Audio EQ: What Is A Low-Pass Filter & How Do LPFs Work?

When studying and practicing music production or audio engineering, you will definitely come across low-pass filters. Low-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 low-pass filters, covering how they work, how they're designed and how they're used, not only in EQ but in other applications pertaining to audio as well.

Upon completing this article, I've realized how deep filter theory goes. In an attempt to keep this article short (it's still over 6,000 words), I've only included the most important information on audio low-pass filters. Please use the table of contents to maneuver your way around this guide!

What Is A Low-Pass Filter?

The initial answer paragraph is a decent definition of a low-pass filter but it leaves a lot to be explained. So then, let's discuss what a low-pass filter is and how it works, beginning with the basics.

So we know that a low-pass filter passes low frequencies below a certain cutoff point, hence the name. Low-pass filters are sometimes referred to as high-cut filters, a title that depicts the cutting out of high frequencies above a certain cutoff point.

The Ideal Low-Pass Filter

Ideally, we would want our low-pass filter to simply cut all the frequencies above its cutoff frequency and leave all the frequencies below its cutoff frequency unaffected. This “brickwall” type of low-pass filter is unobtainable in practice but in theory, it would look like this:

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

Hertz refers to cycles per second. Because audio signals are AC signals, they have cyclical waveforms. When converted to sound waves, these waveforms can be heard as vibrating air molecules. The universally accepted hearing range of humans is defined between 20 Hz and 20,000 Hz. Therefore, most audio signals fall within this range (so as to avoid having an abundance of imperceivable information).

Decibels (a tenth of a Bel) are relative units of measurement used to express the ratio of one quantity to another on a logarithmic scale. In terms of 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).

In the graph above, we have a sharp cutoff frequency at 1 kHz. No frequencies above this cutoff are passed and all frequencies below this cutoff are passed perfectly.

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

In analog LPFs, increasing the filter order will move us closer to the steepness of an ideal filter around the cutoff frequency.

In digital LPFs can also be programmed to approximate such an ideal “brickwall” filter.

More on this later.

Real-World Low-Pass Filters

Though we can get pretty close to ideal LPFs, we'll generally have some sort of roll-off after the cutoff frequency rather than a strict cutoff.

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

We can see in the image that, above a certain frequency the filter begins attenuating/filtering the frequencies by a steady negative slope (amplitude goes down as frequency goes up). We also notice a defined frequency f_H, which is the cutoff frequency (I define it as f_H for “high cutoff frequency” rather than f_C which could be confused with “centre frequency in other filter types).

Notice how the cutoff frequency does not happen immediately as the filtering begins. Rather, the cutoff frequency represents the -3 dB point of the filter's attenuation. 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.

LPF Passband, Stopband & Transition Band

Note that, technically, a low-pass filter will have a passband (the range of frequencies that are passed through) that ranges from 0 Hertz to the cutoff frequency.

The stopband will be at some point past the passband once the attenuation reaches a sufficient point (-50 dB, for example). In an ideal filter, the passband goes up to the cutoff frequency and the stopband is everything above that cutoff frequency. However, real-world low-pass filters work a bit differently.

LPFs will generally have a transition band between the passband and stopband where the filter will effectively roll-off the amplitude of the signal. The bandwidth of the transition band is dependent upon the slope of the roll-off, which is determined by the filter order and type.

Low-Pass Filter Order

Filters are often defined by their order. With simple filters like low-pass and high-pass, the order of a filter largely refers to the slope of the transition band (otherwise known as the roll-off rate).

Technically, the order of a filter is the minimum number of reactive elements used in a circuit. With analog low-pass audio filters, these reactive elements are nearly always going to be capacitors (though inductors may be used in certain situations). We'll discuss this later in the section Analog Vs. Digital Low-Pass Filters.

So then, the order of a low-pass filter is, by definition, an integer number (we cannot have a fraction of a reactive component in a circuit) and it affects the roll-off slope of the filter's transition band.

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.

For now, let's consider the following graph that shows 5 different Butterworth low-pass filters with orders 1 through 5:

The cutoff frequency (the -3 dB point) of each filter is at 1 kHz. The roll-off rate and transition band (which can be limited at the -50 dB attenuation mark) change depending on the order of the filter.

We can see that as the order increases, the low-pass filter gets closer to becoming an ideal filter.

