EQ, filters, and frequency responses are not perfect

Posted by | Monday, October 18th, 2010 | 0 Comments

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When you grab the knobs on a DJ mixer for that essential filter sweep, you probably don’t really care about exact frequencies.  Your ears know the sound and you have an instant response to every change, so you know what you’re getting in the moment.  But when it comes time to evaluate a speaker’s frequency response, set up a crossover, or convert digital recordings from one sample rate to another, it pays to know exactly what’s going on in fairly precise terms.

There are a few common misconceptions about filters and frequency responses, both in the analog and digital realm.  Hopefully it isn’t too hard to unravel the mysteries.  The most common mistake is to assume that EQ, filter, or frequency response numbers represent an all-or-nothing situation.  The second most common mistake is to assume that digital filters are any better at this particular aspect than analog filters.


Starting with an easy example, a low-pass filter pretty much does what it’s name suggests.  A low pass will basically kill high frequencies, but keep low frequencies.  Setting the frequency of the filter selects which frequencies are kept and which are lost.  That description is good enough for most applications, but the deeper details can sometimes be important.  Music usually contains a whole lot of different frequencies, and so the filter controls which frequencies are heard, and which are removed to change the sound.

First of all, analog filters never have an abrupt change in output level by frequency.  When specifying the cutoff frequency of a filter, the near-universal standard is to list the -3 dB point.  What this means for a low-pass filter is that if you look at the lowest music frequencies, they will be pretty much identical in level on input and output.  If the music contains the exact frequency that matches the cutoff frequency, then this frequency will actually drop in volume by 3 dB on output.  Frequencies below the cutoff are supposed to be kept, but in reality you might lose 1 dB, 2 dB or up to 3 dB of volume as the music frequencies get closer to the cutoff.  In addition, above the cutoff you still get some higher frequencies coming through, they might only be 4 dB or 5 dB quieter, which isn’t a drastic change.  There are many types of filters besides lowpass – including shelving, highpass, bandpass, notch, resonant, and more – but the same principles about gradual changes in frequency response still applies.

A related factor to consider is the slope.  Once you understand that the output levels do not change abruptly at the filter frequency, but rather follow a smooth slope, it makes sense that some filters are steep and others are more gentle.  Most analog filters have a slope that is expressed in terms of decibels per octave (dB/8va).  The simplest analog filter has a 6 dB/oct slope.  What this means for a low-pass filter is that one octave above the cutoff frequency, the output volume will only be 6 dB lower than it is at the cutoff.  Since the volume is already at 3 dB down at the exact cutoff, that means the total drop one octave up will be 9 dB.  Two octaves above the cutoff would be -15 dB (two times 6 plus 3).  Other common analog filter curves are made by adding more components to make 12 dB/oct, 24 dB/oct, and other useful slopes.  Often, the goal is to have a very steep slope, because it’s best when the common assumption about all-or-nothing filtering is as close to true as possible.  See below for more details.

We hear in octaves, but most people don’t really think about the mathematics involved.  Each octave holds twice as many frequencies as the octave below, and yet we hear the same twelve equal-tempered notes per octave.  Concert A on a perfectly-tuned piano will be 440 Hz.  One octave below that is 220 Hz, two octaves below is 110 Hz, three octaves below is 55 Hz; meanwhile one octave above is 880 Hz, et cetera.  Note that there are only 55 Hz of frequencies on that lower octave compared to 440 Hz in the upper octave.  If you think about a typical 6 dB per octave filter slope, you can understand how several octaves on the piano would still be heard even though you have a low-pass filter set to “remove” those frequencies.  Since a standard CD has 96 dB of dynamic range, this means a low-pass filter could still be passing some volume at 16 octaves above the cutoff, although they would be much quieter notes.  There are less than 8 octaves on a piano, so that should give some perspective.


