IEEE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY

A Method for Calculating Error Probabilitiesin a Radar Communication System*

RANDALL M. MEALEYt, SENIOR MEMBER, IEEE

Summary-Missfle range instrumentation radars are capable oftransmitting pulse code groups in which some of the pulses in eachgroup can be used to convey information from the ground to a

space vehicle containing a suitable beacon receiver and decoder.Thus, a one-way communication system is provided with only a

small increase in ground and vehicle equipment over that requiredfor the tracking function. Since the system is one way, conmnunica-tion reliability becomes of paramount importance. This paperpresents a method of computing bit error rates, word error rates,and frame error rates as a function of the SNR at the beaconreceiver. The SNR vs range can then be computed by standardmethods to obtain the ground-to-vehicle ranges over which reliablecommunications can be conducted.

INTRODUCTION

/\[ ANY MISSILE range instrumentation radarsused for tracking space vehicles can transmitpulse code groups instead of single ranging pulses

characteristic of standard radars.' These pulse code groupsare used for interrogation coding and for the transmissionof information to a vehicle containing a suitable radarbeacon transponder and decoder. Each pulse code groupconsists of two or more RF pulses. One of these pulses,called the "triggering pulse" or the "beacon rangingpulse," is used for triggering the beacon and is equivalentto the normal single pulse of the radar in the skin-trackingmode. One or more of the other pulses will, in conjunctionwith the triggering pulse, form an "interrogation code."One or more of the pulses remaining in the group, timemodulated with respect to the triggering pulse, or modu-lated on and off, will serve to transmit control commandsvia the beacon. These modulated pulses, with the trigger-ing pulse, are called the "control code." The beacon willreceive the pulse code group and check the interrogationcode for proper pulse spacing. If the interrogation codeis correct, the triggering pulse will trigger the beacon'stransmitter through a fixed delay and pass the controlcode to the command decoder.

Fig. 1 shows a pulse code group containing three pulses.In this example, the first two pulses, I, and I2, are usedin the interrogation code. The I2 pulse is the triggeringpulse and also serves as a reference pulse for command de-coding. The triggering pulse and a data pulse, either R0,R1 or R,o, are used for the command code. The data pulsein the Ro position represents a binary 0, and in the R1

* Received January 4, 1963.t Space Systems Division, Martin Company Baltimore, Md.1 "IRIG System Standards for C-Band (5 cmS Instrumentation

Radars and Beacons," Inter-Range Instrumentation Group,White Sands, N. Mex., Recommendation No. 106-59; 1960.

PULSE CODE GROUP/ I'1 12 R0 R1 R%

ri r - r nI II II I II I IL-J L-.J I.

pp

p

p

p

p

pPPPPPP

ppppppppppppppppp

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p WORD STARTFRAME STARTDATA BIT NO. 15

it It "I 14If if to 13It It 12

1 'of of 11

" " 10it If if 9

8of It 7

6

., If 5,, of if 4it It It

11It110

SAMPLE DATA WORD: 001010010110010

II = FIRST INTERROGATION PULSE

12 SECOND INTERROGATION PULSE AND REFERENCE PULSE

Ro = DATA PULSE IN BINARY 0 POSITION

R1 " DATA PULSE IN BINARY 1 POSITION

Rw = DATA PULSE IN 'WORD START" POSITIONp INDICATES PULSE IS PRESENT- INDICATES NO PULSE IS PRESENT

Fig. 1-Pulse code group and word format.

position, a binary 1. The data pulse in the R. position isused to signify "word start." One pulse code group is re-

quired for each bit of information to be transmitted so

that "n"-pulse code groups are required to transmit an

"n"-bit word. Fig. 1 shows an example of a 17-bit word.An obvious advantage of this coding technique is that

the complement of every digit is inherently transmitted,thus increasing communication reliability. That is, if thedata pulse is in the R, position signifying a 1, both theRo and RW positions must contain no pulses. Similarconditions apply for the data pulse in the RO of the R,,positions. The command decoder, therefore, checks eachpulse code group to ensure that one and only one pulse ispresent in the data positions. If no pulse is present, or ifmore than one pulse is present, then an error has beencaused by noise and the decoder will discard the wordbeing received.When the quantity of data to be transmitted is small,

each word or each frame can be redundantly transmittedto provide more than one chance for correctly receiving

