Digital frequency modulation as a technique for improving telemetry sampling bandwidth utilization

Transcription

1 Digital frequency modulation as a technique for improving telemetry sampling bandwidth utilization by G. E. HEYLGER Martin Marietta Corporation Denver, Colorado NTRODUCTON A hybrid of Time Division Multiplexing (TDM) and Frequency Division Multiplexing (FDM), both wellestablished in theory and practice is described herein. While related to TDM and FDM, the particular combinations of techniques and implementations are novel and, indeed, provide a third alternative for signal multiplexing applications. The essence of the idea is to perform all band translation and filtering via numerical or digital techniques. Signal multiplexing techniques are widely employed as a means of approaching the established theoretical limitations on communication channel capacity. n general, multiplexing techniques allow several signals to be combined in a way which takes better advantage of the channel bandwidth. FDM systems accomplish this by shifting the input signal basebands by means of modulation techniques, and summing the results. Judicious choice of modulation frequencies allows nonoverlapping shifted signal bands, and permits full use of the channel bandwidth. Refinements such as "guard bands" between adjacent signal bands and the use of single sidebands can further affect the system design, but, in general, the arithmetic sum of the individual signal bandwidths must be somewhat less than hah the composite channel bandwidth. TD1V systems achieve full utilization of channel bandwidth in quite a different way. Several signals are periodically sampled, and these samples are interleaved so that the individual signal must be sampled at least twice per cycle for the highest signal frequency present in accordance with Nyquist's sampling theorem. n this case, also, the number of signals that can be combined depends upon the sum of individual signal bandwidths and the bandwidth of the channel itself. The sampling theorem states that only two samples per cycle of the highest frequency component of a strictly band-limited signal are required for complete recovery of that signal. Nevertheless, 5 to 10 samples per cycle are widely employed. There are reasons, practical and otherwise, for the resulting bandwidth extravagance: 1. Many times it is difficult, if not impossible, to place a specific upper limit on "significant'. frequency components. Safe estimates are made. 2. nterpretation of real-time or quick-look plots is simpler and more satisfying if more samples per cycle are available. 3. Aliasing or folding of noise is more severe for relatively low sampling rates and inadequate prefiltering. This paper acknowledges the practice of oversampling but avoids the difficulties previously described. Full use is made of the sampling bandwidth by packing several signals into that bandwitdh utilizing a form of FDM. The novelty lies in the use of FDM and the way modulation is achieved for periodically sampled signals. SYSTEM DESCRPTON Before describing the system, it is useful to briefly consider some theoretical background. The following discussion should clarify the basic ideas. Consider a source signal with the spectrum shown in Figure l(a). t is well known that sampling signals at a frequency is = l/t where T is the time between samples, results in a periodic replication of the original spectrum as shown in Figure l(b). Modulation of the original signal by frequency /0 produces the usual sum and difference frequencies, and sampling then results in the replicated pattern shown in Figure 1 ( c). 275