Digital is supposed to be “better” than analog, right?  Well, that is usually true, up to a point.  An analog 6 dB/oct filter is fairly simple, but it’s slope is too gentle for every use.  As the circuit is enhanced to 12 dB/oct or 24 dB/oct and beyond, more analog components are added.  The problem here is that each analog component adds a small amount of noise.  In addition, each component is affected by temperature and age, and – since the components interact with each other to make the filter work – the cutoff frequency can actually change over time.  Digital filters have the advantage that they do not drift with temperature, they do not age, and they generally do not add as much noise as a complex analog circuit.  Sometimes we like all of those analog imperfections, such as with vintage synthesizers.  Often, though, we need precision, not unpredictability.

Although digital filters can be better than analog filters, a common misconception is that digital filters are all-or-nothing even though analog filters are not.  Perhaps one reason this misconception might persist is that digital is based on 1 and 0, on and off, and thus “all” or “nothing” is expected.  But the truth is that digital filters still usually have exactly the same response as analog filters, just with improvements in noise and accuracy.  When you select a 6 dB per octave digital filter, it uses simple math to mimic the simple circuit, and the slope is still just as gentle.  Where an analog 24 dB/oct filter would require more components, a digital 24 dB/oct would require more math, but the results are still the same slope.  Just keep in mind that the cutoff is not abrupt, even in digital.

The exception to the above is known as a “brick wall” filter.  Because digital filters can usually add math without adding the same amount of noise as analog components, it becomes possible to do some very complex math to get the slope almost to that all-or-nothing ideal.  These filters are almost exclusive used in A/D and D/A conversions or sample rate conversions, and nowhere else.  The thing to keep in mind is that the perfect filter is still not possible even in digital.  What the mathematicians call an “ideal” filter is one which would have full volume below the cutoff and no output whatsoever above the cutoff (we’re still talking about the low-pass here).  The ideal filter doesn’t even have the typical 3 dB point.  But this sort of filter is actually impossible unless it could be based upon an infinite amount of time.  Since the ideal isn’t possible, the real-world brick wall filters have some volume loss below the cutoff, and they still allow some frequencies above the cutoff to bleed through, even at low volumes.

So, digital filters are better than analog when it comes to noise, temperature affects, and aging, but they’re still not perfect all-or-nothing filters.


One application where the imperfections of filters becomes important is with DAC and A/D converters or digital-domain sample rate conversion.  The average person never designs their own digital converters, nor do they write their own sample rate conversion software.  However, it still pays to know what is going on, and what sorts of problems to look for.  You’d be surprised how many programmers in the Digital Audio Workstation software industry do not have a complete understanding of filters.

In the digital world, the sample rate determines the highest frequency.  It’s mathematically impossible to have a 23 kHz frequency in a 44.1 kHz digital recording.  Although that may seem like a perfect filter, it’s important to know what really happens.  Any time a 23 kHz frequency is forced into a 44.1 kHz sample rate, it gets reflected into a different frequency which sounds nothing like the original.  If you’ve heard ring modulation, then that’s the sound, although it might be very faint when it occurs by accident.  These reflected frequencies are the audio equivalent of spinning a wheel under a strobe light, or viewing rotating helicopter blades via video camera.  If you’ve ever seen the effect, then you know that the rate of the strobe or the rate of the video camera can make the wheel or blades to appear to be spinning at a different speed – or even backwards!  Yes, it’s possible to create negative frequencies, but they are just heard as positive frequencies with a different phase.  At any rate, you do not want your music spinning its frequencies backwards – that’s one of the primary causes of the harsh “digital” sound that people generally want to avoid (outside the 8-bit world).

All that said, you want to pay attention to the filters used in your A/D and also with the SRC software.  If the filter does not remove all of the “impossible” frequencies, then they will be reflected into “possible” frequencies and ruin your music (even if it’s only audible on the best systems).  One way to evaluate SRC is by visiting to look for your software.  On that page, the input is a pure sweep of a single, changing frequency.  The output should be a white line on a black background.  Backgrounds which are blue show digital noise, while stripes of faint colors show aliasing (reflected frequencies).  One of the best sample rate converters is the iZotope 64-bit version.

