Ray Greenwald’s e-mail from Feb. 16
(and previous e-mails along the chain)
Kevin's E-mail demonstrates that there are some major misconceptions within our community, but he is probably not the only person who has them. The bandwidths required for pulsed radar and phased-coded pulse radars have been fairly well documented over the years. The important point to note is that the pulse duration is not the sole factor determining the bandwidth of a transmission. Of equal importance is the rise and/or decay time of the transmitted pulse. Moreover, if you are putting out a phase-coded pulse, such as a Barker code, the total pulse length is of no importance. The important pulse length is the duration of an individual chip. For the example given by Kevin, it is 520/13=40 usec. The 40 usec is commensurate with the 6 km range resolution.
In the FCC Frequency Management Manual, expressions are given for the calculation of:
Bn(-20dB) Necessary bandwidth for the operation being proposed
B(-40dB) Bandwidth at which all out-of-band emissions should be down by at least 40 dB, and
B(-60dB) Bandwidth at which all out-of-band emissions should be down by at least 60 dB
See the end of this item for Definitions and references here
Bn(-20dB) = 1.8/sqrt(tr*t)
B(-40dB) = 6.2/sqrt(tr*t)
B(-60db) = 62/sqrt(tr*t)
Here, t is the half-full-amplitude width of the transmitted pulse and tr is the 10%-90% rise time of the pulse for a non-coded pulse. The equivalent width and rise time in a phase-coded pulse is that of an individual chip. Moreover the rise time for a phase coded pulse is half of the transition time from full on with a plus phase to full on with a minus phase. In effect, when a 180 degree phase reversal is applied, the previous phase has to decay to zero power and then recover full power with the opposite phase sign. Also if the decay of a transmitted pulse were to occur on a shorter time scale than the rise time, then one would be required to use the decay time for the pulse rather than the rise time in the calculations above.
The Bn(-20dB) is the parameter that is identified in our licenses. It is something that I entered when the applications were made. However, if there are objections to our operations, the other parameters, particularly B(-40dB) may be checked. Let's consider a few examples.
Let us assume 100, 200, and 300 usec pulses with a rise and decay time of 20 us. We will also assume that the rf pulse reaches half full amplitude is 10us.
Case: 100us 200us 300us
t: 80us 180us 280us
tr: 20us 20us 20us
Bn(-20dB) 45kHz 30kHz 24kHz
B(-40dB) 155kHz 103kHz 83kHz
I should note that Dieter indicated his tx pulse rise times were 10us. Putting this value into the above results increases all of the values by ~1.4.
Now, let's consider the case of the 13-bit Barker code with a chip length of 40 us and assume that the transition time between chips is 10 us. Note that it does not matter if there are no phase transitions between some of the chips, You still have to calculate the bandwidth the same way.
Case: 13-chip Barker code with 10us transitions
Here, if we assume that the transitions were to take 20 us rather than 10 us, the bandwidths would be decreased by a factor of 1.4. For example, Bn(-20) would be close to 100kHz.
I have a bit of good news. I checked the licenses for the mid-latitude radars that we have installed recently and they all have Bn(-20)=60kHz. This means that the non-coded pulse transmission are all compatible with our license as long as we have a reasonable rise time for the pulse (I think we do). However, there is still an issue with the Barker coded pulses. One compromise that would be compatible with the licenses is to run a 5 or 7 chip Barker code with 100us chip lengths and rise times compatible with the uncoded pulse rise times. These transmissions would be compatible with our licenses and offer us 15 km resolution with higher sensitivity due to the coding. I am not sure that Bill will be happy with this, but it is a compromise.
There is one final point that I would like to make. Our licenses are experimental licenses. This means that we do not have rights to any frequency that we might be using. If we come into conflict with another user, we are the ones that have to stop using that frequency. Given that, we are able to operate continuously and at any frequency other than a few that are reserved, we should be happy. Very few others have this freedom and we should not screw it up. I think that if we keep our appetites under control, we won't be forced to leave the table.
