Wide Band Instrumentation

74 IRE TRANSACTIONS ON AERONAUTICAL AND NAVIGATIONAL ELECTRONICS June

fly, and because of lack of accurate stationary tempera- types were delivered for test in May, 1951. Since thatture measurements in the air at various altitudes, and time only minor changes have been made in the vortexlack of definite correlation between the plane's position thermometer, mainly in conformance with recommenda-and that of the stationary temperature references in the tions derived from field tests. Several units have beenatmosphere, if available. For laboratory research and facricated at NRL. These units have been in continualdevelopment of temperature instruments for flight use, demand and have been flown by planes of the U. S.an ideal tvpe of calibrating equipment has been the Navy and Air Force, and on some commercial planes.NRL whirling arm. This equipment has provided speeds To date they have been flown on military planes vary-up to 500 mph at sea level. Bv means of this equipment ing in size from fighters and a Beechcraft to B-36's.it has been possible to determine an accurate reference They have been flown also on Boeing Stratocruisers fortemperature within inches of the flight path at several Pan American Pacific flights and on Lockheed Superpoints around the 100 foot circumference. The effects Constellations. In 1953 a procurement contract was letof speed and meteorological conditions can therefore by the Bureau of Aeronautics for 100 complete systemsbe determined quite accurately. The calibration at of this equipment. Deliveries on this procurement beganhigher airspeeds has not yet been possible with the same in August, 1955. This probe has also been modified tohigh degree of accuracy on any other equipment any- incorporate a hygrometer element in the Navy Aero-where. A transonic whirling arm is under construction graph AN/AMQ-8 now under procurement contract.at NRL to meet this need. Higher altitude studies may Deliveries are expected the summer of 1956 on thisbe possible by.r combinations of accurate calibrations on equipment. Also under procurement contract is a modi-a whirlinig arm and the same instruments being used in fication of this aerograph system for use on helicopters.wind tunnels where cross checks cain be made against In this system the modified vortex thermometer is lo-the whirlinig arm, then extended to higher altitude con- cated in a trailing bomb suspended fifty feet below theditions. helicopter. Future work on this equipment and the vor-

tex thermometer includes de-icing studies which arePRESEiNT STATUS OF THE AXIAL FLOW now under way and an extension of the speed range to

VORTEX THERMOMETER be undertaken when the transonic whirling arm becomesA vortex thermometer research and development available at NRL. Further studies of calibration at

project was begun at the Naval Research Laboratorv higher altitudes and detailed effects of atmospheric con-at the end of 1949. First operation-al suitability proto- ditions will be included.

Wide B3and Instrumentation*RAYMOND L. STRAZZULLAt

Summary-Wide-bandwidth measurement equipment is used A general method for solution of the mixer characteristic is givenwhere both long and short time accuracy is required. Present sensing which indicates that complementary linear filters may be used todevices can be broken into two classes, namely those that function perform the mixing function. The system may be made dynamicallyaccurately over long time measurements and conversely those that exact, that is, the mixer has a unity transfer function and contributesindicate changes but are lacking long time stability. Pressure pick- no error in the absence of noise.offs and radars are examples that fall into the former class of The mixer parameters are chosen on the basis of minimizing thedevices while velocity pickoffs and accelerometers possess only root mean square error in the output considering the noise inputs toshort time accuracy. the system.

This paper presents a technique whereby both types of inputs are An illustrative example is given based on a Baro-Inertial Mixerinjected into a mixer whose output retains the good properties of each using both pressure and inertial data to give accurate wide bandinput while rejecting the inaccurate portions of the input data. The vertical velocity as its output.resultant output of the mixer is superior to either input, since it pos-sesses both high and low frequency accuracy. NEED FOR WIDE-BANDWIDTH EQUIPMENT

X IDE-BANDWIDTH measurements are required* Manuscript received by the PGA.NE, November 25, 1955. This \\JV in many fields, needing instruments with both

paper was presented at the East Coast Conference on Aeronautical VV logadsrtemacucyMnypli-Electronics on November 2, 1955. The technique presented in this logadsrtemacucyMnyppi-report was developed at the Sperry Gyroscope Co., under a subcon- tions only require accuracy when measured for long termtract to Convair, Division of General Dynamics Corp. intervals and considerable smoothing can be applied to

t Sperrv Gyroscope Co., Division of Sperry Rand Corp., Great tedt oeiiaehg rqec os,wieo cNeck, N. Y. ' tedt oeiiaehg rqec os,wieo c

