The Same Everywhere: Exploring System Linearity
The goal is to make sure that everyone in the audience hears and enjoys exactly the same experience...

January 16, 2014, by Pete Soper & Steve Bush

Linearity is reasonably straightforward as a concept, but when applied to sound reinforcement systems – and to large-scale systems in particular – it gets a bit more complicated. Understanding linearity can be extremely helpful when designing, tuning and operating systems for predictable and repeatable results.

It’s a concept that is often either taken for granted, misunderstood or not fully grasped because there are many parameters to consider when evaluating audio systems. We’ll establish some fundamentals before tackling more complex issues.

For a basic definition of linearity, we’ll turn to the world of mathematics. Homogeneity and superposition are two important terms in linear system theory. In the audio world, homogeneity refers to frequency content.

For a system to be completely linear, both of the following statements need to be true:

1) Frequencies that are not present in the stimulus will not be found in the output.
2) Frequencies that are found in the output will always be found in the stimulus.

The rules of superposition require that when two or more signals are summed together, they contain all of the content of both signals, while maintaining a proportional relationship. To develop some simple examples, we’ll introduce two “black box” devices – though here they are blue and green – which represent some part of a mechanism or an interconnecting network. By comparing the input to the output of the devices, we can determine their degree of linearity.

Here’s the most basic example. We input a blue lower case “a” to both devices, and look at what comes out. At the output of the blue box, we see a blue lower case “a,” only it’s bigger. At the output of the green box, we also see a larger letter “a,” but this one is red and upper case. By testing with only the small “a” as an input, we cannot determine whether either device is more or less linear (Figure 1).

We have to keep testing, making the output larger and smaller, and using input sources other than a lower case “a.” We need to keep repeating the tests, both for longer and shorter intervals and under different conditions, such as signal content and input levels.

So let’s say we run multiple tests for the blue box, and each time the input and output always maintain a directly proportional relationship, behaving linearly.  In this example, if we input a blue lower case “d,” we can reasonably expect the output also will be a blue lower case “d,” only enlarged to the same size as the other three letters (Figure 2).

But with the green box, it’s a different story. Yes, the input of the same three letters results in a larger letter at the output, but the colors, case, orientation all change, and one character isn’t even a letter.

Unquestionably, here we have a non-linear device. If you input a blue lower case “d” here, the output will likely be something larger, but it could be just about anything – perhaps even a squirrel or a banana (Figure 3).

Linear devices are very predictable; easy to measure and quantify. This makes them especially attractive to work with when reproducing sound.

Verifying Linearity
Verification of homogeneity and superposition in audio systems can be done by turning to three basic tests: scaling, summing, and timing.

Let’s start with scaling, and here we’ll use examples that bridge us into the world of audio. Let’s say we put a small kick drum into the blue box, and it comes out larger. The only change is in scale. It could get larger or smaller, but everything else is identical. The same thing happens when we put in a piano. The output is proportionally related to the input and has the same scale as the drum (Figure 4).

By comparing the results of these various tests, we can develop a list of meaningful results that are beyond just playing a CD and subjectively being happy or not with the resulting sound.

Below, we see first that the green box has increased the size of the kick drum somewhat, as expected. But when a larger kick drum is the input, we get a smaller one as an output. This behavior fails the scaling rule of linear systems. A linear system would have output a “ginormous” kick drum (Figure 5).

For a superposition (summing) test, we can send the kick drum and the piano at the same time, and the output is the same kick drum and piano, only larger (Figure 6).

Again, the summing proceeds as a linear function. With the green box, alas, when we combine the kick drum and piano at the input, the output becomes a glorious – saxophone? Definitely non-linear (Figure 7).

The third test is timing. An interesting example of timing linearity is the venerable vinyl record. A well-made disk on a good turntable can be a reasonably linear recreation of the original recording.

But what if you put your finger on the edge of the disk? You guessed it, there will be significant changes to the spectral content, which is in violation of homogeneity, and the waveform as a function of time is altered. If you play it backwards, you maintain the same scale and summing information, but the reversed timing makes the signal unrecognizable. (If you happen to hear satanic messages, this nevertheless remains non-linear in relationship to the signal in the intended direction.)

