Telemetry system with a sending station using recursive filter for bandwidth limiting

Before subsampling them for transmission, telemetry data are bandwidth-limited by IIR filtering at the sending station of a telemetry system. The transmitted filtered telemetry data have attendant phase distortion that is compensated for by FIR filtering at the receiving station of the system, so as to obtain overall linear phase response through the system.

The present invention relates to telemetry systems and more particularly to 
systems where the designer finds it desirable to minimize digital hardware 
in a transmitting station of the system even at the expense of increased 
digital hardware in a receiving station of the system. 
BACKGROUND OF THE INVENTION 
This desire is felt when designing certain telemetry systems using wires, 
radio waves, light waves or other medium to link sending and receiving 
stations. This desire is felt, for example, where there is a much greater 
number of sending stations than receiving stations, owing to the overall 
savings of digital hardware that can then be made in the system. 
However, in certain circumstances this desire may be felt irrespective of 
the relative numbers of sending and receiving stations in the telemetry 
system owing to the power, volume or weight restrictions imposed upon the 
sending station being more restrictive than ones for corresponding 
parameters impose upon the receiving station. Examples of this are where 
the sending station is in a missile, in a launch vehicle or in an 
artificial satellite of a planet. 
In order to conserve telemetry bandwidth and possibly to reduce average 
power, it may be desirable to subsample the samples of a digital electric 
signal descriptive of a measured parameter. To avoid objectionable 
aliasing being introduced by such subsampling, it is the common practice 
to filter the digital signal prior to subsampling if the Nyquist rate to 
properly sample the signal exceeds the subsampling rate. After filtering, 
the subsampling rate exceeds the new Nyquist rate for the filtered signal, 
and the subsamples are transmitted to the sending station. 
In some applications, phase distortions introduced by filters with 
non-linear phase responses tend undesirably to obscure features of the 
telemetry signal. In such cases, the filters used in prior art telemetry 
have been of finite-impulse-response (FIR) type in order to secure 
linear-phase filtering. FIR filters are non-recursive and tend to involve 
a larger amount of digital hardware than recursive filters--that is, 
filters of infinite-impulse-response (IIR) type. Extending the number of 
samples in the filter impulse response by recursion allows more abrupt 
cut-off to be obtained for the same computation load. Narrower passbands 
or stopbands can be realized recursively for a fixed amount of power or 
hardware complexity. A bandwidth limiting filter with sharper cut-off 
permits the filter response to be subsampled closer to Nyquist limit 
(i.e., less frequently) without incurring aliasing. 
Recursive filters accumulate samples, adding each new sample as weighted by 
a respective factor less than unity, to an accumulation of past weighted 
samples, which permits their structures to be relatively simple while 
their impulse response is long-extended. Usually only a single multiplier 
is used for each accumulation procedure, and each procedure generates an 
impulse response of extended duration in terms of number of samples. This 
extended impulse response is obtained through short term storage of 
accumulation results. There is no need for extensive delay network and a 
large number of multipliers to obtain such extended duration of response 
as would be the case in an FIR filter. The simpler IIR filter structures 
tend to use less power and have less volume and weight than FIR filter 
structures when impulse responses are required over a large number of 
sample intervals. 
However, the distortion in phase attendant with the use of IIR filters has 
led engineers away from using recursive pre-filters for subsampling 
telemetric data. Correction of these phase distortions at the sending 
station has been done, but the attendant increase in the amount of 
filtering at the sending station also increases the complexity of the 
overall filter. In any case, the desire for maximization of complexity and 
power at the sending station is not achieved. 
SUMMARY OF THE INVENTION 
A telemetry system embodying the invention uses a sending station wherein 
digital signal samples responsive to a measured parameter--i.e., 
descriptive of telemetry data--are applied to a recursive filter for 
limiting the bandwidth of the signal preparatory to subsampling, thereby 
to avoid substantial aliasing. These subsamples are then supplied to a 
transmitter for transmitting information via a medium to a receiving 
station. The recursive filter has a system function with poles (and 
perhaps zeroes) in the complex frequency domain which introduce 
undesirable phase distortion into the telemetry data. The telemetry data 
is transmitted together with attendant phase distortion. 
