|Elliott Sound Products||Frequency & Amplitude Explained|
Frequency, Amplitude & dB
Rod Elliott (ESP)
With Thanks to Lenard Audio 
Sound is carried from the source to our ears or a microphone by means of minute vibrations, which are passed through the air. Sound has two primary components, frequency and intensity. The frequency refers to the pitch of the tone or other sound, and typical sounds have many different frequencies all happening at once. Frequencies are measured in Hertz (Hz), named after the physicist Heinrich Hertz. The old standard (now discontinued almost everywhere) used Cycles per Second (cps) as the standard measurement. Hz and cps are the same thing - both refer to the number of complete cycles of a waveform in one second.
Sound intensity (or amplitude) is measured in decibels (dB). The prefix 'deci' means one tenth. The Bel was invented by engineers of the Bell Telephone Laboratory to quantify the reduction in audio level over a 1,600m (1 mile) length of standard telephone cable, and was originally called the transmission unit or TU. It was renamed in around 1923-4 in honour of the Bell Laboratory's founder Alexander Graham Bell. Because the Bel is too large for general use, the dB became the preferred unit.
The range of frequencies we humans can hear is generally taken as being from 20Hz to 20,000 Hz (20kHz), but the conditions are not usually specified. As we get older, the first to suffer is the high frequencies, and by around 50 years of age, most males will be limited to around 14-15kHz, with females usually suffering less loss. Frequencies below 25Hz are felt rather than heard, but the conditions under which we experience such low frequencies make a big difference to how they are perceived.
Our hearing is most sensitive at around 3.5kHz, as shown in Figure 1. Our hearing, eyes and sensitivity to touch or pain, are all logarithmic functions. This enables us to experience a vast variation with each sense. As the intensity of the sense increases, we automatically compensate by reducing our sensitivity. In this way, we can hear the gentlest rustle of a leaf in a tiny breeze at a sound pressure level (SPL) of 0dB, but are not instantly deafened by a nearby jack-hammer at perhaps 1,000,000,000,000 (1 British billion, 1 US trillion, or 1 x 1012) times the sound power (120dB SPL).
When two frequencies are close to each other, our hearing plays some interesting tricks on us. If one tone is 6dB louder than the other, we will not hear the second tone. This is called acoustic masking, and is used by the MP3 format to remove a great deal of the "redundant" audio information. This reduces the size of the file dramatically, and with some of music the end result may be almost indistinguishable from the original. Material with rich harmonic structure is less successful, with cymbals and harpsichords suffering because there is simply too much information and none of it is actually redundant.
1.1 - Musical Notation
In (western) music, we generally use the equally tempered scale. While not absolutely musically accurate, it does allow musicians to make key changes (moving an entire piece of music up or down the musical scale) without having to re-tune their instruments. This is a vast topic, and requires a great deal more than you will find here if it is to be fully understood. Unless you are a musician, a full understanding is not required.
Musical notation is based on the use of 12 semitones in each octave. An octave is the perfect interval between the 1st and 8th tones of the diatonic scale. See Answers.com if you want more specific information about the diatonic scale.
In western music, each octave is comprised of 12 semitones. An octave is double or half the original frequency, so (for example) one octave from middle A (440Hz) is 880Hz or 220Hz. Both "new" notes are called A. The word octave is derived from "Octo-" (Latin/Greek) meaning eight, because the western octave is divided into 8 "full" tones in the diatonic scale.
Figure 2 shows the range - the keyboard is shown as a reference only, and is not meant to be that of a real piano. Of common musical instruments, open E on a (4 string) bass guitar or double bass has a frequency of 41.2Hz, while a grand piano's bottom A is 27.5Hz. Many instruments can get far lower - examples being pipe organs and electronic synthesisers.
