Understanding Compression 3

In the first instalment of this four-part series we introduced concepts such as threshold and ratio, and introduced terms like ΔInput to represent changes of input signal level above the threshold level, and ΔOutput to represent changes of output signal level above the threshold level. We saw that a downward compressor increases the gain reduction when ΔInput increases, decreases the gain reduction when ΔInput decreases, and removes or applies no gain reduction when ΔInput = 0dB or when the input signal level is below the threshold level. We also saw the simple relationship between input signal level, gain reduction (GR) and output signal level:
Output Signal Level = Input Signal Level – GR
In the second instalment we saw how the compressor’s knee control allows us to shape how the gain reduction is applied. We also demystified the attack and release time controls, learnt about the RC time constant (τ), and found out why we cannot transfer attack and release times from one type of compressor to another and expect the same results.
In this third instalment we’re going to see how the attack and release times shape a sound’s envelope, and learn when and why to use make-up gain.
As with the previous instalments, we’ll assume we’re working in digital systems where 0dBFS is the maximum possible signal level and therefore all signal levels are represented as negative values because they are below 0dBFS. When applying the previous and forthcoming mathematical formulae to analogue processors or digital emulations of analogue processors where 0dBVU is the Nominal Operating Level (typically equivalent to -20dBFS) we need to be vigilant about the calculations we make and where we put the + and – signs, otherwise we’re likely to push the compressor too hard and not get the classic sound we expected.






BEING THE COMPRESSOR
In the previous instalment of this series we saw how the threshold, ratio, knee, attack and release controls work individually to control the gain reduction; now let’s see how they interact to shape the gain reduction. The best way to do that is to ‘be the compressor’, i.e. walk through the compression process from the compressor’s point of view.
The illustration below shows five sound envelopes, A to E, on a horizontal time line that is conveniently divided into intervals of 1τ (i.e. one RC time constant, as explained in the previous instalment). For this example we’re not interested in the absolute durations of those time intervals specified in seconds; by using 1τ as a unit of time we have a relative scale that will remain meaningful regardless of the chosen attack and release times. To keep it simple we will assume the attack and release times are the same, thereby ensuring τ remains the same at all times. This means that five consecutive units on the time line represent the time required for the capacitor to fully charge (attack) from being fully discharged, and also represents the time required for the capacitor to fully discharge (release) from being fully charged.



The best way to do that is to be the compressor…









Each envelope is a simple block shape, allowing us to easily see the effect of the gain reduction and the attack and release times. The first three envelopes, A to C, have the same maximum amplitude of -3dBFS, and the same duration of 5τ so that each is long enough for the RC network to fully charge during its attack time. Envelopes D and E have amplitudes of -9dBFS and -18dBFS respectively, and their durations are intentionally set at 2τ each so that they don’t provide enough time for the RC network to fully charge.
Let’s pass these envelopes through a compressor with a threshold of -18dBFS, a ratio of 3:1, a hard knee, and its attack and release times set to the same duration.













How does the compressor respond to these envelopes? Using the calculations demonstrated in the first instalment of this series, envelopes A, B and C each create a ΔInput of 15dB. With a ratio of 3:1 this results in a ΔOutput of 5dB, and therefore a corresponding output level of -13dBFS. This means the Envelope Follower has instructed the VCA to apply 10dB of gain reduction (GR), reducing the output level from -3dBFS to -13dbFS as shown below.













Similarly, envelope D creates a ΔInput of 9dB which results in a ΔOutput of 3dB and a corresponding output level of -15dBFS. The Envelope Follower has instructed the VCA to apply 6dB of gain reduction, reducing the output level from -9dBFS to -15dBFS.
According to our graph of CV and GR below, we can see that envelopes A, B and C each required a GR of 10dB, which corresponds to a CV of 10V. Envelope D required a GR of 6dB, which corresponds to a CV of 6V.













Envelope E does not exceed the threshold and therefore creates a ΔInput of 0dB and a CV of 0V. It does not require any GR – however, as we’ll see shortly, it gets some anyway…
Integration Function
In the previous instalment we saw how the threshold, ratio and knee controls determine the value of CV1, and how the RC network applies its attack and release times to CV1 to create CV2 – which is the control voltage applied to the VCA.













