Introduction to Analog And Digital Communications Second Edition Simon Haykin, Michael Moher
Chapter 11 System and Noise Calculations 11.1 Electrical Noise 11.2 Noise Figure 11.3 Equivalent Noise Temperature 11.4 Cascade Connection of Two-Port Networks 11.5 Free-Space Link Calculations 11.6 Terrestrial Mobile Radio 11.7 Summary and Discussion
11.1 Electrical Noise We briefly discuss the physical sources of noise in electrical circuits and develop quantitative models for measuring and prediction the presence of noise in a communication system. Lesson1 : Noise in communication systems may be generated by a number of sources, but often the sources are the communication devices themselves. Thermal noise and shot noise are examples of white noise processes generated by electrical circuits. Lesson2 : In a free-space environment, the received signal strength is attenuated propotional to square of the transmission distance. However, signal strength can be enhanced by directional antennas at both the transmitting and receiving sites. Lesson3 : In a terrestrial environment, radio communication may occur many paths. The constructive and destructive interference between the different paths leads, in general, to the so-called multipath phenomenon, which causes much greater propagation losses than predicted by the freespace model. In addition, movement of either the transmitting or receiving terminals results in further variation of the received signal strength. 3
Thermal noise Thermal noise is a ubiquitous source of noise that arises from thermally induced motion of electrons in conducting media. It suffices to say that the power spectral density of thermal noise 2 produced by a resistor is given by 2h f S TN ( f ) = (11.1) exp( h f / kt) 1 T is the absolute temperature in Kelvin, k is Boltzmann s constant, and h is Planck s constant. Note that the power spectral density S TN ( f ) is measured in watts per hertz. For low frequencies defined by kt f << (11.2) h We may use the approximation h f h f exp 1+ kt kt (11.3) 4
Approximate formula for the power spectral density of thermal noise: ( f ) 2kT (11.4) S TN kt h = 6 10 12 Hz (11.5) This upper frequency limit lies in the infrared region of the electromagnetic spectrum that is well above the frequencies encountered in conventional electrical communication systems. The mean-square value of the thermal noise voltage measured across the terminals of the resistor equals E[ V 2 TN ] = 2RB N S = 4kTRB TN N ( f ) volts 2 (11.6) Fig. 11.1 5
Fig. 11.1 Back Next 6
Norton equivalent circuit consisting of a noise current generator in parallel with a noiseless conductance G =1/ R, as in Fig.11.1 (b). The mean-square value of the noise current generator is E[ I 2 TN 1 ] = E[ V 2 R = 4kTGB 2 TN amps (11.7) For the band of frequencies encountered in electrical communication systems, we may model thermal noise as white Gaussian noise of zero mean. N ] 2 7
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Available noise power Thevenin equivalent circuit of Fig.11.1(a) or the Norton equivalent circuit of Fig.11.1(b), we readily find that a noisy resistor produces an available power equal to ktb N watts. 9
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Shot noise Shot noise arises in electronic devices due to the discrete nature of current flow in the device. In a vacuum-tube device, the fluctuations are produced by the random emission of electrons from the cathode. In a semiconductor device, the cause is the random diffusion of electrons or the random recombination of electrons with holes. In a photodiode, it is the random emission of photons. In all these devices, the physical mechanism that controls current flow through the device has built-in statistical fluctuations about some average value. 2 2 E[ I ] 2 amps (11.8) shot = qibn Diode equation The Schottky formula also holds for a semiconductor junction diode. I = I s exp qv kt I s (11.9) Fig. 11.2 11
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The two components of the current I produce statistically independent shot-noise contributions of their own, as shown by qv E[ I 2 ] = shot 2qI s exp BN + 2qI sbn kt = 2q( I + 2I ) B (11.10) s N The model includes the dynamic resistance of the diode, defined by δv r = δ I = q ( I + I s Bipolar junction transistor In junction field-effect transistors In both devices, thermal noise arises from the internal ohmic resistance: base resistance in a bipolar transistor and channel resistance in a field effect transistor. Fig. 11.3 kt ) 13
Fig. 11.3 Back Next 14
