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 RF-man

Decibels

If decibels and logs confuse you, here's one last try at explaining it! :o)

dB this, dB that! Why?

The key to understanding decibels is to get to grips with logs (logarithms), which may appear frightening, but are actually quite harmless. Consider the following table :

1 10
2 100 
3 1000
4 10,000
5 100,000

I'm sure you've spotted what's going on, there. 10 to the power of 2 (i.e. 10 squared) is 100. 10 to the power of 3 (i.e. 10 cubed) is 1000, etc.

Lo and behold, the 'log' (in base 10) of 100 is 2, the log of 1000 is 3, etc. The log of a number is simply a measure of the magnitude of that number, how many powers of 10 are involved, how far removed in steps of multiplying by 10.

Let's have a closer look...

log
-3  0.001
-2  0.01
-1  0.1
 0  1
 1  10
 2  100 
 3  1000
 4  10,000
 5  100,000

... so it goes the other way too, as we keep dividing by 10 the logs decrease into negative values. I can't easily explain how 10 to the power of 0 is 1, but you'll just have to accept that it makes sense to mathmeticians ... it all looks fine when plotted on a graph ... the table above is perfectly reasonable and easy to follow (assuming you're happy enough with decimal numbers!)

(if you were wondering what happening to Square Roots, that is actually the power of 1/2 (half) - try 10 to the power of 0.5 on your calculator, you'll get the same 3.16 result as the square root button. Yes, it IS a nasty concept and I'm as confused as you are, so don't let it put you off!)

So if a graph is plotted of numbers and their log values, the resulting line (a bit of a curve!) can be used to find what the log is for ANY number, not just nice round powers of ten. Of course a calculator is much easier :o)

     log
   1 0
   2 0.3
  10 1
  20 1.3  (10 to power of 1.3 = 20)
  40 1.6
  80 1.9
 100 2    (10 to power of 2 = 100)
 200 2.3
1000 3    (10 to power of 3 = 1000)
2000 3.3

You just have to be willing to accept that 10 to the power of 1.3 is 20 - it's nasty to understand but the graph says so! Note how each time a number is doubled, the log goes up by 0.3 ...

It actually works out rather easily : multiply a number by 10 and its log goes UP by 1, whereas DIVIDE a number by 10 and its log goes DOWN by 1, so it all makes sense really. Multiply by 100, up goes the log by 2.

Here's where decibels come in. If you amplify a signal by 100 times, the log of the amount of power increases by 2. A way of explaining gains and losses to an amount of power in this way (only applies to power *), is to use a unit named a bel (symbol B), the name bel was coined in the early 20th century in honour of the telecommunications pioneer Alexander Graham Bell. The signal is now 2 bels stronger. However, in the real world it turns out to be a lot easier to work in tenths of bels, and as the metric prefix for a tenth is deci, we end up with dB decibels (similarly a dm decimetre is a tenth of a metre). Our signal, 100 times stronger than it was, could be said to have risen by 2B, but you won't find whole bels used - more conventionally we would say it has risen by 20dB (or gained 20dB).

* you may see decibels used with voltages, but the numbers are adjusted such that they correspond to the correct rise/fall in power

We find dB being used when we record our amateur radio power in our logbooks using dBW - the number of dB greater than 1 Watt. 10W is 10 times (1B or 10dB) more than 1W for example, so 10W is 10dBW. 2W is 2 times (0.3B or 3dB) more than 1W, so 2W is 3dBW (remember above where a doubling made a change of 0.3 to the log?!). Half a Watt is 2 times LESS than 1W, so it is -3dBW. 4W is 1W doubled and then doubled again (3dB gain plus another 3dB gain), so 4W is 6dBW. Here is where it starts to make sense that the log of 1 is 0 ... a 1W signal is NO decibels at all (0dB) stronger or weaker than 1W! If you get confused, use the formula :

dBW = 10 x log(Watts)

Or, on a scientific calculator :

enter the Watts ... press [log] ... [X] [1] [0] [=]

