Ohmic Audio

4.1 Understanding Frequency Response and Crossover Settings

🔰 BEGINNER LEVEL: Frequency Response Basics

What is Frequency Response?

Frequency response shows how loud a system plays each frequency.

Labeled frequency response graph comparing a smooth target curve to a peaky real-world response across bass, midrange, and treble
A frequency-response graph is just a map of loudness versus frequency. Smoother curves tend to sound more balanced, while big peaks and dips usually translate into obvious tonal problems.

Ideal response: Flat line (all frequencies equal volume) Real systems: Have peaks and dips

Why flat response matters: - Music sounds balanced - Natural tonality - No frequency emphasis/missing - What artist intended

Reading frequency response graphs:

X-axis: Frequency (Hz) - logarithmic scale - 20 Hz (deep bass) to 20,000 Hz (high treble)

Y-axis: Level (dB) - 0 dB = reference - +6 dB = twice as loud - -6 dB = half as loud

Common problems visible: - Peak at 50 Hz: Too much bass - Dip at 3 kHz: Vocals sound recessed - Rolloff above 10 kHz: Lacks sparkle

What Are Crossovers?

Crossover = Filter that sends specific frequencies to specific speakers.

Why needed:

Tweeters can't play bass - they'll break! Subwoofers can't play highs - they're too slow!

Simple signal-split diagram showing a full-range input dividing into low frequencies for a subwoofer, mid frequencies for a woofer or midbass, and high frequencies for a tweeter, with basic crossover regions labeled.
The point of the crossover is not just protection. It also cleans up overlap so each driver handles the frequency range it is physically suited for.

Types of crossovers:

High-pass filter (HPF): - Passes high frequencies - Blocks low frequencies - Used for tweeters and midrange

Low-pass filter (LPF): - Passes low frequencies - Blocks high frequencies - Used for subwoofers

Band-pass filter: - Passes middle frequencies - Blocks highs and lows - Used for dedicated midrange

Crossover Points

Where to cross over?

Rule of thumb: - Subwoofer: 80 Hz (typical for music) - Midbass to midrange: 250-500 Hz - Midrange to tweeter: 2,500-4,000 Hz

Too low crossover: - Speaker tries to play below its range - Distortion - Possible damage

Too high crossover: - Gap in frequency coverage - Thin, unnatural sound

Start with manufacturer recommendations!

Crossover Slope

Slope = How quickly filter attenuates

Comparison chart showing 6, 12, 24, and 36 dB per octave crossover slopes so the difference between gentle and steep rolloff is easy to see at a glance.
The right slope depends on the driver, the crossover point, and the blend you are chasing. The chart makes the tradeoff visible before you start tuning by feel.

Common slopes: - 6 dB/octave: Gentle, smooth transition - 12 dB/octave: Most common, good balance - 24 dB/octave: Steep, clean separation - 36 dB/octave: Very steep, maximum protection

Steeper slope: - Better speaker protection - Cleaner separation - Less overlap - But can cause phase issues

Gentler slope: - More natural sound - Better integration - More overlap (power handling concern)

🔧 INSTALLER LEVEL: Professional Crossover Configuration

Active vs Passive Crossovers

Passive Crossover:

Instructional passive crossover diagram showing amplifier output feeding a simple two-way network with an inductor for the woofer path, a capacitor for the tweeter path, and an optional resistor pad for level trimming.
A passive crossover does the filtering after amplification, using physical components instead of DSP math. It is simple and reliable, but it also locks you into the network values that were built into it.

Components: - Capacitors (block lows, pass highs) - Inductors (block highs, pass lows) - Resistors (level adjustment)

Location: Between amplifier and speakers

Pros: - No power required - Simple - Reliable - Included with component speakers

Cons: - Power loss in components (heat) - Fixed frequencies (can't adjust) - Component tolerances (±10-20%) - Interacts with speaker impedance

Active Crossover:

Location: Before amplifier

Types: 1. Built into head unit (basic) 2. Built into amplifier (common) 3. Dedicated DSP (professional)

Pros: - Adjustable frequencies - Adjustable slopes - No power loss - Precise filtering - Time alignment possible (DSP)

Cons: - Requires separate amp channels - More complex - More expensive - Requires tuning

Which to use?

