Filters

Shaping the spectrum of audio signals

Lorenz Schwarz
Karlsruhe University of Arts and Design (HfG)

Winter Semester 2024/25
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Examples of a filter

• Vocal tract: Dynamic filtering.
• Human hearing (e.g., pinna, head, torso): Acts as a direction-selective filter.
• Musical instruments (e.g., Helmholtz resonator): Filtering through resonances
• Rooms: Altering the sound waves through boundary behaviors.
• Loudspeakers and microphones: Shape sound through their frequency response.
• ...

Filter

A filter selectively emphasizes or attenuates certain frequencies in a signal:

• commonly a device, electronic circuit or software algorithm
• used to shape the frequency spectrum of an audio signal

→ Phase response is often perceptually less critical, but may be relevant in some contexts.

Electronic filter

Visualization of a filters behavior

Transfer function plots graphically represent how a filter modifies input signals across frequencies.

• Frequency (x-axis): Plotted on a logarithmic scale (in Hz).
• Magnitude (y-axis): Plotted on a logarithmic scale (in dB).
• Phase angle (y-axis): Plotted on a linear scale (in degrees or radians).

Magnitude response of a filter

The Bode magnitude response plot shows the gain (ratio of output amplitude to input amplitude) of a filter as a function of frequency, typically displayed on a logarithmic scale.

Gain versus frequency:

• Vertical axis (magnitude): Logarithmic scale, typically in decibels (dB).
• Horizontal axis (frequency): Logarithmic scale in Hertz (Hz).

▶ Sawtooth 260 Hz unfiltered - filtered

        Input               Output

Phase response of a filter

A Bode phase plot is a graph that shows the phase relationship between a sinusoidal input signal and the output signal of a filter, as a function of frequency.

• Vertical axis (phase): Linear scale, typically in degrees or radians, representing the phase shift introduced by the filter.
• Horizontal axis (frequency): Logarithmic scale in Hertz (Hz).

Phase and time-domain filters

All-pass filter

An all-pass filter changes the phase relationship between frequencies by introducing a frequency-dependent phase shift, while allowing all frequency components to pass with equal amplitude (unity gain).

Applications: Phaser audio effect

Comb filtering in sound (phaser)

A comb filter mixes a signal with a delayed copy of itself:

• causes constructive and destructive interference
• produces characteristic spectral notches (resemble the teeth of a comb)

▶ White noise comb filtered

Natural occurrences of comb filtering

• Early reflections from hard surfaces (floor or wall reflections)
• Multiple microphones capturing the same source at different distances

→ Comb filtering can be an unwanted acoustic artifact or a deliberate effect.

Passband, stopband, transition band, ripple

• Passband: The range of frequencies that pass through the filter with no attenuation.
• Stopband: The range of frequencies that are significantly attenuated or blocked by the filter.
• Transition band: The region between the passband and stopband where attenuation gradually increases.
• Ripple: Deviation from flatness in a filter's magnitude response, showing the ratio of max to min gain in the passband or stopband.

Bandwidth

The range of frequencies passed by a band-pass filter or attenuated by a notch filter, defined as the difference between the upper and lower cutoff frequencies. It represents the section of the frequency spectrum that the filter affects most significantly.

Cutoff or corner frequency

The cutoff frequency (also known as the half-power point) is the frequency at which the output voltage level decreases by 3 dB compared to the input voltage level (0 dB). Beyond this point, the output voltage progressively decreases relative to the input voltage.

The -3 dB level corresponds to the factor:

This means the output voltage is about 0.7071 times the input voltage.

Center frequency f₀

The midpoint center frequency of a band pass filter is the geometric mean between the lower cuttoff frequency and the upper cutoff frequency.

Geometric mean:

Quality factor (Q-factor)

Parameter that describes the selectivity of a filter, defined as the ratio of the center frequency to the bandwidth of the filter

Filter resonance

Filter resonance refers to the phenomenon where a filter exhibits a peak in amplitude at a specific frequency, often near the cut-off frequency.

• A sufficiently high Q-factor can lead to self-oscillation even without an input signal.

▶ Low pass sweep

▶ Low pass sweep with resonance

▶ Low pass self-oscillating

Frequency ratio units

• Octave: Doubling or halving of frequency
• Decade: Tenfold increase or decrease in frequency (factor-of-ten)

Relationship of filter key parameters

Center frequency

Bandwidth

Quality factor

Example: calculating bandwidth

Relation between center frequency (), bandwidth (), and Q-factor ()

Example: ,

• high Q narrow bandwidth
• low Q wide bandwidth

Relation between Q-factor and bandwidth

Roll-off (filter slope)

Roll-off describes the steepness of the transition from the passband to the stopband in a filter's transfer function graph.

• Indicates how rapidly the filter attenuates the signal beyond the cutoff frequency.
• Measured in decibels per octave (dB/octave) or decibels per decade (dB/decade).

▶ White noise corner frequency 500 Hz

Filter order

The filter order refers to the highest power of the variable in the polynomial of the filter's transfer function, which is the algebraic representation of the filter's behavior. The order determines the steepness of the filter's attenuation beyond the cutoff frequency.

For example 3rd order Butterworth polynomial:

→ Higher-order filters produce steeper slopes.

