dBs Made Simple (Jan. 1975)

Home | Audio Magazine | Stereo Review magazine | Good Sound | Troubleshooting

by Herman Burstein

IN THE popular literature for audiophiles, writers have generally skirted the task of explaining the decibel (dB). They have contended that a full explanation involves logarithms and therefore is too technical for pages such as these.

To this I am compelled to reply NOT SO! First, the decibel is well within the intellectual grasp of those who read these pages; it can be explained and understood without re course to logarithms or other mathematical concepts more profound than multiplication. Second, an under standing of the ubiquitous decibel is too important to the audiophile to deny him a full explanation. Where, in any discussion of audio equipment's specifications and performance, does the decibel fail to appear? The essential meaning of the decibel can be summed up in two brief statements, on which we will en large:

1. The decibel denotes a ratio between two amounts of power--electrical or acoustic power.

2. The decibel rests on the concept of multiplication as the way to get from a small magniture (of power) to a large magnitude (of power). Successive additions of decibels denote successive multiplications.

If Harry has twice as much money in the bank as Tom, the ratio between their respective ban accounts is 2. If Jack makes thre: nes as much money this year as last year, 3 is the ratio between his spending power this year and last year.

Now let's talk about audio power. If Amplifier A can produce 10 times as many watts as Amplifier B, the power ratio is 10. If Amplifier C can produce 50 watts at 1,000 Hz but only 20 watts at 30 Hz, the power ratio is 2.5. The notion of a ratio between two amounts of power is straightforward and simple.

In going from a low number to a high number, we can do so slowly by adding. Thus we can go from 2 to 16 by adding: 2+2+2+2+2+2+2+2=16.

Or we can proceed more swiftly by multiplying: 2x2x2x2=16. In the first case we are repeatedly adding a constant factor, 2. In the second case we are repeatedly multiplying by a constant factor, also 2. When we use the decibel, we are multiplying by a constant factor.

Specifically, 10 dB signifies multiplication by a factor of 10. Every 10 dB signifies another multiplication by 10. For example, what does 20 dB signify?

20 dB = 10 dB + 10 dB, which in turn signifies 10 x 10, which equals 100. If Amplifier A can produce 10 dB more power than Amplifier B, this says that A can produce 10 times as much power as B; in brief, the power ratio is 10. If Sound C is 20 dB louder than Sound D, this means that C is producing 100 times as much acoustic power as D; the power ratio is 100.

This simple yet very basic know ledge can be put in the form of Table 1A. In fact, we can put it into the form of a brief rule, which applies when converting 10 dB steps into power ratios: For each 10 dB, add a zero to the number 1. Thus 10 dB represents a power ratio of 10; 20 dB represents a ratio of 100; 30 dB represents a ratio of 1000; etc.

(Another, but seldom used, term for 10 dB is 1 bel. This was named in 1928 in honor of Alexander Graham Bell.)

What power ratio does 1 dB represent? The answer is 1.26 (more exactly, 1.25893). An explanation follows.

Keep in mind that the decibel represents a process of successive multiplications. 100 dB is produced by 10 steps of 10 dB each; correspondingly, a ratio of 10,000,000 is produced by 10 successive multiplications by 10. In parallel fashion, 10 dB is produced by 10 steps of 1 dB each; correspondingly, a ratio of 10 is produced by 10 successive multiplications by 1.26 (If you have doubts, borrow a calculator for a few moments to check the result of 10 successive multiplications by 1.25893.) Our knowledge about the meaning of 1 dB can be put in the form of Table 1B. Together, Tables 1A and 1B fully equip us to translate decibels into power ratios:

* Assume a figure of 52 dB. Table 1A shows that 50 dB represents a power ratio of 100,000, while Table 1B shows that 2 dB represents a power ratio of 1.58. Adding dB signifies that we are multiplying the corresponding ratios.

Therefore we have: 52 dB = 50 db + 2 dB = 100,000 x 1.58 = 158,000. In sum, 52 dB represents a power ratio of 158,000.

* Assume a figure of 47 dB: 47 dB = 40 dB + 7 dB = 10,000 x 5.01 = 50,100 (power ratio).

* Assume a figure of 134 dB. This exceeds the scope of Table 1A. How ever, the table note states that for each 10 dB we add a zero to the number 1. Therefore: 134 dB = 130 dB +, 4 dB = 10,000,000,000,000 x 2.51 = 25,100,000,000,000 (power ratio).

Note how useful the decibel is in succinctly expressing very high power ratios. In the preceding example, 134 dB is a much more compact statement than a ratio of 25,100,000,000,- 000.

Table 1A--Translation of decibels into power ratios in steps of 10 dB.

Table 1B--Translation of decibels into power ratios in steps of 1 dB.

