Audio EQ: What Is A Band-Pass Filter & How Do BPFs Work?

When studying and practicing music production or audio engineering, you will certainly come across band-pass filters. Band-pass filters are powerful tools that are used in equalization and in general audio design.

In this article, we'll have a detailed look at band-pass filters, discussing how they work, how they're designed and how they're used. We'll then study how they're used in EQ and mixing along with any other BPF applications in audio.

The study of electronic filters is rather dense. In this article, we'll get into some theory to help us understand band-pass filters. However, this is by no means meant to be a complete study of band-pass filter. Rather, it's a guide to understanding and using BPFs in the context of audio mixing and production. It's a rather long article so please use the table of contents to skip around.

What Is A Band-Pass Filter?

In the introductory paragraphs I explained, very briefly, what a band-pass filter is. Of course, there's a lot more to know about these filters and how they're used in the world on audio.

We know that a band-pass filter will effectively pass a specified band of frequencies and filter out frequencies below and above this defined band. In this way, band-pass filters can be thought of as a combination of a high-pass filter (which passes high frequencies and attenuates low frequencies) and a low-pass filter (which passes low frequencies and attenuates high frequencies).

So long as the low-pass cutoff frequency is higher than the high-pass cutoff frequency, the result will effectively be a band-pass filter. This is important to keep in mind as we go through this article.

The Ideal Band-Pass Filter

Science and mathematics love to deal with the “ideal world”. In this ideal world, a band-pass filter would completely remove all frequencies below its low cutoff point and above its high cutoff point and pass all frequencies within its passband (between the low and high cutoff points).

This “brickwall” type of BPF is a theoretical idealization and can only be approached in practice. Note that, with digital signal processing and the power of computers, we can get awfully close.

An ideal band-pass filter would look something like this:

In the graph above, we have frequency (in Hertz) on the x-axis and relative amplitude (in decibels) on the y-axis.

Frequency is measured in Hertz (Hz), which refer to cycles per second. Audio signals are AC signals and, therefore, have frequency components. When audio signals are converted to sound waves, we are able to hear the information of the audio signal. Since the human hearing range is universally accepted to be 20 Hz – 20,000 Hz, most audio signals will have information within this range (so as to avoid having an abundance of imperceivable information).

Relative amplitude is measured in decibels (dB), which express the ratio of one quantity to another on a logarithmic scale. When it comes to audio signal amplitude, a 3 dB difference will be a doubling/halving of power quantities (power and ultimately sound intensity) while a 6 dB difference will be a doubling/halving of root power quantities (voltage/current and ultimately sound pressure level).

The graph of the ideal band-pass filter above shows a sharp low cutoff frequency (f_L) at 100 Hz and a sharp high cutoff frequency (f_H) at 2,000 Hz. No frequencies below f_L or above f_H will be present in the output of such an ideal BPF while frequencies between f_L and f_H will be perfectly represented.

Note that this filter also has a centre frequency (f_C) that can be calculated using the two cutoff frequencies with the following equation:

f_C = \sqrt{f_Lf_H}

Though impossible to achieve by analog means, there are ways to approximate this type of band-pass filter.

In analog BPFs, increasing the filter order will move us closer to the steepness of an ideal filter around the cutoff frequencies. In digital BPFs, we can code plugins to pretty much have the same effect as an ideal band-pass filter.

More on analog and digital band-pass filters later.

Real World Band-Pass Filters

So we can get pretty close to an ideal/brickwall band-pass filter with modern technology. However, generally speaking (and always with analog BPFs), we'll have some sort of attenuation roll-off below the low cutoff and above the high cutoff. This is different than the sharp cutoffs present in the ideal BPF.

A typical band-pass filter, then, can be easily visualized in the following EQ graph:

We can see the roll-off in this image. Beyond the passband, the filter will begin to attenuate the frequencies rather steadily. The low-end roll-off will attenuate the output as frequencies go down and the high-end roll-off will attenuate the output as frequencies go up.

