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

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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).

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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).

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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).

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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).

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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).

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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).

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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.)


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