PULSE-FREQUENCY-MODULATION TELEMETRY by

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

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AND . S P A~ E ADMINISTRATION-.I -

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This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

PULSE-FREQUENCY-MODULATION TELEMETRY

By Robert W. RochelleGoddard Space Flight Center Greenbelt,Maryland

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

F o r sale by the Office of Technical Services, Department of Commerce, Washington, D.C. 20230 -- Price$1.50

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

PULSE-FREQUENCY-MODULATION TELEMETRYby Robert W. Rochelle Goddard Space Flight Center

SUMMARY

This report is concernedwiththeheuristicdevelopment of thebasic features of pulse-frequencymodulation,aninformationencodingtechnique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and its ease of generation. A description of the present method of formating and synchronization is presented in order to illustrate the convenient handling of both analog and digital data. The theory of group-codedbinary sequences is derived by using a special type of correlation matrix. By combining particular sequences from a number of these correlation matrices, a new matrix is generated which is identical to the correlation matrix of a set of pulse-frequency modulation words. Thus, it is shown that pulse-frequency modulation has the same communication efficiency as a comparable set of coded binary sequences with an equal number of quantized levels. In the detection process for P F M signals, a set of contiguous unmatched filters is used to enhance the signal-to-noise ratio. To examine the effects of Rayleigh noise on the output of these filters, the word-error probability is derived as a function of the energy per bit, noise power density, and degree of coding. The same development is given for the matched-filter case. The analysis of the excitation of an unmatched filter to a step sinusoid is carried out to indicate the magnitude of the e r r o r s affecting the direct frequency measurement of the pulsed sine wave. The analysis also covers the action of a pulsed sine wave in which the frequency is changing as it passes through the filter. Experimental results from the spectral analyses of satellite recordings perturbed with random noise are presented to illustrate the noise-immunity characteristic of pulse-frequency modulation.

i

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

CONTENTS Summary

................................. INTRODUCTION ............................ The Encoding Problem .....................History of Coded Telemetry CHARACTERISTICS OF PULSE-FREQUENCY MODULATION

i

11 2

..................

........................... General Description ....................... DesignConsiderations ........................7 1011 Group Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . Parity Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . PFM Correlation Table . . . . . . . . . . . . . . . . . . . . .

CHARACTERISTICS OF CODED BINARY SEQUENCES

NOISE ANALYSIS

........................... Unmatched Filter .

. . . . . . . . . . . . . . . . . . . . . . . Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . .

13

14 2228 30 32 34 34 3841

........ Analysis of Unmatched Filters . . . . . . . . . . . . . . . . Matched-Filter Techniques . . . . . . . . . . . . . . . . . . EXPERIMENTAL RESULTS .................... SpectralAnalysis . . . . . . . . . . . . . . . . . . . . . . . . Signal-to-Noise-RatioComparisons ............ CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS ........................ References ................................CONTIGUOUS-FILTER DETECTION SYSTEM Appendix A-Amplitude Density Spectrum of a Pulsed Sine Wave

43 43 45

...........................

Appendix B-Autocorrelation Function for Gaussian Noise Appendix C-Response of a Single-Pole Filter to a Step Sinusoid Appendix D-Response of a Single-Pole Filter to a ChangingFrequency

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47

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49 53

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This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

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I 11111

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

PULSE-FREQUENCY-MODULATION TELEMETRY *by Robert W. Rochelle Goddard Space Flight Center

INTRODUCTIONThe problem of communicating over large distances with minimum power is particularly challenging in view of the advances in information theory during the last 2 decades. With this theory as a guide, a number of interesting types of communication systems can be postulated which, in the limit, approach the theoretical maximum communication efficiency. The encoding methods of some of these systems are exceedingly complex in their implementation. The ideal type of system retains a high communication efficiency with little loss in simplicity. Pulse-frequency modulation is an attempt to fulfill these two basic criteria. Pulse-frequency modulation (PFM) has been successfully employed as the information encoding technique in a number of small (< 200 lb) scientific satellites and space probes where the reduction of spacecraft power and weight was a prime consideration. The use of such systems leaves a greater percentage of the spacecraft available for scientific instrumentation. Since little information has appeared in the literature on the subject, it is hoped that this work will provide a background f o r practical usage, as well as a theoretical derivation, of pulse-frequency modulation.