The undesirable phase distortion is compensated for in a receiving station 
of the telemetry system, which station has a receiver therein for 
recovering a replica of the subsampled filter response. The replica is 
resampled at the original sampling rate with the sample positions empty of 
subsamples being filled with zero values. The compensation is provided for 
by filtering the resampled replica with a further filter having a low-pass 
system function, for suppressing repeats introduced by resampling, which 
system function includes zeroes located at the same place in the Z 
transform domain as the poles of the system function of the recursive 
filter in the sending station. Any zeros introduced at the transmitter are 
left alone, so these become part of the overall response.

DETAILED DESCRIPTION 
The FIG. 1 telemetry system comprises a sending station 2, a receiving 
station 3 and a transmission medium 4 linking them. Sending station 2 
includes a source of digital samples of the telemetry data that describes 
the measured parameter, as obtained from a sensor, for example. A 
recursive IIR digital filter 6 limits the bandwidth of these samples so 
that the sample frequency spectrum and its repeats generated by subsequent 
subsampling or decimation in a subsampler 7 will not intermingle to any 
appreciable degree, so as to avoid aliasing. The subsamples from 
subsampler 7 are supplied to a transmitter 8 appropriate to the 
transmission medium 4 being used--e.g., wires, radio transmission through 
an ether, or light transmission. At the receiving station 3 a receiver 9, 
of a type appropriate to the transmission medium 4 and to transmitter 8, 
recovers a replica of the subsampled filtered telemetry data supplied to 
transmitter 8. This replica exhibits phase distortion owing to the 
non-linear phase characteristics of the sending station filter 6. These 
non-linear phase characteristics are attributable to that filter 6 being 
recursive in nature, so that the filter kernel perforce is asymmetric. To 
suppress by means of compensation this phase distortion, the replica of 
the subsampled filtered telemetry data is subjected to filtering after 
receiver 9. 
It is more convenient to consider such filtering as being done after 
resampling the replica to the original sample rate of the digital signals, 
using re-sampler or expander 10 after receiver 9. Resampling is a 
procedure for generating from a set of original samples a new set of 
samples at a different sampling rate, each new sample being generated by 
an appropriate weighting of closeby samples in the old set. The accepted 
procedure for doing expanding, or resampling to higher sampling rate, on 
an interpolative basis is to place the samples preserved during 
subsampling in their previous cyclic order; to insert zero samples in 
cyclic order where the original samples have not been preserved during 
subsampling, which insertion generates repeat frequency spectra offset by 
respective multiples of sampling frequency. Re-sampler 10 is followed by a 
filter 11 to suppress the repeat frequency spectra by appropriate 
band-limit filtering and for compensating unwanted phase distortions 
introduced by filter 6. To further this goal, filter 11 has zeros in its 
system function located at corresponding points in the Z transform domain 
as the poles in the system function of the recursive filter 6 as well as 
zeros that suppress the repeat frequency spectra. 
Filter 11 is not a complementary filter with respect to the cascade 
connection of recursive filter 6 and subsampler 7, however. That is, it is 
not a filter which has a system function that when connected with the 
system functions of recursive filter 6 and of subsampler 7 results in flat 
response over all frequencies. Recursive filter 6, being a 
bandwidth-limiting filter for avoiding aliasing, provides relatively large 
attenutation in the portions of its stopband next to the skirt of its 
frequency response in order to perform its assigned duty, and concomitant 
phase distortion is great for these skirt frequencies and portions of the 
passband near them. So a complementary filter would have to provide 
relatively large gain at certain frequencies to compensate for the 
relatively large attenuation afforded them by recursive filter 6 and 
relatively miniscule gain at certain other frequencies to compensate for 
the relatively small attenuation afforded them by recursive filter 6. This 
requires substantially greater precision in the sampling and filtering 
procedures at sending station 2 and in the filtering procedure at 
receiving station 3 because the complementary filter response to 
quantizing noise is so exaggerated in the skirt-frequency regions. This 
increased need for precision is incompatible with the desire to reduce the 
complexity of digital hardware at sending station 2. The increased need 
for precision also makes it necessary to increase the number of 
significant taps and the range of tap weights in a complementary filter, 
which procedures one desires to avoid, if possible. Further, the region of 
high gain through the complementary filter extends over the region where 
the signal frequency spectrum and repeat spectra skirts extend and tend to 
intermingle to cause undesirable aliasing. Accordingly, there tends to be 
a stronger requirement for skirt suppression imposed on the recursive 
filter 6, which also is incompatible with the desire for simpler filtering 
hardware at sending station 2. 