2 276 Fall Joint Computer Conference, 1970 rh (a) Original r71 -T2~-~ r (b) Sampled (c) Sampled and Modulated by fo Figure 1-Spectral effects of sampling and modulating Now consider three source signals with the spectra shown in Figure 2(a), all with roughly the same bandwidth. Modulating the second and third signals with the frequencies f8/2 and fs/4, respectively, results in the shifted spectra shown in Figure 2(b). Summing yields the composite spectrum shown in Figure 2(c). This composite signal now makes full use of the sampling bandwidth. Figure 3 shows the inverse process of obtaining the original spectra. Demodulating by the same frequencies used for modulation successively brings each signal band to the origin where low pass filtering eliminates all but the original signal. Since few signals are strictly band-limited, it is evident that crosstalk noise will appear in the received signal. This noise can be controlled by the degree of pre- and postfiltering. For certain relatively inactive signals, the crosstalk may be no penalty at all. n general, however, crosstalk presents the same problems here as with any FDM system. The important point to be made is that tracking of the modi demod oscillators is not relevant since these operations are obtained directly by operating on successive samples, i.e., there are no local oscillators per se. n general, modulation is accomplished by multiplying the signal source by a single sinusoidal frequency or carrier. Sampled signals are modulated in S;"ered by Low-Pass Fllt.rin~ ~ C'\OC'\ ~ do~ A C'\OC'\., :.!, s s 3 2. (a) Combined Spectra of Three Oversamp1ed and Modulated Sources 1 1 1, C"'\ f:,. C'\ c;j ~ f:,. ~ Q C'\ ) 1"'""Jr; 3l1... -t, -12f~ 0 ~ 12f" 's 32f. (b) Demodulated by 1/2 f ; t s rro G""\ rro C" fa) ~rro C\ rro C"' fa) ~ 3!f, -, -12/. 0 12/ SS (c) Demodulated by 1/4 fs Figure 3-Prefiltered separation of combined signals the same way, but the modulating frequency multiplier is required only at successive sample times. Modulation (i.e., multiplication) by integer fractions of the sampling frequency is particularly simple if appropriate sample times are chosen. For example, certain modulation frequency amplitudes are quite easily obtained as shown in Table. The phase shift of 1'(/4 for 1/4 fs was chosen to avoid multiplication by zero yet retain natural symmetry. All the modulation factors may be easily obtained by modifying the sign of the signal magnitude and/or multiplying by a factor of 1/2. Furthermore, the majority of interesting cases are handled by these modulation frequencies, packing two, three, or four FDM channels within the sampling bandwidth. This degree of packing nicely accommodates practical oversampling systems encountered in practice. For particular applications, it may be useful to employ arbitrary modulation frequencies and the corresponding sequence of numerical multipliers (nonrepeating or repeating). A hybrid form of implementation is shown in Figures 4 and 5. Figure 4 is the modulator, and Figure 5 is the demodulator. Not explicitly shown, but implied, (a) Three Oversampled Source Signals t. Table Modulation Factor Modulation Frequency General Expr.ession. k = 0,1,2,... Periodic Sequence t fs cos (t fs 21fT) = cos k1f l, -1,... s 1f ( f) cos k;;- 21fT = cos k2 1 f 4 s 0,, 0, -1,... C"'\ 1 (b) Original Signals Modulated by O. 1/2 fs' and 1/4 fs' Respectively ' C"'\ 0 C'""\ t (c) Combined Spectra (Reduced to Sampling Bandwidth) 1 f b s 1 f 'f ( f) cos k 6' 21fT = cos k3 cos ( [ s) 2 k 321fT = cos k3 1f (No t e Con stan l Amplitude) 1, t, -t, -1, -to t. 1, -t, -to... Figure 2-Combinations of oversampled signals TABLE -Modulation Factor