Low-Pass Filter Q Factor

Some low-pass filters will have a Q factor control. This is particularly the case with parametric EQ plugins and digital EQ units, where the filter is not designed as any particular type (Butterworth, Bessel, Chebyshev, Elliptic, etc.).

The Q factor is somewhat arbitrary. Though it has its definitions, many manufacturers will have their own technical calculations for the Q parameter.

However, in a general sense, increasing the Q factor of a LPF will steepen the roll-off slope while causing a resonance peak to form at and above the cutoff frequency.

Conversely, decreasing the Q factor of a LPF will increase the attenuation at and above the cutoff frequency while making the roll-off slope more gradual.

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

Low-Pass Filters & Phase-Shift

It's critical to note that, in typical analog filters like the standard Butterworth, there will be frequency-dependent phase-shift between the filter/EQ's input signal and its output signal.

Generally speaking, each reactive component in an analog filter will introduce 90º of phase shift in the signal. For analog low-pass filters (and the digital filters that aim to recreate them digitally), this means that there will be 90º of phase-shift per integer increase in filter order.

With standard Butterworth low-pass filters, half of the total phase-shift will happen by the cutoff frequency.

Here is a visual representation of a first-order Butterworth low-pass filter with both amplitude-frequency and phase-frequency graphs:

Analog Vs. Digital Low-Pass Filters

The key difference between analog and digital low-pass filters is that analog filters work with analog audio signals and digital filters work with digital audio signals.

Analog audio LPF circuits utilize analog components such as resistors and capacitors (active LPF circuits use active components such as operational amplifiers). Digital LPFs, on the other hand, are either embedded within digital chip circuits or within software.

Let's discuss each in a bit more detail, shall we?

Analog Low-Pass Filters

Analog filters are more simple to explain since they're made from actual analog circuits that can be relatively easy to understand. Note that I'm not an electrical engineer and digital circuits/programming are beyond my scope of knowledge.

So in this article, I'll do my best to explain how analog low-pass filters work. Note that many digital low-pass filters are designed to recreate the effect of analog LPFs.

In the explanation, there will be plenty of equations to go through to help us understand.

To really understand the basics of how a low pass filter works, we can study a simple passive first-order RC LPF. This filter can be visualized with the following image. Note that “RC” refers to the resistor and capacitor used in the circuit.

The circuit above can be thought of similarly to a voltage divider:

In the above schematic, we derive the following formula:

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

We can infer from this formula that as R₂ increases, V_out increases (assuming R₁ remains constant). Remember this.

In this DC voltage divider equation, R₁ represents the resistance of the resistor that would be in place of the resistor of the RC circuit and R₂ represents the resistance of the resistor that would be in place of the capacitor of the RC circuit. Keep this in mind.

Let's say that the audio signal at V_in has frequency content between 20 Hz and 20,000 Hz (the human range of hearing). This is an AC signal, not a DC 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 (which we'll use to understand RC low-pass filters), the reactance of a resistor is zero and the resistance of a capacitor is zero. The resistor will offer a resistance component to the overall impedance of the audio signal and the capacitor will offer a reactance component to the overall impedance of the audio signal.

So with the following simplified RC low-pass filter schematic:

We'd have the following equation:

V_{\text{out}} = V_{\text{in}} \cdot \frac{X_C}{Z}

Where:
• X_C is the capacitive reactance of the capacitor
• Z is the overall impedance of the circuit

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}} \cdot \frac{X_C}{\sqrt{R^2+X_C^2}}

Look familiar? It's nearly the same as the simple voltage divider.

Okay, so our RC low-pass filter can be likened to a voltage divider but for AC audio signals. As X_C increases, so too does V_out (again, assuming the R remains constant).

How does it actually work as a low-pass filter? Well, the reactive capacitance decreases as the frequency of the input signal increases. The formula for this is as follows:

X_C=\frac{1}{2πfC}

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

So with that we have the following rules of the RC low-pass circuit:

• As the 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)

Basically, as the capacitive reactance decreases (as the frequency increases), more of the signal is sent to ground rather than to the output.

Therefore, generally speaking, the RC low-pass circuit will begin attenuating higher frequencies and as the frequency increases, the circuit will attenuate more.

