Most of us are familiar with the concept of frequency response - I would venture to guess that most of us have looked at the occasional frequency response plot from a speaker, or perhaps the frequency response of a preamp, processor or amplifier. However, the word 'linearity' is often synonymously (and regrettably, erroneously) used to describe 'flat frequency response'. I would like to explain the difference for those who are not necessarily sure of the difference. For now, let's start with frequency response.

Frequency response, as its name suggests, defines how a device passes or reproduces a known signal of a given frequency. There are several stimuli that can be used to test frequency response, but most often, white noise (and depending upon the device, pink noise) is used. Alternately, this cab be accomplished using a sweep (a slowly increasing frequency) or a chirp (a rapidly increasing frequency), and one can also get this from the impulse response (but that's yet another post). White noise is a very 'sane' choice for such a measurement as it differs from other stimuli in that it carries equal power per bandwidth. Its name comes from white light, which is a mix of all visible colors of the spectrum.

OK...now let's talk about linearity...

There is a mathematical definition for linearity (essentially additivity and homogeneity), but for now, let's try to give it a more practical (i.e. less 'nerd-inspired' definition (that's not a slam on nerds, as I consider myself to be one)) definition. Here goes:

A linear system will provide an output in response to an input, that is either the exact same thing as the input, or merely a larger (i.e. scaled) version of the input; it will add or subtract nothing from the input signal. Moreover, said degree of linearity of the output will not depend upon the magnitude of the input to the system.

In other words, if we were testing a power amplifier and feeding it a known signal, all we would ever see, no matter what the input magnitude was, at its output we would see the exact same signal, only at a higher magnitude (due to the amplification). But...we all know this is not possible. Amplifiers are linear...but only when the design limits of the amplifier are respected. Clipping is a perfect example - over-drive an amplifier and the next thing you know, you have all kinds of odd-order harmonics showing up at its output...and these harmonics were not present at the input to the amplifier. Thus, we can say that when operated within its design envelope, the amplifier behaves as a linear device.

Speakers. Speakers are notoriously non-linear. Why?

If you ever look at the mechanical systems that comprise a typical dynamic loudspeaker, you'll start to understand. We all know that the cone can only travel so far without thermal or mechanical damage, but at the same time, the things that define the physical properties of the speaer are themselves a function of cone diaplacement. That's not linear.

However, if (like an amplifier) the speaker is operated in its linear range, we can (sort of) neglect this aspect. However, there's another important point in all of this, namely, that all of the enclosure / driver modeling software out there assumes (requires) that all of the electro-mechanicl parameters of the speaker (i.e. the Thiele-Small parameters) are time-invariant, and inherently linear. That is, the Thiele-Small parameters are not affected by the displacement of the speaker diaphragm...but...this is not the case - they can be very much affected by the speaker cone displacement. Effective, speaker modeling software relies upon what is known as 'small signal analysis', and this is done because under such conditions so that the assumptions of linearity that allow the algebra to work (in the modeling software) are invoked.

However, even under moderate cone excursion, the stiffness (borne primarily of the voice-coil spider(s) and the surround) is a function of cone displacement. This means that the stiffness (or inversely, compliance) literally changes as the cone is displaced further and further from rest position; the greater the displacement, the more air you move. The more air that you move, the louder you perceive the sound to be.

See where we are going with this?

OK, so now let's talk about how one could check for linearity. In other words, are the performance characteristics of my speakers independent from how hard I drive them, and can I define when things start to go south?

Apart from your ears and common sense, the measurement answer lies in taking the complex FRF and plotting a few key things (and there are other methods of doing this, but I like this approach).

First of all, we set up the test bench such that the amplifier we will use has mountains of power capacity (this is to ensure that the amplifier does not corrupt the measurement, i.e that we are measuring the speaker and not the speaker + amplifier).

Next, we set up our analysis system to acquire the input signal going to the amplifier, the output signal coming from the amplifier, and finally, the speaker output (as measured with a suitable measurement microphone). Oh, and we also perform this test in a suitable environment (i.e. anechoic chamber) so that we are measuring only the speaker, and not room affects.

So...here's where it becomes real and practical.

Let's say we drive the speaker with 2.8 VRMS from the amplifier. If we look at the SPL versus frequency, we will see that the speaker produces a certain sound pressure level at that drive voltage. Now...recall that SPL in a speaker is directly proportional to the voltage applied to it.

We now increase the drive voltage (to make the math simple and the example more tangible) to 5.6 V. What should happen?

Well, effectively we should see the exact same SPL curve as we did at 2.8 V, only now, each value observed at 2.8V will be 6 dB higher than it was at 2.8 V. Why? Because speaker SPL is proportional to voltage, and the change from 2.8V to 5.6V is exactly 6.0 dB. Thus, if our system is linear, the curve observed at the 5.6V drive will be an exact replica of that at 2.8V, but just 6 dB higher - because the speaker is doing precisely what it should - if the drive voltage goes up by 6 dB, then the SPL from the speaker has to go up by 6 dB...at least...if it is a linear system. Were you to see changes in the measured frequency response of the speaker as a function of the drive level...that...is textbook non-linearity.

Were you to repeat this exercise at progressively higher drive voltages, you would find a 'knee' at which each dB increase in drive voltage no longer translates 1:1 in the SPL output of the speaker; this is known as 'power compression' in speaker circles, but effectively, it's non-linearity incarnate.

Now, another really good way to determine this is to look at the Coherence function (its formal definition is...ahem...the squared cross-spectral density between x and y, divided by the product of the autospectra of x and y. I know... a lot to get one's brain around...but just hang in there).

Coherence is a remarkably useful tool to assess linearity of a system. For example, were you to measure the complex FRF of a piece of wire (signal at one end versus the other) and to plot the coherence between the input and the output, the coherence would be 1.0 over the entire measurement bandwidth. A value of 1.0 (in coherence) indicates 100%, or complete linearity.

Coherence is almost universally plotted with respect to frequency - especially if one is interested in frequency response and linearity (in contrast to SPL which can be stated as an overall value, or the SPL at a given frequency).

Thus, in our experiment defined above, what would happen in the spaeker as we got closer and closer to power compression (borne of the speaker's non-linearities), we would start to see dips values below 1.0) at various frequencies in the speaker. In other words, in our assumedly well-engineered speaker, at small drive voltages to the speaker, the coherence woul (should be) 1.0 at all frequencies within the bandwith capabilities of the speaker design envelope. However, as the drive voltage is set higher and higher (and measurements performed), the coherence values would no longer be 1.0, and where the speaker was struggling most (in terms of frequency) the coherence values would be the lowest.

in the experiemnt I also mentioned looking at the drive voltage to the speaker (from the amplifier). So if one were to do this when performing a speaker test, one could compute the FRF and the coherence of the amplifier alone, and always know whether or not anomalies observed in the final measurement of the speaker were soloely attributable to the speaker, or were influenced by the amplifier. In other words, at all drive voltages in such a test, the FRF should be a flat line, and the coherence between amplifier input and its output should be 1.0. If they are not, then this would tell us that the amplifier is contributing a degree of non-linearity to the overall result.

Granted, there are many other ways to test, and most of them are very reliable, well-documented, and industry-accepted procedures (i.e. THD, THD+IM, etc), but I wanted to discuss linearity in terms of the tools that one would typically use to assess them.