Instead, filter 11 is a bandwidth-limiting filter with a passband somewhat 
narrower or at least not appreciably broader than that of the recursive 
filter. Such a filter suppresses the repeat spectra generated by zero 
re-insertion in re-sampler 10, so re-sampler 10 itself need not include 
filtering for that purpose and may, for ease of analysis, simply consist 
of means for inserting zero-value samples in appropriate sample positions. 
The fact that all sample positions except every n.sup.th in the output 
signal of re-sampler 10 are zero-valued, when filtering is exclusively in 
filter 11, means filter 11 may be considered as being an interpolative 
filter. Design procedures for interpolative filters have been studied 
extensively in the prior art, particularly in connection with quadrature 
mirror filtering, and are usefully applied to filter 11. 
To obtain a more specific feeling of what type of filter might be used for 
filter 11, consider a filter having a system function that is the system 
function of the previously considered complementary filter, as convolved 
with the system function of a linear-phase window filter having 
appreciable attenuation in the frequency regions where repose the skirts 
of the recursive filter 6. Preferably the bandwidth of the window filter 
should be narrower than that of recursive filter 6, thus to cause the 
bandwidth of filter 11 to be somewhat narrower than that of recursive 
filter 6 (and incidentally provide for the suppression of repeats 
introduced by insertion of zero-value samples in re-sampler 10). Such a 
filter 11 does not exhibit so great a range in the amplitudes of its 
filter coefficients as a complementary filter, and the exaggeration of 
quantization noise effects at passband edges is not incurred. The range of 
tap weights in filter 11 and the number of significant taps are not higher 
than the norm for low-pass digital filters. Customary precisions of twelve 
to sixteen bits in the multiplications involved in the filters 6 and 11 
appear to be sufficient, the inventor has calculated by computer 
simulation. 
These calculations were made assuming a subsampler 7 that selected every 
third sample in a baseband. To avoid aliasing in the subsequent 3:1 
subsampling a nine-pole Chebyshev low-pass filter with cut-off frequency 
at 0.12 sample rate, with attenuation of about 48 dB at 0.17 sample rate 
and with zeroes at sample rate was selected for modification to provide 
recursive filter 6. The modifications made in the filter were replacing 
the numerator of the system function with unity and adjusting filter 
sensitivity to maintain attenuation across band and remove the hump in 
response in the higher frequency portions of pass band. Both the amplitude 
variation and phase variations of such a filter through the passband and 
beyond are notoriously great. The filter 11 was designed convolving the 
complementary filter system function with the system function of a 
linear-phase low-pass filter. 
This linear-phase low-pass filter system function was generated by the 
following procedures. An appropriate sinc function was convolved with a 
Gaussian window of sufficient narrowness of bandwidth to achieve stopband 
attenuation in excess of that afforded by the recursive filter 6. For 
convenience this symmetric filter kernel was truncated to 299 samples and, 
to decrease the likelihood of trunction error, was convolved with itself. 
The resulting low-pass filter with a 599-sample-wide kernel provided over 
200 dB attenuation in a stopband. 
Overall system response to an impulse in a system using the filters 6, 11 
just described was noted to be symmetric with calculations both where 
twelve bits plus sign were preserved throughout calculations and where 
nine bits plus sign were preserved throughout calculations. An additional 
two bits or so resolution may be required in actual digital hardware 
implementations because of the customary practice of constructing the 
filters by sections. 
A more general design procedure for telemetry systems embodying the 
invention employs Z transforms, as follows. An overall system function 
(H(Z) should be factorable into factors H.sub.1 (Z) and H.sub.2 (Z), where 
H.sub.1 (Z) consists of the product of unity times each zero of filter 6, 
and where H.sub.2 (Z) consists of the product of unity times each zero of 
filter 11. One determines G(Z), the product of unity times each pole 
desired in filter 6, to go with the zeroes in H.sub.1 (Z) in order to 
provide a H.sub.1 (Z)/G(Z) system function that affords appropriate 
band-limiting for the subsequent subsampling. This can be done with 
reference to filter tables or by using an appropriate computer program for 
filter design. The system function for filter 11 will then be H.sub.2 (Z) 
G(Z), which is not linear-phase, but is FIR owing to the absence of poles. 