The EncodingProblemTelemetry implies measurement at a distance. The concept of modern telemetry includes the gathering and encoding of information at a remote station, transmitting this encoded signal to the receiving station, decoding the signal at this station, and presenting the measurements in an acceptable form. The transmission path may involve propagation by radio, light, o r even sound waves. Attenuation of the transmitted signal is usually a predominant factor because of the physical separation between the transmitting and receiving apparatus. If attenuation is low, the choices of the encoding and modulation methods are not critical, and methods which lead to simple and reliable sy

stems are usually employed. With the introduction of attenuation in the transmission path, interference in various forms may perturb the signals and cause transmission errors. By proper encoding of the information, the effect of these noise perturbations can be minimized. The amount of

*This report was submitted to the University of Maryland a s partial fulfillment of the requirements for rhe degree of Doctor of Philosophy.

1

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

improvement or immunization to noise that can be accomplishedis governed by Shannon's channel capacity theorem (Reference 1):

where c is the channel capacity (bits per second), B is the channel bandwidth, P is the signal power, and N is the average noise power. As the signal power is decreased, the bandwidth of the encoded signal must be increased to maintain the same channel capacity and probability of error. Various schemes can be devised to take advantage Shannon's channel capacity theorem and of thereby reduce the transmitter power for a given information rate. Kotel'nikov has described the general theory of high efficiency encoding (Reference 2). The problem involved is to approach the Shannon channel-capacity limit as closely as possible. But this must be consistent with the ability to implement the encoding scheme with equipment which is not of undue complexity. A practical example of this problem is the telemetering of scientific data from satellites and space probes. Since spacecraft power is limited, savings in transmitter power due to efficientcoding can extend either the range or information rate or both. The improvement is greater in the small satellite class (< 2001b), where a larger percentage of the satellite weight is devoted to the transmitter power system.

History of Coded TelemetryCoded telemetry began with the early work on frequency modulation. At that time frequency modulation was thought to require less bandwidth than amplitude modulation, since the frequency deviation could be made smaller than the modulating frequency. Carson disproved this theory in 1922 (Reference 3). In 1936 Armstrong demonstrated the major advantage of frequency modulation-the improvement of the signal-to-noise ratio for large frequency deviations (Reference 4). The system traded bandwidth for signal-to-noise-ratio improvement. A recognized deficiency was the sharp threshold or decrease in the signal-to-noise ratio at low signal levels. Pulse-code modulation* developed much more slowly than frequency modulation (Reference 5). It was not until 1948 that Shannon predicted that pulse-code modulation would have error-reducing properties (Reference 1). In 1950 Hamming devised a practical scheme of coding to effect these error-reducing properties (Reference 6). He introduced the error-detecting parity bit for a binary sequence and added bits to these for error-correcting purposes. Since 1950, numerous papers have been written on the subject of coding theory. Viterbi utilized these theories in calculating error probability as a

function of the signal energy per bit for various degrees coding (Reference 7). of With the entry of the United States into space exploration, the need for a lightweight low-power telemetry system developed. Such a system (Reference 8), utilizing the noise-reducing properties of frequency modulation and the error-reducing properties of pulse-code modulation, was incorporated in the first Vanguard scientific space vehicle and ultimately achieved orbit in Vanguard I11*P.M. Rainey, U. S. patent 1,608,527, November 30, 1926, issued to Western Electric Co.,Inc.