1963 37

IEEE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY J

a message. Otherwise, a word verification or a frame veri-fication is sent from the vehicle to the ground via telem-etry. This verification must be received by the groundsystem before a new word or frame is transmitted. If noverification is received, the word or frame is retransmitted.The standard two-pulse beacon interrogation decoder

generates time slots roughly 2 ,usec wider than the beaconpulses being received. The spacing between these slotsis adjustable to accommodate several different codes. Atapped delay line or a multivibrator is commonly usedto generate the time slots. Each slot has an output to acoincidence gate. Incoming interrogation pulses havingthe proper spacing between pulses will appear at the in-puts to the gate in time coincidence and thus provide anoutput to the beacon transmitter and the commanddecoder.The command decoder also generates time slots for

command decoding which are approximately 2 ,usec widerthan the pulses being received. These time slots correspondto the Ro, R,, and RW positions in Fig. 1. The spacingbetween the data pulse and the triggering pulse deter-mines whether the data pulse represents a 0, 1, or "wordstart" bit. Pulse-width discrimination is provided toprevent very wide pulses from entering the decoder andcausing false decoding. In addition, the beacon receiverIF bandwidth is made just wide enough to pass the nar-rowest beacon pulses expected. As a result, noise spikesnarrower than the beacon pulses will be greatly attenuated,and the number of false decodings due to noise will bereduced.

PULSE CODE GROUP ERROR PROBABILITY

A pulse code group error can occur in three differentways, as shown in Fig. 2. In this figure, the horizontalspaces represent the time slots where a pulse can occur,the dotted pulses represent noise pulses, and the solidpulses represent true pulses of the pulse code group. Thetrue pulses are 2 ,sec wide for a three-pulse code group.The slot width is assumed to be 1 ,usec and is thus 2 .secwider than the beacon code pulses. The minimum widthof the noise pulses is assumed to be 4 .tsec and is deter-mined by the receiver bandwidth. In each of the cases,proper identity decoding must occur before the informationcan be recovered from the pulse code group.

Case (a) of Fig. 2 is the occurrence of three noise pulseswith spacing such that they will fall in the decoder timeslots and thus cause an output from the command de-coder. The probability of this event is

PF = 1 -(1 - P,Pf,P,,)M, (1)

where

Pt,

P,3Pfa

PF

1 - (1 - Pf),1 - (1 - Pf)1 - (1 _ pf)Ns

probability of decoding a false informa-tion bit due to noise exceeding thethreshold,

r T

I I

- 1I2s'II. I

IIs

RS

r vf R

F I Ro FrPH R1 F-J!- RWS '-

(a)

2S I, 'R0SItls H-ffl2s Hf-IROs-F

L-J

(b)

IIs I2s Ros

r 1

R WJPH RI L-IS s~i

(c)

HORIZONTAL SPACES REPRESENT TIME SLOTSDOTTED PULSES REPRESENT NOISE PULSESSOLID PULSES REPRESENT CODE GROUP PULSESils u FIRST INTERROGATION PULSE TIME SLOT

'2S e SECOND INTERROGATION PULSE TIME SLOT

Rloe Rl, Rws = DATA PULSE TIME SLOTS, (Rs5 Ros + RIS + Rws)Fig. 2 Pulse code group error possibilities.

Pf = probability of a single noise pulse ex-ceeding the threshold,

Pf,,, Pf5, Pf, = probability of a single noise pulse ex-ceeding the threshold in the I,,, I2.,and R, time slots, respectively (R, =Rog + R1i + Rwj),

N,,N2, N3 = number of independent time intervalsin the Ii,, I2q., and R, time slots, re-spectively,

II = number of independent time intervalsover a group repetition period.

When all of the time slots are of equal duration, i.e.,IIs =12. = R,, (1) can be simplified to

(2)

where

Pf, = probability of a single noise pulse exceeding thethreshold in any time slot,

r = number of pulses in the pulse code group.

The number of independent time intervals in a timeslot N may be expressed as

W.WJP(min) (3)

where

Ws = time slot width,Wv(min) = minimum pulse width that can be passed

by the beacon receiver.

PF = 1 - (1 - Pf )M

38 June

Rls

Mealey: Calculating Error Probabilities in Radar Communication

The number of independent time intervals over a grouprepetition period M may be expressed as

that proper pulse code group decoding hasoccurred;

11 ,-W Wb

M = GRFWP(min)

where

GRFWb

Wb

= pulse code group repetition frequency,- pulse code group width,- receiver blanking time following a beacon reply

transmission.

