Digital Audio - Equalization

One of the earliest techniques one stumbles across in the manipulation of audio is equalization (EQ), both when mixing multiple tracks to create an audio output or when trying to fix up existing recordings. Equalization allows all kinds of magic such as the ability to pull out voice from a lot of background noise, suppress a particualarly noise instrument or clip the tinny higher frequences. But in order to work the magic you have got to know what frequencies the things you want to accentuate (or suppress) work on as well as their harmonic characteristics.

Once you have mastered that lot then the real fun of equalization starts. Those words were written with more than a hint of irony. This page collects information about software based equalization (though most principles remain the same) and its related cousin - frequency analysis.

Serious Confusion Warning: Most equalizers, espcially band equalizers, use the terms Octave , 1/3 Octave, 1/12 Octave etc. This use should not be confused with a musical Octave. A musical Octave has a frequency range from C to B based on a reference (tuning) frequency of A4 = 440 Hz. Equalizer Octaves are normally based on a reference frequency of 1000 Hz (1 KHz - the ISO and ANSI standard). Both have standard Octave property of a 2:1 ratio, thus moving from one octave to the next will result in a doubling of the frequency. The term Decade is sometimes used in equalization meaning a 10:1 ratio between the decades (normally 20 Hz to 200 Hz, 200 - 2Khz, 2KHz to 20 Khz). Finally like much in the audio world most terms have their roots in the analog world - when applied to digital systems many terms are either not meaningful or may have very different properties.

Contents

Equalization - Overview (Octave, 1/3 octave etc.)
Preferred and Center Frequencies

Equalization - Overview

An equalizer allows boosting (or suppression/attenuation) of frequencies between the source of a sound (a microphone or recorded material) and the output of the sound (a loudspeaker system or recording system). Equalizers normally work on groups of frequencies called frequency bands or more commonly just bands. Analog equalizers come in all shapes and sizes with the most common today being what is called a graphic equalizer - a big board with lots of slider controls for individual bands. This page focuses on software based equalizers for manipulating recorded (digital) audio but most of the principles remain the same. First some categorization:

1. Simple: An equalizer which attempts to fulfil some end-user driven function and as such tends to have simplistic effect labeling. It is typified by the 2 band (normally labelled Bass and Treble) and the 3 band equalizer, normally labeled Bass, Mid and Treble which is based on 3 decades (10:1 ratios) of 20Hz - 200Hz (Bass), 200Hz - 2 KHz (Mid) and 2KHz to 20 KHz (treble). Controls are usually software sliders or knobs mimicking their real world counterparts. These equalizers have fixed functionality (or presets) and rarely come with any documentation describing the frequencies being affected. Boost (gain) or suppressing (attenuation) scales tend to be limited to + and -. More feature rich simple equalizers will label the effects, for example, iTunes use of Vocal Booster, Dance etc.. Use of these equalizers requires an act of faith in assuming that the designers/developers selected sensible frequency ranges. Nothing wrong with a simple equalizer if you get the the desired result.

2. Band Equalizers: These equalizers control specific frequency bands and allow fine-grained control over the gain (boost) or suppression (attenuation) within the bands. Gain/Attenuation ranges will vary from +-6 dB to +-24 dB or even greater. The frequency bands cover the full audio range of 20Hz to 20Khz and are typically based on an Octave (9 to 11 bands), 2/3 octave (15 - 17 bands) and 1/3 octave (30 - 31 bands) being the most common. (Note: There are also 17 - 22 band equalizers which may be 1/2 octave or use frequency ranges defined by the supplier). With modern software 1/6, 1/12 and 1/24 octave or even higher equalizers are possible but require some serious thought about the user interface since a 1/24 octave equalizer covering the entire audible region would have more than 200 bands to control! Equalizers that support bands lower than an Octave are frequently called Fractional Octave Equalizers. Most band equalizers label their frequency bands according to the ISO Preferred Frequency standard (ISO 266:1997 or ANSI equivalent S1.6-1984) and use appropriate standard methods for calculation of center. The Preferred Frequency specification contains both a Preferred Frequency value and a Calculated Center value. Either may be used according to the desired accuracy.