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This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

(1959~1). The telemetry encoder had a 48 channel capability, weighed 6 ounces unpotted, and required 12 milliwatts of power. Pulse-frequency modulation, as the system was called, was used again on the Ionosphere Direct Measurements Satellite, Explorer VI11 (1960 51). In April 1961 the space probe ExplorerX (1961~l), which measured the interplanetary magnetic field, was sent to an altitude of 240,000 km and again pulse-frequency modulation provided the encoding technique. Since that time, four more scientific satellites have been orbited with pulse-frequency-modulation telemeters. Explorer XII (1961 ul) provided continuous measurements of the energetic particles in the Van Allen radiation belts out to 80,000 hm; Ariel I (1962 01) was an ionospheric satellite (a joint effort between the United States and the United Kingdom); Explorer XIV (1962/3 yl) was a follow-on to Explorer XII; and Explorer XV (1962/3 X1) is providing a study of the artificial radiation belt. Two satellites to be launched in the near future, the Interplanetary Monitoring Probe and the successor to Ariel I, will use pulse-frequency modulation as the encoding technique.

General DescriptionConsider a time function f 1( t ) which is band-limited between zero and 1/2T0 cps. The function may be completely described by a series of impulses of separation T and area f l(nTo), with, n= - m, ..., -2, -1, 0, 1 2 - -,+ c A sequence of these time samples may be encoded for trans,, o . mission over a telemetry link in several ways. The method considered here is to encode the magnitude of the a r e a of each impulse as the frequency of a pulsed subcarrier, the duration of the pulse being some fraction of the sample time To.A s e t of k analog time functions f m( t ) may be multiplexed by sequentially sampling each function with spacings between samples of T,/k. In order that the spectrum may be utilized more efficiently, each pulse length should be equal to the sampling time, In practice the pulse length has been set at half this value to allow for response-time limitations in the crystal filters used in the detection process. Thus, a train of sequential pulses, the frequency of each being proportional to the amplitude of a sample, comprises the basic configuration for pulse-frequency modulation.

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The signals f m( t ) may not necessarily be analog in nature. A sizable portion of the outputs of experiments aboard spa

cecraft occur as digital signals, and from a signal-to-noise-ratio viewpoint it is desirable to retain the digital character of the signal. The binary digits of the signal are combined and presented as a single digit of a higher order base. The present state of the art of PFM telemetry employs the encoding of three bits as one digit to the base eight; that is, three binary digits are encoded as one of eight frequencies. A special digital pulsed-subcarrier oscillator has been developed for this purpose; it is restricted to oscillation at only one of the eight possible frequencies, the frequency depending on the value of its three-bit input. In this manner the complexity of the switching circuitry in the encoder is materially reduced, since readout of the accumulators and s c a l e r s is accomplished three bits at a time instead of serially. Figure 1 illustrates the manner of commutating the digital oscillator to scan the stored data in the accumulator. Pulses from the experiment are counted and stored as a binary number in the accumulator. The digital oscillator is3

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

EXPERIMENT INPUT

15

15 BITACCUMULATOR

100

-

1

DIGITAL:;::F F 1 OSCILLATOR

0100TIME (rnsec) Figure 1-Digital data readout system.

I

commutated three bits at a time in five steps through the stored fifteen bits in the accumulator. The output of the oscillator, for this case (octal number 47216), is represented, as in Figure 1, by five serial pulses of discrete frequencies. Since both analog and digital signals are encoded as frequencies of pulses, they may be intermixed in any order in multiplexing. Any one channel may be subcommutated to extend the number of signals f m( t ) which may be encoded, with a corresponding reduction in the signal bandwidth. The present format for PFM specifies that sixteen sequential channels comprise a telemetry frame. The first channel is devoted to synchronization, and the remaining fifteen are distributed between the analog and digital data to be telemetered. A group of sixteen sequential frames formsa telemetry sequence of 256 pulses (sixteen of these are devoted to synchronization). Figure 2 shows the frame and sequence formats. Synchronization is assured in two ways. First, the energy in the sync pulse is increased by increasing its length 50 percent. Second, a unique frequency is utilized outside the data band for the sync pulse. To provide a means for identifying the subcommutated data, the frequency of the sync pulse in every other telemetry frame is stepped in sequence through eight frequencies. These frequencies lie in the data band and are the same as the eight frequencies of the digital oscillator. When there is a poor signal-to-noise ratio, the energy at the sync frequency may be stored over a number of frames, and thus the sync signal-to-noise ratio is improved. This, of course, increases the acquisition time but it is a necessary compromise in regions of poor signal-to-noise ratios.