11. 2 Noise Figure The maximum noise power that the two-port device can deliver to an external load is obtained when the load impedance is the complex conjugate of the output impedance of the device-that is, when the resistance is matched and the reactance is tuned out. Noise figure of the two-port device the ratio of the total available output noise power (due to the device and the source) per unit bandwidth to the portion thereof due to the source. Then we may express the noise figure F of the device as S ( ) NO f F( f ) = (11.11) G( f ) S ( NS f ) The noise figure is commonly expressed in decibels-that is, as 10log10( F). 15
Average noise figure F 0 The ratio of the total noise power at the device output to the output noise power due solely to the source. F 0 = S G( f NO ) S ( f NS ) df ( f ) df (11.12) Fig. 11.4 16
Fig. 11.4 Back Next 17
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11.3 Equivalent Noise Temperature The available noise power at the device input is N s = ktb N (11.13) We define as N d N = GkT Then it follows that the total output noise power is N = GN + N = Gk( T + Te ) BN The noise figure of the device is therefore N T + T l e F = = (11.16) GN T The equivalent noise temperature, l d T e s s e B N d = T ( F 1) (11.14) (11.15) (11.17) 19
Noise spectral density A two-port network with equivalent noise temperature (referred to the input) produces the available noise power N = kt B e (11.18) av N We find that the noise may be modeled as additive white Gaussian noise with zero mean and power spectral density / 2, where N 0 N 0 = kte (11.19) Fig. 11.5 20
Fig. 11.5 Back Next 21
11.4 Casecade Connection of Two-Port Networks At the input of the first network, we have a noise power N 1 contributed by the source, plus an equivalent noise power ( F1 1) N contributed by the network itself. We may therefore express the overall noise figure of the cascade connection of Fig.11.6 as FG 1 1N1G2 + ( F2 1) N1G2 F = N1G1G 2 F 1 2 = F1 + (11.20) G 1 1 F = F F2 1 + G 1 F 1 3 F4 1 + G G G G G 1 + + 1 2 1 2 3 (11.21) 22
Correspondingly, we may express the overall equivalent noise temperature of the cascade connection of any number of noisy twoport networks as follows: T e = T T T 2 3 4 1 + + + + G1 G1G 2 G1G 2G3 T (11.22) In a low-noise receiver, extra care is taken in the design of the preamplifier or low-noise amplifier at the very front end of the receiver. Fig. 11.6 23
Fig. 11.6 Back Next 24
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11.5 Free-Space Link Calculations We move on to the issue of signal and noise power calculations for radio links that rely on line-of-sight propagation through space. The satellite, in effect, acts as a repeater in the sky. Another application with line-of-sight propagation characteristics is deepspace communication of information between a spacecraft and a ground station. Calculation of received signal power Let the transmitting source radiate a total power P T. If this power is radiated isotropically (i.e., uniformly in all directions), the power flux 2 2 density at a distance r from the source is P T /(4πr ), where 4πr is the surface area of a sphere of radius r. Thus for a transmitter of total power P T driving a lossless antenna with gain G, the power flux density at distance r T in the direction of the antenna boresight is given by P G Φ = T T (11.23) 4πr 2 26
Effective aperture area of the receiving antenna. A = ηa (11.24) The gain of the receiving antenna G R is defined in terms of the effective of the effective aperture A eff by 4πA eff G R = 2 λ (11.25) 2 G R λ A eff = 4π (11.26) Given the power flux density Φ at the receiving antenna with an effective aperture area A eff, the received power is = A Φ eff (11.27) eff P R Substituting Eqs.(11.23) and (11.26) into (11.27), we obtain the result 2 λ PR = PT GT G R (11.28) 4πr Fig. 11.7 27
Fig. 11.7 Back Next 28
Form Eq.(11.28) we see that for given values of wavelength λ and distance r, the received power may be increased by three methods: 1. The spacecraft-transmitted power is increased. Hence, there is a physical limit on how large a value we can assign to the transmitted power P T. 2. The gain G of the transmitting antenna is increased. The choice of G T is therefore limited by size and weight constraints of the spacecraft. P T T G R 3. The gain of the receiving antenna is increased. Here again, size and weight constraints place a physical limit on the size of the groundstation antenna, although these constraints are far less demanding than those on the spacecraft antenna; we typically have G >> R G T. 29