Notice how both cutoff frequencies are at the -3 dB point of their respective roll-offs. As we've discussed briefly, this is the frequency at which the filter cuts the power of the signal in half. This definition of cutoff frequency is used in low-pass, high-pass, band-pass and other filters.

BPF Passband, Stopband & Transition Band

Band-pass filters allow a specified range/band of frequencies to pass through to the output. This band is known simply as the passband of the filter.

Technically speaking, the passband of a BPF is the band between the low cutoff frequency (f_L) and the high cutoff frequency (f_H). So then, some frequencies will be affected (amplitude and phase) within the passband near the -3 dB cutoff points. However, for the most part, these frequencies are relatively unaffected and make up the passband. Note that the passband is also referred to as the bandwidth of the filter.

A band-pass filter will have two stopbands. In an ideal filter, these stopbands will be from the low cutoff frequency (f_L) to 0 Hz and from the high cutoff frequency (f_H) to ∞ Hz.

However, in real-world band-pass filters, the stopbands will be at some point past the passband once the attenuation reaches a sufficient point (-50 dB, for example).

The transition bands of a BPF are the ranges between the passband and the stopband where the filter rolls off the amplitude of the filter output. The bandwidth of the transition band is dependent upon the slope of the roll-off, which is determined by the filter order and type along with the threshold attenuation of what is considered to be the stopband (the aforementioned -50 dB, as an example).

Band-Pass Filter Order

Band-pass filters are often referred to as “second-order filters”. This is because, at a minimum, they must have two reactive components (typically capacitors) in their design. That's at least one reactive component for the high-pass filter section of the band-pass filter and at least one reactive component for the low-pass filter section of the band-pass filter.

We'll discuss the HPF and LPF portions of the typical BPF circuit shortly.

Technically speaking, the order of a filter is what we've just described: the minimum number of reactive elements used in a circuit. Because band-pass filters need at least two reactive components, they are often called second-order filters. These reactive components are nearly always capacitors, though inductors may be used in certain passive BPF designs.

However, band-pass filters can have more than two reactive components in their design, which would thereby increase the order of their design.

In most band-pass filter designs, there will be an equal number of reactive components in the high-pass and low-pass sections of the circuit. This would mean that, generally speaking, band-pass filters will usually have an even number order. Of course, they can be designed in other ways but these designs are less common.

The order of a typical HPF or LPF determines the roll-off rate or steepness of the roll-off (and the width of the transition band). An increase in order will steepen the roll-off.

For standard Butterworth low-pass filters, each integer increase in order steepens the roll-off by an additional 6 dB per octave or 20 dB per decade.

Note that an octave is defined as a doubling (or halving) of frequency and a decade is defined as a tenfold increase (or decrease) in frequency.

Note, too, that the standard Butterworth filter holds the above relationship between order and roll-off rate true. Other filter types offer different relationships. More on this later.

So then, a second-order BPF would have a 6 dB per octave roll-off at the low-end and high-end. A fourth-order BPF would have 12 dB per octave roll-offs. A sixth-order BPF would have 18 dB/octave roll-offs. So on and so forth.

Let's have a look at a Butterworth high-pass and low-pass filter in the following images. Each image represents orders 1 through 5. Envision both filters making up the BPF, remembering that the order of the BPF will be double that of the matching order of the HPF and LPF:

Notice how the cutoff frequency (which is at 1 kHz for each graph) takes place at the -3 dB attenuation point regardless of filter order.

We can also see, from the graphs above, that the filter's amplitude graph approaches that of an ideal filter as the order of the filter is increased.

Band-Pass Filter Q Factor

Some band-pass filters will have a Q factor control. Q factor parameters are common in EQ plugins and digital EQ units, where the filter is not designed as any particular type (Butterworth, Bessel, Chebyshev, Elliptic, etc.).

A band-pass filter can be defined by its Q factor, which is defined as the reciprocal of the filter's bandwidth/passband. Q is a dimensionless unit.

