Equalizers are probably some of the simplest tools for audio and music production, at least at the surface. Their purpose is clear, and the process of boosting or cutting certain frequency ranges is a comparably clean and easily understandable one. Yet there are vast differences among different equalizer models in both sound and usability. But what exactly are these?

Talking About Equalizers

When you observe discussions about audio gear, you might notice that most of the time very abstract and subjective vocabulary is used. A good equalizer is “musical”, it “sounds open” and it adds the right kind of “shimmer” or “air”. Or some are said to be great for vocals, while others are only suitable for drums. It’s pretty much the same for compressors, microphones and other devices.

It’s fascinating how notoriously imprecise these descriptions are. In contrast, an equalizer is rarely described in terms of its filter shapes, Q factors, parameter ranges or specific nonlinear distortion. The specific requirements for such devices are extremely diffuse, they just have to sound good. The why and how is rarely ever discussed.

There is an interesting article by the legendary Michael Gerzon from 1990, which has the same title as this post. It’s well worth a read, although most of the phenomena he describes don’t really apply to what we typically refer to as equalizers. The article is mostly about surprisingly well audible coloration that is hardly visible at all in typical frequency response measurements. A very valid concern. But interestingly, most of the time we don’t even talk about those sonic features that are very well visible in such measurements.

So let’s start a bit easier and walk through the most important design decisions to make when putting together an equalizer. I’ll divide the topic into two parts. First, I’ll deal with developing an equalizer, which means defining its specific function, parametrization and ideal desired behavior. The second step is the practical implementation of this desired behavior using either analog electronics or digital signal processing. These implementation methods each bring their own special needs to the table, which can strongly influence the outcome compared to the ideal design.

A Glimpse Into Signal Theory

An equalizer is basically a set of separate filters, which change the frequency content of a signal in a specific way. If you have ever used one, you probably know quite well what that means, so I won’t go that much back into the very basics here.

As long as there is no nonlinear distortion or compression present (= the filter is linear), and as long as the parameters don’t change (= the filter is time-invariant), the filter can be conveniently described by means of LTI system theory (LTI = linear and time-invariant).

At the very core, a filter is just a set of equations that govern the relationship between an input and an output signal. If the filter happens to be linear and time-invariant, these equations can be greatly simplified, and we get a nice set of mathematical tools to describe the behavior of these filters. This also enables us to calculate the frequency response of the filter, which tells us which frequency ranges are amplified or damped how much, and also what happens to phase in the process.

These two aspects – the magnitude and phase responses – completely define the behavior of a filter.

How To Build A Filter

In order to implement a non-trivial frequency response, the filter has to have some information about the past of the signal. That means the mathematical equations need to contain at least one element of “memory”. In digital signal processing this is simply a chunk of memory containing one or more past input and/or output samples. In analog signal processing, the typical memory element is an integrator, which essentially accumulates signal energy and releases it back “into the wild” after some time. We’ll see some real-world examples of what such elements can look like later.

Anyway, the number of such elements present in the filter determines the filter order. Simply speaking, the higher the filter order, the more complex the frequency response can be.

But engineers actually hate complexity, so usually they find a way to tame complexity by breaking it down into separate, less complex parts. It’s possible to do the same with complex high-order filters by splitting them up into several first and second order filters. And that’s what practically all typical equalizer filters consist of: a couple of individual filters with order one or two. Usually, we are dealing with second order filters.

Equalizer Filter Design Decisions

So let’s look into what design decisions an engineer faces when building a typical equalizer.

Filter Order And Shape

Of course, the first decision to make is the filter shape and filter order. A first order filter will only yield very simple shapes: the well known lowpass, highpass and shelving filters. In fact, lowpass and highpass filters are special versions of a shelving filter. Look at it this way: with a first order filter, you can achieve different gains for high and low frequencies, with a slope between those two ranges. With highpass or lowpass filters, the gain of either low or high frequencies is simply zero. The steepness of the slope is limited to 6dB gain difference per octave.

Consequently, with a second order filter you can get two such slopes. For a second order shelving, lowpass or highpass filter, you get a 12 dB/oct. slope. Or you combine two opposite slopes to get bandpass or peak/notch filters.