1956 Strazzulla: Wide-Band Instrumentation 75

casion the converse is true, accurate measurements of M1echanical restraint devices that are used to meas-the change in a quantity are desired and therefore low ure force by the deflection of a sensitive element incursfrequency drifts may be removed from the information. the penalty of reduced sensitivity when the bandwidthThis paper specifically deals with simultaneous meas- of the pickoff is increased. Since the sensitivity andurement of both high and low frequency information bandwidth are both related to the mechanical dimen-where simple filtering of the data source is not applicable. sions of the unit, an increase in bandwidth results in aAn example of the need for wide-band measuring loss in sensitivity.

techniques would be a directional reference for an aircraft. Long term heading data is required for naviga- High Frequency Data Sourcestion from one point to another and the output of a Many instruments are capable of measuring short-magnetic compass may be used for this purpose. To term changes in a quantity to great accuracy but areprevent the local variation in the earth's magnetic field deficient when used to measure over long time becausefrom imposing high frequency noise on the heading indi- of the steady state error in the instrument. Acceler-cation, the magnetic heading reference signal is smoothed ometers and gyros are typical of this class of instrument.by a suitable filter. However, the pilot also must know If a high-pass filter is placed at the output of thesehis instantaneous heading for maneuvering the air- devices, the steady state error is removed and all thecraft. Obviously, the smoothed output of the magnetic changes in the acceleration or angular rate can be meas-reference is not suitable since a heading error would be ured to great accuracy. The low frequency errors ofintroduced proportional to the smoothing time and the these devices may be minimized theoretically but inturning rate of aircraft which makes the pilot's opera- practice, this is difficult.tion of the aircraft cumbersome. Therefore, a differenttechnique must be used that gives accurate long term of Ovtigvheading for navigational purposes and short term head- Bandwidth Outptinig to be used by the pilot for aircraft maneuvering.' Each type of sensing element has a band of fre-

Ideally, the problem of wide-band measurements may quencies wherein its performance is optimum. A methodbe solved by selection of a device capable of supplying is presented in this paper whereby wide-band measure-the desired information over a sufficient bandwidth. ments may be performed using conventional inputHowever in practice, this pickoff is seldom available. source-s. This method requires two inputs, one a lowSomething is usually sacrificed to obtain bandwidth frequency measuring instrument and the other an ac-anid this penalty is often too severe. curate short term sensor. These two inputs are intro-

duced into a mixer which combines the inputs in suchAVAILABLE MEASURING EQUIPMENT a manner to give an output superior to either input.

The mixer output retains the good properties of eachinput while rejecting the band of frequencies over whichgories, those that are basically low frequency measure- theiputs do not give optimum accuracy. The mixing

ment devices and those that are designed to have high function is performed by linear filters that are easilyfrequency stabilltv.frequency stability, instrumented and complementary in nature.

Low Frequency Sources MIXER CHARACTERISTICS

Contained in this class are those instruments whose The function of the mixer is to receive data from twodata output should be smoothed before computation. sources, each accurate over different frequency bands,These include pressure pickoffs, radars, and mechanical and combine them in such a manner as to not incur anyrestraint type devices. dynamic error in the mixer output while filtering theThe pressure pickoff is noisy at high frequencies and undesirable portion of each input. The process by which

although its bandwidth may be extended by various the mixer is determined considers first the general formmeans, this often incurs a severe weight penalty. of the mixer and the constraint of dynamic exactness.