We've discussed the cutoff frequency already. It's the point at which the passband turns into the transition band (or the stopband in ideal filters). The cutoff frequency is at the -3 dB point of attenuation. It can be calculated with the following equation:

f_C=\frac{1}{2πRC}

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

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

\phi=-\arctan(2πfRC)

I hope this makes sense. We've look at the most basic form of an analog RC low-pass filter here.

Analog filters are generally simple in design though they increase in complexity as their designs approach the performance of an “ideal filter”. Many digital filters (including EQ plugins) emulate these analog filters.

Remember that, by adding additional RC sets to (increasing the order of) the low-pass filter, we can effectively steepen the roll-off and shorten the transition band.

There are plenty of filter types to be aware of. Thus far, we've focused largely on the popular Butterworth filter. However, there are 3 main filter types (among the many) that we should be aware of when it comes to audio. They are:

• Butterworth filter: 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.
• Bessel filter: 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.
• Chebyshev filter: 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).

These “types” of filters are dependent on the values of the components used in the filter design and the damping factor that comes with the filter design. The study of individual low-pass filter schematics is beyond the scope of this article but these popular types are worth knowing about.

Here is an image from Wikipedia showing the typical differences between Butterworth, Chebyshev I/II and Elliptic low-pass filters:

Note that the elliptic filter (also known as a Cauer filter) is a linear analog filter with equalized ripple in both the passband and the stopband. It offers a very steep transition band. It is achieved by combining a low-pass filter and a band-stop/notch filter.

Digital Low-Pass Filters

Digital filters are often more precise and much more flexible in design due to the expansive nature of digital signal processing (DSP). The exactness of their design makes them much more accurate to their given parameters whereas analog filters are somewhat limited by the accuracy of their components and the signal path at large.

Digital filters also come with the benefits of an improved cost-to-benefit ratio and a more consistent nature in temperature and humidity changes.

Analog filters, of course, benefit from working on a continuous spectrum.

Note that some digital low-pass filters are designed to emulate the performance of analog LPFs. We often find the previously-mentioned filter types (Butterworth, Bessel, Chebyshev, etc.) in digital designs.

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 low-pass 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 low-pass filter options) effectively eliminates any phase-shift within the audio processor.

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

A linear phase EQ (and low-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 Low-Pass Filters

The key difference between active and passive low-pass filters is that active LPFs require power to function and passive LPFs do not.

This is because active LPFs will have some sort of amplifier in their circuit. These amplifiers (often op-amps) will take in power from a source and use it to amplify the signal passing through the low-pass filter or audio equalizer.

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 low-pass filters in greater detail, starting with the simpler of the two: the passive LPF.

Passive Low-Pass Filters

In my explanation of analog low-pass filters I focused exclusively on a passive RC low-pass filter circuit. So we already have a solid understanding of passive low-pass filters.

Once again, the most basic first-order passive low-pass filter looks something like this:

Note that we could increase the roll-off rate of a passive filter by adding poles. However, this comes at the cost of losing signal amplitude (as there are no gain stages in the circuit) and worsens the signal transfer within the circuit due to poor impedance bridging (because there is no buffer between the poles or at the output of the LPF).

Passive low-pass filters are easy to understand. Fortunately, because they're only tasked with cutting frequencies (above the cutoff frequency), they don't necessarily need active amplification.

However, as has been mentioned, a passive LPF may perform poorly as it naturally drops the amplitude of the signal passing through it (even in the low-end). It is also more difficult to find proper impedance bridging between the output of the passive LPF and the next audio device (load).

Passive low-pass filters are still found in certain applications and there are even passive EQ units on the market that, by definition, will have passive LPFs (if they include a low-pass filter).

Note that, with passive EQs, there is an amplification stage for “makeup gain” after the filter circuit(s). It's just that there are no active components within the filter circuit(s).

Active Low-Pass Filters

More often than not we'll have an active low-pass filter.

Active analog low-pass filters typically utilize operation amplifiers. These op-amps are useful for unity gain filters (filters that maintain signal amplitude but do not increase signal amplitude) and filters that do offer a proper gain stage.

This gain allows LPF designers to increase the order of the filter, thereby steepening the roll-off, without the worry of losing overall signal amplitude.

Another huge benefit of active LPF design is the improvement in the filter's output impedance. By including an op-amp, we can set the output impedance low across all frequencies for improved signal transfer between the LPF and the following audio device.