Once you've got the hang of it, you'll see you can often get by (or get close enough) by just memorising a few combinations of gain/loss ratio versus dB gain/loss, i.e. the ratio of 10 and 10dB, 2 and 3dB (and logically enough, 4 and 6dB, 8 and 9dB etc). Multiplying by 5 is very close to gaining 7dB, but the remaining numbers 3,6,7 and 9 as ratios are not close to whole numbers of dB :

multiplication factors as an increase in dB
x 1       0dB
x 1.26    1dB
x 1.58    2dB
x 2       3dB (3.01)
x 2.51    4dB
x 3       4.77dB
x 3.16    5dB
x 4       6dB
x 5       7dB (6.98)
x 6       7.8dB
x 6.31    8dB
x 7       8.45dB
x 8       9dB
x 9       9.54dB
x 10     10dB

x 20     13dB  ( x10 then x2 = 10dB plus 3dB)
x 100    20dB
x 400    26dB  ( x100 then x4 = 20dB plus 6dB)

It's as simple as that! Once you understand the beauty of it, you'll probably find it's easier to say a signal is "150dB weaker" than "1,000,000,000,000,000 times weaker". Make it twice as strong again and you just add 3dB (-147dB now) rather then having to divide 1,000,000,000,000,000 by two in your head!

In the real world it makes all sorts of huge numbers fit into a small range of digits. For example, a test set measuring cell phone base station signal strengths might display -20dBm (100 times less than a milliwatt) right next to a base station, all the way down to -110dBm when the signal is so weak it is not deemed fit to use for GSM. That's a collosal range of 90dB (1,000,000,000) and yet it can be all be displayed with just 4 characters on the display (assuming that the user knows that -102 is meant to mean -102dBm).

Hopefully you now know why putting your power up from 10W to 100W (10dB gain) makes the same RELATIVE difference as from 100W to 1000W.

It is very handy to use dB figures when taking into account the many gains and losses in a system. Instead of working with many multiplications, the nature of the dB means we can simply add and subtract instead.

Suppose you start with a 1W UHF transmitter, boosted by a 10W amplifier. You connect an antenna with x8 forwards gain, with a run of feeder cable that is long enough to lose you half the power. Your signal travels far enough to a distant receiver that the signal is a mere 1,000,000th of what it was at the sending antenna. What a pain to think in terms of multiplying by 10, dividing by 2, multiplying by 8, dividing by 1,000,000 to consider the received signal strength. Much easier to add the dB figures together : +10 -3 +9 -60 dB etc. Any improvements made to the system such as using a shorter run of better coax, or raising the power, mean that you can simply add the extra dB improvement figure directly to the existing total. A proposed system can be easily budgeted by considering all the gains and losses from one end of the link to the other (known as a Link Budget) to see if there is a sufficient overall gain to make communication possible. It is simply easier to work with decibels rather than with multiplication ratios.


dB Volts

dB are only used for POWER related comparisons - you wouldn't say that 2km is 3dBkm. However, you may find dB used with voltages. In this case the number of dB is adjusted to reflect the actual powers involved. Because power is proportional to the SQUARE of voltage *, the voltage decibels are double - 6dB means twice the voltage because this results in 4 times the power (x2x2 = 3dB+3dB). So a 20 appears in formulas, rather than a 10 :

dBuV = 20 x log(Voltage/0.000001)

* remember the formula "power = voltage squared divided by resistance"

As an example, a voltage is amplified 5 times (the power rises 25 times). You can either take the log of 5 (0.7 approx) and multiply by 20 to get 14dB, or you can square 5 to get the power increase straight away... the log of 25 is 1.4 (approx), so it's still a 14dB gain in power either way.

A 10 times voltage increase makes a 20dB gain (100 times more power).