Radar information is again accurate over a long Secondly, the general shape of the filter required to re-term measurement but contains high frequency noise. tain the good frequency band of information from eachVarious techniques may be used to extend the band- source is decided. Finally, the filter parameters neces-width of the information such as increasing the prf, sary to give the minimum error in the output due to-sweep rate, or continuously tracking the target. In noise from each data source are selected.each case, there are penalties incurred such as decreasein range or a weight and size increase and usually theseDeiaonfGnrlFrmfMxrare not consistent with good radar design. Therefore, Consider two inputs to a mixer as shown in Fig. 1(a)radar information falls into the category of a low fre- with OH having good high frequency properties and OL aquency measurement device, good measure of steady state information. The mixer

contains two filters, A(p) through which OH iS passed'The Sperry Gyrosyn Compass essentially solves this problem. and B (p) which operates on 0X.


Certaini colnstraints must be placed on the operation A(p) + B(p) = 1 (5)of the mixer. No dynamic error must result from the which results in the followinig relationshipmixing operation. The output must contain high andlow frequency data from OH and OL respectively. 0o = 0 + A(P).AOH + B(P)A6L. (6)

- - -- -- - - Eq. (6) indicates that the mixer output is perfect inthe absence of input error.The mixer shown in Fig. 1(a) may be simplified to

l A(p) r(p) 8 that shown in Fig. l(b) by substituting (5) in (1) whichOH IHIGH PASS + + LOW PASS gives

l = OH + B(p)[OL - OH]. (7)tMIXER This changes the mixer components to one filter B(p)

L_-____-- - - - -- - -----and two summing elements, from two filters and oniesummer. It must be remembered that these filters musthave complementary characteristics such that the sum

98=A(P)9H+B(P)OL OUTPUT of A (p) and B (p) equals unity. Any mismatch will give698+ AH dynamic errors in the output. The mixer in Fig. l(b)6L=a+ 6L removes the possibility of filter mismatch but adds a80=(A(p)+B(p))e+A(p) 8H+B(p) 9L summer which is usually more easily constructed.

(a) Filter Characteristicsr1 The second constraint to the design of the mixer was

that the output shall contain high and low frequencydata from OH and OL respectively. This may, be stated in

I 11 equationi form as

l . lim-=1 (8)l l | B~B(p) |P- , p OL 8

FILTER lim-= 1. (9)P--- 0 OH

Since

0A AA(P)OH + B(P)OL. (1)

K T MIXER The following restrictionis must be placed on A (p) anid_____;R_~~~~~ ~ ~~~~B(p) &

lim A(p) = 0 lim A(p) = 1 (lOa, b)(b) o P+

Fig. 1-(a) Simple mixer; (b) modified simple mixer. lim B(p) = 1 lim B(p) = 0. (l la, b)P s~o pO

Referrinig to Fig. l(a), the equation relating the out- These relationships indicate that A (p) and B (p) areput to the input is high- and low-pass filters respectively whose sum must

Oo = A(P)OH + B(P)OL. (1) be equal to unity at any frequency.Assume that Since their sum and end points only are fully de-

scribed, many filter pairs could be used to fulfill theseOH = 0 + A0H (2) characteristics such as

OL= 0 + AOL (3) p k

where A (p) = B(p) =---.p+k p+k0 is the desired quantity The time constants of the filter are to be determined by

AOH is the lowv frequency error in OHI minimizing the noise passed through the filter froml theAOL iS the high frequency error in OL. inputs.

Then1 SELECTION OF FILTER PARAMETERS

00= [A (p) + B(p) ]o + A (P)AOH + B(P)AOL. (4) Basis for Parameter SelectionTo satisfy the first constraint that the mixer give no The basis for selection of the filter parameters is thatdynamic error a minimum error results from the mixing operation.


Ideally, the mixer would operate perfectly, regardless of Likewise, the response to noise from OL iSthe parameters of the filters, if noise generators werenot physically present. Obviously, noise will be intro- =B(p) AdL= error inOLduced into the svstem from the OH and OL inputs. These AOL Bare in fact the primary disturbances in the system.Knowledge of the frequency spectrum of the noise is of Ifdthe Powr spectritylo thenosefrom iHprincipal importance, so that the filter can be selected and

tharePu(t ) andPb () respectively, the rms error in

to eliminate this noise without adversely affecting data the output may be determined fromflow. The principle used in selection of the filter param-eters is to minimize the root mean square error in the H02=()I Aoutput for the noise inputs to the mixer. 7

