Manfred Schroeder had a massive impact on the development of modern diffusers. Some say his invention of phase grating diffusion is among the most significant design advancement in diffusion.
Schroeder believed the use of number theory could make for predictable and "optimal" diffusion. He used maximum length, quadratic residue, primitive root, and index sequences; among others. What these all have in common is they all have a series of uniform width wells at varying depths separated by thin fins. Each well radiates sound at the same magnitude, but, due to the varied depths, there are phase differences, creating diffusion.
Maximum Length Sequence (MLS)
MLS diffusers are one of the first Schroeder began working on. This was because they have a flat power spectrum across all frequencies. As far as the design goes, it is a simple one. The sequence of well depths is binary. 0 being the front surface (no depth) and 1 being a well at the determined depth based on a design frequency. Below shows a N=7 (where N is the number of wells) MLS diffuser with a sequence of {0,0,1,0,1,1,1}.
Schroeder believed the use of number theory could make for predictable and "optimal" diffusion. He used maximum length, quadratic residue, primitive root, and index sequences; among others. What these all have in common is they all have a series of uniform width wells at varying depths separated by thin fins. Each well radiates sound at the same magnitude, but, due to the varied depths, there are phase differences, creating diffusion.
Maximum Length Sequence (MLS)
MLS diffusers are one of the first Schroeder began working on. This was because they have a flat power spectrum across all frequencies. As far as the design goes, it is a simple one. The sequence of well depths is binary. 0 being the front surface (no depth) and 1 being a well at the determined depth based on a design frequency. Below shows a N=7 (where N is the number of wells) MLS diffuser with a sequence of {0,0,1,0,1,1,1}.
The problem with MLS diffusers is that the effective bandwidth is very low, however, there is maximum scattering at the design frequency. Because the well depths are based on the design frequency wavelength, phase grating fails at critical frequencies. Say the well depth is a quarter wavelength of the design frequency, the well depth would be half the wavelength an octave above the design frequency; causing the octave to re-radiate in phase. Meaning the surface would act like a flat surface for that frequency, a critical frequency. To avoid this issue, other sequences were sought, such as the quadratic residue sequence.
Quadratic Residue Diffuser (QRD)
The quadratic residue diffuser was introduced as a way to widen the overall effective bandwidth with dispersion characteristic are similar to the MLS. Optimum scattering is achieved at the design frequency and its integer multiples. Between these frequencies diffusion is still fairly effective. Below is a N=7 QRD with a sequence of {0, 1, 4, 2, 2, 4, 1}.
Quadratic Residue Diffuser (QRD)
The quadratic residue diffuser was introduced as a way to widen the overall effective bandwidth with dispersion characteristic are similar to the MLS. Optimum scattering is achieved at the design frequency and its integer multiples. Between these frequencies diffusion is still fairly effective. Below is a N=7 QRD with a sequence of {0, 1, 4, 2, 2, 4, 1}.
Primitive Root Diffuser (PRD)
This is a much more complicated structure. A PRD is designed to have a notch in the scattering pattern. This notch is only effective at discrete frequencies. It is also key to note that there is a difference between notched dispersal and filtering as a PRD is not an effective notch filter. With that said, the performance of a PRD is similar to that of a QRD with the optimal scattering effecting the design frequency and its integer multiples the same.
Index Sequence
The index sequence was formed by a complex Legendre sequence based on the index function. For a given N, finding the sequence set requires a little trial and error. This number sequence has unique reflection factors. The reflection factor for the first well is 0. Therefore, the first well should be filled with absorbent material. Aside from that, the rest of the index sequence diffuser behaves like a PRD, only with more absorption.
The math behind most of these number sequences is far beyond me. A QRD sequence is rather easy to figure out and I will discuss that in another post. Of course, there are downsides to Schroeder diffusers, as with any other, but there are also modifications for combat these downsides. Overall, the QRD is the most widely used Schroeder diffuser. This may be because of the simplicity and ease of design, comparatively, while still achieving excellent performance.
This is a much more complicated structure. A PRD is designed to have a notch in the scattering pattern. This notch is only effective at discrete frequencies. It is also key to note that there is a difference between notched dispersal and filtering as a PRD is not an effective notch filter. With that said, the performance of a PRD is similar to that of a QRD with the optimal scattering effecting the design frequency and its integer multiples the same.
Index Sequence
The index sequence was formed by a complex Legendre sequence based on the index function. For a given N, finding the sequence set requires a little trial and error. This number sequence has unique reflection factors. The reflection factor for the first well is 0. Therefore, the first well should be filled with absorbent material. Aside from that, the rest of the index sequence diffuser behaves like a PRD, only with more absorption.
The math behind most of these number sequences is far beyond me. A QRD sequence is rather easy to figure out and I will discuss that in another post. Of course, there are downsides to Schroeder diffusers, as with any other, but there are also modifications for combat these downsides. Overall, the QRD is the most widely used Schroeder diffuser. This may be because of the simplicity and ease of design, comparatively, while still achieving excellent performance.
Works Cited
Cox, Trevor J., and Peter D'Antonio. Acoustic Absorbers and Diffusers: Theory, Design and Application. London: Taylor & Francis, 2009. N. pag. Print.
Perry, Tim. "The Lean Optimization of Acoustic Diffusers." (2011): 1-89. Arqen.com. Arqen Sonics, 20 Dec. 2011. Web. <http://arqen.com/wp-content/docs/Acoustic-Diffuser-Optimization-Arqen.pdf>.
Cox, Trevor J., and Peter D'Antonio. Acoustic Absorbers and Diffusers: Theory, Design and Application. London: Taylor & Francis, 2009. N. pag. Print.
Perry, Tim. "The Lean Optimization of Acoustic Diffusers." (2011): 1-89. Arqen.com. Arqen Sonics, 20 Dec. 2011. Web. <http://arqen.com/wp-content/docs/Acoustic-Diffuser-Optimization-Arqen.pdf>.