Note that we are using base 10 logs here. A log can be expressed relative to any number, in fact many calculators have a [ln] key for logs in base "e", where e (exponential) is a "natural" constant (Euler's) of about 2.718282 - don't ask! You only hear scary stuff like 'the expression "(1+1/m) to the power of m" approaches e' and 'ln(x) is the integral of 1/x' - yuk!


dB milliwatts : dBm

A useful unit for radio power, as the range from 1,000,000 W all the way down to the lowest level of noise possible (in 1 Hz bandwidth at Absolute Zero) is contained numerically within the range of +90 to -199 dBm.

There are 1000 mW to the Watt, so dBm is always 30dB more than the equivilant dBW figure.

---Effective Radiated Power---  ------1km away------
dBm dBW  Example                W/sq.m *      FS **
+90  60  1MW Broadcast          80mW          5.5V/m
+82  52  160kW Ku satellite TX  12.8mW        2.2V/m
+60  30  1kW                    80 uW         174mV/m
+56  26  400W amateur limit     32 uW         110mV/m
+50  20  100W                   8 uW          55mV/m
+44  14  25W Marine VHF/PMR     2 uW          27.5mV/m
+40  10  10W                    0.8 uW        17.4mV/m
+36   6  4W CB                  0.32 uW       11mV/m
+30   0  1W                     0.08 uW       5.5mV/m
+27  -3  0.5W FRS/PMR446        0.04 uW       3.9mV/m
+10 -20  10mW radiomic / LPD    0.8 nW        0.55mV/m
 0  -30  1mW=OdBm, no change relative to 1mW

---power in feeder at receiver---
dBm
-33    S9+60dB  (VHF/UHF)
-63    S9+30dB
-93    S9
-99    S8
-105   S7
-111   S6
-117   S5
-119   typical NFM sensitivity for 12dB SINAD
-123   S4
-129   S3
-135   S2
-133.6 thermal noise floor (FM 11kHz bandwidth)
-139.5 thermal noise floor (SSB 3kHz bandwidth)
-141   S1
-147   thermal noise floor (CW 500Hz bandwidth)
-174   noise level per Hz at 17C (290K)
-198.6 Boltzmann's Constant: lowest noise level per Hz at abolute zero


*  Power density in Watts per Square Metre
   in freespace, at 1000m distance, due to
   spreading loss, at the given isotropic ERP

** Equivilant Field Strength in Volts per Metre (at 1km)
   (sq.root of: the Power density times 377 Ohms impedance)

The field strength is the root of Power density times the 377 Ohms impedance of freespace (120 x Pi), because
voltage = square root of (power x resistance)

Spreading Loss is all about how the power from a point in freespace is spread over an imaginary sphere. A sphere of radius 1000m has a surface area of 4 x Pi x 1000 x 1000 = 12,566,371 square metres, so the power is divided over this area and consequently any square metre of this 'surface' gets 1/12,566,371 of the power radiated from an isotropic radiator (which radiates equally in all directions), 71dB less than the ERP.

To work out the power density (or field strength) at other distances, consider that a sphere twice the radius (or diameter) of a smaller sphere has FOUR times the surface area - so there is 'square law'. At 10km then, there is 100 times less power per square metre (and so 3.16 times less voltage field strength; the square root of 10), at 10 times closer (100m with our table) there is 100 times more. Frequency is irrelevant, a Watt is a Watt no matter what the wavelength is - however a dipole antenna resonant at a particular freequency will pick up less signal the higher the band, because the dipole will be smaller.

Received signal levels in cable connected to an antenna will thus vary depending upon 'aperture' (or capture area) which is wavelength squared, divided by 4 pi for an isotropic antenna; for example -21dB at 950 MHz, -4.67dB at 145 MHz, +9.6dB at 28 MHz (dB losses rise by 20 times the log of frequency rise). So the power density in dBm/sq.m plus the antenna aperture, gives the received signal strength. Alternatively working in voltages, we can use 'antenna factors' which amounts to the same thing as aperture, taking the antenna size into account and a conversion from the 377 Ohms impedance 'in the air' down to typically 50 Ohms in one go.
Antenna Factor = 20 Log (FreqInMHz) - 29.79 - GainOverdBi
i.e. 11.3dB for a dipole (2.15dBi) at 145MHz.