Here's an example of an active first-order RC low-pass filter with unity gain:

Note that it looks very similar to the aforementioned passive RC filter. The main difference being, of course, the op-amp. In this case, the op-amp does not offer any amplification to the signal. Rather, it maintains unity gain and allows for an appropriate output impedance for the low-pass filter circuit.

Now let's have a look at a look at a simple first-order RC low-pass filter that does offer amplification:

The gain A_V of the non-inverting amplifier is calculated by the following equation including the feedback resistor (R₂) and its corresponding input resistor (R₁):

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

The gain of the overall circuit is frequency-dependent (as the low-pass filter attenuates higher frequencies). This gain can be defined with the following equation:

A=\frac{V_\text{out}}{V_\text{in}}=\frac{A_V}{\sqrt{1+(\frac{f}{f_C})^2}}

With this equation, we can observe the following:

• 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

If plug 0.707 A_V into the following equation for decibels, we can confirm that the cutoff frequency is indeed at -3 dB from unity:

dB=20\log(\frac{V_\text{out}}{V_\text{in}})

If we look at a second-order filter (in a simplified schematic), we'd have the following:

When dealing with second-order filters (and higher), we 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:
  • R_F / R_I = 0.586
  • DF = 1.414
  • A_V = 4 dB
• Bessel:
  • R_F / R_I < 0.586
  • DF > 1.414
  • A_V < 4 dB
• Chebyshev:
  • R_F / R_I > 0.586
  • DF < 1.414
  • A_V > 4 dB

Let's now have a look at a sixth-order RC low-pass filter below:

Perhaps the first thing to note is that for every two resistor-capacitor pairs (for each increase of two in the filter's order), the circuit will have an op-amp. That's standard in order to maintain proper gain-staging and buffering throughout the circuit.

Getting back to the roll-off slope, this low-pass filter would have a slope of 36 dB/octave or 120 dB/decade above the cutoff frequency. This filter could take on the Butterworth, Bessel, Chebyshev or any other possible low-pass filter “type” given the topology. The various R_F / R_I ratios between the 3 sets would be different than those defined above for the second-order filter.

I hope I didn't confuse you. There are plenty of other in-depth resources on filters. The main focus of this article is the design and use of low-pass filters in audio so I'm refraining from going too far down the rabbit hole!

Mixing With Low-Pass Filters

Now that we understand what a low-pass filter is and how it works, let's consider its practical applications when it comes to mixing audio.

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

• Reduce Competition Between Instruments In The High-End
• Reduce Hiss
• Add Depth
• Add Edge With Resonance
• Automate!

Reduce Competition Between Instruments In The High-End

One of the most important jobs for audio EQ is to clean up the frequency spectrum in order to allow instruments to be heard. This means reducing the frequency bands of some tracks so that other tracks can shine through in these same bands.

Low-pass filters can effectively eliminate the high-end frequencies of some select tracks, thereby allowing other track(s) to take up the high-end with improved clarity. This can also reduce the harshness of the overall mix.

The high-end doesn't have all that much “musical” information (harmonics). However, by cutting out the “brilliance” of some instruments, we can enhance the perceived brilliance/airiness of other instruments. There's also nothing stopping us from bringing the LPF cutoff frequency down into the mid-range to begin filtering harmonic content.

Reduce Hiss

If the source material is not recorded properly, or with low-quality equipment, hiss (among other things) can be an unwanted audible result.

Some amount of hiss is inevitable in analog equipment, including microphones, due to the nature of electricity and the electrical components that go into audio equipment design. This is typically referred to as “self-noise“.

Much of what we refer to as “hiss” is located in the high-end of the frequency spectrum. Therefore, using a low-pass filter can help to reduce the level of hiss in the signal. Just be sure to be aware of any effects the LPF will have on the tone as you move the cutoff frequency down.

There are also audio plugins that can help reduce noise without being as invasive on the frequency content of the signal. Waves X-Noise is a great example of such a plugin.

Add Depth

Depth is an important dimension when it comes to mixing. It is essentially the perceived distance of a sound source in the context of a 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 (LPF).

So by reducing the high-end of a source with a low-pass filter (or a high shelf or other EQ), we can give the illusion of the source being further back in the mix.

