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Logarithmic System of Units Appendix A Logarithmic System of Units A.1 Introduction This appendix deals with the concepts and fundamentals of the logarithmic units and their applications in the engineering of radiowaves propagation. Use of these units, expressing the formulas, and illustrating figures/graphs in the logarithmic systems are common and frequently referred in the book. A.2 Definition According to the definition, logarithm of the ratio of two similar quantities in decimal base is called “bel,” and ten times of it is called “decibel.” = p bel log10 pr =[ ]= p decibel dB 10 log10 pr As the above formula suggests, it is a logarithmic scale, and due to the persistent usage of decimal base in the logarithmic units, it is normally omitted from the related symbol for simplicity. Since values in the logarithmic scale are the logarithm of the ratio of two similar quantities, hence they are dimensionless and measured in the same unit. In other words, the unit of denominator, pr, is the base of comparison and includes the required unit. To indicate the unit of the base in the logarithmic system of units, one or few characters are used according to the following examples: dBw: Decibel unit compared to 1 W power A. Ghasemi et al., Propagation Engineering in Radio Links Design, 523 DOI 10.1007/978-1-4614-5314-7, © Springer Science+Business Media New York 2013 524 A Logarithmic System of Units dBm: Decibel unit compared to 1 mW power dBkw: Decibel unit compared to 1 kW power Example A.1. Output power of a transmitter is 2 W, state its power in dBm,dBw, and dBkw. Solution. 2 × 103 P [dB ]=10 log = 33 dB t m 1mw m 2 P [dB ]=10 log = 3dB t w 1w w 2 × 10−3 P [dB w]=10 log = −27 dB w t k 1kw k A.3 Basic Formulas Applying the decibel units necessitates the frequent usage of logarithm; thus, it is essential to review the following basic mathematical formulas: a = ⇐⇒ = P ⇐⇒ = 1. log10 P a 10 Antilog10P a a a an 2. log(a1 × a2×···× an)=log 1 +log 2 +···+ log ( p ) q = p 3. log a q log a ( a )= − 4. log √b log a log b m = 1 5. log a m log a A.4 Common Logarithmic Quantities Most of quantities used in the radio communications are defined in terms of logarithmic units. Main logarithmic units used in this book are indicated in Table A.1. In addition to the aforementioned cases, all of the quantities mentioned in the sequel are normally defined in dB: • Polarization loss, PL • Cross-polarization discrimination Loss, XPD • Antenna front to back ratio, F/B • C/N, S/N, E/N • Gain or loss factors such as FSL, passive reflector gain, feeder loss, and rain loss A Logarithmic System of Units 525 Table A.1 Main logarithmic units No. Quantity Base Sign No. Quantity Base Sign 1 Gain/loss Ratio dB 8 Noise temperature ◦K(kelvin) dB/K 2PowerWattdBw 9 Bandwidth Hertz dBHz 3 Power Milliwatt dBm 10 Bit rate b/s dBb/s 2 4 Power Kilowatt dBkw 11 Power flux Watt/m dBw/m2 5 Antenna gain Isotropic dBi 12 Noise power Watt dBKTB 6 Antenna gain Dipole dBd 13 Field intensity (strength) Microvolt dBμV/m per meter 7 Voltage Microvolt dBμV 14 Boltzmann coefficient Kelvin/Joule dBJ/K A.4 Principles of Logarithmic System of Units Rule 1: The logarithmic system of units are dedicated only for scalar quantities. If this unit is used for a vector quantity, it implies the magnitude of the vector quantity. Rule 2: In the logarithmic system of units, mathematical operations are simplified compared to the ordinary system of units (IS). Rule 3: The logarithmic scale when compared with the ordinary system of units presents large quantities related to the reference unit in a compact form, while its role for small quantities is vice versa as follows: 10 ⇐⇒ 10 dB 0.1 ⇐⇒ − 10 dB 100 ⇐⇒ 20 dB 0.01 ⇐⇒ − 20 dB 1,000 ⇐⇒ 30 dB 0.001 ⇐⇒ − 30 dB . 10n ⇐⇒ 10n dB 10−n ⇐⇒ − 10n dB In radio communications, most of curves, graphs, and plots are normally illustrated on the logarithmic scaled axis for better presentation and simpler interpretation. A.5 Advantages of Logarithmic System of Units Applications of the logarithmic system of units have dominant advantages as specified below: • Simplicity of the mathematical operations • Much better presentation • Better understanding and more simple interpretation 526 A Logarithmic System of Units As an example, the following formula indicates the free-space loss in the ordinary (metric) system of units: (4πd)2 · f 2 FSL = C2 The same formula in the logarithmic system of units is given as below: FSL [dB]=92.4 + 20 log f · d which is more popular in the technical calculations related to the radiowave propagation. It should be noted that every quantity presented in the logarithmic system of units can be easily transformed into the ordinary (metric) system of units or vice versa. Example A.2. Equivalent isotropic radiated power of a transmitter is 43 dBw (i.e., EIRP = 43 dBw), find 1. Equivalent radiated power in terms of kW. 2. Assuming 37 dBi antenna gain, determine antenna directivity in the metric system of units. 