3 Digital Frequency Modulation , 5, ~f'., , S L ~ 1/2 t 1 : Notes: Figure 4-Sampled FD M modulator To Conventional Time Division Multiplexing System 1) fa' is a periodic pulse stream. delayed with respect to f s. the sampling pulse sequence. (See text..) is the use of the combined signal output as a single sampled source for conventional TDM systems. The system diagram assumes the case of four signals of roughly equal bandwidth to be combined into a single signal. Subfunctions such as sampling, counting, digital decoding trees, and operational amplifiers can be implemented in a variety of ways utilizing conventional, commercially available functional blocks or components. Details of the subfunction implementations themselves are incidental to the concept but important to the particular application. Referring to Figure 4, the multiplexer modulator works as follows: Four independent signals (Sl, S2, S3, and S4) are accepted as inputs. One, shown as S1, goes directly to the summing amplifier, A. Each of the other signals is switched periodically under control of the approprlate binary counter which is synchronized and driven by the sampling frequency pulses. As shown, S2 is alternately switched from the first to the second of a two-stage cascade of operational amplifiers. The effect of this chain is to alternately multipiy S2 by the factors plus one and minus one, i.e., the modulation factor cos k7l"; = 0, 1, 2,... in accordance with Table and considering the modulation signal valid at the sample times only. Similarly, S3 is multiplied by the periodic sequence (1, -72, -72) again in accordance with the third line of Table. The effect, considered at sample times only, is to modulate Sa by 1/3 fs. Fip.ally, S4 is modulated by 1/6 F s, by periodically switching this signal to one of six inputs of the operational amplifier chain with the gains (1, 72, -72, -1, -72, 72) in accordance with line four of Table. All four outputs are summed by the operational amplifier A, and the summed signal sampled at the output of A at the sampling frequency, fs. t should be noted that the switching counters can be changed at any time after a sample is taken from the output of A; therefore, the design of the system provides that the pulse driving the counters is delayed slightly more than the aperture time of the sampled output. This mechanization provides ample time for switching operations prior to the subsequent sampling. The sampled output signal, St*, can be used as an input to a conventional TDM system. The demodulator shown in Figure 5 is very similar to the modulator. n fact, within the dotted lines it is identical. Here, the appropriate output from a conventional TDM system, St*, is used as input to all four counter-controlled switches. A sample and hold operation is employed at the input in order to drastically reduce the time response requirements of the operational amplifiers. Again, sequentially switching the input effectively demodulates St by the frequencies 1/6 fs, 1/3 fs, and 1/2 fs. Since this modulation is effective only at the sampling instants, a sample and hold circuit is required at each output. The low-pass filter eliminates components of all but the demodulated signal. Note that for the demodulator, the signals f't should precede fs in phase by the aperture (or pulse width) of St*, to allow a maximum time for change in St* to be accommodated by the amplifier and switching chain. Since fs is derived from St* and is a periodic signal, any desired relative phasing is readily achieved. SYSTEM ADVANTAGES AND CAPABLTES Several useful and interesting features are inherent in the system: 'T" 1. Numerical Modulation of Sampled Signals Because the modulation signal is required only "{"ehron'"} L T c.in.tor f. Figure 5-Sampled FDM demodulator 1/2 " '3 " "

4 278 Fall Joint Computer Conference, 1970 at the sampling instants, a periodic sequence of numerical multipliers substitutes for the local oscillator of conventional frequency modulation systems. Conventional oscillator accuracy and stability problems do not arise, and very low frequency modulation is readily achieved. 2. Coincident Sampling of Several Signals-Conventional TDM systems may combine signals sampled at the same rate, but at different instants of time. This approach provides for combining signals sampled at the same rate and same times. Full use of conventional TDM techniques can be employed on the combined signal. 3. Full Utilization of Sampling Bandwidth The sampling rate chosen defines the unaliased bandwidth in a sampled data system. Here, a way of combining several independent signals is employed so that the total sampling bandwidth can be utilized for transmission of information. 4. Signal ndependent Choice of Sampling Rate As a corollary to 3, this system permits, even promotes, oversampling of individual signals. Oversampling is attractive and widely practiced as previously noted. The system described here avoids the usual oversampling penalties by packing several independent signals within the sampling bandwidth. 5. Noise Aliasing Avoidance-Some source signals must be heavily prefiltered or oversampled in order to avoid the noise signal folding effects of sampled data systems. Again, oversampling can be employed without the usual penalties. t should be noted that wideband noise will of course result in crosstalk among the combined channels. n summary, the system described gives a new dimension in the design of signal multiplexed systems. Combination of these techniques with the conventional TDM and FDM techniques allows the designer to tailor a sampled data system to the peculiarities of a specific set of source signals, while making full use of the available sampled bandwidth. ALTERNATVE MPLEMENTATONS The hybrid system described herein uses pulse amplitude modulation (PAM). However, pulse code modulation (PCM) can be employed as well, in one of several attractive alternative implementations. The following system functions can be identified: 1. Sampling, timing, and switching; 2. Analog/digital (A/D) conversion; 3. Sample modulation/demodulation. The modulator/transmitter also requires an adder for combining the signals, while the demodulator / receiver requires a suitable lowpass filter for each output. Conversion to a digital representation of the signals can be performed at most any point in the system. Following conversion, the subsequent functions are performed via conventional digital arithmetic and logic operations. Exclusively digital implementation As an extreme example, consider an implementation that provides A/D conversion at the source (modulator/transmitter input). Modulation is accomplished by arithmetic multiplication of the source sequence values by the desired modulation sequence, cos ko o where 0 = st. Note that in this case, the modulation sequence need not be a periodic sequence if a means is provided for generating the values cos k 0 0 for all integers, k. ndependent signals are combined after modulation simply by arithmetic addition of corresponding modulated sequence values. The summed sequence is the output. The combined PCM samples are then handled as with a conventional TDM system. At the demodulator/receiver, the input is the digital sampled sequence as derived from a conventional PCMsystem. Demodulation is performed as before; arithmetic multiplication of the input sequence by the appropriate sequence of values, cos k 0 0 Each resulting output must be filtered to eliminate the other signal components. Filtering can be ac complished numerically using either recursive or nonrecursive techniques. The outputs then are available as separate signals corresponding to those first transmitted. The digital output sequence may be used directly for listing, further processing, or as an input to an incremental plotter. Alternatively, D/ A, conversion and hold operations convert the signal to its analog equivalent. Mixed analog/digital implementations Evidently, a number of obvious combinations of PAM and PCM are possible. Thus, operational amplifier (op-amp) modulation can be used in combination

