Audio Shelving EQ: What Are Low Shelf & High Shelf Filters?

From dedicated audio equalizers to tone controls on stereos and instruments, shelving EQ filters are commonplace components in both analog and digital audio devices.

In this article, we'll discuss shelving EQ filters in detail, discussing how they work, how they're designed and how they show up in audio devices. We'll also consider some uses of low shelf filters and high shelf filters in the context of audio mixing.

It's important to note, before we begin, that the study of electronic filters is rather dense. In this article, we'll get into some theory to help us understand shelving filters. However, this is by no means a complete study on filters. Rather, it's a guide to understanding and using shelving EQ/filters in the context of audio mixing and production.

What Is Shelving EQ?

In the opening paragraphs, I gave brief answers as to what defines a low shelf and a high shelf filter. Collectively, these filters make up what is known as “shelving EQ”.

What is shelving equalization? Shelving EQ utilizes high and/or low shelf filters to affect all frequencies above or below a certain cutoff frequency, respectively. Shelving can be used to either boost/amplify or cut/attenuate and affects all frequencies equally beyond defined cutoff frequency points.

Shelving EQ is often listed as it's own type of audio equalizer. These “EQs” are technically restricted to having only low and high shelf controls and you'll typically only ever have them in tone controls. To limit things even further, these tone controls will only have “bass” (low shelf) and “treble” (high shelf) controls.

However, low shelf and high shelf filters can be present in all types of audio equalizers and tone controls (yes, even those with “mid” controls).

And so we'll focus more on the particulars of the low and high shelf filters in this article rather than on shelving EQ itself. I just wanted to start off by discussing this type of audio equalization and distinguishing between the filters and the EQ type.

Low Shelf Vs. High Shelf Filters

The major difference between low and high shelf filters is the name: low shelf filters affect (boost or cut) frequencies at the low-end of the audio signal while high shelf filters affect (boost or cut) frequencies at the high-end of the audio signal.

For perspective, the universally-accepted audible frequency range of human hearing is 20 Hz to 20,000 Hz. Audio signals, therefore, will often only have frequency information within this range so to avoid having any imperceptible information that would take up headroom or storage space or cause other issues (such as aliasing, imaging, etc.).

So a low shelf filter will attenuate or amplify frequencies below a defined cutoff frequency (f₂). The attenuation or amplification will ramp down (or up) to a lower cutoff frequency (f₁), at which point the attenuation/amplification will level out.

That looks something like this (with the low shelf boost/amplification in pink and the low shelf cut/attenuation in blue):

So a high shelf filter will attenuate or amplify frequencies above a defined cutoff frequency (f₁). The attenuation or amplification will ramp down (or up) to a higher cutoff frequency (f₂), at which point the attenuation/amplification will level out.

That looks something like this (with the high shelf boost/amplification in pink and the high shelf cut/attenuation in blue):

Beyond these difference, low shelf and high shelf filters are actually quite similar.

In fact, if we're to take amplification out of the equation and look strictly at relative amplitude, a high shelf boost would have the same practical effect as a low shelf cut (assuming f₁ and f₂ are the same). Of course, the circuits of these two (which we'll get to later) are different and one requires amplification, but the similarity is notable.

Similarly, taking amplification out of the equation and looking strictly at relative amplitude, we'd see that a low shelf boost would have the same practical effect as a high shelf cut (again, assuming f₁ and f₂ are the same).

With that all said, let's first discuss low shelf filters in greater detail followed by high shelf filters.

What Is A Low Shelf Filter?

Let's build upon the short answer of the introductory paragraphs to better understand low shelf EQs/filters.

As mentioned, low shelf filters will effectively boost or cut (raise or lower the amplitude) of frequencies below a defined cutoff frequency by a defined amount. The resulting amplitude-frequency graph, as we'll see, resembles a shelf in the low-end, hence the name “low shelf filter”.

In general, there will be a smooth transition band of frequencies between the original amplitude and the relative amplitude. This band will be defined between a low “cutoff” frequency and a high “cutoff” frequency (we'll call these points f₁ and f₂ throughout this article).

Note that f₁ will mark a 3 dB increase from a low shelf cut's maximum attenuation or a 3 dB decrease from a low shelf boost's maximum amplification as long as the overall cut or boost is greater than 6 dB. If the overall cut/boost is less than 6 dB, the difference between the max cut/boost and f₁ will be some amount smaller than 3 dB.

f₂, similarly, will mark a 3 dB increase or decrease from the higher frequency amplitude plateau (depending on if the low shelf boosts or cuts, respectively). If the overall cut/boost is less than 6 dB, the amplitude difference between f₂ and the higher-end amplitude will be equal to the difference between the max cut/boost and f₁.

This is easier to visualize than to explain with words, so let's look at some graphs to help with our understanding:

Low shelf cut with 6 dB attenuation or more:

Low shelf cut with less than 6 dB attenuation:

Low shelf boost with 6 dB attenuation or more:

Low shelf boost with less than 6 dB attenuation:

Low shelf filters can effectively increase or decrease the energy of the low-end without completely eliminating frequency content from the signal.

Low Shelf Boost Vs. Low Shelf Cut

When it comes to low shelf filters, we can have low shelf boosts and low shelf cuts.

Low shelf boosts effectively increase the relative amplitude in the low-end and require gain. Low shelf cuts effectively decrease the relative amplitude in the low-end and, in general, do not necessarily require gain (though they are generally designed with op-amps and gain).

