PRINCIPLES
`OF COMMUNICATION
`SYSTEMS
`Second Edition
`
`Herbert Taub
`
`Donald L. Schilling
`Professors of Electrical Engineering
`The City College of New York
`
`McGraw-Hill Book Company
`New York St. Louis San Francisco Auckland Bogota Hamburg
`London Madrid Mexico Montreal New Delhi
`Panama 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 byt'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 under
`the United States Copyright Act of 1976, no part of this publication
`may be reproduced or distributed in any form or by any means, or
`stored in a data base or retrieval system, without the prior written
`permission of the publisher.
`
`5 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 communication systems.
`
`(McGraw-Hill series in electrical engineering.
`Communications and signal processing)
`Includes bibliographies.
`1. Telecommunication systems.
`Donald L.
`II. Title.
`III. Series
`TK5101.T28 1986
`621.38
`85-11638
`ISBN 0-07-062955-2 (text)
`ISBN 0-07-062956-0 (solutions manual)
`
`I. Schilling.
`
`RPX-Farmwald Ex. 1042, p 2
`
`

`

`COMMUNICATION SYSTEM AND NOISE CALCULATIONS 615
`
`in which the left-hand member is the average energy stored on the capacitor. This
`result is an example of the famous equipartition theorem of classical statistical
`mechanics. The equipartition theorem states that a system in equilibrium with its
`surroundings, all at a temperature T, shares in the general molecular agitation
`and 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 three
`degrees of freedom and correspondingly has an average kinetic energy which is
`3 x !kT = !kT. At the other extreme, a macroscopic system such as a speck of
`dust suspended in a gas similarly flits about erratically and has an average energy
`associated with this random motion of ikT. Since the dust speck is much more
`massive than an atom, the average velocity of the dust speck will be correspond(cid:173)
`ingly much smaller. As another example, consider a wall galvanometer, which,
`being free only to rotate, has a single degree of freedom. The kinetic energy
`associated with such rotation is !llP where I is the moment of inertia and () is the
`angular velocity. Such a galvanometer shares in the thermal agitation of the air
`in which it is suspended, and f1() 2 = !kT. If the beam of light reflected from the
`galvanometer mirror is brought to focus on a scale sufficiently far removed, the
`slight 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 a
`phenomenon restricted to electrical systems alone, but is a manifestation of, and
`obeys, the same physical laws that characterize the general thermal agitation of
`the entire universe.
`Returning now to the RC circuit of Fig. 14.4-1, we observe that it has one
`degree of freedom, i.e., the circuit has one mesh, and a single current is adequate
`to describe the behavior of the system. On this basis, then, Eq. (14.4-5) is seen to
`be an example of the equipartition theorem.
`
`14.5 AVAILABLE POWER
`
`The available power of a source is defined as the maximum power which may be
`drawn 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 + jX, then maximum power is drawn
`when the load is ZL = R - jX, that is, ZL = z:, the complex conjugate of z ..
`The available power is, therefore,
`
`(14.5-1)
`
`z. = R0 +jX0
`
`L'•
`
`Figure 14.5-1 A source of impedance z. is loaded by
`a complex conjugate impedance ZL = z: in order to
`
`draw maximum power.
`
`RPX-Farmwald Ex. 1042, p 3
`
`

`

`616 PRINCIPLES OF COMMUNICATION SYSTEMS
`
`Note that the available power depends only on the resistive component of the
`source impedance.
`Using Eq. (14.5-1), we have that the available thermal-noise power (actual
`power, not normalized power) of a resistor R in the frequency range df is
`Pa= 4k~; df = kT df
`
`(14.5-2)
`
`The two-sided available thermal-noise power spectral density is
`
`(14.5-3)
`
`Observe that Ga does not depend on the resistance of the resistor but only on the
`physical constant k and on the temperature. If the source consists of a com(cid:173)
`bination of resistors (all at temperature T) together with inductors and capac(cid:173)
`itors, then in Eq. (14.5-2) the R in the numerator and the R in the denominator
`are both replaced by R(f), where R(f) is the (usually frequency-dependent)
`resistive component of the impedance seen looking back into the network. These
`R(f)'s will cancel, as do the R's. Hence, whether the network is a single resistor
`or a complicated RLC network, the available noise-power spectral density is
`Ga= kT/2 quite independently of its component values and circuit configuration.
`Equation (14.5-3) expresses the available noise-power spectral density as pre(cid:173)
`dicted by the principles of classical physics, which also predict that this value of
`Ga 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 which
`were, in part, responsible for the development of the branch of physics called
`quantum mechanics. The quantum mechanical expression for Ga(f) is
`
`hf/2
`Ga(/)= hf{kT
`-
`e
`
`1
`
`(14.5-5)
`
`Ga({)
`
`ltT
`2
`
`0.45/tT
`
`0
`
`4.3xl09 7'
`2.6Xl010 T
`
`I
`
`Figure 14.5-2 Available power spectral density of thermal noise as given by Eq. (14.5-5).
`
`RPX-Farmwald Ex. 1042, p 4
`
`