Design ConsiderationsIn the design of an

y telemetry system, such features as bit rate, accuracy, precision, error rate, and bandwidth must be firmly specified. Proper choices for such factors are products of the application of information theory to the problem.4

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

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Let us investigatea few factors which are necessary in a PFM telemetry system. Let the maximum value of the analog time function f ( t ) be defined as the full-scale value F, of the channel. I F, is divided into N equal parts, the magnitude F,/N represents the precision of the system. f This is the smallest discernible change between two samples of f ( t ) The value of N is determined by the requirements of the experiment being telemetered. For small scientific satellites a precision factor 1/N of 0.01, o r 1 percent of the full-scale value, is normally considered adequate. For experiments, such as the flux-gate magnetometer, which depend on the measuring of the modulation of the magnetic field by the satellite's spin, precisions to 0.1 percent are desirable, although accuracy better than 1percent is not necessary. For analog signals encoded with PFM, the precision is a function of the signal-to-noise ratio. At close range, the precision is an order of magnitude better than at maximum range. This feature, which will be discussed later in more detail, is equivalent to encoding with a variable bit-rate.

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""_DATA CHANNEL 1 SYNC PULSE DATA CHANNEL 2

DATA CHANNEL 15

Figure 2-Pulse-frequency-modulation format: (a) telemetry frame; (b) sequence.

5

I

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

Since the frequency of the pulse contains the signal information, a measurement precision of 1percent, for example, requires that the detection system be able to discern 100 different frequencies. The total frequency band B is therefore divided into 100 equal parts. The separation between frequencies, Af, is B/lOO. The power spectra of the pulses (Appendix A) have zeros of power density on each side of the center frequency at multiples of the reciprocal of T the pulse length. T is determined by the sampling rate necessary to telemeter the highest frequency of f t ) at least better than at the Nyquist rate. The zeros of the power spectrum then should f a l l at multiples of B/lOO, o r A f,= I/T. This gives the relation between the bandwidth, precision, and pulse length,

where

1/N

is the precision constant.

The above analysis was made on the basis of the sampling time of a single channel f (t ) determining the pulse length T. Since time-division multiplexing is used, the pulse length must be shortened in proportion to the number of commutated channels. Therefore the bandwidth B will increase in proportion to the numberof commutated channels.

In effect, the application of Equation 2 predicts that the samplewill assume any one of N, in this case 100, discrete amplitudes. The power spectrum for any one of these amplitudes is centered in one of the 100 equal parts of width A f, in the band B.A first-order solution to the detection p

roblem is to arrange 100 contiguous bandpass filters with bandwidth A f,= 1/~ cover the bandwidth B A maximum-likelihood detector on the outputs to of the filters can select the filter with the greatest output. The signal is thus quantized into one of 100 possible levels. The original amplitude of a sample would not necessarily cause the frequency to fall exactly in the center of any one of the 100 filters. The output would still be greater in one filter than in an adjacent filter. As the amplitude of the sample is changed, the maximum-likelihood detector indicates that the adjacentfilter has acquired the signal only when its response has an amplitude greater than the output of the original filter. With a good signal-to-noise ratio the filter output which is greatest can be gated into a discriminator in order thatthe frequency might be measured with better than 1 percent precision.

.

Rather than samplings only at times nT, the frequency of the pulse can be a continuous representation of the amplitude of the sample for the whole duration of the pulse, T By using a discriminator on the output of the contiguous filters, not only can the average amplitude during the time T be determined but also the rate of change of amplitude. Having the latter information is equivalent to doubling the sampling rate (Reference 9). These topics will be discussed in detail later.

.

For data encoded as digital information, the minimum number of filters required in the detection system is the same as the number of frequency levels encoded. Filters for these frequency levels as well as filters contiguous to these discrete frequencies are tied together a"greatest of" conin figuration as in Figure 3. This allows some latitude in the stability of each discrete frequencytemperature drift or aging of the digital oscillator might cause a discrete frequency to f a l l outside

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This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

its assigned filter. Thus, the"greatest of" detector selects the filter with the greatest signal and transmits to the output a digital number indicative only of the group of filters in which the signal fell.

