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# How we test headphones: distortion

Have you listened to your own voice on a video, over a loudspeaker, or over the echo of a Skype chat? It probably sounds pretty different in all three circumstances. The reasons for this are twofold: Signal distortion, and Frequency Response (FR).

FR is one of the most important aspects of our testing, but signal distortion also has a profound effect on how your voice sounds when played on a given device. And like all distortion, it has to do with altering something from its original form. In this case, the things being altered are the digitally rendered sound waves that live inside your music player.

## Getting Started

Each pure tone you hear in a piece of music corresponds to a frequency. For instance, Middle C is represented by a sine wave with a frequency of 261.6 Hz, or 261.6 cycles per second.

In a recording studio, an artist or musician plays Middle C. In the following video, you will be able to hear Middle C, as well as see its equation and shape through time on a plot. Note the different time scales in each video:

When a recording artist or musician plays that note, the sine wave is one signal that gets rolled in with all of the other signals representing other notes and sounds that are being played at the same time. The sine wave is saved inside your music device, and when you click “play”, the sine wave is summoned by your headphones.

However, it turns out that your headphones cannot perfectly reproduce the original sine wave that was recorded in the studio. This occurs for two reasons: The first is that we do not live in a perfect world where signals are perfectly transmitted all the time, and the second is that the omnipresence of electronics with non-linear loads (i.e. your music device/headphones) means that the digital signal response of a device, when it draws a current, can be flawed and unpredictable.

## Distortion

While your headphones cannot exactly replicate that original sine wave, they do their best to reproduce it.

For example, let’s say your headphones were aiming for Middle C:

But came out with this instead:

The two sine waves don’t look terribly different, but if you listen closely to the two sound clips, you can hear the difference.

Our headphone tests quantify how your headphones distort the original music signal at each individual frequency. In SoundCheck, our headphone testing software, we measure distortion with a frequency sweep that starts with a pure tone at a subset of frequencies from 20 Hz (increase your volume to hear this one):

to 10 kHz (lower your volume for this one):

Knowing the original frequency and height (= amplitude) of the pure tone, we can analyze the sine wave that is actually output by the headphones and quantify how different the two sine waves are.

## Harmonics

So how do we analyze the resulting digital wave produced by your headphones? SoundCheck uses a method called Fast Fourier Transform, or FFT. With FFT, a wave is broken down into a finite number of sine waves with different frequencies and amplitudes. The best way to understand FFT is to imagine the process in reverse. You can take a finite number of sine waves, with known frequencies and amplitudes, and add them together to create a new wave of any shape or size. This process is known as superposition.

At a given point in time, some of those sine waves will cancel one another out to some extent by having opposing high and low values (destructive interference). At other points, some sine waves will add together by having similarly high or low values (constructive interference).

With a Fast Fourier Transform, SoundCheck takes the final digital wave and reduces it into the component sine waves. In this particular case, SoundCheck is only looking for component sine waves that represent harmonics of the original (or “fundamental”) tone.

Harmonic waves are simply waves that have frequencies that are integer multiples of the fundamental frequency. For instance, if the fundamental tone has a frequency of 20 Hz (=20*1), then the first harmonic tone would have a frequency of 40 Hz (=20*2), the second harmonic tone would have a frequency of 60 Hz (=20*3), all the way up the nth harmonic (=20*(n+1)), and so on.

## Total Harmonic Distortion

There are many different types of distortion that can be measured in headphones. Total Harmonic Distortion, or THD, is the most basic kind of distortion. We'll be talking about THD in this post, but if you're curious about other kinds of distortion, check out this link for more information.

After identifying the harmonic tones that add together to create the imperfect fundamental tone, we calculate the THD using the equation shown below. Here the THD is presented as a percentage of the original amplitude of the fundamental sine wave, where Vn represents the amplitudes of the sine waves.

For n > 1, Vn is the amplitude of the nth-1 harmonic sine wave, and for n=1, Vn is the amplitude of the fundamental sine wave.

If the amplitudes of the harmonic sine waves are very close to the amplitude of the fundamental sine wave (high THD percentage values), then the resulting wave will be significantly distorted from the original sine wave. If, on the other hand, the amplitudes of the harmonic sine waves are very small (low THD percentage values), then the fundamental sine wave is better preserved.

Let’s take another look at our distorted wave. In this video, you will first see the shape of the distorted version of Middle C, then the three waveforms that make up that distorted wave: the original fundamental frequency sine wave and two harmonic frequency sine waves. This waveform has a THD of 20%.

Here is a second wave that has also been distorted from the original Middle C fundamental tone. This distorted wave has a THD of 40%.

## Distortion at different frequencies

The human ear tends to have an easier time hearing higher frequency tones. Consequently, the human ear can more easily perceive distortion of higher frequency tones than of lower frequency ones.

For example, listen to a sine wave with a fundamental frequency of 40 Hz (increase your volume for this one):

And here is a version of that sine wave with a THD value of 20%:

By contrast, this is a sine wave with a fundamental frequency of 1000 Hz (=1 kHz) (lower your volume for this one):

Here is a version of that sine wave with a THD value of 20%:

The distorted version of the low frequency tone is practically indistinguishable from the original fundamental tone. With the high frequency tone, the difference is immediately obvious.

Ideally, headphones would not distort the original fundamental tone at any frequency. Headphone manufacturers know that our sensitivity to distortion increases with higher frequencies, so they do their best to minimize the perceivable distortion at higher frequencies, while being somewhat less strict when it comes to THD at lower frequencies. We take this into account when we run our distortion test on a pair of headphones.

## Our headphone testing

When developing our testing, we built an empirical THD curve that mimics the human ear’s sensitivity to distortion. This empirical curve (the shaded blue curve shown below) was defined by previous distortion data from 70+ products we’ve tested over the past few years. (Note the logarithmic, rather than linear scale in the chart.)

In our distortion test, we do a sweep of original fundamental tones with frequencies from 20 Hz to 10,000 Hz. SoundCheck breaks the resulting distorted signals down into their component harmonic sine waves, and calculates the THD for each individual frequency. We plot that THD data (red line), and compare it to our empirical data curve (blue shaded area).

Any part of the product’s THD curve that lies above our empirical data curve results in a point deduction from a perfect score. The more a product’s THD curve resides above our empirical data curve, the more noticeable the distortion is to the listener, resulting in a lower overall distortion score.

Conversely, if a product’s THD curve resides entirely below our empirical data curve, then the distortion score for that product is a perfect ten.

To make a long story short, the less the red line goes above the blue line, the less noticeable any distortion will be to the listener, and the better the product will score.

Our best scoring products have low distortion across the entire range of frequencies. To find out more about distortion in individual headphone products, be sure to check out our ever-updating library of headphone reviews. Click on the "Test Results" tabs in individual product reviews for the greatest detail.

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