Regards to all, Ray
Sent: Wednesday, February 15, 2012 11:25 AM
To: Chisham, Gareth
Cc: Ray Greenwald; 'J. Michael Ruohoniemi'; Nathaniel Frissell; Joseph Baker
Subject: Re: Darn-swg
Potential problems with SuperDARN high-spatial
Kevin Sterne, RF Engineer from Virginia Tech here. In looking at our licenses, we are licensed for a 50 kHz bandwidth at the Kapuskasing site. So the 15 km range separation, and thus 100 microsecond pulse length, should be well within this license.
Also, for clarification, we were not outside of our licensed bandwith when running the special mode that Todd request in Jan. Virginia Tech's older radars (GBR, KAP, & BKS) used a 100 microsecond pulse (10 kHz bandwidth) and the Fort Hays radars (FHE & FHW) used at 520 microsecond pulse (~1.9 kHz bandwidth) with a 13 bit Barker code in order to achieve a 6 km range separation. We should be within our licensing if/when we would run the 6 km range separation mode again.
In short though, Kapuskasing seems well within it's licensing to continue running co-ordinated Cluster observation.
On 2/15/2012 10:20 AM, Chisham, Gareth wrote:
Hi SWG and PIs,
I am just in the process of putting together the SuperDARN schedule for April and it has come to my attention that there may be some problems with some of the modes that we have been running recently, and which are planned for the coming month.
It was recently brought to Ray's attention that recent SuperDARN operations may have been interfering with other HF radar systems in the US (which is obviously a big problem for us as well). Although some/all of this may be due to the special mode that Todd was running in January, Ray has pointed out that the running of high spatial resolution modes results in an increase in transmission bandwidth that may have resulted in some of our operators violating their transmission licences (roughly, changing from 45 to 15km range gates will increase the transmission bandwidth by ~3).
The spacecraft working group has been requesting high-spatial resolution modes (15 km range gates) for about 6 months now for co-ordination with Cluster observations of the auroral acceleration region (these typically involve one or two radars at a time for intervals of about 6 hours, about 3 times a month). Obviously, I still want us to be able to run these modes, given the importance of the science. However, we need to be sure that this increase in resolution is not causing operators to violate their transmission licences through the increased bandwidth.
Before I schedule these modes it would help if someone (PI or SWG rep) from each of the radars involved (KAP, PYK, HAN, SAS, PGR, and KOD) could confirm that there is no problem with these operations. Each of these radars will be operating in this mode for 6 hours of the month (April). A response by the end of this week, if possible, would be helpful.
Dr. Gareth Chisham
British Antarctic Survey
Bill’s email, Feb. 22, 2012:
Hello everyone. Here is the discussion Ray and I had about pulse bandwidth. Attached are the results of calculations and measurements.
The upshot is that with a 40-microsecond baud Barker-Coded pulse, with a 20 microsecond rise time, we were transmitting about 40 kHz of bandwidth (-20 dB bandwidth). The bad news is that radars like Kodiak that have a 22 kHz license was in violation. The good news is that radars like the mid-latitude radars had bandwidth to spare and we will be proposing even higher resolution
Using the pulse we transmit, the bandwidth is about 4./(5*tau). For 22 kHz bandwidth we should limit tau to no shorter than about 70 microseconds, which gives 10.5 km range resolution.
For 60 kHz, we can go a bit shorter than 28 microseconds, which gives 4 km resolution... By the way Gareth, I want to discuss a special mode
Phase coded pulse bandwidth
Ray’s initial estimate of the pulse bandwidth was based upon 1/tau, where tau is the baud length of the phase code. For a 6 km range resolution we used a 40 microsecond baud, then 1/tau is 25 kHz, which is actually a large underestimate of the bandwidth.