Filters 6 and 11 are cascaded insofar as overall system function is 
concerned, so overall system function is the product of their respective 
system functions H.sub.1 (Z)/G(Z) and H.sub.2 (Z) G(Z)--i.e., H.sub.1 
(Z)H.sub.2 (Z) which equals the originally specified H(Z). Known synthesis 
procedures can now be applied for designing filters with the desired 
system functions. 
The filter 6 may include zeroes for a number of different reasons. For 
example, zeroes may be included to reduce the number of poles required to 
obtain desired cut-off characteristics in filter 6. Use of an elliptic 
filter design would be a case in point. Or, for example, zeroes may be 
included to implement subsampling which does not simply select every 
n.sup.th sample of an IIR response, but instead selects every n.sup.th one 
of a linear-phase (and thus FIR) weighted average of 2n+1 successive 
samples of IIR filter response. More particularly, the IIR response may be 
convolved with a triangularly weighted interpolation function, to make 
filter amplitude response that has less high-frequency quantizing noise. 
FIG. 2 shows an alternative telemetry system configuration that can 
sometimes be used in place of the FIG. 1 telemetry system. The receiving 
station 3' of FIG. 3 differs from receiving station 3 of FIG. 1 in that 
the filtering after receiver 9 is not interpolative in nature. The 
subsampled telemetry data recovered by receiver 9 is limited in bandwidth 
and interpolative filtering does not increase the bandwidth of the 
telemetry data per se, of course. In the FIG. 2 telemetry system the 
digital filter 12 is operated at the subsampling rate, rather than at the 
original sampling rate, which subsampling rate is selected in the design 
of sending station 2 to be sufficient to sample the limited-bandwidth 
telemetry data in excess of Nyquist rate. The question of significance is 
whether the filter design at subsampling rate can afford adequate 
compensation of the phase distortion introduced by the poles in filter 6. 
The general nature of the technique involved in answering this question is 
easier to understand for the case where bandwidth limiting is low-pass in 
nature (although extension can be made to band-pass filtering). In 
describing this technique by subsampling example, rate is chosen one 
quarter the original sampling rate because the graphic illustrations of 
frequency scaling of the Z transforms are easier to verify by eye. 
The Z transform allows sampled-data functions to be mapped onto a unit 
circle in a complex coordinate space in a way that conformally maps the 
way continuous functions in the complex-frequency domain are mapped onto 
the real axis of a complex-coordinate space using the Laplace transform. 
In FIGS. 3, 4, 5 and 6 the Z transform is used in such mapping. 
FIG. 3 shows the location of the poles 13 and zeroes 14 of a representative 
filter 6 (of elliptic low-pass type) in the Z transform domain for the 
original sampling rate. The locations of the poles 13 are denoted by X's 
and the locations of the zeroes 14 are denoted by 0's. The zeroes 14 are 
located on the unit circle. Zeroes located on the unit circle are known by 
those skilled in the art of filter design not to introduce departure from 
linearity of phase. Pairs of zeroes having like angular components and 
having radial components exhibiting like percentage differential from unit 
circle also are known by those skilled in the art of filter design not to 
introduce departure from linearity of phase. Other zero locations are 
known to introduce departures from linearity of phase in the filter 
characteristic. 
FIG. 4 shows the locations of the pole-compensating zeroes 15 of a 
non-interpolative filter 12 in the Z transform domain for a subsampling 
rate one-quarter the original sampling rate. Each quadrant of the FIG. 3 
bilinear transform domain at original sampling rate--i.e., the one from 
zero frequency to half Nyquist rate, the one from half Nyquist rate to 
Nyquist rate, the one from Nyquist rate to three-halves Nyquist rate, and 
the one from three-halves Nyquist rate to sampling rate--conformally maps 
to the bilinear transform domain at subsampling rate, with radial 
components of the Z transform vector preserved and with angular 
commponents multiplied by the ratio of original sampling rate to 
subsampling rate. The locations in the FIG. 4 Z transform domain for 
pole-compensating zeroes 15 of the non-interpolative FIR filter 12 
correspond to the poles 13 of recursive filter 6 as subjected to frequency 
scaling. The radial component of each pole-compensating zero 15 of filter 
12 is the same as that of a corresponding pole 13 of filter 6, but the 
angular component of each zero 15 of filter 12 is four times that of the 
corresponding pole 13 of filter 6. That is, the angular component is 
multiplied by the ratio of original sampling rate to subsampling rate. One 
notes the pole-compensating zeroes 15 are not located at the unit circle; 
that is, filter 12 is not linear-phase. 