BANDPASS FILTE-iGRT

OF" DETECTOR

101

CHARACTERISTICS OF CODED BINARY SEQUENCESA somewhat different approach to the theory of codedbinarysequenceswillbe developed in this chapter in order to illustrate the positionof pulse-frequency modulationincodedtelemetry.The idea is to show that PFM is a specialbinary code takenfrom a group of manycodes which can be made up of patterns of binary zeros and ones.

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DIGITIZED

RECEIVED SIGNAL

011

SIGN^

Figure 3-Contiguous-filter bank digital-oscillator mode.

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Pulse-code modulation (PCM) is ordinarily thought of as the representation of sequential samples of a signal by a binary code; however, the original definition of PCM included all codes: binary, ternary, quaternary, etc. Patterns of these code elements make up the quantized amplitude value of the sample. For two reasons the binary code is used almost exclusively: the advantage in

the signal-to-noise-ratio relation, and the ease of generation. If the amplitude of a sample is to be encoded as an n-bit binary sequence, 2" different sequences are available to quantize the amplitude. Shannon has shown by his second theorem that the probability of e r r o r in recognizing any of the transmitted 2" sequences may be reduced by recoding the n-bit sequences or words into selected sequences of larger m-bit sequences (Reference 1); or, conversely, if only selected sequences of the 2" available sequences are allowed, then the probability of error per bit is reduced. Coded n -bit binary sequences are defined as a set of M selected sequences in the M< 2". When M= 2" the set of sequences is said to be uncoded.

Group CodingAdvantages may be gained by allowing the transmission of only selected sequences in the available 2" sequences of an n-bit encoded sample. This, of course, reduces the precision to which the samples may be quantized; however, the precision may be increased back to the original value by increasing n, since this increases the numberof sequences from which a judicious selection may be made. Later we shall discuss how these selections are made. A signal-to-noise advantage is obtained if the selected sequences are detected not bit-by-bit but as an entire sequence orgroup. U s e is made of the fact that signals add directly (since their phases are correlated) and noise adds as the square root of the sum of the squares.7

This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

The detection processis accomplished by cross-correlating each of the selected orallowed sequences with the transmitted signal; the allowed sequence which yields highest the crosscorrelation coefficient when correlated with the signalis selected as the most probable representation of the signal. Let f, ( t ) be one of a s e t of k transmitted sequences and f,( t ) be one of the allowed sequences stored in k correlators. Both f, ( t ) and f,( t ) are zero for t> T and t< 0. The correlator performs the mathematical operation:

wheren

= number of bits in the word,

TT

= length of sequence,= lag time,=

c,

(T

)

unnormalized cross-correlation function.

For the matched or optimum condition the lag time should be zero. The correlator with the greatest value of C,, ( 0 ) is selected as the one having the greatest probability of containing the signal.A z e m may be represented in the transmitted signal by a+1volt and a one by a -1 volt. The selected sequence or stored waveform in the correlator f m ( t ) is normalized so that the units of the cross-correlation coefficient CIm( 0 ) will correspond to the voltage measured at the output of the correlator. When 1= m, the cross-correlation coefficient is n volts. Since the correlator with the highest coefficient is being selected, it is desirable for the coefficient of all the other correlators to between the correct and be as low as possible, so that the largest possible distinction can be made the incorrect values of the coefficient. A s an exa

mple, let u s take all the available sequences of a four-bit word and construct a truth table which gives the values of the cross-correlation coefficients of Equation 3. Table 1 contains such a table; however, the order of the four-bit words was prearranged to bring out some salient features. For I= m the correlation coefficient is+4 volts; when f, ( t ) equals the complement of f, ( t ), the coefficient becomes - 4 volts. In Table l(a) there are two groups of eight sequences each. In either group the cross-correlation coefficient is zero except when the stored waveform matches the signal or is i t s complement. If an attempt w a s made to correlate signals in group 1 with the stored waveforms of group 2, the cross-correlation coefficient would be rt2 volts. Noise could easily perturb a+2 volt output into+4 volts, causing an error. Thus, if the transmitted sequences of a four-bit word are restricted entirely to either group 1 o r group 2 and the stored waveforms in the correlators are of the same group, then an improvement in the signal-to-noise ratio can be realized. These code groups a r e of the Reed-Muller type (Reference 10).