3. Transmitter power in watts. Solution. 1. 43 P = Antilog = 20,000 W ⇒ P = 20 kW e 10 e 2. 37 G [dB]=37 dB ⇒ G = Antilog = 5,000 t i t 10 3. EIRP[dBw]=Pt[dBw]+Gt[dBi] 6 P [dB ]=43 − 37 = 6dB ⇒ P = Antilog = 4W t w w t 10 Due to the nature of mathematical operations in the ordinary and logarithmic systems, it is important to know the method to be used for every expression or formula. Different methods may be employed in technical books and papers. The method which is applied in this book uses brackets [ ] for the logarithmic units as seen in the following examples: Pr[dBw]=Pt[dBw]+Gt[dBi]+Gr[dBi] − FSL[dB] − Lm[dB] A Logarithmic System of Units 527 or EIRP[dBm]=Pt[dBm]+Gt[dBi] − Lt[dB] or Lbf[dB]=Pt[dBw] − E[dBμV/m]+20 log f (GHz)+167.2 Appendix B ITU-R Recommendations P-Series Radiowave propagation P.310 Definitions of terms relating to propagation in non-ionized media P.311 Acquisition, presentation, and analysis of data in studies of tropospheric propagation P.313 Exchange of information for short-term forecasts and transmission of ionospheric disturbance warnings P.341 The concept of transmission loss for radio links P.368 Ground-wave propagation curves for frequencies between 10 KHz and 30 MHz P.369 Reference atmosphere for refraction Note-Suppressed on 24/10/97 (RA-97)-This Recommendation has been replaced by Rec.ITU-R P.453-6 P.370 VHF and UHF propagation curves for the frequency range from 30–1,000 MHz. Broadcasting services Note-Suppressed on 22/10/01 (CACE/233) P.371 Choice of indices for long-term ionospheric predictions P.372 Radio noise P.373 Definitions of maximum and minimum transmission frequencies P.434 ITU-R reference ionospheric characteristics and methods of basic MUF, operational MUF and ray-path prediction Note-Suppressed on 24/10/97 (RA-97). This Recommendation has been replaced by Rec.ITU-R P.1239 and ITU-R P.1240 P.435 Sky-wave field-strength prediction method for the broadcasting service in the frequency range 150–1,600 kHz Note-Suppressed on 20/10/95 (RA-95) P.452 Prediction procedure for the evaluation of microwave interference between stations on the surface of the Earth at frequencies above about 0.7 GHz P.453 The radio refractive index: its formula and refractivity data Note-This Recommendation replaces Rec. ITU-R P.369-6 P.525 Calculation of free-space attenuation P.526 Propagation by diffraction draft revision P.526-9(08/05) P.527 Electrical characteristics of the surface of the Earth P.528 Propagation curves for aeronautical mobile and radionavigation services using the VHF, UHF, and SHF bands P.529 Prediction methods for the terrestrial land mobile service in the VHF and UHF bands Note-Suppressed on 22/10/01 (CACE/233) (continued) A. Ghasemi et al., Propagation Engineering in Radio Links Design, 529 DOI 10.1007/978-1-4614-5314-7, © Springer Science+Business Media New York 2013 530 B ITU-R Recommendations P-Series (continued) Radiowave propagation P.530 Propagation data and prediction methods required for the design of terrestrial line-of-sight systems P.531 Ionospheric propagation data and prediction methods required for the design of satellite services and systems P.532 Ionospheric effects and operational considerations associated with artificial modification of the ionosphere and the radiowave channel P.533 HF propagation prediction method P.534 Method for calculating sporadic-E field strength P.581 The concept of “worst month” P.616 Propagation data for terrestrial maritime mobile services operating at frequencies above 30 MHz Note-Suppressed on 22/10/01 (CACE/233) P.617 Propagation prediction techniques and data required for the design of trans-horizon radio-relay systems P.618 Propagation data and prediction methods required for the design of Earth-space telecommunication systems P.619 Propagation data required for the evaluation of interference between stations in space and those on the surface of the Earth P.620 Propagation data required for the evaluation of coordination distances in the frequency range 100 MHz to 105 GHz P.676 Attenuation by atmospheric gases P.678 Characterization of the natural variability of propagation phenomena P.679 Propagation data required for the design of broadcasting-satellite systems P.680 Propagation data required for the design of Earth-space maritime mobile telecommunication systems P.681 Propagation data required for the design of Earth-space-land mobile telecommunication systems P.682 Propagation data required for the design of Earth-space aeronautical mobile telecommunication systems P.683 Sky-wave field-strength prediction method for propagation to aircraft at about 500KHz.
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