Here are a few more graphs to help illustrate a low shelf cut and boost, respectively:

Notice how, in either case, the visual “shelf” can be seen.

A low-shelf boost looks similar to a low-pass filter except the facts that:

1. The low-end is amplified.
2. The amplitude levels out past f₂ rather than continuing to roll-off.

If we take phase out of the equation (or utilize linear phase EQ), a low shelf boost can be thought of as the summing of a dry (unfiltered) signal and that same signal affected by a low-pass filter.

Similarly, a low shelf cut acts similarly to a high-pass filter except for the fact that the low shelf's attenuation will level out below f₁ while the HPF's roll-off continues indefinitely.

If we take phase out of the equation (or utilize linear phase EQ), a low shelf cut can be thought of as the summing of a dry (unfiltered) signal and that same signal affected by a high-pass filter and then bringing down the overall amplitude to match the original signal's amplitude above f₂.

I hope I didn't confuse you here. We'll discuss gain and amplitude in the following sections. I just wanted to give comparisons between low shelf filters and the other filter types.

The Ideal Low Shelf Filter

When dealing with filters and other electronics, we like to deal with “ideal” circumstances.

In an ideal world, a shelving filter would have a sharp “shelf” in its frequency-amplitude output graph. In other words, there would be no transition period and the aforementioned f₁ and f₂ would be equal.

The ideal low shelf boost would look as follows:

The ideal low shelf cut would look as follows:

These ideal shelves are unobtainable by typical filter design. Some digital EQs can closely approximate these kinds of “brickwall” filters but in general, there will be some sort of transition period.

We generally want a sort of gentle transition with a low (and high) shelf filter in order to attenuate or amplify the low-end (or high-end) in a gentler, more sonically natural fashion.

Real-World Low Shelf Filters

Real-world low shelf filters are subject to the limitations of real-world components and design.

The main point here is that, compared to ideal filters, real-world shelving filters will have a transition band between the two target amplitude levels.

Once again, let's have a look at a typical low shelf cut followed by a low shelf boost to help visualize the transition periods of low shelf filters:

Passive low shelf filters will only be able to cut while active low shelf filters may be capable of applying gain and boosting the low-end.

Low Shelf Filter Transition Band

The transition band of a low shelf filter is the band of frequencies between the aforementioned cutoff frequencies (f₁ and f₂) and is largely defined by the slope of the amplitude change. This is where the order of the filter comes into play.

The order of a filter is defined as the minimum number of reactive components a filter requires in its design. In the vast majority of audio filters, these reactive components are capacitors, though inductors may also be used in certain passive designs.

For each integer increase in order, a shelving filter will experience an additional 6 dB/octave (20 dB/decade) increase in roll-off steepness.

So the transition band, amplitude difference (whether it's a low shelf cut or boost) and order all come into play to determine the shape of the resulting low shelf filter output.

Low Shelf Filter Q Factor

Some low shelf filters (especially in parametric EQs) will have Q controls.

The Q factor is technically the inverse of the bandwidth and, therefore, is best applied to bell-type EQ, band-pass and band-stop filters. However, many other filters, like low shelves, can have controllable Q parameters.

In the case that a low shelf filter does have a Q factor, it will likely also have a centre frequency control (rather than a cutoff frequency control) that marks the mid-point of the maximum cut or boost. The higher the Q, the steeper the transition band will be.

Low Shelf Filters & Phase-Shift

Thus far we've discussed how a low shelf filter will affect the frequency-dependent amplitude of audio signals. However, these filters will also affect the frequency-dependent phase of audio signals.

In typical high-pass and low-pass filters, the amount of each reactive component in an analog filter will introduce 90º of phase shift in the signal.

However, in low-shelf filters, this full 90º potential phase-shift per reactive component isn't reached.

High and low-pass filters simply roll-off the amplitude of the signal below the cutoff point to negative infinity. Shelving filters roll-off to a second defined amplitude. It's this difference in design that also keeps the maximum phase-shift from reaching a full 90º per reactive component.

Rather, the amount of phase shift will be restricted by the overall impedance of the circuit. The maximum amount of phase-shift will happen at the centre frequency (the point between the maximum and minimum amplitude levels of the filter).

Here are examples of amplitude-frequency and phase-frequency graphs for both low shelf boost and low shelf cut filters:

In the above graphs, we're dealing with first-order shelving filters (6 dB/octave slopes). The phase-shift is at a maximum (negative) at the centre frequency of the low shelf boost and at a maximum (positive) at the centre frequency of the low shelf cut. The phase-shift is not quite 90º either way.

What Is A High Shelf Filter?

High shelf filters are effectively the opposite of low shelf filters (though, in design, high shelf boosts share much in common with active low shelf cuts and active high shelf cuts share a lot in common with low shelf cuts).

High shelf filters effectively boost or cut (raise or lower the amplitude) of frequencies above a defined cutoff frequency by a defined amount. The resulting amplitude-frequency graph, as we’ll see, resembles a shelf in the high-end, hence the name “high shelf filter”.

Generally speaking, there will be a smooth transition band of frequencies between the original amplitude and the relative amplitude. This band will be defined between a low “cutoff” frequency and a high “cutoff” frequency (we’ll call these points f₁ and f₂ throughout this article).