Passive: - Simple systems - Component speakers with included crossovers - Budget builds

Active: - Serious systems - Custom installations - Competition - Maximum performance

Crossover Point Selection Methodology

Step-by-step process:

Step 1: Identify speaker capabilities

Check specifications: - Tweeter: Fs = 1,200 Hz → Cross >2,000 Hz - Midbass: Fs = 80 Hz, Usable to 3,000 Hz

Step 2: Determine overlap region

Where both speakers can play: - Tweeter: 2,000 Hz and up - Midbass: 80-3,000 Hz - Overlap: 2,000-3,000 Hz

Step 3: Choose crossover point in overlap

Considerations: - Lower (2 kHz): - Tweeter plays more range (more output, more distortion risk) - Midrange has less work (better power handling)

Typical choice: 2,500-3,000 Hz

Step 4: Select slope

For this overlap: - 12 dB/octave: Good starting point - 24 dB/octave: Better if speakers widely separated

Step 5: Listen and adjust

Subwoofer Integration

Subwoofer crossover is critical!

Frequency selection:

Low crossover (50-63 Hz): - Subwoofer only for deep bass - Front speakers carry more bass - Use when: Large front speakers (6.5"+ midbass)

Medium crossover (80 Hz): - Most common choice - THX standard - Good for most systems

High crossover (100-120 Hz): - Subwoofer does all bass work - Protects small front speakers - Use when: Small front speakers, factory integration

Slope selection:

12 dB/octave: - Gentle integration - More overlap with front speakers - Can cause boom if not careful

24 dB/octave: - Clean separation - Most common - Good all-around choice

36 dB/octave: - Very steep - Maximum separation - Good for sound quality - May sound disconnected if not tuned well

Subsonic filter:

Essential for ported enclosures!

Comparison graphic showing a ported subwoofer system without and with a subsonic filter, illustrating rising cone excursion below box tuning and how the filter reduces wasted motion and protects the driver.
The subsonic filter is a protection tool, not a tone toy. It trims the frequencies where the enclosure stops supporting the woofer, which helps keep excursion under control when the system is driven hard.

Function: - Extreme high-pass filter - Protects subwoofer from over-excursion - Filters infrasonic frequencies (below hearing)

Settings: - Sealed: 20-25 Hz - Ported: 5-10 Hz below tuning frequency

Example: - Box tuned to 32 Hz - Subsonic at 22-25 Hz - Slope: 24 dB/octave minimum

Phase Alignment

Phase = Timing relationship between speakers

In phase: Speakers move together (reinforce) Out of phase: Speakers oppose (cancel)

Waveform comparison showing in-phase left and right signals summing constructively versus out-of-phase signals partially cancelling, with a simple note about the audible result in the crossover region.
Phase alignment is easy to hear once you know what to listen for. When the sources reinforce, the overlap region gets stronger. When they oppose each other, the handoff hollows out.

Phase switch on amplifier: - 0° = Normal - 180° = Inverted

When to invert:

Test by listening: 1. Play bass-heavy music 2. Toggle phase switch 3. More bass: Correct setting 4. Less bass: Wrong setting

Physical explanation:

If subwoofer and front speakers move opposite directions: - At crossover frequency they cancel - Weak midbass response - Inverting phase fixes this

More advanced: DSP can adjust phase continuously (not just 0° or 180°)

⚙️ ENGINEER LEVEL: Filter Theory and Design

Transfer Functions of Filter Types

First-Order High-Pass (6 dB/octave):

Transfer function:

H(s) = s / (s + ω_c)

Magnitude:

|H(jω)| = (ω/ω_c) / √(1 + (ω/ω_c)²)

Phase:

∠H(jω) = 90° - arctan(ω/ω_c)

At cutoff frequency: |H| = -3 dB, Phase = 45°

Second-Order High-Pass (12 dB/octave):

Transfer function (Butterworth):

H(s) = s² / (s² + √2×ω_c×s + ω_c²)

Magnitude:

|H(jω)| = (ω/ω_c)² / √[(1-(ω/ω_c)²)² + 2×(ω/ω_c)²]

Phase:

∠H(jω) = 180° - arctan(√2×(ω/ω_c) / (1-(ω/ω_c)²))

At cutoff: |H| = -3 dB, Phase = 90°

Fourth-Order (24 dB/octave):

Cascade of two second-order sections.

Linkwitz-Riley:

H(s) = [s² / (s² + √2×ω_c×s + ω_c²)]²

Special property: LR4 HPF + LR4 LPF sum to flat response when added acoustically

At cutoff: |H| = -6 dB, Phase = 180°

Crossover Phase Relationships

Phase shift accumulates:

First-order: 90° total phase shift Second-order: 180° total Third-order: 270° total Fourth-order: 360° total

Practical implication:

If tweeter and woofer crossed with 12 dB/octave: - Each has 90° phase shift at crossover - Total: 180° difference - They sum out of phase!