Testing Procedures
Now that we’ve moved into the realm of audio, we need to consider measurement tools and develop some procedures to test for linearity.

At the most basic level, of course, we can simply insert a sine wave test tone and use an oscilloscope to plot the signal amplitude and display the time required to complete one cycle. Or we could use a simple spectrum analyzer, which would display amplitude over the frequency as a line or bar.

A more sophisticated approach employs dual-channel FFT transfer measurements. This is a tool which, when properly used, can reveal shortcomings in any or all three of the main linearity requirements.

Transfer function measurements, using full bandwidth signals, can be used to qualify scaling and superposition (summing) properties of a system as well as the timing aspect of homogeneity. High resolution spectral analysis of single or multi-frequency tones can be used to readily identify system shortcomings in homogeneity which manifest as distortion products.

Theoretically, we could do all our testing using single tones and laboriously plotting the results. But it’s usually much faster to use broadband noise or program content with a spectrum analyzer, which looks at the full range of frequencies in real time.

However, this does not show the time relationship.  By using dual-channel analysis you can look at the complex transfer function of the system, the difference between what was sent and what was delivered (Figure 8).

FFT analysis also lets us look at relative phase differences over a frequency range, to see if we are achieving linearity in the time domain. This is extremely useful in the time alignment of multi-way loudspeaker systems.

Finally, dual channel FFTs let us analyze the relative arrival time of energy. Do all frequencies present at the input arrive at the test microphone at the same instant, or do they spread out over time, with the system producing some frequencies later than others?

If there is a time offset between two signals, this will cause changes of magnitude across the spectrum, as signals at some frequencies will add as others will cancel. A typical example is the comb filtering caused by the relative phase differential between two signals. When two time offset signals are summed, their combined response looks like this (Figure 9).

Spectral analysis reveals other differences between input and output signals, such as the harmonics generated by a loudspeaker. In the example shown here, a 50 Hz signal is being sent and the level ratio of the fundamental versus the harmonics can be quantified. The more non-linear the loudspeaker is, the closer in level the harmonics will be to the fundamental (Figure 10).

In The Field
That concludes our whirlwind tour of the basics of linearity in audio systems. And, fortunately, we know we have good tools at our disposal to isolate and measure deviations from linearity in audio systems. Sound level meters and spectrum analyzers give us good information about the real-time performance of an audio system at one measurement point, whether at a mic location or a point in the signal path, but they don’t make input-to-output comparisons.

It’s really the modern dual-channel FFT analyzer that gives us the powerful tool to make comprehensive comparisons in both frequency and time domains. By applying various test signals – noise, swept sine tones, fixed sine tones, multiple fixed sine tones, and pulses – we can isolate and measure potential non-linearities in an audio system.

Using these concepts and tools, we can set out to design and manufacture linear audio systems, and then configure, tune and operate them to perform in a linear fashion.

But first we need to define the operating parameters. Are we talking about linear performance only in the digital or analog electrical domain, or also in the domain of acoustical sound pressure? And do we need linearity across the entire audio bandwidth, above and below it as well (some say we do), or only a portion of it?

If we stay completely in the realm of signal recording and electrical amplification, audio systems have become remarkably linear in recent decades. Digital recording and mixing systems, when used within their wide but strictly defined operating parameters, exhibit a degree of linearity unheard of 50 years ago. (Outside those parameters, their non-linearity is off the charts.)

Recently, modern audio amplifiers as a rule are exceptionally linear. Progress has also been made to better convert electrical voltage to the acoustic realm of sound pressure in air, but this is an area where the limits of linear behavior are still often encountered.

But before we examine that in detail, let’s consider the elements of our complete sound reinforcement system and set some parameters within which we expect to maintain a linear relationship between signal input and acoustic output.

For our purposes here, we’ll assume that all the “artistic decisions” have been made, and we’re taking the signal from the main output of a mixing console. From here on out, the goal is to transfer that electrical signal into an acoustical signal at the desired level throughout the intended listening area, without changing anything.

To accomplish that, the signal first must pass through some kind of signal distribution, crossover and limiter devices, either in the analog or digital domains. By the time we get to the power amplifier, we must by necessity return to analog, as loudspeaker drivers (and subsequently our ears) remain stubbornly analog.