Finally, one of the primary reasons for digital oversampling is not to capture bat frequencies (although there is some debate about what effects can be heard above the highest 20 kHz pure tone that the average human can detect), but to make the filtering easier.  When converting to 44.1 kHz digital, whether the input is analog or a higher digital sample rate, the audio signal must first be filtered to remove everything above 22.05 kHz.  Designing a filter which has perfect quality at 20 kHz and below, and absolute zero output at 22.05 kHz is actually quite difficult.  As can be seen by the SRC site, what usually happens is that the frequencies near 20 kHz are actually attenuated (they’re near the -3 dB point), or some frequencies above 22.05 kHz are allowed to remain and reflect, or often you’ll find a combination of both effects.  Using higher sample rates like 96 kHz or 192 kHz allow for much better filters which leave everything we can hear perfect, and competely avoid aliased frequencies.  According to experts, about 60 kHz should be the minimum to mimic the best analog tapes, and higher frequencies improve the filters.


Speakers are basically bandpass filters.  Each driver can only handle a rather narrow range.  Speaker designers will use two or three drivers to cover the whole range of human hearing.  The ideal speaker would be “flat” from 20 Hz to 20 kHz.  But here is where misconceptions begin.  Many people assume that if a speaker goes out to 20 kHz, then there isn’t any sound at all above that frequency.  Similarly, if a subwoofer only extends down to 35 Hz, then the assumption is that nothing will be heard at 20 Hz.  The reality is that each driver is acting like a filter, but the response curves are fairly gradual.  This means that 20 kHz might be down by 6 dB or more, but there is still quite a bit of sound at the higher frequencies.  Maybe that’s not important when listening to a CD, but your vinyl has plenty of sound up to 50 kHz or so.  Likewise, a subwoofer might still have a little bit of output power at 20 Hz even though the flat part of the response stops at 35 Hz.  Remember, 20 Hz to 20 kHz is not an all-or-nothing situation, there can still be a lot of loss within that range, as well as residual output levels above and below that range.


Besides frequency response, a second characteristic of filters is phase response.  Phase response is almost always ignored when evaluating filters.  Phase response is actually a deep topic, but a brief analogy might help.  If you’ve ever taken a ride in a vintage seventies car – you know, the ones they call a “tank” – then you know what happens when you hit a bump on the road.  The shocks on that caddy make it feel almost like there was no pothole at all, but you still bounce up and down until the end of the block.  This is an example of a low-pass filter.  The pothole is a burst of noise with a lot of frequencies from low to high.  But since the shock-absorbers filter out the highs, you’re left with only the low frequency aspect.  The phase response of the filter explains why you still feel those low frequencies long after the pothole has been left way behind.  A similar term is group delay, which refers to a group of frequencies being delayed on output based upon their frequency.  Of course, this isn’t a perfect example, because those old caddys would bounce for quite a while, meaning the “filter” has “resonance” – a factor that’s more common in synthesizer filters than in studio, DJ, and mastering filters.

The whole point of mentioning phase response is to briefly explain that when you use multiband filtering, such as bass, mid, and high knobs on your mixer, the final sound can be changed even if you recreate the input volumes at every frequency.  If you were to carefully plan the filter slopes for each band, such that they all added up to 0 dB in the end, you would still have the effect of the phase response or group delay from each individual filter.  Thus, the music coming out could actually sound a bit different than it did on the input, even though you’ve fixed all of the frequency levels.  In other words, the bass knob still lets some mids and highs through, and that signal combines with the other mids and highs such that phase cancellations and even reinforcements can occur.  The latter means that you could even get more volume on output than input.  The latter can occur with MP3 playback – just consider MP3 as a 32-band EQ with noise injected.

This phase reponse issue can be applied to speakers, where you might be tempted to correct for the various slopes from the drivers, crossover, et cetera.  You can measure the volume at every frequency and then boost the frequencies which are quieter coming out of the speaker, however, both the inherent filters and your corrective filters have phase delay, and thus you can’t quite get back to the original sound without some fancy digital tricks.


Hopefully, the above gives a taste of some of the issues when dealing with filters.  As the opening sentence concedes, sometimes you don’t really care about the detailed numbers, especially when you’re going for a wild sound.  But for those times when you are trying to get the best sound possible, and you’re using any kind of filter in the process, then be wary that filters can never be perfect at removing or boosting all of the frequencies in their range.

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