In essence, the RC network applies two integration functions (one for attack, one for release) to CV1. This, in turn, creates CV2 which is applied to the VCA to control the gain reduction. Let’s follow the process to see how it affects the envelopes…
The illustration below shows the five envelopes, A to D, as seen above. Beneath the envelopes is a graph showing the values of CV1 as calculated above in response to envelopes A to D. The vertical axis on the left shows the value of CV1, while the vertical axis on the right shows the corresponding gain reduction it will create in dB (in accordance with our graph of CV vs GR).













CV1 immediately rises to the required voltage when the input signal exceeds the threshold, and immediately falls to 0V when the input signal falls below the threshold. For envelopes A, B and C it rises immediately to 10V to provide 10dB of gain reduction, and for envelope D it rises immediately to 6V to provide 6dB of gain reduction. Envelope E requires no gain reduction so CV1 remains at 0V.













The illustration above shows what the resulting envelopes would look like if CV1 was used to control the VCA and therefore control the gain reduction. The resulting envelopes (dark green) have been superimposed over the original envelopes (light green). In this example the envelopes retain their same basic rectangular shape; the gain reduction simply lowers their overall levels. Envelopes A to C are all reduced by 10dB, envelope D is reduced by 9dB, and envelope E receives no gain reduction because it is below the threshold.
Why do the envelopes retain their shape in this example? Firstly, because the envelopes don’t change level after exceeding the threshold – each individual envelope’s ΔInput value remains constant for the envelope’s duration, which means the applied gain reduction also remains constant. Secondly, we are seeing the effects of using CV1 to control the gain reduction, and CV1 is before the RC network. This means the attack and release times are as fast as possible (theoretically 0s), so therefore the gain reduction is applied and removed instantly and has no opportunity to change the fundamental shape of these simple envelopes. The gain reduction in these examples is really just a very fast level control.
Now let’s introduce the RC network and its attack and release time controls. The illustration below shows the RC network circuit we evolved in the previous instalment.















CV1 (from the threshold, ratio and knee circuit) connects at the top left of the circuit, and CV2 (which controls the VCA) is available at the lower right of the circuit. The comparator circuit monitors CV1 and CV2 to determine if C1 should be charging or discharging. Potentiometer R1 controls how fast C1 charges and is therefore the attack time control, and potentiometer R2 controls how fast C1 discharges and is therefore the release time control. The operation of this circuit was explained in the previous instalment.
It’s important to remember that CV2 is the voltage across capacitor C1, and therefore CV2 rises and falls in accordance with the charging and discharging characteristics of the capacitor.
The illustration below shows how CV2 increases in accordance with the capacitor’s charging curve. The time constant (τ) of the charging circuit is determined by R1 and C1. Starting from zero charge (i.e. theoretically 0 volts), it takes 5τ for CV2 to rise to 99.3% of the value of CV1, which we consider to be fully charged. This is the compressor’s attack characteristic.













The illustration below shows how CV2 decreases in accordance with the capacitor’s discharging curve. The time constant (τ) of the discharging circuit is determined by R2 and C1. Starting from full charge, it takes 5τ for CV2 to fall to 0.7% of the full charge voltage, at which point we consider it to be fully discharged. This is the compressor’s release characteristic.













The illustration below shows how the RC network converts CV1 into CV2. The upper graph is CV1, as we’ve been using in the previous examples; it is the control voltage provided by the threshold and ratio circuit, and is the control voltage required to achieve the target gain reduction.













The lower graph is CV2, which is the result of CV1 passing through the RC network. It is essentially CV1 after the RC network’s attack and release times have been applied, and is the voltage that controls the VCA.
Each graph has a series of markers along the time axis indicating the start and end of each envelope. Let’s ‘be the compressor’ by walking through the gain reduction process and seeing it from the compressor’s point of view.













Marker A1 indicates the point where envelope A rises above the threshold. CV1 rises immediately to 10V, which is the voltage required to reach the target gain reduction of 10dB. CV2, however, takes 5τ to reach 10V in accordance with the charging characteristics of the RC network’s attack circuit. In this example the envelope is just long enough for the charging circuit to fully charge (i.e. 5τ), allowing CV2 to charge to 10V and apply the target gain reduction of 10dB.
Marker A2 indicates the point where envelope A falls below the threshold. CV1 falls immediately to 0V, meaning no gain reduction. CV2, however, takes 5τ to fall to 0V in accordance with the discharging characteristics of the RC network’s release circuit. In this example there is just enough time for the discharging circuit to fully discharge (i.e. 5τ), taking CV2 down to 0V and resulting in no gain reduction.