Then, assuming an active multi-amped system, the power amplifiers outputs connect to the mechanical motors of the loudspeaker drivers. These drivers, in turn, are mounted in cabinets with a specific acoustical environment defined by cabinet volume, baffling, ports and waveguides (Figure 11).

It’s at this transition stage between electrical voltage and acoustic pressure where linearity becomes particularly problematic, and the higher the SPLs involved, the more difficult it is to maintain linear behavior. For contrast, think of a condenser microphone, in which a capsule diaphragm need move only a few micrometers to generate a few millivolts of signal. Even a relatively inexpensive condenser microphone can exhibit excellent linearity.

Compare that to a high-power loudspeaker system, where an amplifier can pump peaks of around 200 volts and 80 amps into a stiffly mounted 18-inch loudspeaker, moving it rapidly several inches. Maintaining linearity here is not easily so easily accomplished, and requires a good bit of creativity along with diligent engineering.

Demands Of Modern Music
So what parameters do we need to set for linear behavior in our desired audio reinforcement system?

First, we need to set bandwidth, and ideally that would be about 30 Hz to 18 kHz, perhaps even wider in the electrical stages.

Now, here’s the critical part. We need the loudspeaker system to respond to the signal demands all across that frequency bandwidth in a linear fashion, regardless of the input signal.

For example, let’s look at spectrum measurements for two pop songs. For each, the top red trace is the peak value, and the white trace is the RMS level, both averaged over one minute.

The top screen is that of a 1970s pop hit, the bottom screen shows a newer pop tune. At 50 Hz, there’s a level difference of almost 15 dB, or almost six times louder.

That means you’d need a little more than 30 times more power to realize linear reproduction of the newer song if the mids and highs for both songs were reproduced at the same levels (Figure 12).

That calculation is important. Remember, Decibels are logarithmic ratios, dB = a X log(x/y), where a is either 10 (power or intensity) or 20 (sound pressure or voltage), and x and y are values like 0.5 volts and 1.0 volt.

For our example, here’s how it works out:

Sound pressure (dB) = 20 log (x)  x = amplitude change factor or ratio
15 dB = 20 X log (x)
x = 5.6 times louder

Power (dB) = 10 log (y)  y = power change factor or ratio
15 dB = 10 X log (y)
y = 31.6 times more power

And if you want to project that level across a distance of 100 meters (approximately 330 feet), you need a prodigious amount of power. With that in mind, it’s easy to see why large-scale reinforcement systems often are driven well into their non-linear regions.

But is that the right approach? Some live sound mixers overdrive certain systems to “get their sound.” Is that a role that should be played by a loudspeaker system? And what if it works for one artist, and not another? Does it really make sense to have one system for heavy metal, another for jazz, and a third for classical? Can you count on a system’s consistent performance when it does not behave in a linear manner?

Art Or Science?
Here we need to acknowledge – and applaud – the legitimate and deliberate creation of audio non-linearity for artistic effect. Certainly the output of Jimi Hendrix’s Marshall stack was not a linear transfer of the string sound on his Fender Strat. And today’s live sound engineers have at their disposal a vast array of tools – from digital console plug-ins to racks of vintage outboard gear – to create entire universes of non-linear sounds to express an artistic vision.

But once that artistic intent is realized, usually at the output of the mixing console, the sound reinforcement system should transmit that sound transparently, coherently and consistently. And regardless of the frequencies and sound levels, the loudspeaker system should not alter the sonic characteristics of the source at any point during a show. The end result should be, to the greatest extent possible, a linear translation of that small electrical input into a very powerful acoustic output.

A linear system puts more of the artistic control in the hands of the artists. So if the artist and live sound mixer decide to change a squirrel-sound into a banana-sound, that’s how it should be. That’s the role of art.

However, the goal of the sound reinforcement system is to make sure that everyone in the audience hears and enjoys exactly the same, undiluted banana-sound experience. And that’s a job for science.

Peter Soper is R&D engineer and Steve Bush is senior technical support specialist for Meyer Sound.

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The Same Everywhere: Exploring System Linearity
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