2.1
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The compressor was invented to control the gain on a recorded track. In the early days this was done by hand. The best way to even out the level of a recorded track is to "ride" the fader manually. When the signal gets too loud, the engineer brings it down. This involved following a score (or working from memory) and making level changes in anticipation of the peaks in the music. And so the compressor was invented to do this automatically. It takes less attention and it can respond to faster changes.
But then came, yes indeed, The Beatles! And they started using that compressor to change sound in a new way. Sly Stone was also very pleased with heavily compressed sounds. Since those days compression is everywhere. Compression can be applied to any signal that varies too much in level, on separate tracks or complete mixes. Or can be used as a way to create a certain sound famously used on Daft Punk albums.
Nowadays it is used extensively in audio recording, production work, noise reduction, and live performance applications. Compression can easily be overdone though, and this leads to music with little or no dynamics. This also created the so-called (not so cool) "loudness war" to which FabFilter Pro-C does not intend to contribute. (More on that in this Wikipedia article
.)
So how does it work?
On an analogue mixer a fader would be a "manually controlled amplifier", but since it is here done automatically that component is shown as DCA (digitally controlled amplifier). To make that fader "go down", the signal is sent to a so-called "side chain" which consists of a level detector and gain control. So the gain control is the actual compressor. It will control the level in time using the attack and release times of the compressor, and will change the volume based on threshold and ratio settings.

When a signal is fed to the level detector that is above the threshold, the gain controller will lower the volume (after the attack time) with an amount based on the ratio setting. This input/output relationship is often described by a simple graph called the transfer function. The horizontal axis corresponds to the input signal level, and the vertical axis is the output level (measured in decibels). The line at 45 degrees corresponds to a ratio of 1 (no compression).

The compressor setting is usually stated as a ratio, such as 2:1, which means that the input level would have to increase by two decibels to create a one decibel increase in the output and thus the effective gain reduction is 1 dB. With a 4:1 setting, the input would need to change by 4 dB for a 1 dB change in the output level and thus the effective gain reduction is 3 dB, and so on. This means the amount of gain reduction is determined by the ratio control in combination with the threshold. Low thresholds mean the compression starts at a lower level, so there will be a lot of gain reduction. High ratios squash signals above the threshold harder, also introducing a lot of gain reduction.
This picture shows what is actually happening:

Here you can see that the compressor will lower the level of a signal above the threshold based on the ratio setting after the attack time. When the input signal falls below the threshold again, the compressor recovers with the release time, back to the original signal. And that's what a compressor is all about.
Last but not least:
Soft knee compression works slightly different. A soft knee brings in the compression more progressively by gradually increasing the compression ratio as the signal level approaches the threshold level. Soft knee compression is more subtle, since the compressor doesn't wait and then suddenly applies the full level of compression at the threshold. Therefore a soft knee is more suitable for compressing whole mixes or gentle sounds that hover around the threshold.

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Next: Advanced compressor techniques
See Also
Dynamic parameters
Time parameters
Expert mode