Generally speaking, a high-Q band-pass filter (10 or above) will have a narrower passband and will be referred to as a narrow-band filter. Conversely, a low-Q band-pass filter (below 10) will have a wide passband and will be referred to as a wide-band filter.

Though Q has a definition (it's defined at the centre frequency divided by the bandwidth/passband), this definition isn't always held true.

What I can say about the Q factor is, regardless of how a manufacturer defines “Q”, increasing this parameter in a band-pass filter will reduce the passband while decreasing this parameter will increase the passband.

The EQs that will offer a Q factor control on the band-pass filter will typically have a graphic to show you how the filter is affecting the signal.

Band-Pass Filters & Phase-Shift

It's important to note that, in addition to affecting the amplitude of the signal, a band-pass filter will also affect the phase. As the BPF attenuates the output of certain frequencies by varying amounts, it will also shift the phase of the frequencies by varying amounts.

As a rule, each reactive component in an analog filter will introduce 90º of total phase shift in the signal. Analog band-pass filters (and the digital filters that aim to recreate them digitally) will have at least 180º of phase-shift since they are, at the very least, second-order filter (remember that they are made of a high-pass and low-pass filter.

With standard Butterworth low-pass and high-pass filters, half of the total phase-shift will happen by the cutoff frequency. The high-pass filter would cause the output signal phase to lead that of the input while the low-pass filter would cause the output signal phase to lag that of the input.

So then, if a BPF was to use a first-order Butterworth HPF and first-order Butterworth LPF, we'd have an amplitude-frequency and phase-frequency graph resembling the following:

We can see, in the graphs above, that the lower transition band of the BPF has an output with a positive phase-shift (the output phase leads that of the input). With a first-order HPF, the shift maxes out at 90º and drops to 45º at the cutoff frequency (f_L) before reaching 0º at the BPF centre frequency (f_C).

Past the centre frequency, the phase-shift becomes negative (the output phase lags that of the input), reaching -45º at the LPF cutoff frequency (f_H) as it approaches a maximum of -90º.

Note that if we were to increase the order of the BPF, we'd not only steepen the amplitude roll-offs but we'd also cause more phase-shift.

In the section on digital band-pass filter, we'll discuss linear phase EQ/filters that are designed to eliminate phase-shift while maintaining the same amplitude filtering of a typical BPF.

The High-Pass Filter Component

Thus far we've discussed the “high-pass” part of a band-pass filter in some detail. Let's now look at high-pass filters by themselves to help us understand the HPF component of a BPF.

Let's start our discussion with a set of graphs representing the amplitude-frequency and phase-frequency relationships of a first-order high-pass filter:

Everything checks out relative to our previous explanations of band-pass filters:

• The cutoff frequency is at the -3 dB attenuation point.
• The phase-shift is positive up to 90º (for a first-order HPF).
• The phase shift at the cutoff frequency (+45º) is half that of the maximum phase shift.
• The roll-off of a first-order filter is 6 dB/octave. The cutoff frequency in the graph above is at 1,000 Hz and the attenuation reached -24 dB four octaves below 1,000 Hz at 62.5 Hz.

Unsurprisingly, the easiest way to explain how a high-pass filter works is to examine the simplest type of high-pass filter. That is the passive first-order RC high-pass filter. It looks like this:

This RC HPF circuit has one resistor (R) and one reactive component: the capacitor (C). It is, therefore, a first-order filter.

Let's quickly compare this to a simple DC voltage divider circuit:

In the above schematic, we have the following equation:

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

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

Though a voltage divider acts on DC voltage, the idea can be translated to AC voltage in our simple RC high-pass filter. With AC signals, we must consider impedance (Z) and not only resistance.

Impedance is made up of two components: resistance and reactance. It's measured in ohms (Ω) just like resistance and can effectively be thought of as “AC resistance”.

So if we have, let's say, an audio signal at V_in with frequency content between 20 Hz and 20,000 Hz (the human range of hearing), then we have an AC signal. AC signals are subject to impedance, which has both phase and magnitude and is made up of the resistance and reactance of a circuit.

