Filter Design

1. Can All Filters Be Defined by f, Q, and Sometimes Gain?

Yes! Most common filters are defined by f (frequency), Q, and optionally gain, depending on the filter type. Here’s a breakdown:

Filters Using f and Q:

• High-Pass/Low-Pass:

  • f = cutoff frequency (f_c), where attenuation begins.

  • Q = controls resonance/peaking near f_c (e.g., Q = 0.707 for Butterworth).

• Band-Pass/Notch:

  • f = center frequency (f₀).

  • Q = defines bandwidth (BW = f₀/Q).

• All-Pass:

  • f = frequency where phase shifts.

  • Q = adjusts phase shift steepness.

Filters Adding Gain:

• Shelving Filters (Low/High Shelf):

  • f = shelf frequency (f_c).

  • Q = shapes the transition slope (gentle vs. abrupt).

  • Gain = boost/cut in dB (e.g., +6 dB bass boost).

• Peaking (Parametric EQ) Filters:

  • f = center frequency (f₀).

  • Q = bandwidth of the boost/cut.

  • Gain = amplitude adjustment at f₀ (e.g., +3 dB at 2 kHz).

---

2. Distinction Between f₀ (Center Frequency) and f_c (Cutoff Frequency)

These terms are often confused but serve distinct roles:

Parameter | Definition | Used In | Example -- | -- | -- | -- f₀ | Center frequency of a band-pass/notch/peaking filter. | Band-pass, notch, parametric EQ. | A notch filter at f₀ = 1 kHz removes 1 kHz. f_c | Cutoff frequency where a high-pass/low-pass/shelving filter begins acting. | High-pass, low-pass, shelving. | A low-pass filter with f_c = 100 Hz attenuates above 100 Hz.