Note that f₁ will mark a 3 dB increase or decrease from the lower frequency amplitude plateau (depending on if the high shelf boosts or cuts, respectively). If the overall cut/boost is less than 6 dB, the amplitude difference between f₁ and the higher-end amplitude will be equal to the difference between the max cut/boost and f₂.

Similarly, f₂ will mark a 3 dB increase from a high shelf cut’s maximum attenuation or a 3 dB decrease from a high shelf boost’s maximum amplification as long as the overall cut or boost is greater than 6 dB. If the overall cut/boost is less than 6 dB, the difference between the max cut/boost and f₁ will be some amount smaller than 3 dB.

This is easier to visualize than to explain with words, so let’s look at some graphs to help with our understanding:

High shelf cut with 6 dB attenuation or more:

High shelf cut with less than 6 dB attenuation:

High shelf boost with 6 dB attenuation or more:

High shelf boost with less than 6 dB attenuation:

High shelf filters can effectively increase or decrease the energy of the low-end without completely eliminating frequency content from the signal.

If the above graphs look familiar, it's because they can also describe low shelf filters. In fact, if we look at basic parameters, we can effectively liken low shelving filter to high shelving filters in the following fashion:

• High shelf cuts are similar to low shelf boosts.
• High shelf boosts are similar to low shelf cuts.

High Shelf Boost Vs. High Shelf Cut

Speaking of boosts and cuts, let's discuss high shelf boosts and cuts.

High shelf boosts effectively increase the relative amplitude in the high-end and require gain. High shelf cuts effectively decrease the relative amplitude in the high-end and, in general, do not necessarily require gain (though they are generally designed with op-amps and gain).

Here are a few more graphs to help illustrate a high shelf cut and boost, respectively:

In either case we can visualize the “shelf” in the high-end.

A high-shelf boost looks similar to a high-pass filter except the facts that:

1. The high-end is amplified.
2. The amplitude levels out below f₁ rather than continuing to roll-off.

If we take phase out of the equation (or utilize linear phase EQ), a high shelf boost can be thought of as the summing of a dry (unfiltered) signal and that same signal affected by a high-pass filter.

Similarly, a high shelf cut acts similarly to a low-pass filter except for the fact that the low shelf’s attenuation will level out above f₂ while the LPF’s roll-off continues indefinitely.

If we take phase out of the equation (or utilize linear phase EQ), a high shelf cut can be thought of as the summing of a dry (unfiltered) signal and that same signal affected by a low-pass filter and then bringing down the overall amplitude to match the original signal’s amplitude below f₁.

The Ideal High Shelf Filter

As we've discussed, in an ideal world, a shelving filter would have a sharp “shelf” in its frequency-amplitude output graph. In other words, there would be no transition period and the aforementioned f₁ and f₂ would be equal.

The ideal high shelf boost would look as follows:

The ideal high shelf cut would look as follows:

These ideal shelves are unobtainable by typical filter design. Some digital EQs can closely approximate these kinds of “brickwall” filters but in general, there will be some sort of transition period.

That being said, the gentle transition that comes with a real high (or low) shelf filter will help to attenuate or amplify the high-end (or low-end) in a gentler, more sonically natural fashion.

Real-World High Shelf Filters

Real-world high shelf filters are subject to the limitations of real-world components and design.

The main point here is that, compared to ideal filters, real-world shelving filters will have a transition band between the two target amplitude levels.

Once again, let’s have a look at a typical high shelf cut followed by a high shelf boost to help visualize the transition periods of low shelf filters:

Passive high shelf filters will only be able to cut while active high shelf filters may be capable of applying gain and boosting the high-end.

High Shelf Filter Transition Band

The transition band of a high shelf filter is the band of frequencies between the aforementioned cutoff frequencies (f₁ and f₂) and is largely defined by the slope of the amplitude change. This is where the order of the filter comes into play.

The order of a filter is defined as the minimum number of reactive components a filter requires in its design. In the vast majority of audio filters, these reactive components are capacitors, though inductors may also be used in certain passive designs.

For each integer increase in order, a shelving filter will experience an additional 6 dB/octave (20 dB/decade) increase in roll-off steepness.

So the transition band, amplitude difference (whether it’s a high shelf cut or boost) and order all come into play to determine the shape of the resulting low shelf filter output.

High Shelf Filter Q Factor

Some high shelf filters (especially in parametric EQs) will have Q controls.

The Q factor is technically the inverse of the bandwidth and, therefore, is best applied to bell-type EQ, band-pass and band-stop filters. However, many other filters, like high shelves, can have controllable Q parameters.

In the case that a high shelf filter does have a Q factor, it will likely also have a centre frequency control (rather than a cutoff frequency control) that marks the mid-point of the maximum cut or boost. The higher the Q, the steeper the transition band will be.

High Shelf Filters & Phase-Shift

High shelving filters will affect the amplitude across a signal's frequency response and they will also affect the frequency-dependent phase of the signal,

In typical high-pass and low-pass filters, the amount of each reactive component in an analog filter will introduce 90º of phase shift in the signal.

However, in high-shelf filters, this full 90º potential phase-shift per reactive component isn’t reached.

High and low-pass filters simply roll-off the amplitude of the signal below the cutoff point to negative infinity. Shelving filters roll-off to a second defined amplitude. It’s this difference in design that also keeps the maximum phase-shift from reaching a full 90º per reactive component.