In an ideal world (remember how much we like idealizations) a resistor will have only resistance and no reactance. Conversely, a capacitor will have only reactance and no resistance.

With a voltage divider, we have a situation where, as R₂ increases, V_out increases (assuming R₁ remains constant). Remember this as it will translate to the simple RC high-pass filter circuit.

If we were to simply swap the resistors of the DC voltage divider with the components of the RC HPF circuit, then R₁ would become the capacitor and R₂ would become the resistor.

Let's revisit our simple passive analog RC high-pass filter schematic:

Now if we have a fresh look at the RC high-pass filter circuit, we can come up with the following equation:

V_\text{out}=V_\text{in}•\frac{Z}{R}

Where:
• Z is the overall impedance of the circuit
• R is the resistance of the resistor

Remember that the impedance is made of the resistance and reactance components of the circuit. The typical impedance formula is:

Z = \sqrt{R^2+(X_L-X_C)^2}

Where X_L is the inductive capacitance. Because there is no inductor in the RC circuit, X_L is equal to zero.

Let's quickly rewrite our RC output voltage with this new information:

V_\text{out}=V_\text{in}•\frac{R}{\sqrt{R^2+X_C^2}}

Now we state the following: as X_C increases, V_out decreases (assuming R remains constant).

Here's where we bring frequency into the equation and begin to understand the high-pass filter design.

The reactive capacitance (X_C) of a capacitor increases as the frequency of the input voltage/signal decreases according to this formula:

X_C = \frac{1}{2πfC}

Where:
• f is the frequency of the signal
• C is the capacitance of the capacitor (constant)

Recapping what we've learned, we can now understand how a high-pass filter works:

• As the input signal frequency decreases, the capacitive reactance increases
• As the capacitive reactance increases, the output signal level decreases relative to the input signal level (assuming the resistance of the circuit remains the same)

Therefore, the circuit attenuates lower frequencies more than higher frequencies. In other words, the circuit is a high-pass filter!

Of course, we haven't discussed the passband in which the circuit will not affect the amplitude of the frequencies. To begin this discussion, let's look at finding the cutoff frequency with the following equation:

f_L = \frac{1}{2πRC}

Where:
• R is the resistance of the resistor
• C is the capacitance of the capacitor

Anything above f_L will be part of the high-pass filter's passband

As an additional equation, we can calculate the aforementioned phase-shift of an RC high-pass filter with the following equation:

\phi=\arctan \frac{1}{2πfRC}

The Low-Pass Filter Component

In addition to a high-pass component, a BPF will also have a low-pass filter component. In this section, we'll focus on how low-pass filters work to help us understand band-pass filters as a whole.

Let's begin, as we had done with the high-pass filter section, with a set of graphs representing the amplitude-frequency and phase-frequency relationships of a first-order low-pass filter:

Again, let's check these graphs to ensure they hold true the rules previously discussed in our explanation of band-pass filters:

• The cutoff frequency is at the -3 dB attenuation point.
• The phase-shift is negative up to 90º (for a first-order LPF).
• The phase shift at the cutoff frequency (-45º) is half that of the maximum phase shift.
• The roll-off of a first-order filter is 6 dB/octave. The cutoff frequency in the graph above is at 1,000 Hz and the attenuation reached -24 dB four octaves above 1,000 Hz at 16,000 Hz.

Knowing what we know from the previous section on high-pass filters, let's have a look at the simplest low-pass filter design: the first-order passive RC LPF:

Likening this circuit to the voltage divider, we'd get the following equation:

V_\text{out}=V_\text{in}•\frac{X_C}{\sqrt{R^2+X_C^2}}

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

Remember that the reactive capacitance of the capacitor increases as the frequency of the input voltage/signal decreases.

So then, we can state the following points about this low-pass RC circuit:

• As the input signal frequency increases, the capacitive reactance decreases
• As the capacitive reactance decreases, the output signal level decreases relative to the input signal level (assuming the resistance of the circuit remains the same)

Therefore, the circuit attenuates higher frequencies more than lower frequencies. In other words, the circuit is a low-pass filter!