For Case (a) of Fig. 2, the probability must be con-sidered over the entire group repetition period, 1/GRF,since the event is equiprobable anywhere in the interval.A group repetition frequency of 640 GPS yields a grouprepetition period of 1562.5 usec. A pulse code group isapproximately 12 gsec long, so that the interval overwhich noise can cause false decoding is reduced to approxi-mately 1550 usec. In addition, the receiver is blanked forabout 50 usec after a beacon reply pulse is transmitted,which further reduces the period to about 1500 ,usec. Thereceiver video bandwidth was assumed to be capable ofpassing '-/Asec pulses. Thus, there are 4 X 1500, or 6000independent time intervals in a group repetition period,i.e., M = 6000. In Cases (b) and (c), time slot discrimina-tion precludes a noise pulse causing an error unless itoccurs in defined time slots.

Case (b) is the occurrence of a noise pulse of oppositepolarity to the code group pulses and in time coincideneewith one of them. The probability of this event is

PM 1 (1 -PP ) (5)

where

PM = probability of decoding a false information bitdue to noise cancelling one or more pulses of thepulse code group,

Pm = probability that a single pulse of the pulse codegroup will be cancelled by noise.

Case (c) is the occurrence of a noise pulse in one of theunoccupied time slots coincident with a proper pulse codegroup. The probability of this event is a conditional proba-bility defined by

Pf, = P(I)P(f/I) (6)

where

PfI = probability of decoding a false information bitdue to noise occurring in an unoccupied timeslot coincident with the occurrence of a properpulse code group;

P(I) = probability of proper pulse code group de-coding, i.e., probability that no pulse codepulses are missed;

PU(/I) = conditional probability of a noise pulse occur-ring in one of the unoccupied time slots given

(4)

P(I) = (1 - Pm)r;

P(f/I) = 1 - (1 - Pf) '

(7)

(8)Thus,

PfI = (1 - Pm) [1 - (1 - pf)(s-r)NI (9)

where s = number of time slots in the pulse code groupdecoders.The pulse code group error probability may then be

defined as

PGB = 1 - (1 - PF)(1 - PM)(1 - PfI) (10)

where PGE = pulse code group error probability.To minimize the pulse code group error probability, the

receiver threshold (bias level) must be correctly set. In abinary communication system, the proper setting is thelevel where the probability of a miss is equal to the proba-bility of a false alarm. The probability of a miss is PMand the probability of a false alarm is 1- (1- PF) (1- PfI).Thus, the optimum threshold setting is the bias level where

PM = 1 - (1 - PF)(1 - PfI). (11)In terms of P1 and Pm, the optimum level is

PM = 1

r ~~(1- PfYfPf.)M,ll[[1 + (1 - P P P )XJ[1 - (1 - p)(s-r)N]

(12)The single pulse miss probability Pm vs the single pulsefalse alarm probability Pf for the optimum bias levelsetting is shown in Fig. 3.The required signal for the optimum threshold setting

is Yb + (S - Yb), where Yb and S- Yb are the biaslevel and signal minus bias level, respectively (see Fig. 4).If the noise distribution in the video section preceding thethreshold device is essentially Gaussian, the values of Yband S -Yb for discrete values of Pf and Pm, respectively,can be obtained from a cumulative normal distributiontable.2 In the receiver types under consideration, the videobandwidth is much narrower than the IF bandwidth;therefore, the presence of Gaussian noise in the IF sectionwill result in approximately a Gaussian distribution in thevideo section.3 This is true if either a linear detector ora quadratic detector is employed for second detection.Fig. 5 shows the false alarm probability Pf or the missprobability Pm as a function of bias level Yb or signalminus bias level S- Yb, respectively. Fig. 6 shows themiss probability or the false alarm probability as a functionof SNR for the optimum bias level setting.

2 "Reference Data for Radio Engineers," I. T. T. Corp., NewYork, N. Y., 4th ed., p. 1117; January, 1957.

3 W. L. Blair and C. R. Ammeman, "Effects of Bandwidth onReceiver Sensitivity," Haller, Raymond and Brown, Inc., StateCollege, Pa.; 1959.

1963 39

IEEE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY

S - Yb

Yb

SIGNAL LEVEL

BIAS LEVEL

RMS NOISE LEVEL

10-3 10-4 10-5 10-6FALSE ALARM PROBABILITY, Pf

Fig. 3-Single pulse miss probability vs single pulse falsealarm probability for optimum bias level.

E

co:0

0-

0-

co

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-r

LI-I

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Fig. 4-Signal and bias level relationship.

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0-

1-

co

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CC-

1-

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Yb OR S- Y1b(UNITS OF RMS NOISE)

Fig. 5-False alarm probability or mis probability as a function ofbias level or signal minus bias level, respectively.

10

10

10

15 16 17

S/N RATIO (db)

Fig. 6-False alarm probability or miss probability vs SNRfor optimum bias levei.

40 June

ECL

-i

co

co

nC=

1963 Mealey: Calculating Error Probabilities in Radar Communication

10o1I

-j

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co

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In-)- - , , _ _ .