3. Harmonic Equalizers: The term is relatively new to acoustics and historically was typically used to describe power and optic rectification systems. In principle the term can be applied to acoustic equalizers having similar properties to band equalizers but which allow the user to control the harmonics (and overtones) using some form of instrument specific profile describing the harmonic relationships. If a band is boosted and the profile is, say, a piano, then the corresponding harmonic (and overtone) frequencies can be boosted (or attenuated) automatically by some proportion based on the instrument's harmonic profile and the detected audio material. Thus if, say, the band 250Hz is boosted by 10dB and C4 (262Hz) is detected in the audio stream then the 2nd harmonic (at 524 Hz) would be boosted by, say, 30% or to a 30% level relative to the adjusted fundamental.

4. Enhancement: These equalizers have similar properties to harmonic equalizers but allow the user to add harmonic (and overtone) material based on some form of typically instrument profile. If a band is boosted and the profile is, say, a saxophone then the corresponding harmonic (and overtone) frequencies are added (if necessary) automatically to some proportion of the fundamental tone based on the profile and the detected audio material. Thus if, say, the band 250Hz is boosted by 10dB and C4 (262Hz) is detected in the audio stream then the 2nd harmonic (at 524 Hz) would be added (if required) or boosted to make it, say, 30% relative to the fundamental (and so on through the various harmonics. Enhancers are clearly controversial since they can add audio material which was not present in the captured recording whereas classic equalizers merely manipulate material that exists in the audio stream.

Preferred and Calculated Center Frequencies

ISO band equalizers normally allocate (and label) the bands based on the ISO Preferred Frequencies (defined in ISO R 266-1997 or ANSI equivalent S1.6-1984). Center frequencies may be Preferred or Calculated (the later occasionally referred to, somewhat misleadingly, as Exact Centers). The Calculated centers for each band are computed, starting from a base frequency of 1,000 Hz, using one of two (base 10 and base 2) standard algorithms and the resulting frequency value is compared with a table of Preferred value to find the closest Preferred frequency match (the tables are Renard number series called R5, R10, R20, R40 or R80 and each is used depending on the fractional octave value). At one level the Preferred value is simply a convenience for we simple mortals since it is typically a nice rounded value but at another level can, at the users discretion, be used for all subsequent computations. Much of the literature suggests that only the Calculated centers should be used for this purpose. This is not what the standards say. However they also say that if serious (up to 5 decimal place) computation is being performed this should be done using the Calculated values. Clearly the centers (Preferred or Calculated) defined are centers of a frequency band. The standards, however, appear entirely silent on the topic of edge/crossover band frequencies and their calculation which seems, on its face, a tad forgetful.

Equalization strategies within the bands can vary significantly. The band can be uniformly boosted across its frequency range which can lead to abrupt changes in the adjacent bands. Alternatively the center of the band can be boosted to the full gain and attenuated toward both edges which can result can result in very peaky equalization. Perhaps with historic analog equipment these were the best possible outcomes. Digital techniques can bring a totally different set of control functions from the simplest which takes into account adjacent settings through harmonic profiles to perhaps automatic equalizers which can react in real time according to a given set of parameters describing what to at different frequencies and dB levels. Figure 1 crudely illustrates some possible strategies.

Figure 1 - EQ Strategies

The strategy labeled Next seeks to take into account neighbouring band EQ values and build attenuation/boost characteristics to ensure a smooth transition between bands.

Frequency Bands

ISO R 266-1997 (and equivalent ANSI S1.6-1984) defines the Preferred Frequencies (and a convenience Band number), their associated band range and center frequency based on a starting point of 1000 Hz (1 KHz). The term Preferred Frequency simply refers to a convenient label for we limited humans to work with whereas the center frequency is a precise value which is used for all computations. Thus, for example, Band 12 has a Preferred Frequency (a label or sometimes referred to as the Nominal Center Frequency) of 16Hz but a computed center frequency of 15.85 Hz.

Warning: The following information is, to the best of our knowledge, correct. If this stuff is vitally important the source documents should always be consulted directly (and require that you pay handsomely for the privilege of doing so). If you do notice and error please take the time - using links at the top or bottom of every page - to let us know.