Here are examples of amplitude-frequency and phase-frequency graphs for both low shelf boost and low shelf cut filters:

In the above graphs, we’re dealing with first-order shelving filters (6 dB/octave slopes). The phase-shift is at a maximum (negative) at the centre frequency of the high shelf boost and at a maximum (positive) at the centre frequency of the high shelf cut. The phase-shift is not quite 90º either way.

Analog Vs. Digital Shelving Filters

Now that we understand what high shelf and low shelf filters are, let's discuss the differences between analog and digital shelving filters.

The obvious answer to this question is that analog shelving filters filter analog (continuous-time) audio signals while digital shelving filters filter digital (discrete-time) audio signals.

Analog filters are made of analog components (resistors, capacitors, operational amplifiers, etc.) while digital filters are either coded in software or embedded in digital circuits.

Analog shelving filters are easier to explain and comprehend and are discussed more thoroughly in the section Basic Low Shelf EQ/Filter Circuits. If we can understand these circuits, we will effectively understand the working principles behind shelving filters.

Note that many digital shelving filters and equalizers are designed to emulate their analog counterparts.

Analog shelving filters generally cost more to manufacture and distribute and are somewhat limited in functionality. For instance, they're sensitive to environmental factors (humidity and temperature, primarily) and the analog components used in their construction will not be absolutely ideal.

It's also true that analog filters are limited in the results they can achieve and can potentially get overly noisy and/or cost prohibitive as they b0come more and more complex.

Digital shelving filters (and EQs), with modern DSP, can be much more powerful, precise and flexible than their analog counterparts. These filters can even get close to approximating ideal filters.

A digital shelving filter will fit into one of two camps:

• Infinite Impulse Response (IIR): An IIR filter is a linear time-invarient analog type of filter (that has been digitized as well) that works with an impulse response that continues indefinitely, never becoming exactly zero. Butterworth, Chebyshev, Bessel and elliptic filters are examples of IIR filters.
• Finite Impulse Response (FIR): An FIR filter is a filter (analog or digital, though nearly always digital) that works with an impulse response of finite duration, settling to zero within some amount of time. It lends itself well to linear phase EQ.

Speaking of linear phase EQ, these specialized equalizers are worth mentioning here as well.

A linear phase EQ (which will almost certainly always have shelving filter options) effectively eliminates any phase-shift within the audio processor.

A linear phase EQ uses digital signal processing (DSP) to analyze the frequency content of a signal and apply gain to the appropriate frequencies via FIR (finite impulse response) filters in order to eliminate any phase-shifting that arises.

Active Vs. Passive Shelving Filters

Active and passive filters differ in one key way: active filters have active components that require power to function and passive filters do not.

In the case of shelving filters, these active components are nearly always operation amplifiers. The passive components are the resistors, capacitors and, in some instances, inductors.

Operational amplifiers require power to function but offer a myriad of benefits circuit, including:

• Signal amplification
• Allows for higher-order filters to be constructed without a worsening signal-to-noise ratio due to added components
• Improved output impedance for driving loads
• Improved impedance between gain stages in higher-order filters (buffering)

Note that the “active” and “passive” labels generally only apply to analog filters. Digital filters, by the nature of their design, are active (this is true of hardware, which is built with transistors and software, which requires a computation).

Passive shelving filters cannot apply any boost within the circuit itself. However, they could be used with a separate gain stage to achieve boosts. Note that low-pass shelving filters would produce high shelf boosts and high-pass shelving filters would produce low shelf boosts.

Passive filters may suffer from poorer signal-to-noise specifications and lower voltage transfer due to the potentially poor impedance bridging between the circuit and the following audio device.

Active filters utilize amplifiers. These amps can be used to boost or cut depending on the design of the circuit. Shelving EQ units will nearly all have potentiometers to control the amount of cut and/or boost in the low shelf and high shelf filters.

Basic Low Shelf & High Shelf EQ/Filter Circuits

In this section, we'll take a look at a few basic low shelf and high shelf circuits.

In particular, we'll discuss the following (note that the names do refer specifically to the type of shelf):

• Passive RC Low-Pass Shelving Filter
• Passive RC High-Pass Shelving Filter
• Non-Inverting Low-Pass Shelving Filter
• Inverting Low-Pass Shelving Filter
• Non-Inverting High-Pass Shelving Filter
• Inverting High-Pass Shelving Filter
• Basic-Adjustable-Shelving-EQ-Filter

These circuits are largely referred to as either high-pass shelving filters or low-pass shelving filters. They are similar to typical high-pass and low-pass filters, respectively, other than the fact that they level off the attenuation at a set amplitude difference (rather than rolling off to negative infinity).

Here's our basic amplitude-frequency graph for the high-pass shelf filter type:

And here's our basic amplitude-frequency graph for the low-pass shelf filter type:

I should reiterate an important note about the cutoff frequencies f₁ and f₂. When the shelving filter cuts or boosts more than 6 dB, the cutoff frequencies will be 3 dB from their respective amplitudes.

In other words, the amplitude at f₁ will be 3 dB greater than G₁ and the amplitude f₂ will be 3 dB less than G₂ in the high-pass-style shelving EQ.

Conversely, the amplitude at f₁ will be 3 dB less than G₁ and the amplitude f₂ will be 3 dB greater than G₂ in the low-pass-style shelving EQ.

