February 1945 QST
Wax nostalgic about and learn from the history of early electronics. See articles
QST, published December 1915 - present. All copyrights hereby acknowledged.
In 1945, when this article was published in QST,
radar was still in its infancy. Engineers were already aware of the need to shape pulse
waveforms from experience with CW keying and the need to mitigate the effects of "chirping."
A perfectly rectangular pulse in the time domain, as we learned in our signals and systems
courses, creates a sin (x)/x
response in the frequency domain. The Fourier transform shows that a perfectly square
pulse in the time domain is the summation of an infinite number of odd harmonics of the
fundamental (1st harmonic). The first few harmonics are audible to the CW copier as higher
frequency "chirps." To reduce the annoyance (and wasted transmitted energy), time constants
were added to the leading and trailing edges of the waveform to remove the higher frequencies,
while leaving the pulse shape sufficiently rectangular to achieve its goal. The same
type issue applies to radar pulses.
Bandwidth Requirements for Pulse-Type Transmissions
A Discussions of Wave Shape as a Guide in Frequency Allocations
By W. W. Hansen (Research Engineer, Sperry Gyroscope Co., Inc., Garden City,
IN MAKING suggestions as to suitable allocations of wave bands in the microwave region
for the future, due caution should be exercised in considering any proposed system which
uses grossly more bandwidth than the minimum required by the information communicated.
It may be that wave bands can be allotted more liberally in the microwave region than
in the long-wave region, simply because there is so much band space available that it
is difficult to see at present how all of it can be used. Nevertheless, the history of
the spark transmitter suggests that caution is in order; at least serious thought should
be given before systems using excessive spectrum space are permitted. It is the object
of this article to present some information relative to one family of such systems and
to make some suggestions as to possible regulations which might usefully be imposed on
such types of transmissions.
|Pulse technique in which the carrier is broken up at regular intervals before modulation
probably is one of the most important radio developments since the beginning of the war.
lts possible applications still are far from being fully explored. While it is apparent
that greater band-widths are required for transmissions of this type, certain advantages
may be gained if it is possible to operate under conditions where band-width is not of
too great importance. This, of course, points to applications which make use of much
higher frequencies than those heretofore employed by amateurs. Among the advantages claimed
for pulse-type transmissions are a considerable improvement, in signal-to-noise ratio
and the simplification of equipment. Since there may be ways in which this technique
can be used to advantage in amateur work of the future, this discussion of pulse shape
in relation to spectrum economy should be of unusual interest.
The type of systems we have in mind perhaps may be illustrated by means of an example.
Suppose the information we wish to transmit has frequencies up to 10 kc. Then assume
that we choose some higher frequency, say 20 kc. as a "subcarrier." The transmitter then
is turned on and off at a 20-kc. rate. This is called discontinuous modulation. When
the signal input voltage is zero, the transmitter is on half the time and off half the
time. In other words, the microwave modulation envelope is a 20-kc. square wave with
equal" on" and "off" periods. Then to modulate upward, one increases the time the transmitter
is on and decreases the time it is off, 100 percent modulation occurring when the transmitter
is on continuously. To modulate downward, one decreases the fraction of the time the
transmitter is on. More generally, the unmodulated condition can correspond to the transmitter
being on less (or more) than half the time. For example, one microsecond pulses at a
rate of 1000 pulses per second might correspond to no modulation and two-microsecond
pulses to 100 percent upward modulation. Another possible system is one having all pulses
of the same duration but varying in the number per second in accordance with the information-bearing
modulation. Still another method would be to vary the phase of the pulses in accordance
with the signal modulation.
The advantage of such systems is that they
will work with power sources that cannot be modulated linearly in a continuous manner.
Another point is that a great deal of suitable technique is available as a result of
war-inspired research. Incidentally, one of these systems was used in the early microwave
link across the English channel developed by I.T.T. about ten years ago.
Let us now inquire into the frequency spectrum required by such a system. Fundamental
to the problem is the frequency spectrum corresponding to a single typical pulse. This
depends, of course, upon the shape of the pulse. We shall give results for two pulse
shapes which, it would seem, constitute a sort of upper and lower boundary for pulses
which may be realized in practice.
Consider first, then, a simple flat-topped pulse of duration to. The frequency
spectrum corresponding to this pulse is and the distance between 71-percent
points on a frequency scale is easily found to be 0.88/to. While the distance
between 71-percent points is a good measure of the frequency interval containing most
of the energy, it should be noted that the above function drops off rather slowly with
ω so that considerable intensity exists at high values of ω. Specifically the envelope
of the function is 1/πft0
so that, for example, when f is 10 times the value corresponding to the 71-percent
point, the amplitude is down only about fourteen times near one of the peaks of . In this matter of a rather
slow decrease of amplitude with increasing w, the flat-topped pulse is the worst function
likely to be encountered in practice.
Consider next a pulse in the form the numerical factor being so chosen that the time between
71-percent points is to. Then one finds the frequency spectrum to be of the form and from this we find the separation of the 71-percent points
on frequency to be 0.44/to. Also, at a frequency ten times the frequency at
the 71-percent point the function is down by about 15 powers of ten or 300 decibels in
power. This is to be compared with 26 decibels for the square pulse.
Actual pulse forms which may be used will fall between the above two limits. The frequency
spectrum cannot fall off as slowly as that first considered, because the start and finish
of the pulse cannot be perfectly abrupt, as assumed. On the other hand, a pulse as smooth
as the Gauss-error type discussed above is not likely in practice.