Then we may restate the Friis transmission formula in the form P R = EIRP + G R L P (11.29) EIRP = 10log ( P G 10 T T ) (11.30) G R L P 4πA eff = 10log10 2 λ (11.31) 4πr = 20log10 λ (11.32) Then we may modify the expression for the received signal power as PR = EIRP + GR LP L0 (11.33) P R The received power is commonly called the carrier power. 30
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Carrier-to-noise ratio Carrier-to-noise ratio (CNR) As the ratio of carrier power to the available noise power PR CNR = (11.34) kt B s N The carrier-to-noise ratio is often the same as the pre-detection SNR discussed in Chapter 9. 33
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11.6 Terrestrial Mobile Radio With terrestrial communications, both antennas are usually relatively close to the ground. With these additional modes of propagation, there are a multitude of possible propagation paths between the transmitter and receiver, and the receiver often receives a significant signal from more than one path. There are three basic propagation modes that apply to terrestrial propagation: free-space, reflection, and diffraction. Fig. 11.8 36
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Free-space propagation A rule of thumb is that a volume known as the first Fresnel zone must be kept clear of objects for approximate free-space propagation. The radius of the first Fresnel zone varies with the position between the transmitting and receiving antenna; it is given by λd1d2 h = (11.35) d1 + d2 Reflection The bouncing of electromagnetic waves from surrounding objects such as buildings, mountains, and passing vehicles. Diffraction The bending of electromagnetic waves around objects such as buildings or terrain such as hills, and through objects such as trees and other forms of vegetation. Fig. 11.9 38
Fig. 11.9 Back Next 39
Thus multipath transmission may have quite different properties from free-space propagation. Measurements indicate that terrestrial propagation can be broken down into several components. 40
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Median path loss The measurement of the field strength in various environments as a function of the distance r, from the transmitter to the receiver motivates a simple propagation model for median path loss having the form PR β = (11.36) n P r T Path-loss exponent n ranges from 2 to 5 depending on the propagation environment. The right-hand side of Eq.(11.36) is sometimes written in the equivalent from P P R T = β0 ( r / r0 ) n (11.37) Table. 11.1 42
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Numerous propagation studies have been carried out trying to closely identify the different environmental effects. This model for median path loss is quite flexible and is intended for analytical study of problems, as it allows us to parameterize performance of various system-related factors. Fig. 11.10 44
Fig. 11.10 Back Next 45
Random path losses The median path loss is simply that: the median attenuation as a function of distance; 50 percent of locations will have greater loss and 50 percent will have less. These fast variations of the signal strength are due to reflections from local objects that rapidly change the carrier phase over small distances. The probability that the amplitude is below a given level r is given by the Rayleigh distribution function P[ ] 1 exp 2 r R < r = 2 R rms (11.38) Fig. 11.11 46
Fig. 11.11 Back Next 47
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11.7 Summary and Discussion 1. Several sources of thermal noise were identified, and we characterized their power spectral densities through the simple relation N 0 = kt, where k is Boltzmann s constant and T is the absolute temperature in Kelvin. 2. The related concepts of noise figure and equivalent noise temperature were defined and used to characterize the contributions that various electronic devices or noise sources make to the overall noise. We then showed how the overall noise is calculated in a cascaded two-port system. 3. The Friis equation was developed to mathematically model the relationship between transmitted and received signal strengths as a function of transmitting and receiving antenna gains and path loss. 4. Propagation losses ranging from free-space conditions, typical of satellite channels, to those typical of terrestrial mobile applications were described. Variations about median path loss were identified as an important consideration in terrestrial propagation. 51