However, things change when the cut or boost of the shelf is less than 6 dB. The cutoff frequencies will not exhibit a 3 dB amplitude difference from their respective extremes (so to avoid rippling in the resulting amplitude-frequency graph). Each cutoff will have a relative amplitude less than 3 dB from where it levels off but the dB values of each cutoff will still be equal. This can be calculated using the equations I'll be sharing below.

In other words, when the difference between G₁ and G₂ is less than 6 dB (2x gain), the amplitude at f₁ will be X dB greater than G₁ and the amplitude f₂ will be X dB less than G₂ in the high-pass-style shelving EQ (where X < 3 dB). Note that X will also have to be less than 1/2 Y where Y is the gain difference between G₁ and G₂ in decibels (Y < 6).

Similarly, when the difference between G₁ and G₂ is less than 6 dB (2x gain), the amplitude at f₁ will be 3 dB less than G₁ and the amplitude f₂ will be 3 dB greater than G₂ in the low-pass-style shelving EQ (where X < 3 dB). Note that X will also have to be less than 1/2 Y where Y is the gain difference between G₁ and G₂ in decibels (Y < 6).

Hopefully that's not overly confusing. What I'm basically saying is this: a shelving filter cutoff frequency will be 3 dB below or above where the filter flattens out so long as the filter causes more than 6 dB of amplitude difference between the main band and the shelved band.

With that confusing but important part out of the way, let's have a look at some shelving circuits:

Passive RC Low-Pass Shelving Filter

Beginning with the most basic pass RC shelving filter, let's have a look at the low-pass shelving cut. This filter can be used as a high shelf cut even as a low shelf boost if we include a gain stage after the filter.

The passive RC low-pass shelf circuit look like this:

Note that it's a first-order circuit as there is only one reactive component (the capacitor) and so the slope of the roll-off will be 6 dB per octave (20 dB per decade).

This circuit can be likened to a simple DC voltage divider with a capacitor added to the circuit or, alternatively, to a low-pass filter with an additional resistor (R₂).

To further our understanding of this basic low-pass shelving circuit, let's actually compare it to these two circuits, starting with the voltage divider.

In the above schematic, we have the following equation:

V_{\text{out}} = V_{\text{in}} \cdot \frac{R_2}{R_1 + R_2}

As R₂ decreases, V_out decreases (assuming R₁ remains constant).

We can effectively translate this DC voltage divider to AC (for audio signals) if we take impedance (resistance and reactance) into account rather than just resistance. Impedance can be thought of as “AC resistance”.

Now if we move our sights to a basic low-pass filter, we can use the following equation in place of the voltage divider equation:

V_{\text{out}} = V_{\text{in}} \cdot \frac{X_c}{\sqrt{R^2 + X_c^2}}

Where X_C is the capacitive reactance of the capacitor.

In the cases of the low-pass filter, we have a situation where, as X_C decreases, V_out decreases (assuming R remains constant).

The capacitive reactance of the capacitor decreases as the frequency of the input voltage/signal increases. Therefore, as the frequency increases, V_out decreases and we have a low-pass filter.

The cut-off frequency of this passive RC low-pass circuit (at -3 dB) is calculated as:

f_L = \frac{1}{2\pi RC}

With that primer, let's look at the passive RC low-pass shelving filter once again:

Combining what we've learned about voltage dividers and low-pass filters with Laplace transform mathematics (beyond the scope of this article but here's a link to the subject on Wikipedia), we can derive our equations of the low-pass shelving filter.

Simply put, the Laplace transform effectively allows us to add the complex impedance of components more linearly (rather than having to add resistance and reactance together, which are not technically the same).

Using the Laplace transform, the complex impedance of a resistor is simply its resistance (since it doesn't have reactance). The complex impedance of a capacitor is defined as 1/sC where C is the capacitance and s is a complex variable of the s-plane in which the Laplace transforms are graphed.

Again, the proofs and theory of these mathematics are beyond the scope of this article. However, we'll use their results in order to have nice, clean equations.

Let's first have a look at a basic graph that will lay out the parameters in a visual fashion and then I'll have the equations listed:

f_1 = \frac{1}{2\pi C_1 (R_1 + R_2)}

f_2 = \frac{1}{2\pi R_2 C_1}

\frac{V_{\text{out}}}{V_{\text{in}}} = \frac{1 + sC_2R_2}{1 + sC_2 (R_1 + R_2)}

\alpha = 1 + \frac{R_1}{R_2} = \frac{f_2}{f_1} = 10^{\frac{A}{20}}

A = 20 \log(\alpha)

Now if we would plug in real numbers, we'd see that the transfer function (V_out/V_in) will effectively level out below f1 and above f2. The greatest change in voltage transfer will have between f1 and f2 with the maximum slope happening at the centre frequency, which is defined as follows:

f_C = \sqrt{f_1 \cdot f_2}

This circuit is a first-order circuit (it only have one reactive component) and so its maximum slope is 6 dB/octave (20 dB/decade). Of course, the cutoff frequencies are subject to a 3 dB difference from the respective levelled amplitude (unless the overall amplitude difference is under 6 dB).

Here's a simple graph to explain the typical 3 dB difference. Replace 3 dB with X dB (where X < 3) when the total relative amplitude difference is less than 6 dB).

Passive RC High-Pass Shelving Filter

Now that we've seen the basic low-pass shelving filter, let's have a look at the basic passive RC high-pass shelving filter:

This circuit can be likened to a simple DC voltage divider with a capacitor added to the circuit or, alternatively, to a high-pass filter with an additional resistor (R₁) in parallel with the capacitor.