Add Edge With Resonance

As we've discussed previously, the passband of a low-pass filter (especially near the cutoff frequency) is not always perfectly flat. In many cases, there will be some sort of resonant peak or EQ boost near/below the cutoff frequency.

Therefore, we can actually use some low-pass filters to boost certain resonance bands to give some edge to a track just before the point at which the high-end frequencies are filtered out.

To get the most “edge” out of a sound source, it's typically best to have the resonance and cutoff in the mid-range where there is notable harmonic content in the signal.

Automate!

Automating a low-pass 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 the low-pass filter can generate cool results.

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

When it comes to standalone low-pass filters, we can extend these effects to any sound source by automating the low-pass filter (particularly the cutoff frequency parameter).

We can also use automation to effectively increase or decrease the perceived depth of the track and also to reduce competition in the high-end as other tracks are introduced to (or taken out of) the arrangement.

Other Uses Of Low-Pass Filters In Audio

In addition to mixing, low-pass filters are utilized in many other audio standards and equipment.

Low-pass filters are used in the following ways when it comes to general audio:

• Anti-Aliasing & Reconstruction Filters
• De-Emphasis Filters
• Subwoofer Crossovers
• Inclusion In Band-Pass Filters
• Inclusion In Band-Stop Filters

Anti-Aliasing & Reconstruction Filters

If you've been interested in audio for a while, you'll know that audio signals can be either analog or digital. While analog signals are typically involved with transducers (loudspeakers, headphones, microphones, etc.) and some storage methods (vinyl, tape, etc.) it's digital audio storage that is generally used in modern cases (inside DAWs, streaming, cloud storage, hard drive storage, etc.).

Whether we're recording with microphones or analog instruments into a digital audio workstation or we're playing back digital audio through speakers or headphones, we'll need to convert between analog and digital audio.

This conversion is done with aptly named analog-to-digital converters (ADCs) and digital-to-analog converters (DACs).

When going from analog to digital, the ADC will sample the audio at a fast sample rate and assign an amplitude (within a set bit-depth) to each sample in an effort to model the waveform of the analog signal.

When going from digital to analog, the DAC will attempt to produce a smooth, continuous-time signal based on the samples of the digital signal.

Analog low-pass filters are used in both converters.

Anti-Aliasing Filter

With ADCs, the LPF is referred to as a anti-aliasing filter. The anti-aliasing filter, as the name suggests, filters the analog signal before the sampling/conversion takes place in an effort to avoid aliasing.

Aliasing is a sampling error that happens when a sampling rate is too slow to properly identify the frequency of the input signal. When aliasing occurs, the sampled signal ends up having a lower frequency than the input signal.

Note that typical audio signals are not simple sine waves and rather have a wide range of frequencies. Aliasing, then, introduces distortion and other artifacts to the digital audio signal (rather than simply altering the frequency of the signal).

That being said, it's easiest to visualize aliasing with a simple sine wave. Let's have a look at a few illustrations to help us understand aliasing:

In the following image, we have a 12 kHz sine wave being sampled at a rate of 48 kHz. The dots represent each sample point and the red waveform represents the sampled waveform (note that it is overlaid on top of the original waveform in black). In other words, the ADC converts the signal effectively from analog to digital.

In this next image, we have a 36 kHz input signal sampled at the same rate of 48 kHz. The dots represent each sample point and the red waveform represents the sampled waveform. Notice that in order to produce a waveform that passes through each sample point (without going through a cycle first), the sampled waveform must take on a different waveform, this time with a frequency of 6 kHz. This is essentially what aliasing is.

Normally digital audio is sampled at a rate of 44.1 kHz or 48 kHz, though higher rates of 88.2, 96, 176.4 and 192 kHz are also common.

The Nyquist-Shannon sampling theorem essentially states that, in order to avoid aliasing, a digital sampling system must have a sample rate at least twice as high as that of the highest audio frequency being sampled.

The audio range of human hearing is between 20 Hz and 20 kHz so we can effectively low-pass anything above 20 kHz without overly affecting what we hear. Note that, if aliasing was to occur, frequencies above the hearing range would cause distortion and artifacts in the hearing range.

So with the lowest common sample rate of 44.1 kHz, we'd need the highest frequency in the audio signal to be at 22.05 kHz or 22,050 Hz. This gives us a bit of space on the frequency spectrum to roll-off frequencies between (ideally) 20 kHz and 22.05 kHz.