`

`COMMUNICATION SYSTEM AND NOISE CALCULATIONS 617
`
`in which h = 6.62 x 10- 34 J/s is Planck's constant. Equation (14.5-5) yields a
`finite value for Pa and reduces to Eq. (14.5-3) when hf~ 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 whenf~ 4.3 x 109 T , which
`f ~ 1.3 x 1012 =
`at
`room
`corresponds
`to
`temperature, T0 ~ 290°K,
`1.3 x 103 GHz. Hence we may certainly use Ga= kT/2 at radio and even micro(cid:173)
`wave frequencies ( ~ 10 GHz). Note that a microwave receiver may employ a
`maser amplifier operating at a temperature as low as 4°K in order to minimize
`the noise due to the amplifier. Even at these low temperatures, it is still appropri(cid:173)
`ate to assume that the noise is white. At optical frequencies this assumption is no
`longer valid, and Eq. (14.5-5) must be employed.
`
`14.6 NOISE 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 due
`entirely 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 con(cid:173)
`nected to a more general type of circuit, including possibly active devices.
`Suppose that we measure the available power at the terminals and find that the
`noise is white, i.e., the available power Pa increases in proportion to the band(cid:173)
`width, 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 tem(cid:173)
`perature 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, is
`connected to the input terminals of an amplifier of gain A. We assume that the
`input impedance of the amplifier is infinite and assume further, for simplicity, that
`the amplifier output resistance is a noiseless resistor R0 • Then the noise power, in
`a frequency range df, available at the amplifier output terminals is
`
`kTRA 2 df
`v2
`p =--2..=----
`a Ro
`Ro
`
`(14.6-2)
`
`R
`
`Figure 14.6-1 Illustrating that the noise temperature seen looking
`back into a set of terminals a - b may assume any value.
`
`RPX-Farmwald Ex. 1042, p 5
`
`

`

`622 PRINCIPLES OF COMMUNICATION SYSTEMS
`
`14.10 NOISE FIGURE
`
`Let us assume that the noise present at the input to a two-port may be represent(cid:173)
`ed as being due to a resistor at the two-port input, the resistor being at room
`temperature T0 (usually taken to be T0 = 290°K). If the two-port itself were
`entirely noiseless, the output available noise-power spectral density would be
`G~0 = ga(f)(kT0/2). However, the actual output noise-power spectral density is
`Gao, which is greater than G~0 • The ratio GaJG~0 = F is the noise figure of the
`two-port, that is,
`F(f) = Gao =
`Gao
`ga(f)(kTo/2)
`G~0
`If 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 F
`and the effective temperature T,, are related by
`T,, = T0(F - 1)
`
`(14.10-1)
`
`(14.10-2)
`
`or
`
`(14.10-3)
`
`F=l+T,,=T,,+To
`To
`To
`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.
`If 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 is
`related to F(f) by
`
`F =
`
`(14.10-4)
`
`_ f :2
`ga(f)F(f) df
`f 12ga(f) df
`l,
`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 tem(cid:173)
`perature T,, is generally more convenient when the noise temperature is not
`near T0 •
`When following a signal through a two-port, we are not so much interested
`in the noise level as in the signal-to-noise ratio. Consider, then, the situation indi(cid:173)
`cated in Fig. 14.10-1. Here the noise at the two-port input is represented as being
`
`R
`
`Ba(fJ
`
`Figure 14.10-J A signal v, and a noise source
`are superimposed and applied at the input of
`a two-port of available gain g0(f).
`
`RPX-Farmwald Ex. 1042, p 6
`
`

`

`COMMUNICATION SYSTEM AND NOISE CALCULATIONS 623
`
`due to a resistor R so that the available input-noise-power spectral density is
`G~~> = kT/2. A signal is also present at the input with available power spectral
`density 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 output(cid:173)
`noise spectral density is
`
`(14.10-6}
`
`Combining Eqs. (14.10-5) and (14.10-6), we have an alternative interpretation of
`the spot noise figure, that is,
`
`(14.10-7}
`
`Thus F is a ratio of ratios. The numerator in Eq. (14.10-7} is the input-signal-to(cid:173)
`noise power spectral density ratio, while the denominator is the output-signal-to(cid:173)
`noise power spectral density ratio.
`Let us assume that in a frequency range fromf1 to f 2 the power spectral den(cid:173)
`sities of 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 the
`significance
`
`(14.10-8}
`
`where S1 and N; are, respectively, the total input available signal and noise
`powers in the frequency range / 1 to f 2 , and similarly Sa and Na are the total
`output available signal and noise powers.
`The noise figure F (or F) may be expressed in a number of alternative forms
`which are of interest. If the available gain g0 is constant over the frequency range
`of interest, so that F = F, then S0 = g0 S1 • In this case Eq. (14.10-8} may be
`written
`
`(14.10-9)
`
`Further, the output noise Na is
`
`Na= gaNi + N,p
`(14.10-10}
`where g0 N 1 is the output noise due to the noise present at the input, and N,P is
`the additional noise due to the two-port itself. Combining Eqs. (14.10-9} and
`(14.10-10), we have
`
`F=l+~
`gaNi
`or, the noise due to the two-port itself may be written, from Eq. (14.10-11), as
`
`(14.10-11}
`
`(14.10-12)
`
`RPX-Farmwald Ex. 1042, p 7
`
`