Filter order and corresponding roll-off rates

Each increase in filter order results in a roll-off rate increase of 6 dB per octave.

Filter types

• Low-pass filter
• High-pass filter
• Shelving filters
• Band-pass filter
• Band-stop filter
• Notch filter
• Peak filter

Spectrum-limiting filters

Low-pass and high-pass are the most fundamental filter types. They define a boundary frequency and progressively attenuate everything on one side of it.

Low-pass filter (LPF) - high cut

A low-pass filter is the opposite of a high-pass filter:

• It allows low-frequency components to pass while attenuating frequencies above the cutoff frequency.
• Commonly used to remove high-frequency noise.

Applications of low-pass filtering

Low-pass filters attenuate high-frequency content and are widely used in audio processing and sound design.

• Removing high-frequency hiss or noise from recordings
• Creating a muffled sound to simulate distance or obstruction
• Subtractive synthesis by shaping timbre from harmonically rich waveforms
• Anti-aliasing before digital sampling

High-pass filter (HPF) - low cut

A high-pass filter is the opposite of a low-pass filter:

• It allows high-frequency components to pass while attenuating frequencies below the cutoff frequency.
• Commonly used to remove low-frequency noise or rumble from audio signals.

Applications of high-pass filtering

High-pass filters attenuate low-frequency content and are commonly used to improve clarity and technical efficiency in audio systems.

• Removing rumble or handling noise from recordings
• Filtering instruments that do not require bass content
• Preventing low frequencies from wasting amplifier headroom

Shelving filters

Unlike low-pass and high-pass filters, shelving filters boost or cut frequencies to a fixed level and then plateau rather than continuing to attenuate.

• Low-shelf filters affect frequencies below a cutoff frequency
• High-shelf filters affect frequencies above a cutoff frequency
• Commonly used for general tone control in audio systems and mixing desks

Creative use of shelving EQ

Shelving equalizers are commonly used for broad, musical tonal shaping rather than precise corrective filtering.

• High-shelf boost: adds “air” or presence (≈ 10–12 kHz and above)
• High-shelf cut: tames harshness; can act as a gentle de-essing alternative
• Low-shelf boost: adds warmth and weight (use carefully to avoid muddiness)
• Low-shelf cut: reduces boominess and proximity effect

Band-pass filters

Band-pass filters target a specific frequency range rather than everything above or below a single cutoff, making them useful for isolating or emphasizing selected spectral content.

• Defined by a lower and an upper cutoff frequency
• Frequencies outside this band are attenuated
• Commonly used to isolate or remove specific frequency components

Applications of band-pass filtering

Band-pass filters isolate a defined frequency range.

• Creating “telephone” or “radio” voice effects (narrow band-pass ≈ 300 Hz–3 kHz)
• Formant filtering in vocoders
• Wah-wah effects using a swept band-pass filter

Band-stop

A band-stop filter is the opposite of a band-pass filter:

• Frequencies within a specified range (the filter band) around the center frequency are suppressed, while frequencies outside this range are allowed to pass.

Notch filter

A notch filter is an extremely narrow banded type of band-stop filter, designed to attenuate a very specific frequency or a small range of frequencies while leaving other frequencies unaffected.

• Commonly used to eliminate unwanted sounds such as power line hum (e.g., 50 Hz or 60 Hz) or unwanted resonances (e.g., feedback).

Notch filters

Notch filters remove a very narrow frequency band while leaving the rest of the spectrum largely unaffected.

• Eliminating mains hum (50 Hz in Europe, 60 Hz in North America) and its harmonics
• Feedback suppression in live sound by targeting problematic resonant frequencies
• Removing room modes in live sound mixing or on location recordings

→ Notch filters enable precise removal of unwanted frequencies without altering overall timbre.

Peak filter

A peak filter boosts or attenuates frequencies within a specified range around a center frequency, forming a "bell-shaped" response.

• Commonly used in multi-band EQs to target specific frequency bands for precise adjustments without affecting surrounding frequencies.

Peak (bell) filters

Peak (bell) filters are the primary building blocks of parametric equalizers, allowing localized gain adjustments around a center frequency.

• Used for corrective (problem-focused) and tonal shaping
• Narrow bandwidth (high Q): precise attenuation of resonances
• Wide bandwidth (low Q): broad spectral shaping

Parametric EQ

Parametric equalizers are more versatile than graphic equalizers:

• Multiple bands can be adjusted independently.
• Variable Parameter such as center frequency, gain (boost or cut), bandwidth, and Q-factor of each band can be controlled.

Graphic EQ

A graphic equalizer allows amplification or attenuation of predetermined frequency bands using adjustable faders.

• Position of the faders visually represent the frequency curve of the filter
• The width of each frequency band (Q) remains constant.

Parametric vs. graphic EQ

Common filter types

FIR and IIR filters (digital filters)

The impulse response (IR) describes a filter’s output when excited by a single-sample impulse.

Biquad filter

Copyright and Licensing

Original content: © 2025 Lorenz Schwarz
Licensed under CC BY 4.0. Attribution required for all reuse.

Includes: text, diagrams, illustrations, photos, videos, and audio.

Third-party materials: Copyright respective owners, educational use.

Contact: lschwarz@hfg-karlsruhe.de

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