Intuitively, if the signal-to-noise ratio were unity fora single bit of the four-bit word, the output signal-to-noise ratio would be 2/1 (the bits of the signal add directly, but the noise adds as the

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This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

Table 1 Correlation Coefficients: (a) Four-Bit Sequences; (b) One-Bit Sequences; (c) Two-BitSequences. GROUP 2

\0011 0101

SIGNAL - fl( t )

88;*3$$$8 3 0% 2 0 0 0 0

GROUP 1

0000+4

00

0

0 -4+2+2+2 -2+2 -2 -2 -2

0+4 00 0

0 -4 0 c2+2 -2+2 -2+2 -2 -2 0+2 -2 0 -2+2 0+2 -2+2+2 -2 -2+2 - 2

0+4 0

0 -4 0 0 0 0 0

- Qcv

s?

g

0110 1001 0 1010 1100 1111

00

0+4, -40 -4+4

+2+2 -2 -2 -2+2 -2 -2+2+2+2 -2-2+2+2 -2

0

c

E

0

0 -4 0 00

0+4

0 -2+2 -2 0 -2 -2

+2

a

e2 aa

'

0 -4 0

00

0+4 0

+2 -2+2 -2+2+2+2 -2+2+2+20 00 0

-4 0

0

0+4 -2 -2 -2-2 t 4

0001+2+2+2 -2

+2 -2 -2

0

00

0

0-4

"0N

\

0010+2+2 -2+2 -2 0100+2 -2+2+2 -23

+2 -2 -2 -2+2 -2

0+4

0-4

0 0 0 0 00h

0

0+4

0

0-400

00+201I

a 0111 -2+2+2+2 -2 -2 -2

+2 0

0 0

0+4-40-4+4

0 0 0

E

1000

+2 -2 -2 -2+2+2+2 -2+2

0

0

101

mfl(t)

0

0+2 -2 0 0 -2+2

I

1011 -2+2 -2 - 2+2+2 -2 1101 -2 -2+2 -2

0

0-4

00

0+4

O 1111

-2 0

0+2

+2 -2+2+2+2+2

I

0-4

0

00

0+4

1110 -2 -2 -2+2 -2

+e

-4

0

0

0

0

0+4

square root of the sum of the squares). The signal-to-noise ratio has doubled and the information rate has reduced by only one-fourth. This is a good improvement; unfortunately, it cannot be realized. True, the signal-to-noise ratio out of the correlator is improved over the bit-by-bit detection, but the outputs of the correlators must be compared to determine the largest coefficient. It is probable that noise in another correlator willbe greater than the coefficient in the correct correlator. This probability increases as the numb

er of correlators increases. The situation is discussed in detail later. The group code method still provides a substantial improvement in the signal-to-noise ratio over bit-by-bit detection. The number of correlators in group 1 can be reduced from eight to four if the - 4 volt output also is utilized. Since each correlator responds only to its matched signal and its complement, a single correlator can be used to detect two sequences. The correlator with the greatest absolute magnitude is selected as the one having the best probability of containing the signal. The sign of the coefficient determines whether the signal was in the first or second four sequences of a group. A code of this type is said to be bi-orthogonal since negative as well as positive cross-correlation coefficients are utilized. When only the positive outputs are used the code is orthogonal.