Marker B1 indicates the point where envelope B rises above the threshold. CV1 rises immediately to 10V, the voltage required to reach the target gain reduction. CV2, however, takes 5τ to reach 10V in accordance with the charging characteristics of the RC network’s attack circuit. This is the same process as seen for marker A1, but things are about to change.
Marker B2 indicates the point where envelope B falls below the threshold. CV1 falls immediately to 0V, meaning no gain reduction is required. CV2 requires 5τ to fall to 0V and therefore reach no gain reduction, but it never gets there because…













Marker C1 shows that envelope C rises above the threshold just 1τ after envelope B falls below the threshold. CV1 rises immediately to 10V to reach envelope C’s target gain reduction of 10dB, as it did with envelopes A and B. Things are different for CV2, however, because the 5τ required for the RC network to fully discharge from B2 have not elapsed. There has only been 1τ, which means CV2 has fallen to 36.8% of its prior voltage, instead of 0V. This means that envelope C begins with some ‘leftover’ gain reduction from envelope B. Because CV2 had only discharged to 36.8% of its prior value of 10V, CV2 has a value of 3.68V (instead of 0V) when envelope C begins. According to our graph of CV vs GR, if CV2 has a voltage of 3.68V it means that envelope C begins with 3.68dB of gain reduction. Also, because the RC network has not fully discharged when envelope C exceeds the threshold (Marker C1), it is already partially charged and will require less than 5τ to reach full charge – which therefore results in a marginally faster attack time. We can see that envelope C reaches its target gain reduction at approximately 4.5τ.
Marker C2 indicates the point where envelope C falls below the threshold. CV1 falls immediately to 0V as usual, the voltage required to reach no gain reduction. As with the previous example, CV2 requires 5τ to fall to 0V but it never gets there because…













Marker D1 shows that envelope D rises above the threshold 2τ after envelope C falls below the threshold. CV1 rises immediately to 6V to reach envelope D’s required gain reduction of 6dB. Again, things are different for CV2 because only 2τ have elapsed between when envelope C falls below the threshold (C2) and when envelope D rises above the threshold. The RC network has discharged to approximately 14% of its prior charge, resulting in a CV2 of approximately 1.4V which, according to our graph of CV vs GR, corresponds to a gain reduction of 1.4dB. So envelope D begins with 1.4dB of ‘leftover’ gain reduction from envelope C.
Although CV2 is supposed to reach 6V, the short duration of envelope D (2τ) means the RC network doesn’t get the 5τ required to fully charge. So instead of reaching 6V and applying the target gain reduction of 6dB, it reaches just over 5V and therefore applies just over 5dB of gain reduction.
Marker D2 indicates the point where envelope D falls below the threshold. CV1 falls immediately to 0V as usual, aiming for no gain reduction. However, the RC network will take 5τ to fully discharge and bring CV2 down to 0V. And now things take an interesting twist…













Marker E1 shows the start of envelope E, which does not exceed the threshold and therefore should not receive any gain reduction. CV1 remains at 0V. However, because envelope E begins 1τ after envelope D falls below the threshold, the RC network has not fully discharged its prior charge from envelope D; CV2 has fallen to approximately 1.9V, instead of 0V, which means that envelope E starts with approximately 1.9dB of ‘leftover’ gain reduction from envelope D.
Marker E2 indicates the end of envelope E. Because envelope E has a very short duration (2τ) and begins just 1T after envelope D, it starts and ends fully within envelope D’s 5τ release time. Envelope E is a signal that remains below the threshold and should have no gain reduction, but it is affected throughout its duration by ‘leftover’ gain reduction from the previous envelope; approximately 1.9dB of gain reduction at the start, reducing to approximately 0.7dB of gain reduction at the end.
Pushing The Envelope
It’s important to remember that the graph we’ve just been analysing shows the rises and falls of CV2, which is the voltage that controls the gain reduction; it is not the gain reduction itself. For our theoretical examples the VCA’s gain reduction increases in proportion to CV2, as shown earlier in our graph of CV vs GR. When CV2 increases the VCA applies more gain reduction and therefore the amplitude of the envelope decreases, and when CV2 decreases the VCA applies less gain reduction and therefore the amplitude of the envelope increases. The illustration below shows how the gain reduction curves shown in the graph above alter envelopes A to E. The outlines of the uncompressed envelopes are shown in green.