Let's have another quick look at the basic DC voltage divider circuit and transfer function:

In the above schematic, we have the following equation:

V_{\text{out}} = V_{\text{in}} \cdot \frac{R_2}{R_1 + R_2}

As R₂ decreases, V_out decreases (assuming R₁ remains constant).

As we did before, we'll translate this theory to a high-pass filter. Note that, when dealing with DC, we're only concerned with resistance. When dealing with AC, we're concerned with impedance (which is made of resistance and reactance). Resistors have resistance and capacitors have capacitive reactance that varies with frequency.

Here's a diagram of a basic RC high-pass filter circuit followed by its transfer function:

V_{\text{out}} = V_{\text{in}} \cdot \frac{R}{\sqrt{R^2 + X_c^2}}

Where X_C is the capacitive reactance of the capacitor.

In the cases of the high-pass filter, we have a situation where, as X_C increases, V_out decreases (assuming R remains constant).

The capacitive reactance of the capacitor increases as the frequency of the input voltage/signal decreases. Therefore, as the frequency decreases, V_out decreases and we have a high-pass filter.

The cut-off frequency of this passive RC high-pass circuit (at -3 dB) is calculated as:

f_H = \frac{1}{2\pi RC}

And with that, let's return to our simplified schematic of a passive high-pass shelving filter:

Things are a bit more complicated since we have R₁ and C₁ in parallel. As was the case with the passive low-pass shelving filter, the equations of this circuit will be subject to the Laplace Transform for simplification/

Let's begin with a basic graph showing the parameters in a visual fashion and then I'll have the equations listed:

f_1 = \frac{1}{2\pi R_1 C_1}

f_2 = \frac{1}{2\pi C_1 \left(\frac{R_1 R_2}{R_1 + R_2}\right)}

\frac{V_{\text{out}}}{V_{\text{in}}} = \frac{1 + sC_2R_2}{1 + sC_2 (R_1 + R_2)}

\alpha = 1 + \frac{R_1}{R_2} = \frac{f_2}{f_1} = 10^{\frac{A}{20}}

\frac{V_{\text{out}}}{V_{\text{in}}} = \frac{R_2}{R_1 + R_2} \cdot \frac{1 + sC_1R_1}{1 + sC_1\left(\frac{R_1R_2}{R_1 + R_2}\right)}

A = 20 \log(\alpha)

If we would plug in real numbers, we'd see that the transfer function (V_out/V_in) will effectively level out below f1 and above f2. The greatest change in voltage transfer will have between f1 and f2 with the maximum slope happening at the centre frequency, which is defined as follows:

f_C = \sqrt{f_1 \cdot f_2}

This circuit is a first-order circuit (it only have one reactive component) and so its maximum slope is 6 dB/octave (20 dB/decade). Of course, the cutoff frequencies are subject to a 3 dB difference from the respective levelled amplitude (unless the overall amplitude difference is under 6 dB).

Here's a simple graph to explain the typical 3 dB difference. Replace 3 dB with X dB (where X < 3) when the total relative amplitude difference is less than 6 dB).

Non-Inverting Low-Pass Shelving Filter

Now that we've discussed the basic passive shelving filters, let's turn our sights on the active filters. Again, it's important to remember that this is not an exhaustive list. I've only included a few circuits to help deepen our understanding.

In this section, we'll have a look at a non-inverting low-pass shelving EQ filter (it can be used as a low shelf boost or, perhaps, a high shelf cut):

This active circuit is complete with a non-inverting op-amp with a negative feedback loop. With this circuit, we can actually apply gain to the circuit and achieve boost.

This particular filter is beholden to the following graph and equations:

G_1 = 1 + \frac{R_2}{R_1}

G_2 = 1

Therefore, G₂, which is lower on the graph, is less than G₁.

f_1 = \frac{1}{2\pi R_2 C_1}

f_2 = \frac{1}{2\pi C_1 \left(\frac{R_1 R_2}{R_1 + R_2}\right)}

Therefore f₂ must be higher than f₁, as seen in the graph.

G_1 = \frac{f_2}{f_1}

This tells us that what we'd expect of a first-order filter. The filter has a fixed slope of 6 dB/octave. Therefore, the gain (relative amplitude difference) is directly proportionate to the “transition band” between f₁ and f₂.

A = \frac{V_{\text{out}}}{V_{\text{in}}} = G_1 \sqrt{\frac{1 + \left(\frac{f}{f_2}\right)^2}{1 + \left(\frac{f}{f_1}\right)^2}}

This equation relates the transfer function to a given frequency. Plugging in numbers, we'd see how A tends to level out below f₁ and above f₂.

Notice how, at no point, the equations mention the “3 dB points” of f₁ and f₂. Again, if the gain is greater than 6 dB (G₂ > 2), then this would be the case. Otherwise, the cutoff frequencies would be at less than 3 dB a piece.

Inverting Low-Pass Shelving Filter

Now let's have a look at an inverting low-pass shelving EQ filter (it can be used as a low shelf boost or, perhaps, a high shelf cut):

This active circuit is complete with an inverting op-amp with a feedback loop. With this circuit, we can actually apply gain to the circuit and achieve boost.