Fig. 1- Graph of modulation envelope with pulse. type transmission.
Next, what happens if a series of pulses is used to modulate a carrier? If they are
evenly spaced and of uniform intensity, as we shall assume for a moment, then the frequency
spectrum is as illustrated qualitatively in the graph of Fig. 1 which is drawn for the
Gauss-error-curve type of pulse.
Here the origin corresponds to the carrier frequency and the various peaks are spaced
f1 apart, where f1 is the subcarrier frequency.
The dotted line, which is the envelope, has the same shape as the spectrum of a single
pulse. Strictly speaking, the various peaks, which have been drawn with a small but finite
width, should be infinitely narrow and infinitely high but with a finite area corresponding
to the dotted-envelope curve.
If, finally, we vary the height of the various pulses according to some signal voltage,
each peak spreads out to a width corresponding to the frequencies contained in the signal
voltage. We will call the highest frequency contained in the signal voltage f2.
Actually, the modulation is not done by varying the height, but the width or the frequency
or the phase of the pulses. This complicates the analysis too much for discussion here
but one point, and it is the essential one, remains unchanged. Namely, the frequency
spectrum follows roughly the spectrum corresponding to a single pulse; or, stated more
exactly, the envelope of the frequency spectrum follows the spectrum of a single pulse.
From the above we see that the amount of spectrum used is of the order of 1/to,
whereas one could transmit the information with a band 2f2 (or
f2, if a single-sideband transmission were used). Thus, one uses
roughly 1/f2to times as much spectrum as need be.
How big is this factor?
If one makes the subcarrier, f, only slightly greater than f2
and makes to = 1/2f1 (i.e, uses a square wave as an unmodulated
signal), then the factor 1/f2to is not significantly different
from two, and there is little, if any, waste of frequency spectrum. (This statement will
be subject to some qualification later.) If, on the other hand, for some reason to is
made quite small, say, for example 10-6 seconds, while f2 is, say
104 cycles per second, then one uses about 100 times more spectrum than necessary.
In some cases some of this waste can be recovered while still using the same general
scheme of modulation. For example, a number of stations can be assigned the same carrier
frequency provided they are assigned different subcarriers. Then a band-pass filter in
the receiver will separate the signals from various transmitters.
In the above we have considered the band used as that between the 71-percent points.
But, although most of the energy usually will lie in this region, this is not the whole
story when it comes to interference. What we want to know is over how wide a band will
there be enough energy to cause interference. Plainly, this is a question which is difficult
to answer quantitatively. Besides the obvious arbitrariness involved in deciding how
much energy will cause interference, etc., there is the very important matter of pulse
shape. Indeed, as the calculations above show, this is probably the most important single
factor. Thus, whereas a smooth pulse of Gauss-error form of about one microsecond duration
will cause no appreciable interference outside a band a megacycle or two wide, a flat-topped
pulse with perfectly square corners would cause interference that would probably be called
important over 20 Mc. or more.
What conclusions are to be drawn from the above? There follow certain suggestions
and opinions of the author which may form a partial answer to this question.
If the subcarrier frequency is not much higher than the highest information frequency
and the average pulse length not much shorter than a half cycle of the subcarrier frequency,
there is no essential waste of frequency spectrum. But to avoid interference because
of tails of the frequency spectrum, the regulations should call for some means of reducing
the harmonics of the subcarrier frequency; in other words, rounding the corners of the
pulses. Some ideas on this point will be suggested later.
If the pulse length is markedly short compared to the reciprocal of the highest information
frequency, necessarily there is a waste of frequency spectrum, unless the purpose is
multiplex transmission, and it should be considered carefully whether this is warranted.
For example, with onemicrosecond pulses, there would be room for rather less than 300
stations between 9 cm. and 10 cm. In the author's opinion, probably there are enough
available frequencies to allow such waste, provided certain conditions are imposed.
Use of Filters
Some means must be provided to round the corners of the transmitted pulses, as mentioned
before, so avoiding an additional wastage of frequency spectrum by a factor which may
amount to ten or more. Rounding the corners of the d.c. voltage pulse will not be permissible
in some cases, since many of the tubes on which this system will be used have a strong
tendency toward frequency modulation. Besides, this defeats the main purpose of discontinuous
modulation. The simplest and best method would appear to be the requirement of a filter
in the antenna line. This appears to be a thoroughly practical scheme. For example, with
one-microsecond pulses one could use one or more resonators with bandwidths of about
one megacycle between the transmitter and the antenna. How many stages of filter should
be required is a question to be answered by the conditions prevailing in each case. The
author would suggest that two would be sufficient in most cases.
Use of the modulation system suggested should be confined to certain restricted bands,
leaving other bands where more normal systems will be free from what might perhaps be
styled "super monkey chatter." This should present no difficulty since advocates of this
system will no doubt claim that it does not cause undue interference. They should, therefore,
be quite pleased to have various interference-free regions of the spectrum to themselves.
Finally, the author would like to add that almost all the above applies to pulsed
radar systems, except that in this case the use of short pulses is often a real necessity,
not a matter of real or fancied convenience. In the author's opinion much trouble would
be avoided if all pulse systems were put in a segregated band, and if output filters
to cut off the frequency tails were required. As to the first, there is certainly no
reason to burden television and other communications services using continuous modulation
with the difficult problem of putting up with pulse interference caused by discontinuous
modulation. As to the second, such filters need not interfere with the performance of
a system. They are cheap and easy to apply, and will greatly reduce interference potentialities.
Posted January 14, 2019 (original 6/23/2011)