Note that the compressed envelopes shown are simply the uncompressed envelopes minus the gain reduction.
Because all of these graphs are drawn to the same scale we can turn the CV2 graph upside down, align its oV point with the gain reduction’s 0dB point, and the resulting image represents the gain reduction. This is shown in the illustration below with signal amplitude on the left vertical scale in dBFS (green), and gain reduction (GR) on the right vertical scale (red). The red line that formerly represented CV2 has been inverted and now represents the gain reduction.













At any point on the graph, the compressed signal’s amplitude is always equal to the uncompressed signal’s amplitude minus the gain reduction – which brings us back to the formula given at the start of the first instalment:
Output Signal Level = Input Signal Level – GR
Throughout the examples above we have seen how the attack and release times work, how they affect each other, and how they can change the shape of a sound’s envelope. Every time ΔInput increases a new attack begins and gain reduction increases, and every time ΔInput decreases a new release begins and gain reduction decreases. Every new attack or release changes the amount of gain reduction, and that change begins at the level of the existing gain reduction and occurs in accordance with the appropriate attack or release RC time constant. This is demonstrated in the previous illustrations where envelopes C, D and E are each affected in differing amounts by ‘leftover’ gain reduction from the previous envelope.
Leftover GR
From the illustration below we can see that in envelopes A to D the compressor’s attack time has altered the shape of the envelope significantly, giving each envelope a percussive attack transient that might or might not be a preferable sound, while also bringing envelopes A, B and C closer than they were to envelope D in terms of perceived level.
At the same time, envelopes C, D and E are each affected by ‘leftover’ gain reduction from the envelope that preceded them. Envelopes C and D suffer a reduction in their peak level but are still given an exaggerated attack transient. Envelope E, however, is affected quite differently even though it is below threshold and should not incur any gain reduction. Rather than getting an exaggerated transient its attack has been rounded off, which will make it sound quite different to envelopes A to D – perhaps sounding like a poorly performed note or a musical after-thought.













For these examples we can solve the ‘leftover’ gain reduction problem by using a faster release time, a choice that is easily rationalised by the shapes of the envelopes; they all end suddenly and therefore the release time has no effect other than creating ‘leftover’ gain reduction that might affect the beginning of the following envelope. If the release time was fast enough to return the VCA to 0dB of gain reduction within one vertical time division on the graph, envelope B’s release would not affect envelope C’s attack, envelope C’s release would not affect envelope D’s attack, and envelope D’s release would not affect envelope E’s attack or its overall level.
If you’ve made it this far, give yourself a pat on the attention span…
OUTPUT GAIN
The process of downward compression often results in the output signal having a lower overall metered and/or perceived level than the input signal. This is illustrated below, where the shaded red areas indicate the portion of each envelope that has been ‘removed’ due to the gain reduction.













Visually, envelopes A to D each lose a significant portion of their individual surface areas in the illustration. This visual indication tells us that each envelope probably loses a significant portion of its overall energy and therefore we should not be surprised if each envelope has a lower metered and/or perceived level after applying downward compression.
In cases where it is important to maintain the signal’s metered and/or perceived level, the compressor’s output gain is used to make-up for the loss – hence it is sometimes referred to as make-up gain.













The illustration above shows the typical compressor block diagram we’ve been using throughout this series, but now we have added an amplifier at the output. This amplifier’s gain is adjusted manually using the Output control.
For the purposes of this explanation we have added a fourth meter, VCA Output, to show the signal level coming out of the VCA before the output gain is added to the compressed signal. The Output meter shows the signal level after the output gain is added, of course. So for this demonstration we have meters showing the input signal level, the gain reduction, the output of the VCA, and the output of the compressor after the make-up gain has been applied.
Applying Make-Up Gain
The illustration below shows how the output amplifier is used. For this example we are reducing the signal’s musical dynamic range, i.e. the difference between the levels of the loudest and softest moments in the performance. This is marked in yellow on the Input meter, where we can see that the softest moment in the performance reaches -18dBFS and the loudest moment in the performance reaches -6dB. This gives the performance a musical dynamic range of 12dB (-6dBFS – -18dBFS = 12dB).