This specific filter is governed by the following graph and equations:

G_1 = \frac{R_2}{R_1}

G_2 = \frac{1}{R_1}•\frac{R_2R_3}{R_2 + R_3}

In this case, G₁ must be greater than G₂, as can be seen on the graph.

f_1 = \frac{1}{2πC_1(R_2+R_3)}

f_2 = \frac{1}{2πC_1R_3}

Since R₂ cannot be negative, this tells us that f₂ must be greater than f₁.

\frac{G_1}{G_2} = \frac{f_2}{f_1}

Again, this first-order filter is defined as have a 6 dB/octave (20 dB/decade) slope. These is a direct correlation, then, between the relative gain ratio and the relative cutoff frequency ratio.

A = \frac{V_{\text{out}}}{V_{\text{in}}} = G_1 \sqrt{\frac{1 + \left(\frac{f}{f_2}\right)^2}{1 + \left(\frac{f}{f_1}\right)^2}}

This equation just shows us the transfer function for a given frequency. Plugging in numbers, we'd see how A tends to level out below f₁ and above f₂.

Once again, at no point do we mention any “3 dB points” of f₁ and f₂. If the gain is greater than 6 dB (G₁ > 2G₂), then this would be the case. Otherwise, the cutoff frequencies would be at less than 3 dB a piece.

Non-Inverting High-Pass Shelving Filter

In this section, we'll have a look at a non-inverting high-pass shelving EQ filter (it can be used as a high shelf boost or, perhaps, a low shelf cut):

This active circuit is complete with a non-inverting op-amp with a negative feedback loop. With this circuit, we can actually apply gain to the circuit and achieve boost.

This exact circuit will yield the following graph and equations:

G_1 = 1

G_2 = 1 + \frac{R_2}{R_1}

G_2 = \frac{f_2}{f_1}

By these equations, we can tell that G₂ will be greater than G₁. We can also infer that the gain ratio is directly proportional to the cutoff frequency ratio, meaning that the transition slope between G₁/f₁ and G₂/f₂ is fixed (in the case of first-order filters, it's 6 dB/octave or 20 dB/decade).

f_1 = \frac{1}{2πC_1(R_1+R_2)}

f_2 = \frac{1}{2πC_1R_1}

These equations show us that f₁ must be less than f₂ (resistance can't be negative), as is seen in the graph above.

A = \frac{V_{\text{out}}}{V_{\text{in}}} = G_1 \sqrt{\frac{1 + \left(\frac{f}{f_1}\right)^2}{1 + \left(\frac{f}{f_2}\right)^2}}

Here we seethe transfer function according to the frequency. If we were to plug in numbers, we'd see how A tends to level out below f₁ and above f₂.

It bears repeating that there's no mention of “3 dB points” of f₁ and f₂. Again, if the gain is greater than 6 dB (G₂ > 2), then this would be the case. Otherwise, the cutoff frequencies would be at less than 3 dB a piece.

Inverting High-Pass Shelving Filter

Now let's have a quick look at an inverting high-pass shelving EQ filter (it can be used as a high shelf boost or, perhaps, a low shelf cut):

This active circuit contains an inverting op-amp with a feedback loop. We can apply gain to the circuit and achieve boost.

This defined filter can be understood with the following graph and equations:

G_1 = \frac{R_2}{R_1}

G_2 = R_2\frac{R_1+R_3}{R_1R_3}

So we have a situation where G₂ is greater than G₁.

f_1 = \frac{1}{2πC_1(R_1+R_3)}

f_2 = \frac{1}{2πC_1R_3}

These formulae tell us that f₂ is higher than f₁.

\frac{G_2}{G_1} = \frac{f_2}{f_1}

As expected, the gain ratio is directly proportional to the frequency ratio since the roll-off slope is fixed (with first-order filters like this, it's 6 dB/octave or 20 dB/decade). This expectation is confirmed with the above equation.

A = \frac{V_{\text{out}}}{V_{\text{in}}} = G_1 \sqrt{\frac{1 + \left(\frac{f}{f_1}\right)^2}{1 + \left(\frac{f}{f_2}\right)^2}}

Finally, we have our transfer function with frequency as a variable. Using this formula, we can calculate how A tends to level out below f₁ and above f₂.

Once again, at no point do we mention any “3 dB points” of f₁ and f₂. If the gain is greater than 6 dB (G₁ > 2G₂), then this would be the case. Otherwise, the cutoff frequencies would be at less than 3 dB a piece.

Basic Adjustable Shelving EQ Filter

Here's an example of a variable shelving filter with both low-pass shelving and high-pass shelving capabilities.

In this example, we have two variable resistors (potentiometers), labelled as R_bass (which controls the low-shelf/high-pass-shelving side of the circuit) and R_treble (which controls the high-shelf/low-pass-shelving side of the circuit).

At high frequencies, R_bass gets shorted out by C₁, allowing high frequencies to pass unaffected. Adjusting the resistance of R_bass will adjust the effective cutoff frequency of the bass and cause frequencies below the cutoff to be affected by cut or boost.

At higher frequencies, the reactance (impedance) of C₂ is also lower, which ultimately provides more gain (or cut) to the R_treble potentiometer.

Mixing With Low Shelf Filters

Now that we know what low shelf filters are and how they work, let's look at the ways in which they're used in mixing audio.