Audio dB

Most people associate the dB with loudness - the POWER of audio. When someone with a handheld measuring device tells you your test shout was 90dB this is technically meaningless because this is simply saying that it was 1,000,000,000 times louder than something, something not actually specified by just saying xx dB without saying xx dB relative to what. A dB figure is a ratio, a comparison, and is only ever properly used to compare one power amount to another. In audio terms it's actually all relative to the quietest sound anyone can detect if made in an otherwise absolutely silent room, but the proper dBA term is shortened to just dB.

194  Maximum possible (from atmospheric 14.7psi down to 0psi)
177  Record for car audio!
170  Shotgun blast up close
160  Perforation of eardrum
140  Jet Aircraft Taking Off
120  Human Threshold of Pain - 1 watt/sq. meter
120  Loud Rock Concert
110  Moderate rock concert, dance club
100  Motorcycle
 -- extended listening above 85-90dB leads to hearing loss --
 90  Lawnmower, loud home stereo
 85  Jackhammer at 15 meters (50 feet)
 80  Moderate home stereo, ringing telephone
 75  Average City Street
 70  Freeway traffic, TV audio
 60  Normal Conversation
 50  Large office background noise
 40  Quiet office or residential area
 30  Whisper at 3 meters (10 feet), Very soft music
 20  "Silent" TV Studio, Whisper at 1 meter, Quiet living room
 10  Soft rustling of leaves
  0  Human threshold of hearing (youths)
       10 to the -12  Watts per square metre


Light

Light isn't usually measured in dB, but it's still a POWER, and it may be interesting to see the range involved. Light is measured in all sorts of complicated ways, with Lumens of power being radiated from light sources being not as important as what falls upon, or is reflected from, surfaces. Unless you take a lot of pictures of flames, lamps or take ill-advised risks with the sun, you'll more usually be dealing with light levels falling onto other objects that reflect the light back. Luminance is the light coming FROM a surface, which is ILLuminance (light falling ON to a surface) times the reflective factor.

Illuminance is the equivilance of power density in radio terms - the power falling upon on a square metre. The amusing old term Footcandle was 1 Lumen on a square foot of surface, now metricated into Lux (1 Lumen on 1 square metre) with 10.76 FC = 1 Lux because there are 10.76 square feet in one square metre. I'll stop at around this level of complexity because if you try to understand candelas, steradians and the like, it may make your head hurt at the first attempt (as it still does for me).

Out of interest, apparently the equivalent of 1 Lux unit in Watts is a matter of human perception; the response of our eyes varies with wavelength. A webpage says that 0.0029 Watts of green light, 0.0015 Watts of yellow light, or a whopping .015 Watts of red light per square metre will provide an illuminance of 1 Lux. The strength of the Sun (white light) at planet Earth's distance is 62,840,000 Watts per square meter, with a Watts-Lux factor of about 0.002 giving 127,000 Lux. Then I found "In very approximate terms it can be assumed that 270 Lux equates to 1 Watt/M2 light power over a wide spectrum." ... argh!!

As for typical levels of light in Lux, not all webpages agree. The following table lists what seems to be a general consensus.

