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
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.
February 1945 QST
articles are scanned and OCRed from old editions of the ARRL's QST magazine. Here is a list
of the QST articles I have already posted. All copyrights are hereby acknowledged.
Bandwidth Requirements for Pulse-Type
A Discussions of Wave Shape as a Guide in Frequency
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 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
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.
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
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
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
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
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
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.
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.