Let’s assume this 12dB of musical dynamic range is too large to place the sound comfortably alongside other sounds in the mix without relying on automation to increase the levels of the softest moments and/or decrease the levels of the loudest moments – both of which mess with the performer’s expression.
Compression can solve this problem more intelligently because it reduces the musical dynamic range while maintaining the performer’s expression by increasing its impact. Louder moments get more compression, and therefore more impact, than softer moments. When done right, this can preserve the performer’s expression and avoid the expression inversion that occurs when using fader automation to reduce a performer’s level when they’re trying to be loud and increase a performer’s level when they’re trying to be soft. Once heard, expression inversion cannot be unheard.
Reducing Dynamic Range
For the purposes of this demonstration let’s say we wanted to reduce the musical dynamic range to 4dB. We’ll start by setting the threshold to -18dBFS, which is the level of the softest note in the performance. Why? For this type of corrective compression (i.e. putting the sound’s musical dynamic range into the correct dynamic perspective for the mix) we don’t want to compress anything below the level of the softest note because a) we want to maintain the subtlety of the softest played note, and b) compressing signals below the softest note played will simply be adding impact to noise, spill and other unwanted sounds captured during the performance.
A threshold of -18dBFS and a peak level of -6dBFS gives us a ΔInput of 12dB. We want to convert that into a ΔOutput of 4dB, so that when the input goes 12dB above threshold (the level of the loudest note in the performance) the output only goes 4dB above threshold. This requires a ratio of 12:4, which we can simplify to 3:1. This will turn a musical dynamic range of 12dB into a musical dynamic range of 4dB, and will result in a maximum of 8dB (12dB – 4dB) of gain reduction.
These figures are shown on the meters in the previous illustration. We can see that the input signal level goes 12dB above the threshold of -18dBFS, the VCA applies 8dB of gain reduction, and the resulting signal at the output of the VCA goes only 4dB above the threshold. This reduces the performance’s musical dynamic range to 4dB, as required, but now the overall signal level is too low. Before compression the input signal level was sitting comfortably around -12dBFS (±6dB) as shown on the input meter, and we want to get the output signal level sitting around -12dBFS as well. By adding 4dB of make-up gain to the output, we are able to bring the output signal level up to somewhere around -12dBFS again. By comparing the input signal level to the output signal level we can see that both signals sit comfortably around -12dBFS, but the output signal has a considerably reduced dynamic range.













In the illustrations above and below we can see how the right choice of threshold and ratio has reduced the signal’s musical dynamic range, and the right choice of output gain has returned the output signal to the desired metered and/or perceived level. Problem solved!













When Does Make-Up Gain Matter?
Maintaining a signal’s metered and/or perceived level is particularly important in analogue systems where there is a Nominal Operating Level (NOL) of 0dBVU, which is a recommended signal level considered high enough to overcome tape hiss but low enough to allow sufficient headroom in the mixing console for peaks and transients that go above 0dBVU. A signal captured with a nominal level of 0dBVU could fall to -3dBVU or lower after compression. When this occurred in the analogue audio world it was standard practice to use the compressor’s Output Gain control to restore the signal to its original nominal level of 0dBVU and thereby maintain the analogue system’s gain structure. The compression reduced the signal’s dynamic range as required, but in the process it had reduced the signal’s nominal level. The Output Gain was used to restore this loss – as demonstrated in the previous example.
The considerably lower noise floor of digital audio systems makes the Output Gain control less necessary, although it is still good practice – especially when a) using corrective compression with the goal of transparently reducing a signal’s dynamic range as discussed above, and b) chaining two or more dynamic processing plug-ins together, whereby a last minute tweak to the first processor might alter its overall output level and will thereby affect how the second processor responds.













BRINGING IT ALL TOGETHER
So what have we learnt so far? In the first instalment we learnt how to use the threshold and ratio controls to achieve the desired gain reduction. In the second instalment we learnt how the knee control eases the ratio in and out, and how the RC time constant determines the attack and release times. In this instalment we saw how the attack and release settings ease the gain reduction in and out (thereby altering the envelope), and we learnt when and why to use output gain.
From this knowledge we can derive a strategic approach to solving any downward compression problem. In the next and final instalment we’re going to see how to do that, because nobody cares about the theoretically correct GR if it sounds bad…



Next instalment: Get On With It (coming soon)…
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