Examples of the Decibels Use

* It has been stated that the dynamic range of a symphony orchestra is about 110 dB. This denotes the ratio between the loudest and softest passages played. The corresponding power ratio is 100,000,000,- 000!

* A tape deck claims a signal-to noise ratio (S/N) of 63 dB. Specific- ally, if a 400 Hz tone is recorded..` a level resulting in 3 percent harmonic distortion on the tape, then in play back the desired audio signal is 63 dB above the undesired noise produced by the tape system. 63 dB denotes a power ratio of 2,000,000. (In low-price tape decks, S/N ratios of 50 dB are not uncommon. Noise tends to be quite apparent in such decks, even though the audio signal contains 100,000 times as much power as does the noise.)

* Frequency response of a speaker is stated to be 7 dB down at 30 Hz relative to 1,000 Hz. If equal amounts of electrical power are fed to the speaker at 30 and 1,000 Hz, the acoustic power produced by the speaker is 7 dB less at 30 Hz. (Stated conversely, it produces 1/5th as much power at 30 Hz as at 1,000 Hz.)

* Phono discs contain a large amount of treble boost. The RIAA phono equalization standard re quires the playback amplifier to supp ly a prescribed amount of compensating treble cut, starting at about 2,100 Hz and increasing steadily there after. A table or graph shows RIAA treble cut reaching nearly 14 dB at 10,000 Hz. In terms of electrical or acoustic power, this denotes approximately 25 times as much power at 1,000 Hz as at 10,000 Hz; or, conversely, 1/25th as much power at 10,000 Hz as at 1,000 Hz.

How Loud Is a Decibel?

For most program material, such as rock, pop, or classical music, a power increase of 1 dB to 1.26 times its starting level-tends to be inaudible.

It has been observed that volume must be increased about 3 dB in order for the human ear to have a definite impression of an increase in loudness. Even so, a 3-dB volume increase produces only a slight rise in apparent loudness.

Yet 3 dB represents a doubling of power. The lesson is that great in creases in power are required to pro duce substantial increases in apparent loudness. If one considers a 30-watt amplifier to have insufficient power and replaces it with a 60-watt amplifier of otherwise equal quality, one can achieve but a slight rise in maximum undistorted sound level. For a hefty lift in sound level, one might have to go to an amplifier of 300 watts or more. This would amount to a 10-dB increase over the 30-watter, yet still would not permit a "great" change in apparent loudness. A rise of 10 dB sounds to the human ear more like a doubling of the sound level than like a multiplication by 10.

The decibel appropriately describes how the human ear responds to changes in acoustic level. The ear interprets equal increases in decibels as approximately equal increases in apparent loudness. Going from 1 watt to 2, from 2 to 4, and from 4 to 8-in each case an increase of 3 dB, or doubling of power-tends to sound like a series of equal increases in loudness. But going from 1 watt to 2, from 2 to 3, and from 3 to 4--in each case an increase of 1 watt--would sound like successively smaller increases in loudness. The increase from 3 watts to 4 might well be inaudible. Not long after, an increase of 2 watts would be inaudible; then one of 5 watts; etc.

Negative Decibels

Sometimes decibels are presented as negative numbers. For example, a preamplifier's specifications might state that noise is -70 dB for high level inputs (such as tuner). The negative figure merely indicates that one is comparing the smaller with the larger power, rather than the other way around. In our example, we are in formed that noise produced by the preamp is 70 dB less than that of the desired audio signal. 70 dB means a power ratio of 10,000,000 between the audio signal and the noise.-70 dB means that the noise power is 1/10,-000,000th as great as the signal power.

Another way of viewing the negative decibel is to consider it as representing division instead of multiplication. If 10 dB means multiplying by 10, then -10 dB means dividing by 10, so that power is reduced to 1/10th its original level. In the preceding example, -70 dB signifies that in order to arrive at the power of the noise, the power of the audio signal is divided by 10,000,000.

Voltage Ratios

Ultimately we are concerned with the acoustic power produced by the speaker; and before that, the electrical power produced by the amplifier. Therefore the relationship of primary interest to the audiophile is the one between decibels and power ratios. However, in earlier stages of the sound chain (tuner, tape deck, preamp, etc.) electrical considerations and measurements tend to be primarily in terms of voltage rather than power. Hence the need arises to interpret decibels as voltage ratios, particularly on the part of the engineer and technician, but also by the audiophile who assembles kits or otherwise tinkers with equipment.

Electrical power involves both voltage and current: power = voltage x current. When voltage in creases, current tends to increase proportionately. If voltage doubles, current also doubles. But power goes up four-fold, since 2 x 2 = 4. This illustrates a basic phenomenon: power varies with the square of the voltage change. Thus, if voltage increases by a factor of 10, power increases by a factor of 100-the square of 10. Conversely, the increase in voltage is the square root of the increase in power.