By adding more such slopes (thus increasing filter order) it is pretty much possible to create any kind of frequency response. Also note that apart from the “slope stuff”, second order filters can resonate, which first order filters can’t. That means that the output of such resonant filters can contain a ringing tone with an exponentially decaying amplitude. This is equivalent to a sharp peak in the frequency response. The sharpness of such a peak is determined by the Q factor, which many equalizers allow the user to choose.

Parametrization

Another important decision is the parametrization of equalizer filters. Is the filter shape fixed or flexible? Can the user control the Q factor? What’s the gain range in which the user can influence shelving and peak filters? What’s the frequency range?

An important thing to note is that the less parameters are actually available for the user to control, the more decisions about the usage are made a priori by the developer. This can be either a good thing or a bad thing, depending on what you want to achieve. For example an EQ with pre-set frequencies and Q factors suitable for typical vocal treatment is probably much easier to use, as fewer decisions need to be made. On the other hand, it would be far more flexible and universal if every parameter could be individually chosen.

Q Factor Treatment

The Q factor is a very special parameter. In many implementations, it is fixed. Being a critical factor that influences the coloration of sound, the way the Q factor is defined is critical to the individual character of an equalizer filter.

Even if the Q factor is adjustable by the user, there are still some options in how to handle the setting in the end. Typically, the Q factor needs to be adjusted internally for peak filters with negative gain to get a symmetric response in case of positive vs. negative gain. Even more, some equalizers have their Q factor increase with peak filter gain, a feature called “proportional Q”. Obviously there is a lot to say about this topic alone.

Type of Phase Response

There are different types of phase responses that equalizers can have. Most equalizers will have a minimum phase response, and I would personally recommend using minimum phase filters (almost) always. These filters have the minimum latency between input and output, but it’s not constant over all frequencies.

Linear phase filters have a constant latency for all frequencies at the cost of pre-ringing. This can seriously affect the perception of transient sounds.

User Interface

This last one is maybe easiest overlooked. I already described some factors above like parameter ranges and proportional Q. But there are even more ways in which the user interface can have a huge influence on the subjective quality of an equalizer.

Parameter ranges are one thing, but another question is how the parameter reacts to knob or fader positions between the extremes. There’s a lot of variety there especially with analog equalizers.

Sometimes the parameters can’t even be continuously controlled, but only in fixed steps. In this case, fine resolution is traded for better recall of precise settings, which comes in handy in mastering applications.

Another factor is visual feedback. A graphical display of the equalizer curve can be a nice aid, but also distract from the actual sound. The same goes for numerical display, which might make the user gravitate towards nice-looking values.

You see that already the very basic design requires numerous decisions. They all determine the fitness of a design for a given purpose. It gets even worse when limitations and special properties of the chosen implementation method chime in. Next we’ll look at how limitations of analog electronics and digital signal processing influence the result so let’s get serious and look at how decisions regarding their real-life implementation affect the outcome.

Building An Equaliser

Talking about specific qualities and non-qualities of equipment all too often ends up in a mess of abstract buzzwords. The problem with that is that these buzzwords like “open”, “musical” or “tight” have absolutely no properly agreed-upon meaning. Somehow it seems like talking about audio is quite hard.

On the other hand, there’s a widespread skepticism against the use of more technical, clearly defined properties such as frequency response, harmonic distortion and so on. The common belief is that these technical measurements cannot fully describe the sound of a piece of equipment.

That’s true. But isn’t it better than nothing? I leave that thought with you for a moment.

Anyway let’s not get derailed too much right in the beginning. Where were we?

We were looking at some of the decisions that a developer faces when building even a simple equalizer. Or more precisely, even before he or she can start to actually build it.

These basic decisions about the general concept and user interface matter infinitely more than the choice of exotic premium capacitor or the wordlength used in a DSP algorithm. They often make a world of a difference that is immediately noticed in practical use. And it’s easily measurable as well.

But the choice of technology to implement these concepts also has a huge impact on the result. The reason is that each technology brings its own limitations and constraints to the table. So let’s look at what these are for electronics and digital signal processing.

Analog Equalizers

Let’s presume we’ve put together a nice concept for an equalizer and choose to implement it using electronic components. As we learned last week, to create an analog filter we need components that have a “memory”, a knowledge about the recent past of the signal we are processing.