ISO 1/3 Octave Frequency Bands

The ISO 1/3 Octave Preferred Frequency Table is shown below:

Band No.	Preferred (Hz)	Calculated Center (Hz)	Band Range	Notes
1	1.25	1.26	1.12 - 1.41
2	1.6	1.58	1.41 - 1.78
3	2.0	2.0	1.78 - 2.24
4	2.5	2.51	2.24 - 2.82
5	3.15	3.16	2.82 - 3.55
6	4	3.98	3.55 - 4.4
7	5	5.01	4 - 6
8	6.3	6.31	6 - 7
9	8	7.94	7 - 9
10	10	10.0	9 - 11
11	12.5	12.59	11 - 14
12	16	15.85	14 - 18
13	20	19.95	18 - 22	Start of audible range
14	25	25.12	22 - 28
15	31.5	31.62	28 - 35
16	40	39.81	35 - 45
17	50	50.12	45 - 56
18	63	63.10	56 - 71
19	80	79.43	71 - 90
20	100	100.0	90 - 112
21	125	125.89	112 - 140
22	160	158.49	140 - 179
23	200	199.53	179 - 224
24	250	251.19	224 - 282
25	315	316.23	282 - 353
26	400	398.11	353 - 448
27	500	501.19	448 - 560
28	630	630.96	560 - 706
29	800	794.33	706 - 897
30	1000	1000.0	897 - 1121	Base for ISO Octaves
31	1250	1258.9	1121 - 1401
32	1600	1584.9	1401 - 1794
33	2000	1995.3	1794 - 2242
34	2500	2511.9	2242 - 2803
35	3150	3162.3	2803 - 3531
36	4000	3981.1	3531 - 4484
37	5000	5011.9	4484 - 5605
38	6300	6309.6	5605 - 7062
39	8000	7943.3	7062 - 8908
40	10000	10000	8908 - 11210
41	125	12589.3	11210 - 14012
42	16000	15848.3	14012 - 17936
43	20000	19952.6	17936 - 22421	Highest Audible

Notes:

1. Octaves: The blue bands show the start of each ISO Octave. Each Octave is twice the frequency of the previous one.

2. Base Frequency:Band 30 (1,000 Hz or 1 KHz) is the base frequency for the ISO (and ANSI, BSI etc) Octaves rather than the - perhaps - more obvious 1 Hz starting point. The reason being that frequencies around the 1 KHz range (~400 Hz to ~5 Khz) are more sensitive acoustically and therefore require the greatest accuracy.

3. Audible Range:The nominal Audible range starts with Band 13 and finishes with Band 43. The audible frequency range is covered by just over 10 Octaves.

4. Extracting 2/3 Octave Centers: The 2/3 Octave center frequency sequence can be extracted from this table by starting from 1000 Hz and taking every 2nd entry giving a set of Band Numbers of 14, 16, 18,20,22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42 to cover the audible range. Obviously the band range will have to be re-computed as discussed below (or use this Acoustic Calculator).

5. Calculating Additional Fractional Octaves: 1/6, 1/12 and 1/24 center frequencies can be derived for this table (or by using this calculator) by applying the formula from any start frequency (Freg):

   Freq / 10 ^ (3/(10 * N)) = Center of next lower band OR

   Freq * 10 ^ (3/(10 * N)) = Center of next higher band

   Where N is the fractional Octave value, for example, 3 for 1/3 Octave, 6 for 1/6 Octave. etc.. Example calculations:

   # calculate center of the first 1/3 octave below 1000
   Equation: Freq / 10 ^ (3/(10 * N))
   Substituting: 1000 / 10 ^ (3/ (10 * 3)) = 794.33
   # this places it in the preferred Band 29
   
   # calculate center of next 1/3 octave below band 29
   Equation: Freq / 10 ^ (3/(10 * N))
   Substituting: 794.33 / 10 ^ (3/ (10 * 3)) = 630.95
   # this places it in the preferred Band 28
   
   # calculate center of the first 1/12 octave above 1000
   Equation: Freq / 10 ^ (3/(10 * N))
   Substituting: 1000 / 10 ^ (3/ (10 * 12)) = 1059.25
   # there is no defined Band in the above table (only 1/3 octaves)
   # Preferred Frequency is taken from the R40 series (not shown) = 1060
   

6. Frequency Band Edges: When using the frequencies defined in this table, ISO 266 states that either the Preferred Center or the Calculated Center may be used depending on the required level of accuracy (where the calculated center is used up to 5 decimal places). The above table, legitimately, uses the Preferred Center value to calculate the band edges. Much of the literature, erroneously, insists on use of the Calculated (or exact) center. We would argue that for analog purposes and modest digital systems the Preferred Center provides sufficient accuracy. On the other hand precision instruments or advanced digital processing systems should probably use the Calculated Centers. For further discussion on this topic and additional problems arising from this seemingly modest topic, see note 7 of the Center Frequency Calculator.

---

Problems, comments, suggestions, corrections (including broken links) or something to add? Please take the time from a busy life to 'mail us' (at top of screen), the webmaster (below) or info-support at zytrax. You will have a warm inner glow for the rest of the day.