High frequencies are more complex. Any note is made up from the fundamental (usually taken as the lowest frequency component of the sound - the first harmonic) and a series of harmonics above this (usually at octave intervals). While many instruments produce harmonics that are exact multiples of the fundamental, others do not. A flute also contains wind noise, reed instruments often have very complex harmonic relationships, and percussion instruments can have harmonics that are not related, but extend to well beyond our hearing range (snare drums, cymbals, etc). With many plucked or struck stringed instruments the second harmonic is dominant (louder than the fundamental). This is especially noticeable with guitar, but is apparent with many other instruments too.
The division of an octave into 12 equally spaced tones is done using the 12th root of 2 (approximately 1.0594631). If you multiply 440 by the full version of this number 12 times, you get 880 - exactly one octave (depending on your calculator). The same method may be used to divide an octave into any number of divisions - for example, 3 divisions are used for 1/3 octave band graphic equalisers. The third root of 2 is approximately 1.26 in case you were wondering
A decade (one tenth or ten times the frequency) is approximately 3.2 octaves (3.1623 or the square root of 10). Decades are sometimes used instead of octaves in engineering, although current practice prefers to use octaves.
Frequency and amplitude are inextricably coupled in the real world, with both playing an equally important role. It is only in test and measurement where these two functions are separated, and that is so we can see how one affects the other to ensure that a reasonable standard is achieved.
1.2 - Wavelength
The wavelength of any signal depends on the form of the signal (acoustic or electrical), the transmission velocity in the medium (air, concrete, an electrical wire) and the frequency. For audio, we are generally only concerned with the wavelength in air. While the wavelength of RF (radio frequency) signals in cables is usually very important, the wavelengths at audio frequencies are very large indeed. A 20kHz signal has a theoretical wavelength of 15,000 metres (15 km) as an electrical signal, ignoring other effects such as velocity factor (look it up if you are interested).
Sound in air at 22°C and at sea level has a velocity of 345m/s . The speed of sound varies markedly with temperature and is proportional to temperature, but the Hyperphysics calculator will work it out for you if you need to know exactly.
The formula to convert frequency to wavelength (commonly written as λ - the Greek letter (lower case) lambda) is ...
λ = c / f where c is velocity of sound and f is frequency
It is also useful to remember that sound travels at about 345mm / ms (both metres and 1 second divided by 1,000). Our hearing mechanism is carefully refined to ensure that sounds we hear are made as clear as possible, so we automatically reject repeat sounds (echoes) that arrive within about 30ms of the original. This allows us to hear clearly even in a reverberant room (or a cave a few millennia ago). 30ms means a distance of around 11.5 metres, meaning a
cave room of about 5 metres square. Such a room will sound somewhat odd, but speech is still clear. Larger rooms (with longer delays) can cause a significant loss of intelligibility.
Being able to calculate wavelength is very important for anyone designing loudspeakers, as there are many characteristics of a speaker box design and room placement that rely heavily on knowledge of wavelength and time delay. These topics are covered in countless white papers, articles and books, and are not relevant to the material in this article.
Most beginners in electronics find dB very confusing. This is understandable, but it is easy to learn, and is every bit as important as Ohm's law when working with electronics or loudspeakers. The main thing to remember is that 1dB remains 1dB, regardless of the context. Likewise, 6dB remains 6dB. Let's look at the formulae first (no, they are not hard - calculators do almost all the work). For those who prefer not to use a calculator, see the Lenard Audio  website.
dB = 20 * log (V1 / V2 )Where V1 and V2 are any two voltages, and P1 and P2 are any two powers (in Watts).
dB = 10 * log ( P1 / P2 )
But why are there different formulae? This is simple - power into a given impedance or resistance is determined by the square of the voltage. If 1 Volt into 1 Ohm gives 1 Watt, 2V into 1Ω gives not 2W, but 4W ( P = V² / R ). The multiplication by 10 or 20 takes this into account, so it doesn't matter if you work with power or voltage, you get the same answer in dB.
