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PRINCIPLESO COMMUNICATIONSYSTEMS

Second Edition

Herbert Taub

Donald L. SchillingProfessors of Electrical Engineering

The ity ollege o f ew York

McGraw Hill Book CompanyNew York St Louis San Francisco Auckland Bogota Hamburg

London Madrid Mexico Montreal New DelhiPanama Paris Siio Paulo Singapore Sydney Tokyo Toronto

RPX-Farmwald Ex. 1042, p 1



This book was set in Times Roman .The editor was Sanjeev Rao;the cover was designed by ohn Hite;the production supervisor was Marietta Breitwieser .Project supervision was done by Santype International Limited.

PRINCIPLES OF COMMUNICATION SYSTEMS

Copyright © 1986, 1971 by McGraw-Hill , Inc . All rights reserved .Printed in the United States of America. Except as permitted underthe United States Copyright Act of 1976, no part of this publicationmay be reproduced or distributed in any form or by any means, orstored in a data base or retrieval system, without the prior writtenpermission of the publisher .

6 7 8 9 0 BRBBRB 8 9

ISBN a o? 062955 2

Ubrary of Coagress Cataloglna In Publlcadon Data

Taub, Herbert, 1918-Principles of communicat ion systems .

McGraw-Hill series in electrical engineering.

Communications and signal processing)Includes bibliographies.1 Telecommunicat ion systems. I . Schilling.

Donald L . II. Title. III. SeriesTK5101.T28 1986 621.38 85-11638ISBN 0-07-062955-2 text)ISBN 0-07-062956-0 solutions manual)




COMMUNICATION SYSTEM AND NOISE CALCULATIONS 615

in which the left-hand member is the average energy stored on the capacitor. Thisresult is an example of the famous equipartition theorem of classical statisticalmechanics. The equipartition theorem states that a system in equilibrium with its

surroundings all at a temperature T shares in the general molecular agitationand has an average energy which is k T for each degree of freedom of the system.Thus an atom of a gas which is free to move in three directions has threedegrees of freedom and correspondingly has an average kinetic energy which is3 x kT = kT. At the other extreme a macroscopic system such as a speck ofdust suspended in a gas similarly flits about erratically and has an average energyassociated with this random motion of ikT. Since the dust speck is much moremassive than an atom the average velocity of the dust speck will be correspondingly much smaller. As another example consider a wall galvanometer whichbeing free only to rotate has a single degree of freedom. The kinetic energyassociated with such rotation is llP where I is the moment of inertia and ) is theangular velocity . Such a galvanometer shares in the thermal agitation of the airin which it is suspended and f1()2 = kT. I f the beam of light reflected from thegalvanometer mirror is brought to focus on a scale sufficiently far removed theslight random rotation of the galvanometer may be observed with the naked eye.Altogether it is interesting to note that the noise generated by a resistor is not aphenomenon restricted to electrical systems alone but is a manifestation of, andobeys the same physical laws that characterize the general thermal agitation ofthe entire universe.

Returning now to the R circuit of Fig. 14.4-1, we observe that it has onedegree of freedom i.e., the circuit has one mesh and a single current is adequateto describe the behavior of the system. On this basis then Eq. 14.4-5) is seen tobe an example of the equipartition theorem.

14 5 AVAILABLE POW R

The available power of a source is defined as the maximum power which may bedrawn from the source. If, as in Fig. 14.5-1, the source consists of a generator v.in series with a source impedance z. = R X then maximum power is drawnwhen the load is ZL = R - jX that is ZL = z: the complex conjugate of z .The available power is, therefore

z. = R 0 +jX 0

L •

14.5-1)

Figure 14.5-1 A source of impedance z is loaded bya complex conjugate impedance ZL = z: in order todraw maximum power .




616 PRINCIPLES OF COMMUNICATION SYSTEMS

Note that the available power depends only on the resistive component of thesource impedance.

Using Eq. (14.5-1), we have that the available thermal-noise power actualpower, not normalized power) of a resistor R in the frequency range df is

Pa= k ~ ;df = kT df (14.5-2)

The two-sided available thermal-noise power spectral density is

(14.5-3)

Observe thata

does not depend on the resistance of the resistor but only on thephysical constant k and on the temperature. f the source consists of a combination of resistors all at temperature T together with inductors and capacitors, then in Eq. (14.5-2) the R in the numerator and the R in the denominatorare both replaced by R(f), where R(f) is the usually frequency-dependent)resistive component of the impedance seen looking back into the network. TheseR(f) s will cancel, as do the R s. Hence, whether the network is a single resistoror a complicated RLC network, the available noise-power spectral density isGa= kT / 2 quite independently of its component values and circuit configuration.

Equation (14.5-3) expresses the available noise-power spectral density as pre

dicted by the principles of classical physics, which also predict that this value ofa applies at all frequencies; i.e., the noise is white. This result is manifestly

untenable, since it predicts that the total available power

Pa= J_Ga(/) df (14.5-4)

is infinite. This prediction was one of a series of similar inconsistencies whichwere, in part, responsible for the development of the branch of physics calledquantum mechanics. The quantum mechanical expression for Ga(f) is

hf/ 2Ga(/)= h f{ kTe -

Ga {)

lt

20.45/tT

0 4.3xl0 9 7 I2.6Xl0

10 T

Figure 14.5-2 Available power spectral density of thermal noise as given by Eq. 14.5-5).