The cutoff frequency of such a low-pass circuit is calculated with the same formula as the aforementioned high-pass circuit:

f_H = \frac{1}{2πRC}

Where:
• R is the resistance of the resistor
• C is the capacitance of the capacitor

Anything below f_H will be part of the low-pass filter’s passband

As for phase, it can be calculated with the following equation:

\phi=-\arctan(2πfRC)

A Note On Band-Pass Filters In Audio Equalizers

Because band-pass filters can be produced by simply combining a high-pass and low-pass filter together, some manufacturers will do just that.

In some EQ units, the band-pass filter option will simply engage the high-pass and low-pass filters together and control them through auxiliary paramaters.

That being said, we'll focus the rest of this article on actual band-pass filter circuits and designs.

Analog Vs. Digital Band-Pass Filters

Now that we've understood the basics of band-pass filters and the building blocks (high-pass and low-pass filters) that go into making them, let's consider the difference between analog and digital BPFs.

The obvious difference between analog and digital filters is that analog filters are made of analog components (resistors, capacitors, operational amplifiers, etc.) and filter analog audio signals while digital filters are either coded in software or embedded in digital circuits and act on digital audio signals.

Analog BPFs are easier to understand and so the main focus will be on analog filters. Note than many digital BPF and equalizers are designed to emulate their analog counterparts.

Let's discuss each in greater detail in the following sections.

Analog Band-Pass Filters

We've already has a proper look at analog high-pass and low-pass filters in this article. Understanding analog band-pass filters will come easy as we simply combine these two filter types into one.

Starting with the most basic second-order passive band-pass filter, we'd have a circuit that would resemble the following:

This BPF is simply a first-order RC high-pass filter cascading into a first-order low-pass RC filter. We already know how these filters work.

The cutoff frequency of the high-pass filter (f_L) will be calculated with this formula:

f_L = \frac{1}{2πR_1C_1}

The cutoff frequency of the low-pass filter (f_H) will be calculated with this formula:

f_H = \frac{1}{2πR_2C_2}

And each roll-off will be at a rate of 6 dB/octave.

We'll discuss higher-order and active analog band-pass filters in the section on active band-pass filters.

Digital Band-Pass Filters

Digital signal processing (DSP) and improvements in computer processing power have made it possible to design incredibly precise and versatile digital filters. This extends to band-pass filter in audio devices including, particularly, digital equalizers.

Digital filters benefit from improved accuracy and flexibility; improved temperature and humidity resistance, and a lower cost of manufacturing.

Note that some digital band-pass filters are designed to emulate the performance of analog BPFs.

Rather than using analog components (capacitors, resistors, operational amplifiers, etc.), digital circuits will be embedded in digital chips (with adders, subtractors, delays, etc.) or, alternatively, be programmed into audio plugins.

A digital band-pass filter will fit into one of two camps:

• Infinite Impulse Response (IIR)
• Finite Impulse Response (FIR)

What is an infinite impulse response filter in audio? 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.

What is a finite impulse response filter in audio? 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 band-pass filter options) effectively eliminates any phase-shift within the audio processor.

Recall in the section Band-Pass Filters & Phase Shift how we discussed the inevitable phase-shift of analog BPFs (90º of phase-shift for every reactive component in the circuit).

A linear phase EQ (and band-pass filter) 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.

The Blue Cat’s Liny EQ is a great example of a linear phase EQ plugin:

Active Vs. Passive Band-Pass 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 analog band-pass filters, these active components are typically 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 to a band-pass 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 band-pass 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).

With that primer, let's discuss active and passive band-pass filters in greater detail, starting with the simpler of the two: the passive BPF.

Passive Band-Pass Filters

We've already discussed passive RC band-pass filters at length so I'll keep this short.

Let's quickly review the basic schematic of a passive second-order band-pass filter:

We can add resistor-capacitor pairs to the circuit above in an attempt to steepen the roll-off rate of the filter. However, we'd be doing so at the expense of output amplitude and signal-to-noise ratio as each component will drain some amount of power and add some amount of noise.

