whole signal between them. Conservation of frequency content.

Bode plot

Asymptotic magnitude: starts at $-\infty \text{dB}$ at $\omega =0$, rising at $+20\text{dB/decade}$ at low frequencies due to the zero at origin, then flattening at $0\text{dB}$ past $\omega ={\omega }_c$ where the pole’s $-20\text{dB/decade}$ slope cancels the zero’s $+20$. So below ${\omega }_c$, magnitude rises; above ${\omega }_c$, magnitude is flat at $0\text{dB}$.

Phase: starts at $+90°$ at very low frequencies (from the zero at origin), transitions through $+45°$ at $\omega ={\omega }_c$, and asymptotes to $0°$ at high frequencies. Decreasing from $+90°$ to $0°$ over the transition.

Where it’s used

• Audio coupling: a highpass blocks DC offsets between amplifier stages. Common in microphone preamps and signal-conditioning circuits.
• AC-coupling capacitors: a series capacitor in a circuit forms a highpass with the load resistance, blocking DC bias while passing the AC signal.
• Rumble filter: a low-frequency rumble (e.g. footsteps in a recording studio) is removed by a highpass at ~30 Hz.
• DC removal in measurement systems: a highpass strips off DC drift while preserving the AC signal of interest.

Circuit realisation (Electronics I)

The concrete first-order highpass in electronics is the RC highpass filter: a series capacitor feeding a shunt resistor, output across the resistor, with Cutoff frequency $f_c=1/(2\pi RC)$ and a $+20\text{dB/decade}$ rise below it. This is exactly the Coupling capacitor used between amplifier stages — it blocks the DC bias while passing the AC signal, and its cutoff sets the low-frequency end of the amplifier’s useful range.