chapter 11 system and noise calculations
TRANSCRIPT
Introduction to Analog And Digital Communications
Second Edition
Simon Haykin, Michael Moher
Chapter 11 System and Noise Calculations
11.1 Electrical Noise11.2 Noise Figure11.3 Equivalent Noise Temperature11.4 Cascade Connection of Two-Port Networks11.5 Free-Space Link Calculations11.6 Terrestrial Mobile Radio11.7 Summary and Discussion
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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 free-space model. In addition, movement of either the transmitting or receiving terminals results in further variation of the received signal strength.
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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
produced by a resistor is given
T is the absolute temperature in Kelvin, is Boltzmann’s constant, and is Planck’s constant. Note that the power spectral density is
measured in watts per hertz. For “low” frequencies defined by
We may use the approximation
)1.11(1)/||exp(
||2)(−
=kTfh
fhfSTN
)2.11(h
kTf <<
)3.11(||1||expkT
fhkT
fh+≈
2by
kh )( fSTN
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Approximate formula for the power spectral density of thermal noise:
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
Fig. 11.1
)5.11(Hz106 12×=h
kT
)6.11(volts4 )(2][E
2
2
N
TNNTN
kTRBfSRBV
=
=
)4.11(2)( kTfSTN ≈
6
Fig. 11.1 Back Next
7
Norton equivalent circuit consisting of a noise current generator in parallel with a noiseless conductance , as in Fig.11.1 (b). The mean-square value of the noise current generator is
For the band of frequencies encountered in electrical communication systems, we may model thermal noise as white Gaussian noise of zero mean.
)7.11(amps4
][E1][E
2
22
2
N
TNTN
kTGB
VR
I
=
=
RG /1=
8
9
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 watts.NkTB
10
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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.
Diode equation The Schottky formula also holds for a semiconductor junction diode.
Fig. 11.2
)8.11(amps2][E 22shot NqIBI =
)9.11(exp ss IkTqVII −
=
12
Fig. 11.2 Back Next
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The two components of the current I produce statistically independent shot-noise contributions of their own, as shown by
The model includes the dynamic resistance of the diode, defined by
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
)10.11( )2(2
2exp2][E 2shot
Ns
NsNs
BIIq
BqIBkTqVqII
+=
+
=
)( sIIqkT
IVr
+==
δδ
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Fig. 11.3 Back Next
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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
The noise figure is commonly expressed in decibels-that is, as
)11.11()()(
)()(fSfG
fSfFNS
NO=
).(log10 10 F
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Average noise figure The ratio of the total noise power at the device output to the output
noise power due solely to the source.
Fig. 11.4
0F
)12.11()()(
)(0
∫∫
∞
∞−
∞
∞−=dffSfG
dffSF
NS
NO
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Fig. 11.4 Back Next
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11.3 Equivalent Noise Temperature
The available noise power at the device input is
We define as
Then it follows that the total output noise power is
The noise figure of the device is therefore
The equivalent noise temperature,
dN
)15.11()( Ne
dsl
BTTGkNGNN+=+=
)14.11(Ned BGkTN =
)13.11(Ns kTBN =
)16.11(T
TTGNNF e
s
l +==
)17.11()1( −= FTTe
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Noise spectral density A two-port network with equivalent noise temperature (referred to the
input) produces the available noise power
We find that the noise may be modeled as additive white Gaussian noise with zero mean and power spectral density , where
Fig. 11.5
)18.11(av NeBkTN =
2/0N
)19.11(0 ekTN =
21
Fig. 11.5 Back Next
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11.4 Casecade Connection of Two-Port Networks
At the input of the first network, we have a noise power contributed by the source, plus an equivalent noise power contributed by the network itself.
We may therefore express the overall noise figure of the cascade connection of Fig.11.6 as
1N
11 )1( NF −
)20.11( 1
)1(
1
21
211
2122111
GFF
GGNGNFGNGFF
−+=
−+=
)21.11( 111321
4
21
3
1
21 ⋅⋅⋅+
−+
−+
−+=
GGGF
GGF
GFFF
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Correspondingly, we may express the overall equivalent noise temperature of the cascade connection of any number of noisy two-port networks as follows:
In a low-noise receiver, extra care is taken in the design of the pre-amplifier or low-noise amplifier at the very front end of the receiver.
Fig. 11.6
)22.11( 321
4
21
3
1
21 ⋅⋅⋅++++=
GGGT
GGT
GTTTe
24
Fig. 11.6 Back Next
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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 deep-space communication of information between a spacecraft and a ground station.
Calculation of received signal power Let the transmitting source radiate a total power . If this power is
radiated isotropically (i.e., uniformly in all directions), the power flux density at a distance from the source is , where is the surface area of a sphere of radius .
Thus for a transmitter of total power driving a lossless antenna with gain , the power flux density at distance in the direction of the antenna boresight is given by
TP
r )4/( 2rPT π 24 rπr
TPTG r
)23.11( 4 2r
GP TT
π=Φ
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Effective aperture area of the receiving antenna.
The gain of the receiving antenna is defined in terms of the effective of the effective aperture by
Given the power flux density at the receiving antenna with an effective aperture area , the received power is
Substituting Eqs.(11.23) and (11.26) into (11.27), we obtain the result
Fig. 11.7
)24.11( eff AA η=
)25.11( 42
eff
λπAGR =
RGeffA
)26.11( 4
2
eff πλRGA =
)27.11( effΦ= APR
)28.11(r4
G2
R
=πλ
TTR GPP
effAΦ
28
Fig. 11.7 Back Next
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Form Eq.(11.28) we see that for given values of wavelength and distance , 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
2. The gain of the transmitting antenna is increased. The choice of is therefore limited by size and weight constraints of the spacecraft.
3. The gain of the receiving antenna is increased. Here again, size and weight constraints place a physical limit on the size of the ground-station antenna, although these constraints are far less demanding than those on the spacecraft antenna; we typically have
λr
TP
.TP
TG TG
RG
.TR GG >>
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Then we may restate the Friis transmission formula in the form
Then we may modify the expression for the received signal power as
The received power is commonly called the carrier power.
)29.11(EIRP PRR LGP −+=
)30.11()(log10EIRP 10 TTGP=
)31.11(4log10 2eff
10
=
λπAGR
)32.11(4log20 10
=λπrLP
)33.11(EIRP 0LLGP PRR −−+=
RP
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32
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Carrier-to-noise ratio Carrier-to-noise ratio (CNR) As the ratio of carrier power to the available noise power
The carrier-to-noise ratio is often the same as the pre-detection SNR discussed in Chapter 9.
)34.11(CNRNs
R
BkTP
=
34
35
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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
37
Fig. 11.8 Back Next
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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
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.
)35.11(21
21
ddddh+
=λ
Fig. 11.9
39
Fig. 11.9 Back Next
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Thus multipath transmission may have quite different properties from free-space propagation. Measurements indicate that terrestrial propagation can be broken down into several components.
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Median path loss
The measurement of the field strength in various environments as a function of the distance , from the transmitter to the receiver motivates a simple propagation model for median path loss having the form
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
r
)36.11(nT
R
rPP β
=
)37.11()/( 0
0n
T
R
rrPP β
=Table. 11.1
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Table 11.1 Back Next
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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
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Fig. 11.10 Back Next
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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 is given by the Rayleigh distribution function
r
)38.11(exp1][P 2rms
2
−−=<
RrrR
Fig. 11.11
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Fig. 11.11 Back Next
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50
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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 , where 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.
kTN =0 k