The pulse repetition frequency (PRF) in radar systems
Background
The pulse repetition frequency and pulse interval
The pulse repetition frequency, or PRF, is governed mainly by two conflicting
factors. The first is the maximum range required of the radar system. This
is because it is necessary not only to be able to detect pulses returning
from distant targets, but also as it is necessary to allow these pulses
time to return to the radar receiver before the next pulse is transmitted.
For example, if a given radar is to have a range of 50 nm (nautical miles)
or 92.6 km, at least 617 m s must be allowed
between successive pulses; this period is called the pulse interval.
This is because, from the speed of light, (299,792.50 km/s = 161,874.977
nautical miles/s), it takes 6.17 m s to travel
1 nautical mile ( 1 nautical mile = 1.150779 statute miles = 1.852 km),
so for 50 nautical miles it will take 308.5 m
s. However, the pulse must return to the receiver before the next
pulse is transmitted, hence another 308.5 m
s must elapse to allow this to happen. The total time from a transmitter
to a target and from the target back to the receiver is thus, 617 m
s. Ambiguities will result if this time between pulses is not allowed.
For example, if only 500 m s is used as the
pulse interval, then 117 m s after the second
pulse is transmitted, the first pulse will be received back into the receiver.
However, the receiver is not able to give the pulses labels, so it may
think that the received pulse is from the second one sent 117 m
s ago. This means that the round trip has taken 117 m
s from an apparent object, which is 117 m s/2
away from the transmitter. That is, 117/(2 x 6.17) = 9.48 nautical miles
away, where in actual fact the object was 50 nautical miles away and in
reality took (500 + 120)m s to return. It is
the subject of this applet to show the different errors that occur due
to the pulse interval being too short. From this point of view, it is seen
that the pulse interval should be as large as possible, and certainly the
minimum pulse interval is twice the distance of the object multiplied by
6.17 m s/nm. The second factor, is that the
greater the number of pulses reflected from a target, the greater the probability
of distinguishing this target from noise. This is because on a radar screen,
the noise sparkles on and off in a random fashion, but the pulse reflected
from a target will always remain at the same position on the screen. The
more times this position on the screen is updated the more obvious the
point is from a target and not from noise. Because of this integrating
effect of the screen, the received signal power can equal that of the noise,
allowing a received signal-to-noise ratio (S/N) of 0 dB. Since the antenna
moves at a significant speed in many radar systems, and as it is necessary
to receive several pulses from a given target, a lower limit on the pulse
repetition frequency (which is the inverse of the pulse interval) clearly
exists. Values of PRF ranging from 200 to 10,000 per second are commonly
used in practice. This corresponds to pulse intervals ranging from 5000
to 100 m s. This applet has a PRF range of 4000
to an upper value of 1,000,000 per second, corresponding to 250 m
s down to 1m s and a distance range of up to
20 nautical miles. When targets are very distant, such as satellites and
space probes for example, lower PRFs down to 30 pulses per second are used.
Pulse shape
The pulse interval (the mark plus the space) includes the pulse (the mark)
which usually has a duration much shorter than the pulse interval. The
ramifications of the size of the pulse duration are discussed below. The
shape of the pulse, however, is also of importance. Pulses used in radar
systems ideally should have vertical sides and flat tops. The leading edge
of the transmitted pulse must be vertical to ensure that the leading edge
of the received pulse is also close to vertical. Otherwise ambiguity will
exist as to the precise instant at which the pulse has been received. This
ambiguity, if it exists, will allow inaccuracies of the exact measurement
of the target range. A flat top is required for the voltage pulse applied
to the magnetron electrode to prevent the magnetron from being frequency
modulated. Alternatively, a flat top is needed because the efficiency of
the magnetron, multicavity klystron or other amplifier, will drop significantly
if the input signal voltage drops. Finally, a steep trailing edge is required
for the transmitter pulse, so that the duplexer in the radar system can
switch the receiver over to the antenna as soon as the body of the transmitted
pulse has passed. This will not happen if the pulse decays slowly, since
there will be sufficient pulse energy present to keep the transmitter switch
ionized. The energy transmitted during the decay of a pulse does not contribute
significantly to the total transmitted power. Thus, a pulse trailing edge
which is not steep has the effect of lengthening the period of time which
the receiver is disconnected from the antenna, and therefore limits the
minimum range of the radar.