Using dB provides a convenient way to indicate very large numbers, and in a way that directly relates to the way we hear. For example, it is standard practice to measure frequency response of amplifiers, speakers and many other things at the -3dB points. Speakers are commonly quoted as (for example) 40Hz - 20kHz ±3dB. 3dB means half or double the power, or a voltage ratio of 1.414:1
That last number is a good one to remember - the square root of 2 ( √2 ) is 1.414, and it is used in many electronics calculations.
Figure 3 shows the range generally accepted as the minimum dynamic range in audio. As you can see it is vast, covering a span of 1 million to one. The total range that is of interest spans 120dB, being the dynamic range of typical good quality analogue and digital equipment. A microphone preamp may be quoted as having an equivalent input noise of -127dBm ... feel free to calculate the noise level in millivolts (it will actually be microvolts). Using dB to express such small numbers is far more intuitive than specifying the noise level as 0.346uV, which although impressively small, tells us nothing about its audibility.
Here are a couple of very useful dB facts that are worth remembering ...
3dB = half or double the power
10dB = half or twice as loud 10dB = one tenth or ten times the power
Perceived loudness is what you hear as the change, and means that if you have a 100W amplifier and you want the sound to be twice as loud, you need to use a 1kW (1,000W) amplifier to do so. Note that doubling the power results in a 3dB increase, and although audible it is not dramatic. It was determined long ago that 1dB is the smallest change that the average listener can hear. While open to some dispute at regular intervals, it still holds if the test is done with a single tone under ideal conditions.
2.1 - dB Reference Levels
While it is sometimes believed that dB is either some absolute value or a "dimensionless number", neither is correct. Many standards exist to refer to specific levels, both with sound and electrically.
2.2 - Weighting Curves
When sound level readings are taken, it is common to apply what is known as A-weighting (see Project 17 for a design and frequency response of an A-weighting filter). The A-weighting curve is designed to allow for the fact that out hearing is less sensitive at low and high frequencies, but fails to account for the actual SPL. When sound is above 100dB SPL, our hearing response is reasonably flat (see Figure 1), and the use of A-weighting is inappropriate. Under these conditions, the C-weighting curve should be used, which has an essentially flat response over the audio band.
A-weighting is also often used for measuring amplifier noise, and because this is normally only ever at very low volume, the use of the A-weighting filter is appropriate. Personally I prefer not to use it at all, but most do.
A frequency response curve is an example of the use of both frequency and amplitude, with frequency being shown on the X (horizontal) axis, and amplitude on the Y (vertical) axis.
Figure 4 shows an example of a frequency response curve, in this case taken from my Clio analyser. The source material was an FM radio tuner, and the program was set up to show the highest peaks over a 15 minute period. Note that the chart includes any equalisation applied by the radio station (I used radio Triple J as the source - they do not play advertisements, thus eliminating pollution caused by the often radical EQ and compression that is used in ads to make them sound "loud". The 19kHz FM stereo pilot tone is just visible on the right side of the graph, and you can see that the FM bandwidth is limited to 15kHz. (The pilot tone is used to identify a stereo transmission, and is used by the stereo decoder to obtain separate left and right channels.)
It is generally accepted that the overall energy distribution of music looks like that shown in Figure 5. That there will be variations is obvious, and while interesting and potentially useful, you cannot rely on any simple graph to determine how much power you need. Loudspeaker efficiency and peak to average ratio of the signal must also be considered.
Peak to average ratio is an important topic itself. Because music has dynamics (loud and soft passages), and because of the nature of a complex audio waveform, the RMS (root mean squared) voltage is useful only to get an idea of the average power delivered to a speaker. The RMS value of a sinewave is 0.707 of the peak voltage, as shown below.
You may recall that I said earlier that one should remember the number √2 (1.414). The RMS value of a sinewave is determined by dividing the peak value by 1.414, or you may multiply by 0.707 (the reciprocal of 1.414 ... i.e. 1 / 1.414 ). In Figure 6, the peak value of the sinewave is 1V, and the RMS value is 707.1mV. Most meters display the RMS voltage, but only those called "True RMS" will get the value right for a complex waveform such as that shown in Figure 7. Not that the waveform is especially complex - it is made up from 3 sinewaves, at 1kHz, 2kHz and 4kHz, all with a peak voltage of 1V.
