Obtain low-distortion sine waves from loudspeaker

Fixing distortion in audio amplifiers is relatively simple—apply negative feedback (notwithstanding the objections from a sizeable number of audiophiles who hate negative feedback, but we are not going to fan those flame wars). Doing the same thing to a mechanical or electromechanical system such as a loudspeaker is far from easy.

There have been several commercial audio products that have attempted to do this in the past, with varying degrees of success. What characterized most or all of these was the fact that they operated on a real-time music signal. In this article we look at a limited but nevertheless very relevant version of the problem – the production of a steady-state single-frequency sine wave from a bass driver operating at low frequency and high volume – i.e., under circumstances that can be expected to cause high distortion. Such a setup is very useful for calibrating a speaker driver as well as for room equalisation under steady state-conditions, the latter involving detection of peaks representing standing wave antinodes.

Another common application is in investigations of the acoustic response of the human ear to low frequencies, where distortion products at higher harmonics (where the ear is more sensitive) would influence the outcome. How well can we cancel distortion products in such a setup where we have the luxury of steady state operation? Quite well, if we go by a recent paper entitled "Producing undistorted acoustic sine waves" by Henri Boutin, John Smith, and Joe Wolfe of the School of Physics, The University of New South Wales, Sydney, published in the Journal of the Acoustical Society of America 135, 1665 (2014) (Letters to the Editor).

Experimental setup
For experimental evaluation of their method the authors went with an unbaffled 155mm driver operating at 52.49Hz, and generating an SPL of 107dB at 50mm, where a near-field measurement was taken by a B&K 4944A pressure-field microphone. With no correction, the first three partials measured 20dB, 33dB, and 38dB below the fundamental respectively. Rather distorted, we would agree.

The method
The method is disarmingly simple and intuitive. It consists of adding harmonics with appropriate magnitude and phase to the input in order to remove the harmonics in the output signal. Being a steady-state situation, this is done over multiple iterations, cancelling only one harmonic per iteration. This allows the harmonics to be measured anew at each iteration.

Suppose the input to the speaker is a pure sine wave and the output measured from the computer is

where k represents the k^th harmonic, Hk is the gain of the amplifier and loudspeaker, and Gk is the complex gain of the microphone. To cancel the second harmonic in the output in the first iteration, it is evident that we need to subtract from the input the component .

In the next iteration the process is repeated for the third harmonic, and so on. Results appear in the figure, where you can see that after 100 iterations, the overall distortion is down 65dB below the fundamental, which is quite impressive.

Conclusion
To be honest, I was quite surprised when I ran across this method, published so recently, in a peer-reviewed Letters section of a reputed journal. My first reaction was, "But isn't this already well-researched?" The fact is that in the obsession with distortion correction for real-time music signals, the problem of correcting a steady-state sine signal has been ignored. This paper exploited that specific opportunity and made a research contribution. This is an important lesson for those of us engaged in research in general, and acoustics research in particular. There are many real-world problems which tend to get ignored because of our obsession with "popular" problems – which obsession may sometimes be driven by audiophile mania. As the paper by Boutin et al. shows, there are nice opportunities out there to make a significant, focused research contribution if one tends to widen one's field of view.

About the author
Ramkumar Ramaswamy earned his M.Sc in Physics from Delhi University in 1990 and his PhD in Operations Research from the Indian Institute of Management Calcutta in 1994. He built his first audio amp at the age of 12 using the popular TBA810 which, much to his delight then, instantly became a local MW receiver too. While as a pragmatist he believes that digital technologies have replaced analogue largely for the better, he rues the fact that some of the best principles of analogue design, such as the principle of minimalism which it implicitly embodies, are slowly dying away.