Pulse duration
Short pulse durations
If a short minimum range is required, then short pulses must be transmitted.
This is really a continuation of the argument in favour of a vertical trailing
edge for the transmitted pulse. Since the receiver is disconnected from
the antenna for the duration of the pulse being transmitted (in all radars
using duplexers), it follows that the echoes returned during this period
cannot be received. For instance, if the total pulse duration is 2m
s, then no pulses can be received during this period. That is, no echoes
can be received from targets closer than 2/(2x6.17) =0.162 nautical miles
(300 metres) away. And so, this is the minimum range of the radar. Another
argument in favour of short pulse durations is that they improve the range
resolution, which is the ability to separate targets whose distance
from the transmitter differs only slightly. If a pulse duration of 1m
s is used, echoes returning from separate targets that are 1m
s apart in time (0.162 nautical miles) will merge into one returned pulse
and will not be discerned as two separate targets. In this case, the range
resolution is no less than 0.162 nautical miles.
Long pulse durations
The main argument for keeping the pulse durations long is to reduce the
bandwidth of the receiver. Short pulse durations comprise many Fourier
components, which tend to become equal in amplitude as the pulse with reduces
and which spread out into the frequency spectrum. As the pulse duration
is increased, the fundamental frequency of the pulse chain increases above
the sidebands, and the sidebands further out in the frequency spectrum
become smaller in amplitude, that is they can be truncated without undue
loss in the total energy contained in the pulse. From the applet on the
bistatic radar equation, it can be noticed that as the bandwidth of the
IF frequency (which is close to the RF bandwidth in value) is decreased,
by having a longer pulse duration, the range of the radar system is increased.
The shorter the pulse, the wider the IF bandwidth becomes, and more noise
is admitted into the system, which reduces the radar system's range. This
can to some extent be overcome by increasing the peak pulse power, at the
expense of cost, size and power consumption. Actually, from the bistatic
radar equation, it can be seen that the maximum range of a radar system
depends on the pulse energy rather than on its peak power, due to the term
Pt/d f in this equation. The bandwidth
d f, is inversely proportional to the pulse
duration. Therefore, it is true to say that the range of a radar system
depends on the product of the transmitter peak pulse power Pt
, times the pulse duration and that this product equals the pulse energy.
By increasing the pulse duration while keeping a constant PRF has the effect
of increasing the duty cycle of the output amplifier tube of the transmitter
and therefore its average power. As the name implies, the duty cycle is
the fraction of time that the output tube is on. For example, if the PRF
is 1200 and the pulse duration is 1.5m s, then
the period of time actually occupied by the transmission of pulses is 1200
pulses/s x 1.5m s/pulse = 1800 m
s/s = 0.18 % . Increasing the duty cycle thus increases the dissipation
of the output tube. It may also have the effect of forcing a reduction
in the peak power, because the peak power and average powers are closely
related for any type of wave tube. For large duty cycles, it is worth considering
travelling wave tube amplifiers, because they are capable of accommodating
a duty cycle in excess of 0.02 or 2 %.
Theory
As mentioned at the beginning of the background section, if the pulse interval
is not sufficiently long there will be a virtual object seen by the radar
due to the first pulse arriving at the receiver after the second pulse
has been transmitted. Because the radar system cannot label the pulses,
the radar is "fooled" into believing that the second pulse has been reflected
off of a target, and entered the receiver, when in fact is really was the
first pulse which arrived back. If the distance to the target is d nautical
miles, then the time in m s for the pulse to
travel this distance is 6.17d m s, and for the
reflected signal to travel back to the radar set, is another 6.17d m
s. Thus the total distance, out and back is 2d nautical miles, which takes
12.34d m s. Thus, the pulse interval for no
virtual targets to appear (PI)o, is given by:
(PI)o = 12.34 d m s
If the pulse interval (PI) is too short but longer than the time for a
signal to travel out to the target and back some way, that is; (PI)o
> (PI) > 6.17 d m s, the time x m
s, that a pulse still must travel when a second pulse is transmitted is
given by; x = [(PI)o - (PI)] m s.