The real RMS voltage of the waveform in Figure 7 is 1.225V. If one uses the calculated RMS voltage (based on the peak voltage of 2.33V), the answer is 1.566V - an error of almost +22% (+2.13dB). Most meters are average reading, RMS calibrated, meaning that the signal is rectified and averaged, but the meter scale is calibrated to read RMS. Such a meter will give a reading of 1.014V, a -12% error (-1.65dB). It is very easy to introduce serious errors into any calculation that involves complex waveforms, and this is one of many reasons that a reasonably pure sinewave is specified for most test procedures. While so-called "True RMS" multimeters are more accurate, most do not handle high crest factors well. The crest factor is the ratio of the peak and actual RMS values of a waveform, and to work well with high crest factors, some serious maths is generally needed. Digital oscilloscopes with voltage readouts compute the value, and will usually get it right.
|Driver Type||Minimum Frequency||Maximum Frequency|
The above table is not intended to be absolute. There are a great many factors that influence the way a driver can (or should) be used, and these are not relevant to this article. The crossover network is also subject to many variations. Apart from the choice of frequency, there is also the choice of slope (the rate of attenuation with frequency), some networks are deliberately designed to be asymmetrical, having different slopes for the high-pass and low-pass sections.
Filters are divided into three different types ...
No filter simply stops all signals above or below the specified frequency. As the selected frequency is approached, the signal level starts to reduce, and the filter frequency is usually taken as that frequency where the signal level is 3dB below the passband. There are exceptions, and these will usually be explained in the description of the network.
In order to obtain different rolloff slopes, filter "building blocks" can be connected in series to obtain a greater rate of attenuation. The commonly used filter orders are as shown below. The simplest filter is a first order, and uses one reactive component (a capacitor or an inductor). A second order filter uses two reactive elements, and so on.
|Filter Order||Rolloff Slope||Reactive Elements|
|First||6dB / octave||1|
|Second||12dB / octave||2|
|Third||18dB / octave||3|
|Fourth||24dB / octave||4|
Active filters require power - they are called "active" because they use active components, such as opamps, transistors or sometimes valves. Passive filters use only passive components - capacitors and inductors. Passive filters always have losses (especially resistance in inductors), so not all the amp power gets to the speakers. At high power levels the losses can become very high, reducing the available power for the speakers.
Active filters require a separate power amplifier for each loudspeaker driver, while passive networks use a single amp. There is a tradeoff - do we use large and expensive passive components and a single (relatively) large power amplifier, or an active crossover and a number of smaller power amps?
It depends on what we are trying to achieve, the expected performance and the budget. It would be silly to use an active crossover and separate amps for a cheap PC speaker, and equally silly to use passive crossovers in a large sound reinforcement system running at perhaps 5,000W or more. All filters (whether active or passive) will provide a rolloff slope based on the filter order. With passive crossovers, it is usually necessary to compromise because high-order filters become too expensive and consume excessive power.
All filters cause phase shift - it is a characteristic of how they function in the analogue world.
All of the examples in this section show a combination of frequency and amplitude. It must be stressed that a full and complete understanding of these topics is essential to your understanding of audio as a whole. Without that understanding, you are left wondering what certain terms really mean. You may also become less likely to believe some of the outrageous drivel that is spouted by some manufacturers - they rely on a lack of understanding to baffle people with pseudo-science.
|Copyright Notice. This article, including but not limited to all text and diagrams, is the intellectual property of Rod Elliott, and is Copyright © 2006. Reproduction or re-publication by any means whatsoever, whether electronic, mechanical or electro- mechanical, is strictly prohibited under International Copyright laws. The author (Rod Elliott) grants the reader the right to use this information for personal use only, and further allows that one (1) copy may be made for reference. Commercial use is prohibited without express written authorisation from Rod Elliott. Some parts of this article are copyright © John Burnett (Lenard Audio).|