5 Digital Frequency Modulation 279 with a time-shared AD converter and arithmetic summation with the result handled as a conventional PCM signal. Similarly, at the receiver, D A conversion may take place at the output of the PCM arithmetic modulator, and the result passed through a conventionallow-pass analog filter for signal recovery. A nalog system simplifications Figure 4 presents the system in a way that aids description and understanding. Good design practice would permit combination of the modulation and summing functions in a single op-amp stage. Similarly, various combinations of cascades 2- and 3-way switches might be advantageous instead of the single stage 6-way switch shown in Figure 4. Modulation sequence considerations The op-amp modulator implementation requires that the modulation sequence, cos k eo, be a repeating or periodic sequence. From a practical point of view, only a small number of modulation values should be employed, since each requires additional switching and input to the op-amp. While the only theoretical limitation on the number of values is that eo be some rational fraction of 211"', the simple ratios of the examples shown should prove most useful in practice. Arithmetic implementation of the modulation and demodulation function imposes no constraint on the number of distinct modulation values, cos k eo. Successive values may be generated arithmetically using some equivalent of the following algorithm: sinkeo = cos(k - l)e o sinoo + sin(k - 1)0 0 coseo coskeo = cos(k - 1)0 0 coseo - sin(k - l)e o sineo Only the initial values cos 0 0 and sin eo are required to start. f eo is some rational fraction of 211"', the sequence will be repeating; otherwise, not. n this case any desired modulation frequency (wo) may be realized. Bandwidth packing variations While roughly equal bandwidths were assumed for the combined signals of the system described, the fundamental constraint is that the sum of, signal bandwidths plus guard bands must be less than the sampling frequency. As usual with FDM systems, both upper and lower sidebands for each signal must be included in this consideration. Choice of a suitable modulation frequency then depends upon the placement of each signal band within the sampling bandwidth. Clearly, many variations of center frequencies and bandwidth are feasible and useful. Variations in digital system A general purpose digital computer can perform all operations required for modulating, summing, demodulating, and filtering. Where such a computer is already employed in the data system for switching, comparison, calibration, and control, the additional functions described here become particularly attractive. Standard programming practices can be used to perform the essential functions described here. Alternatively, for the system example the arithmetjc operations required are quite simple. Multiplications of Y2 and -1 are readily realized by right shift and sign change operations, respectively. A special purpose digital computer with few storage registers and capability for "right shift," "add," "sign change," and conventional register transfers, will provide the required functions. CONCLUSON The digital frequency modulation technique described herein permits combination of several signals into a single signal having a sampled bandwidth equal to the sum of the original signal bandwidths. Utilization of this technique to reduce the penalties of oversampled telemetry channels appears particularly attractive.

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