Passive filters like this are also at the disadvantage of relying on the passive components to maintain an output impedance. An op-amp can effectively drop the output impedance to improve signal transfer between the filter and the load (the next audio device). Passive circuits do not have this luxury.

However, passive BPFs can still work just fine since they're tasked only with cutting frequencies rather than boosting them.

Active Band-Pass Filters

The majority of band-pass filters will be active.

With analog band-pass filters, an active circuit will have at least one operational amplifier (op-amp). These op-amps can be used to amplify the signal, maintain unity gain, improve output impedance, buffer the signal within the circuit, maintain a proper damping factor, and more. By that list alone, we can see how beneficial op-amps are in BPF design.

There are plenty of circuit topologies that can be used to achieve an active band-pass filter. If we recall our discussion on BPF Q factors, we'll remember that band-pass filters are often considered to be either wide-band (with a Q less the 10) or narrow-band (with a Q factor of 10 or greater).

A wide band-pass filter can be formed by simply cascading a high-pass and low-pass filter together. This could look something like this:

In this circuit, the signal is high-passed; amplified (typically back up to unity in the passband); low-passed; and then amplified at the output.

Here's a look at an active high-pass filter and active low-pass filter for comparison. We'll start with the active first-order RC high-pass filter:

In this active first-order RC HPF, the gain of the op-amp is calculated as follows:

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

And the gain of the circuit gives us the following conditions:

• At low frequencies (f < f_C): A = V_out/V_in < A_V
• At the cutoff frequency (f = f_C): A = V_out/V_in = A_V/√2 = 0.707 A_V
• At high frequencies (f > f_C): A = V_out/V_in ≈ A_V

Now let's move onto the active first-order RC low-pass filter:

In this active first-order RC LPF, the gain of the op-amp is calculated as follows:

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

And the gain of the circuit gives us the following conditions:

• At low frequencies (f < f_C): A = V_out/V_in = A_V/{small number} ≈ A_V
• At the cutoff frequency (f = f_C): A = V_out/V_in = A_V/√2 = 0.707 A_V
• At high frequencies (f > f_C): A = V_out/V_in = A_V/{large number} « A_V

Turning this style of BPF into a fourth-order filter would require a second-order high-pass and a second-order low-pass. It would look something like this (using Sallen-Key filter topology):

Again, the high-pass stage comes first and the low-pass stage comes second. Each stage has its own op-amp for amplification, damping and impedance bridging.

To further our understanding, we'll quickly run through a second-order high-pass filter and second-order low-pass filter (Sallen-Key topology). Let's begin with the high-pass filter:

The gain of this circuit is defined as:

A_V = 1 + \frac{R_\text{F1}}{R_\text{I1}}

When dealing with second-order filters (and higher), we also have a damping factor in the circuit. The damping factor of this simple Sallen-Key filter topology is:

DF = 2-\frac{R_\text{F1}}{R_\text{I1}}

So the R_F and R_I values are involved in determining the gain and damping factor of the circuit. The R_F and R_I also determine whether we have a Butterworth, Bessel or Chebyshev filter. Note that the following only applies to a second-order filter:

• Butterworth: A Butterworth filter (maximally flat magnitude filter) is a linear analog filter designed to have a frequency response as flat as possible in the passband. Butterworth filters do not offer an overly steep roll-off and are often used in low/high-pass and low/high shelf filters.
  • R_F / R_I = 0.586
  • DF = 1.414
  • A_V = 4 dB
• Bessel: A Bessel filter is a linear analog filter with a maximally flat group or phase response to preserve the wave shapes of signals within the passband. Bessel filters provide a gentle frequency roll-off beyond the cutoff frequency and are mainly designed for linear phase response with little overshoot.
  • R_F / R_I < 0.586
  • DF > 1.414
  • A_V < 4 dB
• Chebyshev: A Chebyshev filter is a linear analog filter designed to have a very steep roll-off at the expense of passband ripple (type I) or stopband ripple (type II/inverse).
  • R_F / R_I > 0.586
  • DF < 1.414
  • A_V > 4 dB

Note that, in higher-order HPF, the ratio between R_F / R_I will be different in order to obtain each of these filter types.

