PRD Diffuser Design Guide

Education · 11 min read · ResonAia Editorial

Discover PRD diffusers. Offering wider bandwidth than QRD with asymmetric scattering. Learn the mathematics, design process, and where they fit best.

What Makes PRD Different?

Primitive Root Diffusers (PRD) are based on number theory sequences that offer distinct advantages over Quadratic Residue Diffusers (QRD). While QRD uses k² mod N, PRD uses powers of a primitive root: s_k = g^k mod P.

The key difference: PRD sequences have better autocorrelation properties across the full period, potentially offering wider effective bandwidth and more uniform scattering.

The Mathematics Behind PRD

Primitive Roots

A primitive root g modulo a prime P is a number whose powers generate all non-zero residues from 1 to P-1. Not every number is a primitive root. It must have "full order" equal to P-1.

For example, with P=7:

Powers of 3: 3¹=3, 3²=2, 3³=6, 3⁴=4, 3⁵=5, 3⁶=1 (mod 7)
This cycles through all values 1-6, so 3 is a primitive root of 7

Sequence Generation

The PRD sequence for prime P with primitive root g:

s_k = g^k mod P for k = 0 to P-2

This produces P-1 elements (one fewer than QRD).

Example: P=7, g=3

k: 0, 1, 2, 3, 4, 5 g^k: 1, 3, 9, 27, 81, 243 mod7: 1, 3, 2, 6, 4, 5

Sequence: 1, 3, 2, 6, 4, 5

Well Depth Calculation

d_k = (s_k × λ) / (2P)

Where λ = c/f₀ (wavelength at design frequency)

PRD vs QRD: Performance Comparison

Advantages of PRD

Wider Effective Bandwidth PRD sequences maintain better autocorrelation properties at frequencies away from the design frequency, potentially extending useful bandwidth by half an octave or more.

Asymmetric Scattering PRD inherently scatters more energy in one direction. This can be advantageous when you want directional control. Aim strong scattering away from listening position.

Reduced Back-Scattering Less energy returns directly toward the source, reducing comb filtering with nearby sources.

Disadvantages of PRD

Fewer Available Primes Not every prime has easy-to-compute primitive roots. Design options are somewhat more limited than QRD.

One Fewer Well A prime-7 PRD has 6 wells vs 7 for QRD. Slight reduction in coverage area.

More Complex Mathematics Calculating powers mod P requires more computation than k² mod N.

Design Parameters

Choosing Your Prime

Common choices:

P=7 (6 wells): Compact design, good for testing
P=11 (10 wells): Moderate size, professional quality
P=13 (12 wells): Excellent performance, standard choice
P=17 (16 wells): Maximum bandwidth, larger panels

Finding Primitive Roots

Prime P	Primitive Roots
5	2, 3
7	3, 5
11	2, 6, 7, 8
13	2, 6, 7, 11
17	3, 5, 6, 7, 10, 11, 12, 14

Different primitive roots give different sequences but equivalent acoustic performance.

Design Frequency

Same considerations as QRD:

400-500 Hz: General studio use
300-400 Hz: Extended low-frequency control
600-1000 Hz: Compact, space-efficient designs

When to Choose PRD Over QRD

PRD is Better When:

You need directional scattering: The asymmetric pattern can aim reflections away from critical listening areas.

Maximum bandwidth matters: For broadband applications (full-range monitoring), PRD's extended range helps.

You want less direct backscatter: Recording situations where the source is near the diffuser benefit from reduced back-reflection.

QRD is Better When:

Symmetric scattering is desired: Rear wall applications often want even scattering in all directions.

Simpler calculations: QRD math is more straightforward for DIY designers.

Standard solutions work: QRD is the default choice and plenty effective for most applications.

Construction Notes

Orientation Matters

Because PRD scatters asymmetrically, panel orientation affects performance:

Install with sequence reading left-to-right or right-to-left depending on desired scatter direction
Document orientation for consistent multi-panel installations

Arraying Multiple Panels

When tiling PRD panels:

Maintain consistent orientation
Avoid reflective symmetry (don't mirror adjacent panels)
Consider using different primes for adjacent panels to reduce periodicity

Well Width and Dividers

Same guidelines as QRD:

Well widths 25-50mm for HF extension
Thin, rigid dividers (3-6mm plywood or MDF)
Consistent widths across all wells

Applications

Recording Studio Rear Wall

PRD's directional scattering can aim energy toward side walls rather than back toward the console. Orient panels so minimal scattering points toward mix position.

Live Room Boundaries

Asymmetric scattering creates more complex room response, beneficial for recording natural ambience.

Ceiling Clouds

When overhead reflections should scatter toward walls rather than directly back to source, PRD orientation provides control.

Broadcast and Voice

Voice frequencies (200-4000 Hz) fall well within PRD's effective range. The extended bandwidth handles both fundamentals and formants.

Practical Design Workflow

Determine target frequency based on room problems and available depth
Select prime based on available panel width
Find primitive root (use table above or calculate)
Generate sequence using s_k = g^k mod P
Calculate well depths from sequence values
Verify maximum depth fits available space
Generate build files or G-code in your own CAM software from the calculated dimensions

Our PRD designer preview automates steps 3-6, letting you focus on the acoustic decisions. Build-file and G-code export (step 7) is coming soon via early access.

Verification and Testing

After installation:

Listening test: Clap or snap at the reflection point. You should hear diffuse scatter rather than distinct echo.

Measurement: Impulse response should show spread energy rather than discrete spike.

A/B comparison: If possible, compare to flat panel or absorption at same location.

PRD diffusers offer sophisticated acoustic control when directional scattering and extended bandwidth matter. For most applications, the improvement over QRD is subtle but measurable.

PRD, Diffuser Design, Primitive Root, Acoustics

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