In electronics, we can get such behavior using capacitors and/or inductors. A capacitor stores energy in an electrical field while an inductor stores it in a magnetic field. And of course, they release this energy back into the wild at some point in time. The third basic component we need is a resistor, which cannot store, but dissipate energy and convert it into heat. We can use these energy-storing and energy-dissipating properties to implement the previously defined mathematical description of a filter by connecting them in certain ways.

I’ll leave out the boring details of different circuit structures for now. The important thing to understand is that by connecting some basic elements in certain ways, we are actually implementing a set of mathematical equations. In essence, we let these components compute stuff for us.

Passive Filters

If we only use these three basic components, we are a bit limited in what we can do, because we don’t have a component yet that is able to boost energy. We can only redistribute it in time or get rid of it in the form of heat. That’s called a passive filter.

In the early days of audio electronics, this was the common way to build an equalizer, since active components were still quite expensive and impractical to use in larger numbers. The good old Pultec equalizer is a standard example. And also the tonestacks you’ll find in guitar amplifiers are almost exclusively passive.

The fact that passive electronics can only take energy away from the signal results in an inevitable level loss when going through such a circuit. Usually, an amplifier stage after the passive filter circuit makes up for this level loss. But since the filtering itself happens without any active component, we still call it passive.

Due to all these limitations, the possibilities with passive filters are really quite limited. The biggest issue is that without active components right inside the filter circuit, it’s impossible to decouple different parts of the circuit from each other. Consequently, changing a component value (for example using a potentiometer) will have more effect than we’d like.

For example, adjusting the level of the treble band might not only change the level of high frequencies, but also the corner frequency of the filter. Or the mid band can behave differently depending on the setting of the low band.

This rather touchy behavior of passive equalizers is indeed quite common and makes much of the character of these old devices. They never quite behave as the labelling of the knobs suggests. For example, on most guitar amplifier tonestacks, you get the flattest possible response if you set the mid band to 12 o’clock and both bass and treble bands to their lowest setting.

Active Filters

With the advent of transistors and especially integrated circuits, it became much easier and cheaper to build filter circuits closer to spec. Almost all the hardware equalizers you find anywhere are active. With these, you don’t have these nasty interactions and side effects, and the treble band is really independent from the mid band and so on. You can expect the actual behavior of the unit to be much closer to the labelling printed on the front panel.

Apart from that, however, an important difference between passive and active technology is their distortion behavior. While the makeup amplifier in a passive equalizer applies its distortion characteristics to the output signal as a whole, the case is much more complicated for active circuits. As the active components sit right inside the actual filter implementation, the distortion might be applied to only part of the signal. Sometimes even exactly the part of the signal that is filtered out. As a result, the THD of – say – a lowpass filter can be much higher above the cutoff frequency while it’s rather clean below the cutoff.

Other Issues

Generally, electronic circuits often require compromises due to limited capabilities of components. Especially variation in component values is a big issue that can make it hard to exactly match a specification. Temperature variation and aging are issues as well.

As a result, it’s well possible that one unit of the same equalizer model sounds dramatically different from another one at similar settings. However, this can often be alleviated by readjusting the settings to better match.

However, what these component variations do not change is the general behavior of the box. That means, how it reacts to knob adjustments and how it treats certain signals. And that’s what – for me – constitutes the real character of an audio device.

Digital Equalizers

With computers, we don’t have any component variations. Everything is nice and predictable. And after all, computers are for computing. And what we want to do is to have something compute some equations for us. Perfect, right?

Yes and no. It is true that we can use digital technology to implement filters much more precisely. And much more predictably. But we have different limitations here that lead to totally different problems and solutions.

First of all, there’s one thing we can do with electronics but we can’t with computers, and that’s doing math with continuous signals. We need a set of individual numbers to type into the pocket calculator. Obviously, the solution is sampling.

The bad news is: there’s no way to exactly transfer an analog filter design into digital domain without introducing problems.

There are mainly two issues with digital filters in comparison to analog ones: frequency warping or ‘cramping’ and rounding errors. The former leads to the frequency response being squeezed towards higher frequencies. The latter leads to low-level noisy artefacts especially at low frequencies.

As you can see we’re still only barely scratching the surface. There’s a great deal to equalisers and hopefully this deep dive into the design of these tools has been useful to you.