Bandwidth Requirements for Pulse-Type Transmissions
February 1945 QST Article
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,
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.
of Contents]These articles are scanned and OCRed from old editions of the
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QST articles I have already posted. As time permits, I will
be glad to scan articles for you. All copyrights (if any) are hereby acknowledged.
vintage QST articles.
Bandwidth Requirements for Pulse-Type Transmissions
A Discussions of Wave Shape as a Guide in Frequency Allocations
By W. W.
(Research Engineer, Sperry Gyroscope Co., Inc., Garden City, N.Y.)
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
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.
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.
Fig. 1- Graph of modulation envelope with pulse. type transmission.
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
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.
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
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.