dB and dBm theory
Definition of dB:
The dB is a logarithmic unit used to describe a ratio. The ratio may be power, sound pressure, voltage or intensity or several other things. The definition of decibel abbreviated dB, is ⎛P
a = 10 * log10 ⎜⎜ 1
⎝ P2
⎞
⎟⎟dB ⎠
Example: For example if P1= 20W and P2=5W, then P1 is four times larger than P2. In a decibel scale, we have dB = 10 * log(20 / 5)
= 10 * log(4)
= 10 * 0.602
= 6dB
Of course it is possible to convert decibels back to their liner values. We must first convert dB to Bel by dividing the value of 10. Then we must raise the number of 10 to this power. P1
= 10
P2
a / dB
10
Example: a = 6 dB what is P1/P2 ? After first computing 6/10 = 0.6 we obtain: P1/P2 = 100.6 = 4 The following power ratios and their decibel values are quite helpful, and the ones marked with asterisk should be memorized as shown in table 1. Ratio *1 *2 *3 4 *5 6 *7
*8 *9 * 10 100 103 104 dB 0 dB 3 dB 5 dB 6 dB 7 dB 8 dB 8.5 dB
9 dB 9.5 dB 10 dB 20 dB 30 dB 40 dB Figure 1 show how decibel scales are constructed. Figure 1 It should be emphasized that unlike watts, volts and amperes, the dB is not a physical quantity. Rather, a dB represents a ratio of two physical quantities, typical power. There are some rules or theorems that are useful in the manipulation of decibel quantities. Theorem 1: The product of two pure numbers or ratios A and B is equivalent to their sum when their values are expressed in dB. Example: A =1 0 dB B=2 3 dB A*B = 1*2 0 dB + 3 db = 2 = 3 dB Theorem 2: The division of two pure numbers or ratios A and B is equivalent to their difference when their values are expressed in dB. Example: A = 4 6 dB B = 2 3 dB A/B = 4/2 6 dB – 3 dB = 2 = 3 dB Fractional Numbers in Decibels:
Figure 2 shows the layout of the fractional numbers and their corresponding decibel values. As we go downward, each block decreases in size by a factor of 10. To show that 0.1 is ‐10dB, we invoke Theorem 2 in the following manner. 1 Block
Figure 2 Since 0.1 = 1/10 We know that 1 = 0dB and 10 = 10dB Apply Theorem 2: 0.1 =1/10 = 0dB – 10dB =‐10dB Similarly we can show that 10‐2, 10‐3 etc are ‐20dB, ‐30dB and so on. We should now be able to find the dB equivalent of any positive fractional numbers. Remember that negative numbers cannot be expressed in dB. Example: Number = 0.2 = 2/10 = 3dB – 10dB = ‐7dB Definition of dBm: If we refer an arbitrary power level to a fixed reference quantity, we obtain an absolute quantity from the logarithmic power ratio. The reference quantity most commonly used in telecommunications and radio frequency engineering is a power of 1mW. This reference quantity is designated by appending an m (for mW) to dB to give dBm. The general power ration P1 to P2 now becomes a ratio of P1 to 1mW, indicated in dBm. ⎡ P
⎤
dBm = 10 * log ⎢ watt ⎥ ⎣ 0.001watt ⎦
Example: 50W in dBm 10*log(50/0.001) = 10*log5000 = 47dBm Combining dBm and dB:
Just like voltage and current, power can be amplified by a power amplifier. The amplification factor is known as power gain, G. For instance, an input power of 1mW to an amplifier of gain 100 results in an output power of 1mW*100 = 100mW. Power amplification can be described in terms of dBm and dB, with the former referring to the input and output power, and the latter to the gain of the amplifier. Hence, dBm + dB have physical meaning, namely amplification of power by some numerical factor over its initial level. The resulting quantity is, therefore, also a power level, namely dBm. dBm + dB = dBm Power Amplification Just like the division of a certain voltage by a resistor network, power can be divided (or attenuated) as well. When expressed in terms of dBm and dB, power attenuation is dBm – dB. Because an attenuated power is still power, we have the following important relation: dBm ‐ dB = dBm Power Attenuation Table 2 summarizes all possible combinations of dB and dBm. Operation dB + dB dB – dB dBm + dBm dBm ‐ dBm dBm + dB dBm – dB Resulting Unit dB dB XX dB dBm dBm Physical meaning Product of two numbers comparing of two numbers Multiplying of two powers comparing two powers Power amplification Power attenuation Allowed? Yes Yes No Yes Yes yes Link Budget Analysis
In evaluating system performance, the quantity of greatest interest is the signal‐to‐noise ratio (SNR). This is because the basic system design centers detect the signal, with an acceptable error probability, in the presence of noise. Since the desired signal here is demodulated carrier waveform we often speak of the average carrier power‐to‐noise ratio (C/N) or (Pr/N) as the SNR of particular interest. We obtain equation Pr/N by dividing equation of power received by non isotropic antenna. EIRP * Gr * λ2 EIRP * Gr
(1) Pr =
=
Ls
(4πd ) 2
Where Effective Isotropic Radiated Power (EIRP) = Transmitted Power (Pt)* Gain of the Transmitting Antenna (Gt) λ = Wave length of the carrier Gr = receiving antenna gain Ls = Path loss or free space loss Pr EIRP *
=
N
Ls
Gr
N
(2)
Equation (2) applies to any one‐way RF link. With analog receivers, the noise bandwidth seen by the demodulator is usually greater than the signal bandwidth, and Pr/N is the main parameter for measuring signal detectability and performance quality. With digital receiver correlators or matched filters are usually implemented and signal bandwidth is taken to be equal to noise bandwidth. Rather than consider input noise power, a common formulation for digital links is to replace noise power with noise power spectral density. We can use equation of maximum single‐ sided noise power spectral density. No =
N
= k * T o Watts/hertz (3) W
Where N = maximum thermal noise power To = Temperature in Kelvin k = Boltzmann´s constant = W/Kelvin – Hz Where the system effective temperature, To is a function of the noise radiated into the antenna and the thermal noise generated within the first stages of the receiver. Note that the receiving antenna gain Gr and system temperature To are grouped together which is sometimes called the receiver sensitivity. Lo represents all other losses. If we assume that all the received power is in the modulating signal, we can write from basic SNR equation. S = average modulating signal power T = bit time duration R = 1/T = bit rate N = No*W S is the average modulating signal power, the bit energy per noise power spectral density, and R the bit rate. Link Margin
To facilitate calculating a margin or a safety factor M, we introduce two parameters which are required and the actual or received . We can rewrite equation 6 as The difference in decibels between and yields the link margin: The parameter reflects the differences from one system design to another. These might be due to differences in modulation or coding schemes. Combining equations (4) and (7) and solving for the link margin M yields: Since the link budget analysis is typically calculated in decibels we can express Equation 9 as Transmitted signal power, EIRP, is expressed in decibel‐watts (dBW), noise power spectral density is in decibel‐watts per hertz (dBW/Hz), antenna gain is in decibels references to isotropic gain (dBi), data rate R is in decibels referenced to 1 bits/s (dB‐bits/s) and all other terms are in decibels (dB). The numerical values of the equation 10 parameters constitute the link budget, a useful tool for allocating communications resources. References
1. W.Stephen Cheung and George T. Gillies, “Mathematics of Decibel Scale” 2. Theodore S. Rappaport, “Wireless Communication Principles and Practice”, Second Edition