Evaluation of Root Mean Square Error + PfL(w) B(f,) 12dW. (13)Computation of errors due to noise passing through

a linear filter are dependent upon the nature of the Now the parameters of the filters have an effect onnoise and the characteristics of the filter. the amount of noise passed by each filter. However, itA short presentation of the application of the root must be remembered that A(p) and B(p) are related,

mean square criterion with respect to random noise so that varying the parameters of one filter also alterspassing through linear filters is given in Appendix I. the time constants associated with the other.The general expression for the output mean square Since OL is a good low frequency source of informa-

error in terms of the input noise and the filter charac- tion, PL(,,) contains large magnitude high frequencyteristics is components. Conversely PH(,.) exhibits low noise at

1oc high frequencies but larger amounts at low frequencies.

62= - ep 2dco (12) In the second part of the preceding section, it was de-r j0 termined that A (p) and B (p) were high- and low-pass

wThere filters respectively. Therefore, increasing the band-passof B(p) will increase the output error due to AOL. How-

e is the rms error output ever

P(W) is the power spectral density of the input noise A(p) + B(p) = 1 (5)

H(,) is the Fourier transform of the filter through which indicates that an increase in the band-pass ofwhich P(,,,) is passed. B(p) increases the error due to AOL but decreases the

Obviously P(X) is determined by the characteristics error due to AOH, since the band-pass of the high-passof the input. However, H(),,, is dependent upon the gen- filter A (p) has been narrowed.eral shape and parameters of the filter which operates Obviously, the parameters of the filter may be opti-on the noise. Therefore the error in the output is a func- mized to give the minimum rms error in the output astion of the frequency response of the transfer function a function of the filter bandwidths for fixed noise inputs.which operates on the noise generator. This is the process by which the parameters of the filtersWith regard to the specific determination of the filter are selected.

parameters of a mixer, the power spectral density ofthe input noise for each data source must be known as Order of the Filtera function of frequency. The general shape of the filters Many filter shapes may be used that satisfy the con-in the mixer has been determined in the manner out- straints of the mixer. In cases where a low-pass filter islined in the preceding section, then the specific cutoff needed, the filter may befrequencies of the filters may be determined by observ-ing which parameters give the minimum total error in 1 1the output due to both noise sources. p+1 (rp +1)3As an illustration, the simple mixer previously de-

rived is used. The response of the system to an error The selection of the order or degree of cutoff of the filterin OH may be somewhat dependent upon the noise error in

A00 the output, but usually this effect is small. Therefore,-= A(p) practical instrumentation considerations may be used

AOH as criteria for the selection of the order of the filter.w,xhere Engineering judgement should be exercised in weigh-

ing the possible decrease in system error versus the addi-A00 = output error tional instrumentation complexity incurred by using aAOE = error in Off. higher order or sharper cutoff filter.


EXAMPLE THE BARO-INERTIAL MIXER r1

An instrument that accurately measures the vertical VHvelocity of an aircraft is needed to measure the instan- PRESSUREtaneous and steady state rate of climb. The develop- ISOURCEment of such an instrument is used to serve as an illus- VERTICAL MIEtrative example of the mixing principles previously L--CELEROMETE1 L-- --jderived. vOThe two inputs to the Baro-Inertial Mixer are alti- VERTICAL

tude rate as derived from a pressure sensitive element, VELOCITYOUTPUT

and an accelerometer which is mounted such that itssensitive axis senses accelerations solely along the ver- (a)tical. This accelerometer is biased for 32 foot/second2acceleration to counteract the acceleration of gravity - land therefore only kinematic accelerations of the air- VH VH VLcraft are sensed by the device. P PRESSURI

IVERTICAL SOURCI

Description of Sensing Elements IACCELEROMETERIVertical Accelerometer: The vertical accelerometer

output when integrated yields an excellent indication B(p)of the change in vertical velocity, but possesses poor LOW PASS

FILTERlong term stability.