Now if we move onto the second-order Sellen-Key low-pass filter, we have the following:

The gain and damping factor, along with the filter type categorization, are calculated the same way in this LPF circuit as they were in the previous HPF circuit.

Narrow-band band-pass filters are generally designed with a multiple-feedback circuit. These circuits have two (or more) feedback paths for the op-amp (hence the name) and use an inverted amplifier. Here is an example of a second-order narrow-band RC band-pass filter:

These types of BPFs are generally designed for specific centre frequency (f_C) and Q/bandwidth values.

A_V = -\frac{R_2}{R_1}

f_\text{C1} = \frac{1}{2πR_1C_1}

f_\text{C2} = \frac{1}{2πR_2C_2}

And, of course, the bandwidth is the difference between f_C1 and f_C2 while this Q is the reciprocal of the bandwidth.

The infinite gain multiple feedback (IGMF) band-pass filter is also popular, allowing for an improved passband response over the aforementioned narrow-band BPF circuit. This IGMF band-pass filter (in second-order) looks something like this:

This filter type offers high-gain and high selectivity of the Q/bandwidth. It's governed by the following formulae:

The centre frequency (f_C)is dependent on each resistor and capacitor in the following equation:

f_C=\frac{1}{2π\sqrt{R_1R_2C_1C_2}}

The Q (inverse of the bandwidth) is dependent of the resistance of the resistors:

Q = \frac{1}{2}\sqrt{\frac{R_2}{R_2}}

As is the maximum gain (A_V) of the circuit:

A_V = -\frac{R_2}{2R_1} = -2Q^2

Increasing the order of such filters vastly increases the complexity but can be done. We've completed what we'll be discussing in terms of filter design theory in this article. If you're interested in learning more, there are plenty of superb resources on the internet.

Mixing With Band-Pass Filters

Now that we understand how band-pass filters work, let's turn our attention their role in audio mixing.

Band-pass filters are used for mixing in the following ways:

• Adjusting Perceived Depth
• Cutting Problem Frequencies
• Accentuating Characteristic Frequencies With Resonances
• Automate!

Adjusting Perceived Depth

Band-pass filters can help to improve the important dimension of perceived depth in a mix. This is particularly due to the high-end roll-off (low-pass filter) portion of the band-pass filter.

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 (BPF/LPF).

By filtering out the high-end of a signal with a band-pass filter, we can effectively push a track further back in the depth dimension of a mix. This, of course, can be achieved with a band-pass filter.

Cutting Problem Frequencies

Whether it's eliminating low-end rumble; reducing the presence of hiss in the signal, or simply filtering out resonances or problem frequencies, a band-pass filter can help us at both ends of the audible frequency spectrum.

Low-end rumble can really eat up headroom and cause congestion in the low-end of a mix. A band-pass filter can effectively filter this out.

Similarly, self-noise, hiss and other high-end interference can be removed with the high-end filter of a band-pass filter if need be.

Accentuating Characteristic Frequencies With Resonances

As we've discussed, a band-pass filter will not always have a perfectly flat passband. Sometimes there are resonance peaks or ripples in the passband near the cutoff frequencies.

We can use these boosts in the frequency spectrum/EQ to accentuate certain frequencies while filtering out the frequencies above or below them.

By boosting a certain narrow band with a resonance peak, we can accentuate the character of a particular track while also removing problem frequencies above or below these characteristic frequencies.

Automate!

Automating a bandpass filter can be used to great effect in creating sonic interest in a sound source.

If you're into synthesizers, you're likely aware of how automating or otherwise modulating a bandpass filter can generate cool results.

Wah-wah and envelope filter effects pedals can also modulate a bandpass filter to achieve their sonic effect, especially when there's a resonance peak near the cutoffs.