Low shelf filters are used for mixing in the following ways:

• Correct The Response Of A Microphone
• Filtering Out Low-End Rumble
• Accentuating Characteristic Frequencies

Correct The Response Of A Microphone

Microphones are transducers that convert mechanical wave energy (sound waves) into electrical energy (analog audio signals). Due to design constraints, many microphones will have frequency responses that do not perfectly represent the physical sound waves in their output audio signals.

This is to be expected. Even our ears and sense of hearing are more sensitive to some frequencies than others.

That being said, some microphones can either over-accentuate the low-end or under-emphasize these low-end frequencies. A low shelf filter can help to either attenuate the low-end or boost the low-end to help get a more accurate response from a microphone signal in less-than-ideal situations.

Of course, we want to get it right at the source. However, a low shelf filter can be an easy way to improve the overall EQ/frequency response of a microphone signal by refining the low-end.

Filtering Out Low-End Rumble

When mixing, engineers will typically reach for a high-pass filter when the need to eliminate low-end rumble arises. However, when using a high-pass filter, we virtually eliminate everything below a cutoff point (ideally, though there will be a transition band).

Sometimes we have low-end rumble that need to be attenuated but there's also some important information in the low-end. In these situations, it may be best to at least try a low shelf cut before going straight for the HPF.

The low shelf cut can potentially attenuate the low-end noise without completely eliminating the low-end information of the signal.

Accentuating Characteristic Frequencies

Other times there are low-end frequencies that require accentuating. Think of bass instruments like kick drums, bass guitar and tuba.

Simply boosting the low shelf filter below a defined cutoff frequency can work, though it may enhance some frequencies that will compete with other bass instruments and have a negative impact on the overall mix.

Perhaps a better idea is to use a low shelf filter cut with a resonance peak at the cutoff frequency that will effectively boost the frequencies near the cutoff while attenuating everthing below. That way we can accentuate the characteristic frequencies of a track (the fundamental “thump” of a kick drum, for example) and have the added benefit of reducing the low-end below that is largely noise, anyway.

Mixing With High Shelf Filters

Now that we've discussed mixing with low shelf filters, let's look at the ways in which high shelf filters are used in mixing audio.

High shelf filters are used for mixing in the following ways:

• Correct The Response Of A Microphone
• Adjusting Perceived Depth
• Cutting Problem Frequencies
• Accentuating Characteristic Frequencies

Correct The Response Of A Microphone

Some microphones can either over-accentuate the high-end or under-emphasize these high-end frequencies. A high shelf filter can help to either attenuate the high-end or boost the high-end to help get a more accurate response from a microphone signal in less-than-ideal situations.

Of course, we want to get it right at the source. However, a high shelf filter can be an easy way to improve the overall EQ/frequency response of a microphone signal by refining the high-end.

Adjusting Perceived Depth

High shelf filters can help to improve the important dimension of perceived depth in a mix. By cutting the high-end with a shelving filter, we can move a track toward the back of the mix. Conversely, by boosting the high-end with a shelving filter, we can bring a track toward the front of the mix.

In the real world of acoustics, increasing the distance between a sound source and the listener will cause a few things to happen. I'll add the audio effects that help to mimic this psychoacoustic perceived depth in brackets:

• The sound will be quieter (volume/gain).
• The sound will arrive at the listener's ears later (delay).
• The sound will likely reflect off other surfaces in the acoustic space and reach the listener's ears at varying times (delay and reverb).
• The sound will be less focused (modulation such as chorus).
• The sound will have less high-end as the higher frequency sound waves lose energy first due to the friction of the medium/air (high shelf cut).

Cutting Problem Frequencies

Individual tracks; effects sends; busses, or even entire mixes can sound overly bright and harsh in the high-end. A high shelf filter set to attenuate the high-end can help to alleviate this harshness.

As mentioned, high shelf filters can reduce problem frequencies in everything from a solo track to a master.

Accentuating Characteristic Frequencies

Sometimes we have some high-end information that requires accentuating. This could simply be the non-musical “airiness” or “brilliance” of a signal or even the upper harmonic content of a track.

Simply boosting the high shelf filter above a defined cutoff frequency can work. However, sometimes this brings us to the situation we had before where the high-end in the mix becomes overly harsh.

So rather than boosting everything above a certain point like a typical high shelf filter would, it may be best to choose a high shelf that will cut the high frequencies while offering a resonant peak near the cutoff frequency.

This way, we can accentuate some important harmonic information in the upper mid-range while reducing the brightness of the overall track.

Low Shelf & High Shelf Filters In Audio Tone Controls

I suppose this section ties into mixing but is different enough that I'll give it its own section.

Many audio devices will have a basic equalizer/EQ typically referred to as “tone control”. Tone controls will generally have a bass control, treble control and sometimes a mid control or two.

Think of guitar/bass amplifiers; home theatre amplifiers/receivers; audio playback applications, etc.

The bass controls in such units are based on low shelf filters. These filters often have a fixed bandwidth/cutoff frequency and simply boost the low-end as the bass control is turned up and cut the low-end as the bass control is turned down.

The treble controls in such units are based on high shelf filters. These filters often have a fixed bandwidth/cutoff frequency and simply boost the high-end as the treble control is turned up and cut the high-end as the treble control is turned down.

Related Questions

What is audio equalization? EQ is the process of adjusting the balance between frequencies within an audio signal. This process increases or decreases the relative amplitudes of some frequency bands compared to other bands with filters, boosts and cuts. EQ is used in mixing, tone shaping, crossovers, feedback control and more.