PULSE-FREQUENCY-MODULATION TELEMETRY by(5)

6

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.

-""~-

DIGITIZED

RECEIVED SIGNAL

011

SIGN^

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

in

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).

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