(14.5-5)




COMMUNICATION SYSTEM AND NOISE CALCULATIONS 617

in which h 6.62 x 10 - 3 4 J/s is Planck s constant. Equation (14.5-5) yields afinite value for a and reduces to Eq. (14.5-3) when h f ~kT

The power spectral density of Eq. 14 .5-5) is plotted in Fig. 14.5-2 . Note that

the density is lower than kT /2 by 1 dB or more only w h n f ~4.3 x 109

T whichat room temperature, T 0 290 °K, corresponds to f 1.3 x 10 2

1.3 x 10 3 GHz. Hence we may certainly use Ga= kT / 2 at radio and even microwave frequencies ( 10 GHz). Note that a microwave receiver may employ amaser amplifier operating at a temperature as low as 4 °K in order to minimizethe noise due to the amplifier. Even at these low temperatures, it is still appropriate to assume that the noise is white. At optical frequencies this assumption is nolonger valid, and Eq. (14.5-5) must be employed.

14.6 N OISE TEMPERATURE

Solving Eq . (14.5-2) for T we have

(14.6-1)

When we apply Eq. (14.6-1) to a passive RLC circuit in which the noise is dueentirely to the resistors, then T is the actual common temperature of the resistors.Consider, however , the noise which may appear across a set of terminals connected to a more general type of circuit, including possibly active devices.Suppose that we measure the available power at the terminals and find that thenoise is white, i.e., the available power a increases in proportion to the bandwidth , so that P Jdf is the same at all frequencies. We may then take Eq. (14.6-1)to be the definition of the noise temperature of the network. The noise temperature of the network need not be the temperature of any part of the network.

Consider, for example, the simple idealized situation represented in Fig.14.6-1. Here a resistor R which is a thermal-noise source at a temperature T isconnected to the input terminals of an amplifier of gain A. We assume that the

input impedanceof

the amplifier is infinite and assume further, for simplicity, thatthe amplifier output resistance is a noiseless resistor R0 • Then the noise power, ina frequency range df available at the amplifier output terminals is

R

v2 kTRA 2 dfp = 2 =

a Ro Ro14 .6-2)

Figure 14.6-1 Illustrating that the noise temperature seen lookingback into a set of terminals a - b may assume any value .




622 PRINCIPLES OF COMMUNICATION SYSTEMS

14.10 NOISE F IGUR E

Let us assume that the noise present at the input to a two-port may be represented as being due to a resistor at the two-port input, the resistor being at roomtemperature T0 (usually taken to be T0 = 290°K). I f the two-port itself wereentirely noiseless, the output available noise-power spectra l density would be

~ = ga f) kT 0 / 2). However, the actual output noise-power spectral density isGao, which is greater than ~ The ratio ~ = is the noise figure of thetwo-port, that is,

F f) = ao = Gao~ ga f) kTo /2)

14.10-1)

I f the two-port were noiseless, we would have F = 1 0 dB). Otherwise F > 1.Using Eq. 14.9-2) with T = T0 , and Eq. 14.10-1), we find that the noise figure Fand the effective temperature T are related by

T = T0 F - 1 14.10-2)

or F=l+T, ,=T, ,+ToTo To

14.10-3)

The noise figure as defined by Eq. 14.10-1) is referred to as the spot noise.figure,since it refers to the noise figure at a particular spot in the frequency spectrum.I f we should be interested in the average noise figure over a frequency range from

1 to f 2 , then, as may be verified (Prob. 14.10-3), this average noise figure F isrelated to F f) by

_ 2

ga f)F f) df

F = f12ga f) dfl

14.10-4)

Two-ports are most commonly characterized in terms of noise figure when the

driving noise source is at or near T0 , while the concept of effective noise temperature T is generally more convenient when the noise temperature is notnear T0 •

When following a signal through a two-port, we are not so much interestedin the noise level as in the signal-to-noise ratio. Consider, then, the situation indicated in Fig. 14.10-1. Here the noise at the two-port input is represented as being

R

Ba(fJFigure 14.10-J A signal v, and a noise sourceare superimposed and applied at the input ofa two-port of available gain g 0 f).




COMMUNICATION SYSTEM AND NOISE CALCULATIONS 6 3

due to a resistor R so that the available input-noise-power spectral density isG ~ ~ >= kT/2. A signal is also present at the input with available power spectraldensity G ~ ~ >The available output-signal-power spectral density is

14.10-5}

However, because of the noise added by the two-port itself, the available outputnoise spectral density is

14.10-6}

Combining Eqs. 14.10-5) and 14.10-6), we have an alternative interpretation o

the spot noise figure, that is,

14.10-7}

Thus F is a ratio of ratios . The numerator in Eq. 14.10- } is the input-signal-tonoise power spectral density ratio, while the denominator is the output-signal-tonoise power spectral density ratio.

Let us assume that in a frequency range rom 1 to f 2 the power spectral densities o signal and noise are uniform. In this case it may be verified Prob.14.10-5) that the average noise figure F defined by Eq. 14.10-4) has thesignificance

14.10-8}

where S 1 and N; are, respectively, the total input available signal and noisepowers in the frequency range / 1 to f 2 , and similarly Sa and Na are the totaloutput available signal and noise powers.

The noise figure F or F may be expressed in a number of alternative formswhich are of interest. f the available gain g0 is constant over the frequency rangeof interest, so that F = F then S0 = g 0 S1 • In this case Eq. 14.10-8} may bewritten

14.10-9)

Further, the output noise Na is

Na gaNi + N p 14.10-10}

where g 0 N is the output noise due to the noise present at the input, and N P isthe additional noise due to the two-port itself. Combining Eqs. 14.10-9} and14.10-10), we have

F = l ~gaNi

14.10-11}

or, the noise due to the two-port itself may be written, from Eq. 14 .10-11), as

14.10-12)


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