It must be remembered that an error in the output MIXERof this device represents acceleration error and will + +give a continuously increasing velocity error if it weresimply integrated. Therefore this device should onlybe used for its high frequency properties. VERTICAL VELOCITY

Pressure Altitude Rate: A standard pressure altimeteris used to give altitude rate. This device contains poor (b)high frequency information since it is very noisy at Fig. 2-(a) General Baro-Inertial Mixer; (b) simplest form of Baro-high frequencies. Most pressure devices also exhibit a Inertial Mixer.time lag in reproducing data. For purposes of simplic-ity this example will consider the pressure instrument is used to eliminate the possibility of integrator satura-as though it has no inherent lag. If the delay were taken tion in the presence of steady state errors of iWH. Theinto account in the derivation of the mixer character- system in Fig. 3(b) was finally derived which satisfiesistics, a simple filter to match the lag of the pressure this constraint as well as all the others.instrument would have to be added in the path of the The general form of the mixer as derived in Appenidixinertial information. II resulted in

Derivation of Baro-Inertial Allfixer Characteristics lim A(p) lim t > 0 (30, 46)The two inputs to the mixer shown in Fig. 2(a) are P1o C(p) P C(p)

TIH from the accelerometer and VL from the pressure B(p) = 1 - A(p) (22)instrument. The general mixer characteristics are de- lim A(p) = a finite lim A(p) = 1 (33, 41)rived in Appendix II, and the following constraints were w yo quantity +

placed on the mixer design. lim C(p) > 0 lim C(p) = a finite (34, 39)P-40pO *x quantity.

Constraints to Design:1) System dynamically exact; To satisfy the low frequency characteristic of

2) Output contains high frequency information from A (p)VH; C(-J)

3) Output contains low frequency data from VL; and Cp4) Integrator output must not increase for a steady one would either makie A (p) a high-pass filter or C(P)

A VH input. contain integration or possibly both. Now B(P) iS deter-Constraint 4 results from the fact that the mixer as mined by the choice made for A (p) . If A (p) is chosen as

derived in its simplest form [Fig. 2(b)] contains an a high-pass filter, B(p) must be its complementary low-integrator which may saturate if a constant error in pass filter and a wide latitude is available for C(p). How-the accelerometer output is present. This would cause ever, if C(p) is made to contain integration, then A(p)the mixer to malfunction. Therefore, a feedback loop can be made unity and therefore B(p) must be zero.


VERTICALI r R are indicated by the VH input. Now if VH contained aIACCELEROMETER, I steady state error, this would appear as an increasing

IVH A1P) error in Vo which would no longer match VL. This causes

the feedback loop to function. The integrator in thel | feedback path would assume such a value as to eliminate

++ C(p) 1 VL_ the effect of any steady state error in VT.-SOURCE Data from VL should only be indicated in the output

on a long time basis since this is the prime source of lowl D^p~ frequency vertical velocity information. If a step of VL

| enters the system, the feedback loop causes the Vii in-tegrator to function such that VO= VL in the steady

VO state. Now if there is high frequency noise in VL, theVERTICAL action of the integrators in the path of this noise beforeVELOCITY it reaches VO is such as to greatly attenuate high fre-OUTPUT **,(a) quency noise in VL.

'ACCELEROMETER1 MIXER

VH j r7l IVHIA( ~~~IV11VERTICAL V4 *V ACIALERXERA L V

VELOCITY ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~ELCT

OUTPUT_ _ _JII I OUTPUT

(b) Seecion of PrmTesothe Baro-Inertial Mlixer.

OUTPUT ~ SORC

Fig. 3-(a) Practical Baro-Inertial Mixer; (b) final Baro- The transfer functions of the system for AVH andInertial Mixer configulration. AK VL inputs respectively are

This results in the simplest Baro-Inertial 1Miixer con- zFVo4PTh BaroInert _ (50)figuration. Mixe p2+ Kdp -p kg

Therefore the filters chosen were___ Kdp + Kg

A(p) = 1 (47) AVL p2+Kdp+Kg (51)

B(p) = 0 (48) SubstitutingKg

C(p) = - + Kd. (49) c0,2 = Kg (52)P 2Pcon = Kd (53)

The Baro-Inertial Mixer Loop becomes that shownin Fig. 4 which contains an integrator to convert accel- and converting to the Fourier Transformeration data from VH to velocity and integral plus pro-portional error feedback. =HiO ( ) = -(______(_4)