EV (aprx) Lux 
22 10,480,000 Extremely bright. Rarely encountered in nature. (22-23 EV)
17    328,000 Rarely encountered in nature, some man made lighting. (17-21 EV) 
16    164,000 Subjects in bright daylight on sand or snow. Maximum sun light = 127,000 Lux
15     82,000 Subjects in bright or hazy sun (Sunny f/16 rule). 
14     41,000 Full moon (long lens). Subjects in weak, hazy sun. UK daylight = 50,000 Lux
13     20,000 Gibbous moon (long lens). Subjects in cloudy-bright light (no shadows). 
12     10,000 Half moon (long lens). Subject in heavy overcast. Overcast day 1,000-10,000 Lux
11      5,000 Sunsets. Subjects in open shade. 
10      2,600 Landscapes and skylines immediately after sunset. Crescent moon (long lens). 
 9      1,300 Landscapes, city skylines 10 mins after sunset. Neon lights, spotlit subjects. TV studio 1000 Lux
 8        640 Bright cities at night. Store windows. Fires, night sports. Offices 500 Lux (florescents)
 7        320 Bottom of rainforest canopy. Brightly lit nighttime streets. Indoor sports. Stage shows, circuses. 
 6        160 Brightly lit home interiors at night. Fairs, amusement parks. 
 5         80 Night home interiors, average light. Auditoriums. Subjects lit by fires. 100 Lux 1m from a 100W bulb
 4         40 Candle lit close-ups. Christmas lights, floodlit buildings, monuments. Hallway lighting
 3         20 Fireworks (with time exposure). Streetlights
 2         10 Lightning (with time exposure). Total eclipse of moon. 
 1          5 Distant view of lit skyline. 
 0        2.5 Subjects lit by dim ambient artificial light.  
-1        1.3 Subjects lit by dim ambient artificial light. 
-2        0.6 Rural night, snowscape under full moon.
-3        0.3 Rural night, subject under full moon. 
-4       0.15 Rural night, subject under half moon. Meteors (during showers, with time exposure). 
-5       0.08 Rural night, subject under crescent moon. 
-6       0.04 Rural night, subject under starlight only. 
    1/100,000 starlight only?

This is a very basic table, with very vague categories, after just an hour or two of limited research... so make of it what you will. One thing's for sure, the human eye copes with a range of light levels over a million to one - although I've yet to find what range it's comfortable with within a viewed scene at any one instant. 1000:1? (60dB?!!! argh...)

I'm still researching this (actually I've given up for now) so here are some pertinent things and quotes for further confusion ... when I'm prepared to try and fathom it out.. would it make sense IN ANY WAY to think in terms of dB light levels? dB Lux?

"The typical range of brightness levels of adjacent areas that can be distinguished by eye is about 100:1 (with dark areas being swamped by bright areas). The human visual system has a very large dynamic range, but limited precision within that range."

As far as photography is concerned, a 'stop' is a basic level of adjustment that amounts to a doubling or halving of light levels.. i.e. investigate f stops, film speeds such as 100,200,400 etc., this will be a 3dB step? Exposure values (EV) also rise in sequence with a doubling of light level; Lux = 2.5 x 2 ^(EV)

"Assumptions are that for the method of reflection measurement the important parts of the scene will reflect 1/6 of (17%) of the incident light."

"The human eye does not perceive twice the intensity as being twice as bright. For a common example, photographers use their light meters on a "standard gray card" made to reflect 18% of the light falling on it. Metering from that card is used to calibrate middle gray (50% to our eye) in the hypothetical "average" scene. We see that 18% intensity as apparent 50% brightness."

Unlike film, video gain is expressed in decibels or dB, not f-stops: 6 dB equals one f-stop. Each 6db of gain will double your sensitivity, therefore double your ASA. (??!)

"A printed magazine image has a dynamic range well less than 2.0, maybe half of that (1.7). The blackest ink still reflects some light, the white paper is not so bright that it blinds us, and the difference is relatively small. Photographic color prints have a dynamic range of less than 2.0 too. Film negatives might have a range up near 2.8. Slides may be near 3.2. These are not precise numbers."

"Slides have less photographic range than negatives. Slides have perhaps about 5 or 6 f-stops of total scenic range, compared to perhaps 9 or 10 f-stops for negatives. Expose a slide half a stop off and the results are objectionable. Do that with negatives, and you never even realize it."

"But slide film itself has more contrast, a steeper gamma curve, and while the captured scenic tonal range may be less, the density extremes on the film can be greater. The extremes of slides are more likely to be clear or black, and contain greater dynamic range as seen at the scanner."


Any corrections or assistance? Please let me know!

28 Apr 2004

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