Haus, Hermann A.

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VII. MICROWAVE ELECTRONICSProf. L. J. Chu D. L. Bobroff C. FriedProf. H. A. Haus T. J. Connor A. J. LichtenbergL. D. Smullin R. S. Eng C. MorgenrothA. Bers W. SchwabA. NOISE IN ELECTRON BEAMSIn the April 15, 1955, Quarterly Progress Report, it was shown that Pierce's modelof a transverse field tube permits the definition of a small signal kinetic power flow inthe beam of the same significance as the kinetic power flow defined by Chu for longitu-dinal beam tubes (1). The electromagnetic power extracted from the beam by a sur-rounding structure can be computed, on the basis of the small-signal assumptions, fromthe state of excitation of the beam at the entry into, and exit from, the structure. Theelectromagnetic power extracted from the beam is equal to the difference between thekinetic power leaving the structure and the kinetic power entering it. The combinedsystem, the electron beam-microwave structure, can then be treated as passive (orlossless).In this report we shall outline briefly how the minimum obtainable noise figure canbe evaluated for a tube in whose electron beam a kinetic power can be defined. Thiswork has been done jointly with F. N. H. Robinson, temporarily at Bell TelephoneLaboratories (2) .The excitation in the beam with n degrees of freedom entering the amplifier struc-ture can be expressed in terms of the normalized amplitudes al to a n of the modes ofexcitation in the beam. These modes are orthogonal with respect to their kinetic powerflow. The normalized amplitudes of the incoming waves into the input and output of theamplifier will be denoted by an+l and an+2Z respectively. The normalized amplitudesof the modes in the beam leaving the amplifier will be denoted by bl to b n , the reflectedwaves at the input and output terminals of the amplifier by bn+ 1 and bn+2 . (SeeFig. VII-1.) The action of the amplifier can be described in terms of a matrix S with(n+2) Z elements which gives the bk's in terms of the ak's. The amplitudes ak and bkBEAMMODESCIRCUITMODESIFig. VII1The modes of the beam and amplifier (arrows indicatedirection of power flow).-26-(VII. MICROWAVE ELECTRONICS)are arranged into column matrices a and b. We have thenb = Sa (1)We shall confine ourselves to amplifiers with lossless microwave structures. The lossycase is more complicated and does not add to our understanding (2). The conservationof the sum of the kinetic and electromagnetic power then givesn+2 n+2: Pk I ak Z k Ibk I (2)k=l k=lwhere the Pk's will be referred to as the "parities" of the modes; pk equals +1 if thepower (kinetic or electromagnetic) carried by the mode is positive, and pk equals -1 ifit is negative. The restriction of Eq. 2 imposes upon S the matrix relation (3)S + P S = P (3)where P is a diagonal matrix formed of the pk's, P = diag (P1' 2'...' n+2), and(S)jk = Skj. S is the Hermitian conjugate of S.The noise in the beam is defined sufficiently, once all the power spectralim a.ak/4 f j = 1 . n; k = 1., nAf-o0 Jare given. The a.'s are obtained from a Fourier analysis of the noise process over aperiod T = 1/Af. The bar indicates an ensemble average. The noise parameters canbe arranged into a matrix A.A = lim (aa+)/4rAf (4)where a is the row matrix with the elements a I to an. The transformation of the noiseby a general beam transducer characterized by the n-by-n matrix M leads the noisematrix A over into a noise matrix B.B = M A M+ (5)It has been shown by Robinson (2) that the B matrix can be diagonalized by a properchoice of M, the latter satisfying the condition of a lossless beam transducerM + PM=Pwhere P = diag (p 1 ' P 2 ... ' Pn ) .Equation 5 impliesA = N B N + (6)-27-(VII. MICROWAVE ELECTRONICS)with N = M-1, satisfying the condition N+ P N = P.With M so chosen that B is diagonal, we have proved the following statement: Anynoise process can be represented by a diagonal noise process (a noise process with zerocorrelation among the different beam modes) followed by a lossless beam transducer.This discovery allows one, using a beam with a noise matrix A, to find in a simpleway the amplifier which gives the lowest noise figure. The noise with the matrix A canbe given in terms of the diagonal noise process B followed by the proper lossless beamtransducer of matrix N. The beam transducer can be included in the amplifier with thematrix S so that a new structure, characterized by a matrix G, is formed.(N 0 )G= 0 1 0The available power gain of the over-all amplifier, IG n+ 2, is equal to the avail-able power gain before the inclusion of the beam transducer, IS n+ I 2 If the noisepower fed into the amplifier from the output end in the wave an+2 is neglected, we findfor the noise figure of the amplifiernF1 n + 4r I k 2 Bkk (7)kT IG 2 n+, k kkn+2, n+1 k=lThe matrix G has to satisfy the conditionG + PG = P (8)This condition also implies (see ref. 3)GPG =PThe n+2, n+2 element of the matrix relation given above readsn+2Z Gn+2, k 2 Pk 1  (9)k=lNoting that Pn+l = Pn+Z = + 1, we see from Eq. 9 that the existence of a large gain,IGn+, n+1 2, requires at least one correspondingly large Gn+2, k element of negativeparity, pk = 1.The noise figure of Eq. 7 can be minimized in the following obvious way:First, the mode of negative parity with the smallest value of Bkk is chosen. Letthis be the j mode. Then, all elements Gn+2, k are made equal to zero, exceptIGn+2, n+ll 2 (the available gain) and Gn+ 2 , j . We have from Eq. 9-28-(VII. MICROWAVE ELECTRONICS)2Gn+2, =1- 1n+, n+1 iGn+ 2 , n+land the minimum obtainable noise figure isF = 1 + 4B -  1 (10)kT = (I IGn+Z,n+l 1ZThis minimization has the following physical interpretation.The output power of the amplifier is gained at the expense of the kinetic power in thebeam. An amplifier structure has to couple strongly to at least one mode in the beamwith negative parity. The amplifier is optimized with regard to the noise figure whenit is made to couple only to the "least noisy" mode of negative parity.It is not hard to show that the minimization procedure described above, when appliedto a longitudinal beam amplifier (n = 2), yields the familiar result (4)F = + ( 1 )(S-II)mi n  kT gainH. A. HausReferences1. L. J. Chu, Paper delivered at the Conference of the Professional Group on ElectronDevices, Orono, Maine, June 1951.2. F. N. H. Robinson and H. A. Haus, unpublished paper.3. H. A. Haus and F. N. H. Robinson, Paper to be published in the Proceedings of theI. R. E.4. H. A. Haus, Trans. I.R.E. (PGED) 1, 238 (1954).-29-

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