A\VH (jc@) 2 +V 2PCn(jCw) + (J (54)Operation of the Baro-Inertial Mixer

The operation of the Baro-Inertial Mixer Loop in = 2wn(jw) + (55)Fig. 4 may be explained by considering an input from AVIL (fr)2 ± 2¢)fl(jc) +- wn2 (5

VH. . .* . The error due to noise in each input is given byConsider a rapid change inl vertical velocity. This

would be sensed byr V1 and integrated to immediately 1 rXgive a change of vertical velocity as indicated by VO. A\V02 =-J PH(w) !Hi(cX) |2d~If anyJ difference exists between V0 and VL, the feed-backi loop is slowx operating and no correction would be + -F Pw(.) H2(w) |idw (56)fed back. Therefore, rapid changes in vertical velocity 1


where 3.0the pt

PC(X) is the power spectral density of the noisefrom I7H 2.5

-PL(o.) is the power spectral density of the noise,1from. VL,.

Since IVH is fundamentally the source of high fre- 2.0quency information, the power spectral denisity of itsnoise contenit: contains large amounts of low frequency 1.5 t-=7 _|\llcomponents and lesser amounts of high frequency com-ponents. This noise is therefore assumed to be of the '°.0form 1.0

PH(b,) = ------- (ft/sec.2)2/rad/sec. (57) 51 _ ______10OW2 + 1 ~~~~~~~~~~~~~~.5

The noise contributed by the pressure instrument isassumed to be white noise which has a constant noise 0power per unit bandwidth and therefore has a large 0 .04 .08 .1 .2 .4 .8 1.0 2.0error at high frequencies. The value of noise of this WN RAD/SECsource is assumed to be Fig. 5-Error in vertical velocity due to acceleromneter noise,

PL(w) = .6 (ft/sec)2/rad/sec. 3.0PHu(,) and PL() are independent noise sources. (58)

The quantities D and wn are to be varied to determine 2.5which parameters result in the minimum square errorin vertical velocity. 2.0

Noise from the accelerometer contributes an errorin the output and may be evaluated from

.5 1.5VH2 f + d (59) ::'+

.125 1.0EVH2- ~c100Wn,2+2 -n1 (ft/sec)2 (60)

Pxnt~~~~~~~~~~~~~~~lOco2 -.5n )where ~5V

EVH is the error in the output vertical velocity due toaccelerometer noise. 0 0 .04 .08 .1 .2 .4 .8 .0 2.0

As can be seen by the plot of this function in Fig. 5, wN RAD/SECthe error decreases as con, which is related to the band- Fig. 6-Error in vertical velocity due to pressure system noise.width, is increased.The error contribution due to white noise from the The resultant error

pressure system is given by

1 X .6(4¢2Wn2W2 + cn4) AV02 = VVH2 + EVL2 (63)EVL2=- _dw (61)V r L2 - 2 + 2vwn(jw) + wn2 d ( is plotted in Fig. 7 and indicates that a minimum of

.15wn(4~2 + 1) .475 foot/second error in vertical velocity is reachedEV 2 = + (ft/sec)2 (62) when

¢ x~~~~~~~~~~~~~~cn =.25 rad/sec.where ¢= .7

e VL iS the error in the output due to pressure system Ti en hta piu ytmi tandinoise.

Fig. 6 indicates that the contribution of error due to Kg = .0625 sec.-2pressure noise grows when co)n is increased. Kd = .350 sec.-l


3.0 Stationary random noise is that which can be char-acterized by a complete set of probability distributions

2.s ______ ______ which are invariant under shifts in time.If the function f(t) is ergodic; i.e., ensemble average

equal to the time average, then its autocorrelation2.0 E 7?;5\ \ _ 1 1 function is given by0w +

_____ _____ 4(r)~~~~~~O= lim f(t)f(t + 7)dt0I 5 -__ T-,o 2T -Tand its power spectral density is the Fourier Transform1.0 of the autocorrelation function.

1 f_.,e-iwrTo(T)dr

o where.04 .08 .1 .2 .4 .8 1.0 2.0 4.0

ON RAD/SEC ¢(Xo) is the power spectral density of the function f(t),Fig. 7-Total error in vertical velocity. r is the parameter of the autocorrelation function,

Conclusions 45(r) is the autocorrelation function of f(t).