FIG. 5 shows the location of all zeroes of filter 12. In addition to the 
pole-compensating zeroes 15 of FIG. 4 there are additional zeroes 16 in 
stopband of filter 12, which are on the unit circle. These zeroes 16 are 
those which define a linear-phase low-pass filter function that is 
convolved with the pole-compensating filter function. 
FIG. 6 shows the overall system function response in the Z transform domain 
at original sampling rate. The poles 13 of the overall system function are 
those of filter 6 and they are overlain with respective ones of the 
pole-compensating zeroes 15 of filter 12, as transformed by frequency 
scaling. A pole and a correspondingly located zero are known to those 
skilled in the art of filter design to make no contribution to overall 
filter phase characteristic, and therefore they do not compromise 
linearity of phase of the overall filter. The zeroes 15 of filter 6 are 
located so as not to introduce non-linearity of phasing, as noted 
previously, and so are the zeroes 16 of filter 12 not used for pole 
compensation and their repeats 17. 
The possibility for problem arises in the repeats 18 of the 
pole-compensating zeroes of filter 12 as they appear in the Z transform 
domain at original sampling rate. These repeats 18 are around half Nyquist 
rate, Nyquist rate and three-halves Nyquist rate points on the unit 
circle. These repeats are generated by the necessity to traverse four 
complete revolutions of the Z trasform domain at subsampling rate to map 
conformally one complete revolution of the Z transform domain at original 
sampling rate. The repeats 18 of pole-compensating zeroes lie off the unit 
circle and tend to introduce non-linearity of phase response, increasingly 
so as their frequency ranges are approached. This is a problem to the 
extent that there is still appreciable amplitude response in the overall 
system function in the skirt regions near one-quarter Nyquist rate and 
seven-quarters Nyquist rate, approaching the regions where the 
pole-compensating zeroes without corresponding poles lie. If the zeroes 14 
of filter 6 and if the zeroes 16 of filter 12 and their repeats 19 reduce 
amplitude response sufficiently in these skirt regions, the effects of 
phase distortion from zero repeats 18 will not be an appreciable problem. 
One can also arrange to reduce departure from linear phase response, as 
caused by the repeats 18 in the regions near half and three-halves Nyquist 
rate, by including zeroes in filter 6 which lie outside the unit circle. 
However, this tends to compromise the desire for simpler filtering at 
sending station 2. 
The type of analysis incorporating frequency scaling which has been 
described in regard to non-interpolative low-pass filters in the receiving 
station can be extended to instances where there are non-interpolative 
band-pass filters in the receiving station to cooperate with band-pass 
recursive filters in the sending station in other embodiments of the 
invention. This type of analysis incorporating frequency scaling can also 
be extended to permit design of receiving stations in accordance with the 
invention which use a resampler 10, but resample to a sample rate that is 
higher than subsampling rate and that is not equal to original sampling 
rate. 
When the interpolative filter 11 is used after a re-sampler 10 resampling 
to the original sampling rate in a telemetry system configured per FIG. 1, 
there are no repeats of pole-compensating zeroes in contrast with the 
telemetry system of FIG. 2 where filter 12 is a non-interpolative. Thus, 
one avoids the problems of the repeats of the pole-compensating zeroes 
being off the unit circle, so as to compromise phase linearity. The FIG. 1 
embodiment of the invention, where resampling to original sampling rate is 
followed by interpolative filtering, provides essentially perfect pole 
compensation within the limits of resolution of the digital calculations. 
While only certain preferred features of the invention have been 
illustrated and described herein, many modifications and changes will 
occur to those skilled in the art and acquainted with the foregoing 
disclosure. It is, therefore, to be understood that the appended claims 
are intended to cover all such modifications and changes as fall within 
the true spirit of the invention.