Remember that low-pass filters will have some transition period to take into account. A drop-off of 40 dB is generally considered enough to make aliasing “insignificant”. By that metric, we'd need a very high-order filter approximating a brickwall/ideal filter.

Reconstruction Filter

With DACs, the LPF is referred to as a reconstruction of “anti-imaging” filter.

When the digital signal is converted to analog, it's not a continuous-time signal. Rather, it has discrete changes in voltages at the given sample rate. By low-passing the converted signal, we can effectively smooth out of this discrete signal in the high frequencies to achieve a typicaly continuous-time analog signal.

By removing the high-frequency components of the signal, we can get rid of any distortion or imaging in the signal.

Note that, ideally, these low-pass filters should be ideal, meaning they should be brickwall filters. This is typically achieved (approximately) by a LPF with a sinc impulse response.

De-Emphasis Filters

De-emphasis filters are used in systems where pre-emphasis and de-emphasis are needed for improved signal transfer. These applications are, most notably, FM radio and vinyl recording/playback.

Pre-emphasis filters are generally high-pass, low shelf cut or high shelf boost filters. They are used in order to improve signal-to-noise ratio in the high-end (with FM radio) or improve storage (vinyl is notoriously poor at storing low-end information in its grooves).

A de-emphasis filter is then needed at playback to undo the effect of the pre-emphasis filter, bringing the signal back to its original frequency response.

Since pre-emphasis filters are of the high-pass (or similar) variety, de-emphasis filters are of the low-pass (or similar) variety.

To help visualize, here is an image of a de-emphasis filter (in blue) and pre-emphasis filter (in pink) for FM radio (time constant of 75 µs and a cutoff frequency of 2,122 Hz):

Similarly, the RIAA equalization standard is a pre/de-emphasis EQ for the recording and playback of phonograph/vinyl records. It is represented by the image below with the blue line representing the playback (de-emphasis) EQ and the pink line representing the recording (pre-emphasis) EQ:

Subwoofer Crossovers

Subwoofers are loudspeakers designed specifically to produce the low-frequency sound waves (generally from 20 Hz to 200 Hz) of the audio signal.

These speakers are important in systems designed to produce the entire range of audible frequencies since most speakers cannot accurately produce this low-end information (if at all).

More that allowing us to hear the low-end, subwoofers allow us to feel the low-end of the audio.

In systems that have subwoofers, these specialized speakers are generally sent a defined band of frequencies of the overall audio signal.

The speaker crossover (whether it's a standalone unit or part of a power amplifier) will effectively low-pass the signal that will be sent to the subwoofer so to not send any mid/high-end information. Sending signals with frequencies beyond the dedicated subwoofer range can lead to non-ideal and “muddy” performance from the sub.

Consumer-grade subwoofers, like those found in automobiles, typically reproduce 20 Hz – 200 Hz while professional live sound reinforcement subwoofers are designed to produce sound under 100 Hz. THX-approved systems are designed to produce under 80 Hz.

Inclusion In Band-Pass Filters

What is a band-pass filter in audio? A band-pass filter “passes” a band of frequencies (a defined range above a low cutoff and below a high cutoff) while progressively attenuating frequencies below the low cutoff and above the high cutoff.

Bandpass filters can be thought of as high-pass and low-pass filters in series/cascade. The high-pass filter cutoff frequency (f_H) will be lower than the low-pass filter cutoff frequency (f_L).

Here's a visual representation of a bandpass filter frequency plot:

And here's a simplified schematic representing an analog bandpass filter with a first-order high-pass and first-order low-pass filter:

Inclusion In Band-Stop Filters

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.

Bandstop filters can be thought of as high-pass and low-pass filters in parallel. The high-pass filter cutoff frequency (f_H) will be greater than the low-pass filter cutoff frequency (f_L).

Here's a visual representation of a bandstop filter frequency plot:

And here's a simplified schematic representing an analog bandstop filter with a first-order high-pass and first-order low-pass filter:

Related Questions

What is a high-pass filter in audio EQ? A high-pass filter (HPF) is an audio signal processor that removes unwanted frequencies from a signal below a determined cutoff frequency. It progressively filters out (attenuates) the low-end below its cutoff frequency while allowing the high-end to pass through, ideally without any changes.

What is a shelving EQ? 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 a certain point.