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This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

In the preceding example the prearranged order the four-bit sequences was most important. of A few fundamental rules have been developed to generate codes with bit lengths of n= 2k, where k= 0, 1, 2, 3, These rules are amenable to computer programming but are somewhatdifferent from those found in the literature. Consider first the correlation table for a one-bit word as in Table l(b). Only one correlator is required if t h e bi-orthogonal properties of the signal are utilized. This table and the correlation table for the two-bit sequences of Table l(c) can be written by inspection. The first word of the four-bit table is generated by combining the first word, 00, of the two-bit table with itself to form 0000. The next word is made by combining the first word, 00, with its complement, 11, to form 0011. The same process is carried out with the second word of the two-bit table to form the next two words. The complements of these four words are now taken in reverse order; this generates the code of group 1. The group 2 code is started by combining the first word of the two-bit table with the second. The next word is obtained by combining the first word of the two-bit table with the complement of the second. For the third and fourth words the twobit second word is combined with the first word and then with its complement. The next four words are the complements of the newly generated words in reverse order.

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The eight-bit correlation table has 256 sequences which form a matrix of 65,536 correlation coefficients. A digital computer program for generating the 8-bit table w a s written by utilizing the four-bit table and the rules for forming the table from the preceding paragraph. The program was written in FAP (Fortran Assembly Program) s o that the bit patterns of the signals f, ( t ) and stored waveforms f,( t ) could be cyclically varied. That is, the first bit of all the eight-bit sequences was moved to the last-bit position anda new s e t of correlation coefficients computed. The program is arranged so that a print-out contains the correlation between the first sixteen signals and

the first sixteen stored waveforms. Table 2 is a print-out of the first page of the code book for the program. On the next page a r e t h e first sixteen signals correlated with the second sixteen correlators, and so on. There are sixteen code groups of the type in Table 2. Any one of these sixteen code groups could be used to encode four bits of data for a signal-to-noise-ratio improvement over a standard four-bit code. Table 3 illustrates the correlation between groups. It is interesting to note that a cyclic variation of the four-bit sequences does not generate any new codes. It only rearranges the order in the two code groups. In the eight-bit case, however, new codes are generated; in fact, a tremendous number of new codes can be made not only by cyclically varying the bit locationsof the signal and stored waveformsbut also by interchanging the order of the bits in all possible combinations. The process may be carried to the sixteen-bit case, for which there are over 4 x lo9 crosscorrelation coefficients. Sanders has described the Digilock coded binary PCM telemeter, which encodes sixteen-bit words to represent five-bit samples (Reference 11). This system has been flown on an Air Force Blue Scout rocket.,

Parity CodesA simple error-detecting parity code is constructed by adding one bit to an n-bit sequence to indicate whether the total word has an even or an odd number of 2eYo.s It is most interesting to

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This report is concerned with the heuristic development of the basic features of pulse-frequency modulation, an information encoding technique which has been used in a number of spacecraft. The primary advantages are its noise-immunity characteristics and

note that group 1 of the four-bit correiaLon table can be made from a three-bit sequence by adding an even parity bit; group 2 is generated by adding an odd parity bit. The eight-bit code of Table 2 may be thought of as a four-bit code with four bits for error-detecting and error-correcting.

PFM Correlation TableIn the eight-bit correlation table (Table 2) the second sequence, 00001111, is the same as the second sequence of the four-bit table of Table l(a), 0011, if the words of each are of the same duration. The second word of the two-bit table of Table l(c), 01, is also the same as these two if the time T for the words is the same. Since identical words can be found on different correlation tables, the possibility might be considered that a new table could be constructed from sequences taken from a number of tables. These sequences should have orthogonal properties. The most obvious one is

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to take 01010101 from the eight-bit table, 0101 from the four-bit table, and 01 from the two-bit table and construct a new table. From a six-bit table we could use 010101. All of these sequences are orthogonal to each other when substituted into Equation 3. Also, the alternating ZeYo and one code may be taken from higher order tables in constructing this new table. Pulse frequency modulation does exactly this in constructing a code. The code is made up of alternating zeYos and ones taken from higher order codes. For PFM signals in the frequency range from 5 to 15 kc, the 5.0 kc sequence would be 100 bits of 50 zeyos and 50 ones in an alternating pattern; the 5.1 kc sequence would be 102 bits in this same alternating pattern. Table 4 is such a table. Each frequency contains an integral number of cycles because the pulse length, derived from Equation 3, is constant. The

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