10°

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41

au I I I 10 ~~~~I14 15 16 17 18 19 20 1u4 15 16 17 18 19 20SIN RATIO (db) S/N RATIO (db)

Fig. 7-Pulse code group error probability vs SNR for optimum Fig. 8-Word error probability vs SNR for optimum bias level.bias level.

iol

CL

coco:r

De

1.0-

0.9-

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0.7-

0.6-

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0.4-

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SIN RATIO fdb)

Fig. 9-Frame error probability vs SNR for optimum bias level Fig. 10-Probability(no redundant transmissions).

t

I

I I I 1 115 16 17 18 19 20

S/N RATIO (db)

of correctly receiving a message vs SNR foroptimum bias level.

in 1

n, . s .. I

--A

IEEE TRANSACTIONS ON SPACE ELECTRONICS AND TELEMETRY

The pulse code group error probability is expressed by(10). In terms of Pf and Pm, the equation is

POE = [1 (1 - PfPf~Pfj3)'(1 - Pm) ]

-(1 pm r + (1 Pm)r(I _ pf) (sr)N] (13)

A unique value of Pf and Pm can be obtained from Fig. 6for each value of SNR. If these values are substituted in(13), the pulse code group error probability vs SNR forthe optimum bias level setting is obtained. This functionis plotted in Fig. 7. Since one pulse code group is requiredto transmit one bit of infonnation, bit error probability isidentical to code group error probability. Thus, from thegraph, a SNR of 18.58 db is required to achieve a biterror rate of 1 in 104.

WORD ERROR PROBABILITY

The word error probability is given by

PWE =1F (1 - PGE) (14)

where

PWE = word error probability,B = number of bits in a word.

Fig. 8 shows the word error probability as a functionof SNR for a 17-bit word. For a SNR of 18.58 db, corre-

sponding to a bit error rate of 1 in 104, the word errorprobability is 1.7 X 103. This amounts to a word errorrate of 1 in 588.

FRAME ERROR PROBABILITYThe frame error probability is given by

P(Fr)E 1 - (1 - PWE) (15)where

P (Fr) E =- frame error probability,W = number of words in a frame.

Fig. 9 shows the frame error probability as a functionof SNR for a 7-word frame without redundant transmis-sions. For a SNR of 18.58 db, corresponding to a bit errorrate of 1 in 104, the frame error probability is 3.5 X 102.This amounts to a frame error rate of 1 in 29.The improvement in error rates to be gained by redund-

ant transmissions is shown in Fig. 10. Two cases areshown: one for a single transmission and one for five re-dundant transmissions. It is seen that only about 2-dbdifference in the SNR is required to obtain equal proba-bility in the two cases. Furthermore, it is seen that whenthe SNR is greater than 19 db and 17 db, respectively,the probability of correctly receiving a message is veryhigh while message reliability decreases sharply when theSNR falls below 19 db and 17 db, respectively, for thetwo cases.

A Boolean-Function-Multiplexed Telemetry System*ROBERT C. TITSWORTHt, MEMBER, IEEE

Summary-By inserting encoded data from several sources intoa Boolean function at the transmitter, and using correlation tech-niques at the receiver, it is shown that a very efficient telemetry ormultiple-address system can be devised. The optimum logic isderived and proper channel codes are indicated.

INTRODUCTION

56p HIS PAPER describes a telemetry system in whichinformation from several data sources is sent simul-taneously over a single channel, not using time or

frequency-multiplexing, but in which the encoded input

* Received January 4, 1963; revised manuscript received January16, 1963. This paper presents results of one phase of research carriedout at the Jet Propulsion Laboratory, California Institute ofTechnology, under Contract No. NAS 7-100, sponsored by theNational Aeronautics and Space Administration.

t Jet Propulsion Laboratory, California Institute of Technology,Pasadena, Calif.

data are inserted simultaneously into a Boolean function,as shown in Fig. 1. This system can also be used equallywell as a multiple-address system, in which one trans-mitter simultaneously sends messages to several receiversin a single code stream. The fact that the transmittedsignal is binary enhances its use in satellite repeaters, wherelimiters are often used for signal-to-noise improvement.Word and frame-synchronization information can alsobe carried in the signal very simply, a feat which might notbe easy to implement in other binary schemes (such astime-division multiplex).

Briefly, according as the ith data source is 0 or 1, eithera code word a"', L symbols long, or its complement is fedinto the logic network.The receiver is a correlating device of the form shown

in Fig. 2.The analysis of this system is based on the fact that it

is possible to write the output sequence as a sum of terms

42 June