Bandwidth Requirements for Pulse-Type Transmissions
February 1945 QST Article
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
February 1945 QST
of Contents]These articles are scanned and OCRed from old editions of the
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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, N.Y.)
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
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
Consider first, then, a simple flat-topped pulse of
duration to. The frequency spectrum corresponding to this
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
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
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
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 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.
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.
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.