PS Indepth

DSP Doesn’t Change the Laws of Physics & Math

Principles for Configuring a Sound System for the Best Possible Results

By Mark Radu

Mark-Radu

Archimedes, Galileo, Isaac Newton, Albert Einstein… These are some of the greatest minds in physics – people who helped us understand the world around us and the natural laws that govern it, who theorized and experimented, substantiated and documented, who defined the rules and generated the mathematical equations to both predict it and comprehend it.

While I may not be in the same cerebral category as the aforementioned group of intellects, I can attest to the fact that I am smart enough to abide by the mathematics and laws they so generously made available to me.

With the advancements in digital signal processing (DSP), the resources available to calibrate a loudspeaker system are abundant, and to the uninitiated, they can be overwhelming. While you’re scratching your head trying to define an all-pass filter to try and disperse the low-mid buildup in the centre of your line array coverage, keep one thing in mind: Regardless of the processing tools, analog or digital, old or new, two things will always remain constant: you cannot change the laws of physics and the math still applies.

Basic Calculations for Best Results

I had the good fortune of ending up at Abbey Road Studios years back. One of the stories told that day was about how legendary producer George Martin used to spend hours trying different microphones and positionings on the drum kit, trying to find that perfect combination. In essence, he wasn’t relying on the EQ capabilities of the console to try and emulate the sound he was after; he was going straight to the source, defining the sound he was after at its very core.

For anybody that deploys loudspeaker systems for a living, whether for permanently installed, touring, or one-off applications, think about George Martin. As the overall performance of a loudspeaker system is inevitably governed by the math and physics at its core, its layout and design also need to be governed by those same principles. Understanding and utilizing those principles are essential in any loudspeaker system design.

As an example, we can calculate the spacing of delay speakers in order to attain seamless coverage across the horizontal plane. The loudest focus of a speaker is directly on-axis. As you stray off-axis, the level decreases logarithmically – the higher the frequency, the quicker the decrease. The point at which the off-axis response is -6dB down in comparison to its on-axis level is typically defined as the coverage pattern, both in the horizontal and vertical planes. Decibels, as they relate to amplitude, are simply a ratio defined by the Log Base 10 scale. Based on that scale, we can utilize two key elements.

Firstly, we can calculate how the amplitude of two sources in the offaxis level of our delay speakers will sum together. In attempting to create a seamless transition in coverage between a row of delay speakers, we need to ensure that the off-axis amplitude between speakers is summed to equal that of the on-axis point. Based on the ratios derived from the Log Base 10 scale, two equal sources summed in amplitude will provide a 3dB increase in level. With that knowledge now in mind, we can assert that the off-axis point between each delay row speaker needs to intersect the targeted listening area -3dB down in level.

Secondly, we need to determine where that -3dB off-axis point in level occurs in our delay speaker. We know that the -6dB down point in horizontal coverage is defined in the loudspeakers’ published specifications. Using the Log Base 10 formula, -6dB represents a ratio of .501, 50 per cent of the reference level. Using the same formula, -3dB represents a ratio of .708, approximately 70 per cent of the reference level. So, if our delay speakers have a published horizontal coverage of 90 degrees, we can calculate the -3dB point as 70 per cent of 90 degrees - or 63 degrees.

Using only math and physics, we were able surmise that overlapping the coverage of our delay speakers so that they intersect the targeted listening area 32 degrees off-axis will provide us seamless SPL coverage across the horizontal plane – something that cannot be accomplished using DSP.

The delay row scenario is just a small example of designing a fundamentally correct loudspeaker system – one that is correct at its core. While we focused on SPL, decibels, and the Base Log 10 formulas in our design of a delay row, there are math and physics concepts to deal with every aspect of a loudspeaker system design. From time and phase alignment to air absorption and the inverse square law, every aspect of a loudspeaker system can be made fundamentally correct. While DSP is a great tool to calibrate and finesse a loudspeaker system, it cannot correct the fundamentals of math and physics. With DSP, you can’t manipulate how and where loudspeaker sources sum, or EQ frequency response issues caused by time and phase.

In closing, while we have incredibly advanced processing power and algorithms available within today's DSP platforms, through it all, Archimedes, Galileo, Newton, and Einstein still rule…

Mark Radu is the Senior Systems Designer at Solotech.

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