This svstemiscapabeofivingnodyami. Since 4(co) is a power density, the mean square errorem is capable of giving no dynamicerror,may be computed by summing the total error power

has no steady state integrator drift rate in the presencew~~~~~~~~passed through the filter.of an accelerometer error, and delivers the minimumpossible error for the noise inputs inherent to the data 1 0

sources.2

7rvo

EXTENSION OF MIXING TECHNIQUE where

The theory of mixing inputs to recover output data H(w,) is the frequency response of the filter throughmore accurate than any individual input may be ex- whichf(t) is passed.tended to the use of more inputs than the two usedthroughout this paper. For example, one input of good APPENDIX II-DERIVATION OF BARO-INERTIALlow frequency characteristics, another that contains MIXER CHARACTERISTICSaccurate information over some intermediate band- The Baro-Inertial Mixer contains one input that iswidth, and a third that exhibits excellent high frequency the time derivative of the desired output. This source isdata may be all mixed to yield one output that is su- used to give high frequency vertical velocity informa-perior to anv input. The constraints to the design oft. v . . ~~~~~~~tion.Low frequency data iS to be recovered from asuch a unit may be satisfied by the methods previously pressure instrument that directly measures verticaloutlined but the results would be somewhat complex.Many equipment designs may be enhanced by the y

mixinig technique. Audio amplifiers, home heating sys- Derivation of General Form of Mixertems, and manv other instrumentation problems may beaided by wide-ban1d measurements achieved through Fig. 2(a) shows a mixer whose inputs are VHand VLmixinlg high and low frequency inlformation1 giving an vwhere VH is the time rate of change of VH. The desiredoutput superior toeith enpt output Vo is to be comprised of high frequency data

from I7H and low frequency information from VL.APPENDIX I-EVALUATION OF RMS ERROR The output of this mixer is

DUE TO RANDOM NOISE

A short description of the application of harmonic V0 = A(p)VH + B(p)V. (14)analysis to the evaluation of rms errors for random Using the substitutionsnoise inputs is presented below. The derivation of thistheory and its conditions for validity are contained in VL = V + A\VL (15)other publications.2-4 V= pV +[ ALWH (16)

2 N. W;\einer, "The Extrapolation, Interpolation and Smoothing ofStationary Times Series,"^ John WViley & Sons, Inc., New York, N. Y., where1949.

3H. M. James, N. B. Nichols, and R. S. Phillips, "Theory of Servo- V = true vertical velocitymechanisms," McGraw-Hill Book Co. Inc., New York, N. Y.; 1947.

Y. XVJ. Lee, "Application of Statistical Methods to Communica- A VL = error in VLtion Problems," M.I.T. Research Lab. of Electronics, Tech. Rpt. No.=eroinK18I; September, 1, 1950. V=eroinH


results in Substituting (22) in (23)

v0= [pA(p) + B(p)]V + A(P)AVH + B(p)AVL (17) [1 - A(p)]C(p) -A(p)D(p) = C(p)

To make this system dynamically exact C(P) = -D(p) (24)

pA(p) + B(p) = 1 (18) This simplifies (21) to

which then substituted in (17) gives A(P)AH B(p)p + C(p)

Vo= V + A(P)AVH + B(p)AVL. (19) - p + C(p) p + C(p)The system may be rearranged by noting that Filter Characteristics

A(p) =1 - B(p) (18) Eq. (20) relates the output to VH and VLP Vo A (p)

Substituting this equation in (14) results in VH p + C(p)

VO = P + B(P) VVL V B(p)p+ C(p) (27)p ~ Lp] VL p + C(p)

This indicates that VH must be integrated before it is Constraint 2 statespassed through the filter as shown in Fig. 2(b). VOne serious drawback to the system in Fig. 2(b) is linm . = 0 (28)

that any steady state error in VH would eventually P0 VHsaturate the VH integrator since most integrators in V0general have a limited output range. This would cause im- = 1. (29)the mixer to malfunction. Therefore, a means must be p Vsought to make the integrator attain a nonincreasing Satisfying (28) anld (29)output in the presence of a steady state error in VH. 4(p)This is accomplished by modifying the basic loop to lim = 0 (30)that shown in Fig. 3(a). The low frequency signal is p o C(p)used to monitor the integrator output and the error is pB(p) + C(p)fed back to compensate for any steady A IH inputs. Jim p 1 (31)Therefore, no danger of integrator saturation exists and p o C(P)another constraint to the design of the mixer is estab- which giveslished when one input is the derivative of the other.