As the second pulse has travelled for this time when the first pulse has
arrived back, the radar thinks that this time is equivalent for the second
pulse to travel out and be reflected back. Hence the distance from the
transmitter that a virtual target appears is half of the time for this
second pulse to travel out and half the time for it to be reflected back.
Thus the virtual target distance, or error range e
, is given by;
If the pulse interval (PI) is too short but, on this occasion, shorter
or equal to the time for a signal to travel out to the target only, that
is; (PI) < 6.17 d m s, there will be many
pulses transmitted before the first returns to the receiver from the target.
This will lead to many virtual objects, because when the first pulse returns,
the second, third, fourth, fifth pulses or more are still travelling out
and back from the target. Take the case where there are five pulses in
space when the first pulse arrives back. The time the fifth pulse, say
has travelled when the first pulse returns, will lead to an error range
for the time difference between the fifth and first pulse, divided by two
times the distance conversion factor 6.17 m
s/nautical mile, similar to the above derived equation. The time the fourth
pulse has travelled when the first pulse arrives back divided by 2 times
the 6.17 conversion factor. The third pulse and the second pulse have already
been reflected and are on their way back. Thus, the maximum number of error
ranges nmax , which can occur are given by the integer number
of pulses which can occur in the region between the transmitter and the
target when the first pulse arrives back. That is, for nmax
an integer;
The actual error ranges are the minimum error range, or the error range
for the fifth pulse (as used in the above description) plus half of the
pulse interval converted to a distance for the next error range, plus again
half of the pulse interval converted to a distance for the next error range.
This continues on until the next addition would bring us greater than the
distance of the target, that is we have exceeded nmax. This
cannot happen, as we cannot have an error range greater than the target
distance. Thus, the equation representing each of the error ranges can
be expressed as:
where;
It is all the above equations that are used in this applet, to permit the
error ranges of the virtual targets to be calculated for different pulse
intervals. As the background discussion indicated, the pulse duration is
not involved in the determination of the error ranges. Thus, this applet
does not use pulse duration.
The applet
Two scrollbars are used for this applet. The first scrollbar permits the
pulse interval (PI) m s to be entered and the
second is to permit the target range d in nautical miles, to be entered.
Once the target range is set up, the applet calculates the value for the
correct pulse interval (PI)o m s
for this range and displays it in blue on the bottom right-hand side of
the screen, together with the appropriate pulse repetition frequency (PRF).
The range is limited to 20 nautical miles. The target (a green box) moves
as the distance is changed, with 20 nautical miles scaled from the antenna
to the far right hand side of the applet canvas. As the first scrollbar,
representing the pulse interval (PI) is changed, a black line originating
at the antenna moves across the screen and bounces off of the target, returning
back to the antenna, stopping at a distance which is the maximum error
range from the antenna. If the pulse interval is greater in time than the
time for the pulse to travel to the target, the returned signal is blue
and the error range, which follows the error range, is displayed in a blue
font. If the pulse interval is less than the time for the pulse to travel
to the target, then the returned signal turns red and the various error
ranges are displayed in a red font on the lower part of the left hand side
of the canvas. However, the maximum error range, continues to follow the
returned signal. As more virtual targets appear due to the reduction of
the pulse interval, their error ranges are displayed by a red marker on
the forward part of the signal indicating their scaled position from the
antenna (the transmitter/receiver). To get an initial feel for the applet,
first try making the pulse interval (PI) equal to the correct pulse interval
(PI)o m s, as displayed in blue on
the bottom right hand side of the canvas, for a chosen range of 16 nautical
miles, and note what happens. Reduce the pulse interval to half this value
and note what happens. Reduce it again to a quarter and then to an eighth.
The source code (version Rev.1 98/08/04) is available
according to the GNU Public License.
Tony Townsend tonyart@ieee.org