Solutions:

1. Linkwitz-Riley 24 dB/octave: - Each speaker: 180° shift - Difference: 360° = 0° (in phase!) - Sums to flat response

2. Physical alignment: - Tweeter behind or ahead of woofer - Time delay compensates phase - Complex to implement

3. All-pass networks: - Add phase shift without affecting magnitude - Can align arbitrary crossovers - Used in passive crossovers

Group Delay:

Frequency-dependent time delay:

τ_g(ω) = -dφ/dω

For Butterworth second-order at cutoff:

τ_g(ω_c) = 1/ω_c

Example: 80 Hz subwoofer crossover

τ_g = 1/(2π×80) = 2.0 ms

Audibility:

Group delay <10 ms generally inaudible Crossovers typically 1-5 ms - acceptable

Passive Crossover Design

Component values for Butterworth 12 dB/octave:

High-pass section (tweeter):

C = 1 / (√2 × π × f_c × R)
L = (√2 × R) / (2π × f_c)

Low-pass section (woofer):

L = (√2 × R) / (2π × f_c)
C = 1 / (√2 × π × f_c × R)

Where R = speaker impedance

Example: 3000 Hz crossover, 4Ω speakers

Tweeter capacitor:

C = 1 / (√2 × π × 3000 × 4)
C = 1 / 53,308 = 18.8 μF

Standard value: 18 μF or 20 μF

Tweeter inductor:

L = (√2 × 4) / (2π × 3000)
L = 5.66 / 18,850 = 0.30 mH = 300 μH

Woofer inductor:

L = 300 μH (same)

Woofer capacitor:

C = 18.8 μF (same)

Zobel network for impedance flattening:

Component values:

R_zobel = 1.25 × R_e
C_zobel = L_e / (R_zobel)²

Where: - Re = DC voice coil resistance - Le = voice coil inductance

Example: Tweeter with Re = 3.2Ω, Le = 0.08 mH

R_zobel = 1.25 × 3.2 = 4.0Ω
C_zobel = 0.08×10⁻³ / 16 = 5 μF

Active Crossover Implementation

Analog active crossover:

Op-amp Sallen-Key topology:

Readable schematic-style diagram of a second-order Sallen-Key active filter showing the op-amp stage, two resistors, two capacitors, the feedback path, and a Butterworth design note for equal-value tuning.
The Sallen-Key block is one of the classic analog filter building blocks. It is useful here because it shows how an active crossover section can be realized in hardware before you ever move the same idea into DSP coefficients.

Transfer function:

H(s) = K / (1 + (C₁×R₁ + C₂×R₂ + C₁×R₂×(1-K))×s + C₁×C₂×R₁×R₂×s²)

For Butterworth (Q = 0.707): - Set R₁ = R₂ = R - Set C₂ = 2×C₁ = 2C - Set K = 1.586 (gain)

Cutoff frequency:

f_c = 1 / (2π × R × C × √2)

Digital crossover (DSP):

IIR (Infinite Impulse Response) biquad:

Transfer function:

H(z) = (b₀ + b₁×z⁻¹ + b₂×z⁻²) / (1 + a₁×z⁻¹ + a₂×z⁻²)

Butterworth HPF coefficients:

ω_c = 2π × f_c / f_s
Q = 0.707 (Butterworth)

α = sin(ω_c) / (2×Q)
b₀ = (1 + cos(ω_c)) / 2
b₁ = -(1 + cos(ω_c))
b₂ = (1 + cos(ω_c)) / 2
a₁ = -2×cos(ω_c)
a₂ = 1 - α

Normalize: divide all by (1 + α)

Implementation:

y[n] = b₀×x[n] + b₁×x[n-1] + b₂×x[n-2] - a₁×y[n-1] - a₂×y[n-2]

Computational cost: 5 multiplies + 4 adds per sample

FIR (Finite Impulse Response):

Transfer function:

H(z) = Σ(h[n] × z⁻ⁿ)

Advantages: - Linear phase (constant group delay) - Always stable - Precise control

Disadvantages: - Many taps required (hundreds for sharp crossovers) - High CPU cost - More latency

Typical modern DSP: - Uses IIR for crossovers (efficient) - Uses FIR for EQ (accuracy) - Hybrid approach