If power increases 16-fold, voltage increases 4-fold. (All the foregoing assumes no other changes in the electrical circuit.)

Assume a 20 dB increase in power, representing a power ratio of 100. But the corresponding voltage ratio, namely the square root of 100, is only 10. Thus a voltage ratio of 10 corresponds to 20 dB. Every 20 dB de notes multiplication of voltage by 10.

Similarly, if 1 dB represents a power ratio of 1.26, 1 dB represents a volt age ratio of 1.12 (more accurately, 1.12202); 1.12 is the square root of 1.26. We can put all this together in Tables 2A and 2B, which fully equip us to translate decibels into voltage ratios:

* Assume a figure of 52 dB. 52 dB = 40 dB + 12 dB = 100 x 3.98 = 398 (voltage ratio).

* Assume a figure of 135 dB. 135 dB = 120 dB+15dB = 1,000,000 x 5.62 = 5,620,000 (voltage ratio).

Converting Ratios Into Decibels

Ordinarily the audiophile is more concerned with translating decibels into equivalent power ratios than the other way around. However, conversion of power ratios into decibels can be easily done, using Tables 1A and 1B:

* Assume a power ratio of 200. What is the corresponding number of dB?

First, 200 = 100 x 2. Since a power ratio of 100 corresponds to 20 dB (Table 1A), and a ratio of 2 corresponds to 3 dB (Table 1B), we have: 200 = 100 x 2 = 20 dB + 3 dB = 23 dB.

* Assume a power ratio of 12,000.

12,000 = 10,000 x 1.2 = 40 dB + 1 dB= 41 dB. (This is an approximate answer rather than an exact one, because 1 dB represents a power ratio of 1.26 rather than 1.2. But the error is not serious; the exact answer would be 40.79 dB.) Similarly, one can convert voltage ratios into dB, using Tables 2A and 2B:

* Assume a voltage ratio of 700. 700 = 100 x 7 = 40 dB + 17 dB = 57 dB.

(The exact answer is 56.90 dB.)

Table 2A-Translation of decibels into voltage ratios in steps of 20 dB.

Table 2B-Translation of decibels into voltage ratios in steps of 1 dB.

Accuracy of the Tables

Tables 1 and 2 permit one to translate between decibels and ratios with sufficient accuracy for most practical purposes. Their compactness and ease of use compensate for the slight inaccuracy that may arise.

However, should the audiophile insist, he can achieve greater accuracy by either of two methods: He can obtain and learn to use a table of logarithms. Or he can construct tables similar to 1B and 2B, following the same principles as used to construct those tables, except that the new tables are in steps of 0.1 dB.

To do the latter, he need only equip himself with a calculator that can do multiplication, and note two essential items of information:

1. 0.1 dB corresponds to a power ratio of 1.023293, so that 10 successive multiplications by this number result in 1.25893 (power ratio represented by 1 dB).

2. 0.1 dB corresponds to a voltage ratio of 1.01158, so that 10 successive multiplications by this number result in 1.12202 (voltage ratio represented by 1 dB).

The "Absolute" Decibel Sometimes we encounter statements that suggest the decibel is an absolute measurement in the way that watts, volts, amperes, etc. are absolute measurements. Thus, we are told that the sound level is about 90 dB in a noisy factory, about 110 dB on an orchestral peak, about 120 dB on a rock concert peak (or throughout much of the concert), about 45 dB in a typical residence, etc.

In fact, though, the decibel has no absolute meaning. It does not refer to a specific sound level. As stated at the outset, it denotes a ratio between two amounts of power.

The statements about sound levels in such places as the home, factory, etc. are based on an implied ratio, namely between the cited sound level and a standard reference level. The reference is the sound level at the lower threshold of human hearing.

The reference is a sound wave with an intensity of 0.000,000,000,000,000,1 watt per square-centimeter. That is, one-tenth of one-quadrillionth of a watt barely escapes detection by the human ear.

A sound level of 90 dB, as in a factory, refers to acoustic power that is 1,000,000,000 times as great as the reference level of power; in other words, .0000001 watt per square-centimeter. A sound level of 130 dB, which represents the upper threshold of human hearing (higher levels are felt rather than heard), denotes .001 watt.

This seems very little, yet relatively brief exposure to a level of 130 dB, as can happen near a jet aircraft or in an up-front seat at a rock concert, can temporarily or permanently impair one's hearing.

What Does 0 dB Mean?

0 dB does not signify the absence of power. It means that power is unchanged, or that two amounts of power (other than zero) are equal.

To indicate the absence of power, we simply state that power is zero.

Decibels do not get into the act in this case.

(Source: Audio magazine, Jan 1975)

Also see: Speaker Tests--Phase Response (by Richard C. Heyser) (Dec. 1974)

= = = =

Prev. | Next

Top of Page    Home

Updated: Monday, 2018-02-19 8:19 PST