Jim B(p) # ± G (32)Constraint 1-System dynamically exact. P0

Constraint 2-Output contains low frequency data since A (p) = 1 -B (p)from VL.

Constraint 3-Output contains high frequency lim A(p) 7 (33)data from VH.

Constraint 4-Integrator output must not increase For stability, the characteristic equation of thefor a steady A VH input. system

The output of the mixer in Fig. 3 (a) is p + C(P) = 0

A(p)V, B(p)p+B(p)C(p)-A(p)D(p) VL. (20) must have all real positive coefficients present for allp+C(p) p+C(p) orders of p. This is a restriction placed on C(p) and this

fact coupled with the result of (30) givesSubstituting (15) and (16) in (20)

lim C(p) > 0 (34)[A(p)+B(p) ]p+B(p)C(p)-A(p)D(p) Po

p+C(p) sinceA(p)AVH ±B(p)p+B(p)C(p)-A(p)D(PK)L (1 D(p) = - C(p) (24)

p+FC(p) p+c(p) Jim D(p) < 0. (35)

Two conditions are required to make this system dy-namically exact, However, it must be remembered that

A(p) +I B(p) = 1 (22) AI(P) =0(0

B(p)C(p) -A(p)D(p) = C(p). (23) p-o C(p)


which placed definite restrictionis on A (p) and C(p) such lim = 45that either A (p) =0 or C(p) = x or both when p->O. P A08

For satisfaction of Constraint 3 that the high fre- lim C(p) > 0. (34)quency information be derived from V1H the following P +

must be trueThis constraint had already been satisfied in order to

lim . = - (36) obtain low frequency information from VL and still haveP-oo VK p a stable system.

VO To summarize, for Vo to be composed of high fre-lim-= O. (37) quency information from V1H and low frequency datap+ VL from VL, the characteristics of the mixer are defined as

This gives from (26) follows.

A(p) lim A(p) - +± lim A(p) = 1 (33, 41)p-O

lim = (38) lim B(p) F + lim B(p) = 0 (32, 42-)C(p) p Po PxO

1+-p lim C(p) > 0 lim C(p) X ± (34, 39)

o ,~~~~~~~~~P+00

lim C(P) x ± G. (39)p-4 00 lim D(p) < 0 lim D(p) # +o (35, 40)

Since D(p) =-C(p) lim A(p) A(p) (30, 46)=im0 lm > 0 0 6lim D(p) F- + oo. (40) Po0 C(p) P >/ C(p)P--+00 A(p) + B(p) = 1 (22)

Alsolim A(p) = 1. (41) C(p) = -D(p). (24)

This completelv bounds the filter characteristics ofSince B(p)=1-A(p)I the mixer and the mixer may be redrawn to that shown

lim B(p) = 0. (42) in Fig. 3(b). It must be remembered that the character-P--+00 istics of A(p), B (p), and C(p) are related as p-+O such

Eq. (37) is automatically validated by the restrictions that if A (p) is a high-pass filter B(p) is a low-pass filter

placed on B(p) and C(p) by (39) and (42). and C(p) may be a constant or possess integrationproperties. However, if A4(p) is choseni to be unity, B(p)

B(p) + C(p) becomes a zero transfer function, and C(p) must havep integration. Wider latitude is allowed for the selection

lim = 0. (43) of the filter shapes for dynamic exactness.P--+OO C(p)

1+ A(p) + B(p) = 1p

Now Constraint 4 states that the integrator should not C(p) = -D(p).continually integrate in the presence of a steady error in The selection of the various filters of the system mayVT11 be made on the basis of the minimum of complexity or

e =_ __P (AA\ that which are easilv instrumented since more filtersAVH p + C(P) may be varied to give the same output result.

c7\A5

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Wide Band Instrumentation