Other Uses Of Band-Pass Filters In Audio

In addition to mixing, band-pass filters are found in other audio technologies.

Other uses of band-pass filters include:

• Crossover Networks
• Vocoders
• Multiband Compressors

Crossover Networks

A speaker crossover is a network of filters that separate bands of frequencies of an input audio signal. Each band is then outputted to the driver(s) best-suited to reproduce it. For example, low-frequencies are sent to the woofer, mids are sent to the mid-range speaker and highs are sent to the tweeter.

Tweeters can generally be driven with signals passed through a high-pass filter and subwoofers can be driven with signals passed through a low-pass filter. The mid-range speakers (and even the tweeters and subwoofers in some cases) are driven by signals passed through band-pass filters.

Note that these crossover filters are often of the Bessel type, meaning they have a maximally flat group/phase delay (maximally linear phase response), which preserves the wave shape of filtered signals in the passband.

Vocoders

Vocoding is the process of analyzing and synthesizing the human voice (or another modulator signal) for audio transformation. A vocoder splits the modulator signal (typically the voice) into frequency bands and a carrier signal is filtered according to the level of the modulator in each of these frequency bands.

The term vocoder is a portmanteau of voice and encoder and vocoders are used as instruments in music production to achieve “robotic” voices and other effects.

A vocoder works by analyzing the modulator (the vocal/voice) signal. It does so by measuring the amplitudes of the signal within a defined set of frequency bands. These bands are largely defined by band-pass filters, though the first and last band may be defined by a low-pass and high-pass filter, respectively.

A sort of amplitude envelope is generated for each band. The energy (voltage in analog vocoder) of each modulated band is then sent to an identical set of bands/filters the govern the carrier signal. The level at which each of the carrier signal bands is outputted from the vocoder is modulated by the energy/voltage of the corresponding modulator band.

Let’s have a look at a vocoder diagram to help with our explanation (note that this is an analog vocoder with voltage-controlled amplifiers but the general design is universal):

In the above vocoder diagram, we have 10 bands with 8 band-pass filters acting upon the modulator signal and 10 identical bands acting upon the carrier signal.

The amplitude/voltage of each modulator band is used to control the VCA (voltage-controlled amplifier) of each matching carrier band. This means that the amplitude of the modulator bands controls the output level of the carrier bands.

By using vocal/voice signals as the modulator and some sort of synth patch as the carrier, we can modulate the synth to take on a characteristic frequency output of a vocal/voice signal while maintaining the character of the patch itself.

Let’s have a look at a few examples of vocoders:

• Notable vocoder: Korg microKorg
• Notable 19″ rack mount vocoder unit: Roland SVC-350
• Notable vocoder effect pedal: Boss VO-1
• Notable vocoder Eurorack module: L-1 Vocoder
• Notable vocoder plugin: Image-Line Vocodex

Multiband Compressors

Multiband dynamic range compression is a type of compression that splits the frequency spectrum into different bands (with band-pass filters) and compresses each band by its own unique compression settings.

A multiband compressor can be thought of as several compressors in one with each compressor acting on its own defined band of frequencies. Each band will generally have its own set of parameters including threshold, ratio, attack, release and makeup gain.

Let’s have a look at a few examples of multiband compressors:

• Notable 500 Series multi-band compressor: SM Pro Audio MBC502
• Notable 19″ rack unit multi-band compressor: Maselec MLA-4
• Notable multi-band compressor effect pedal: EBS MultiComp Blue Label
• Notable multi-band compressor plugin: Vengeance Sound Multiband Compressor

For more information on compressors, check out the following articles:
• What Is Multiband Compression & How Do MB Compressors Work?
• What Are Compressor Pedals (Guitar/Bass) & How Do They Work?
• Top Best Compressor Pedals For Guitar & Bass

Related Questions

What is a band-stop filter in audio? A band-stop filter (aka a notch filter or band-reject filter) works by removing frequencies in a specified band within the overall frequency spectrum. It allows frequencies below the low cutoff point to pass along with frequencies above the high cutoff point.

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.