The Fourier Transform of a simple pulse is a sinc function with the nulls of the function separated by 1/tau. The peak of the first and second side lobes of the sinc are above 20 dB down from the peak. Hence, the full 20 dB bandwidth of an unmodulated pulse is just a bit less than 2*3/tau. For a 40-microsecond pulse, that comes to 150 kHz.
Instead of an unmodulated pulse, if we transmit a pulse with a finite rise time the spectrum is narrowed. In the DDS we use a Gaussian profile to smooth the edges of the pulses. Hence the spectrum of whatever we transmit is weighted by the spectrum of that Gaussian, which is a Gaussian with a width that is the reciprocal of the time profile.
The pulse we were transmitting was in fact a 13-baud Barker coded 520-microsecond pulse, with a rise time equal to half of the shortest baud. Below is a plot of the waveform:
The power spectrum of this pulse is the FFT of the ACF:
Zooming in and centering you get:
I know the axes aren’t labeled so give me an F for the assignment. The vertical axis is dB. The horizontal axis is point number in the FFT. The spectrum still looks like a sinc function, with first zeros at 1/tau, but the side lobes are suppressed. You can see that all side lobes are below 20 dB down from the central lobe. The central lobe crosses below -20 dB at about 4/5*tau, giving a 20 dB bandwidth of 8/5*tau. For a 40-microsecond baud this is 40 kHz.
To make sure that my calculations are correct I had Jef set up the DDS in the basement to put out the Barker coded pulse, and measured the spectrum on the spectrum analyzer. Here are the results:
The pulse on the scope:
Sorry you can’t read the scales. The scope was set with 100 microseconds per division, which gives the 520-microsecond pulse. You can see that the shortest bauds just come to full amplitude before changing phase (rise time = half baud).
Output on Spectrum Analyzer:
It is a little fuzzy but I think you can read it. The table below the spectrum gives the marker positions and amplitudes. Marker 1 is at 10 MHz and has an amplitude of -38 dBm, Marker 2 is at 10.021 MHz and has an amplitude of -58.12 dBm. Markers 3 and 4 are at the peaks of the 1st and 2nd sidelobes, which are at -67.8 dbM and -92.7 dBm. Hence, Marker 2 is at the -20 dB point, 20 kHz off of the central peak, giving a 20 dB bandwidth of 40 kHz, just as the math said. (Yay! Math works again!)
For reference, here is our basic 300 microsecond pulse and its spectrum:
Marker 2 at first null, 3.3 kHz off of center, Marker 3 at peak of 1st sidelobe, 5 kHz off center, 13dB down, Marker 4 at peak of 2nd sidelobe, 9 kHz off center, 18 dB down.
Ray’s following e-mail from Feb. 23
Needless to say, I have some problems with Bill's latest message. Before going into some picky details about what Bill presented, I wanted to discuss some general principles about how pulse modulation impacts spectral width. Of particular importance is the impact of the rise and fall time of the pulse on the width and decay of the spectral response.
You can think of a radar pulse as the product of a modulating pulse and an rf signal. In the frequency domain the spectral response is the convolution production of the respective Fourier transforms. This convolution product effectively transforms the pulse modulation spectra to and about the frequency of the rf signal.
In the attached figure, I have plotted the spectral response for rectangular pulse modulation and for triangular pulse modulation. The two pulses each have a width of 100 microseconds. This width is not the half power width for the triangular pulse, but rather the width from the start to the end of the transmission. The rectangular pulse turns from off to full on immediately. The spectrum that you see is half of the spectrum of the transmitted pulse when it is translated to the RF frequency. The other half is a mirror image of the half you see.
The triangular pulse has a half power width that is about 30% of the half-power width of the rectangular pulse, so you might expect that it would have a broader spectral response than the rectangular pulse. Actually, it drops below the -20 dB level first and it remains well below this level in all of the side lobes. If you were to plot this response on a log-log plot, you would see that the sidelobes have a 1/offset_freq^2 dependence. In contrast, the rectangular pulse does not remain below -20 dB until the third sidelobe, but even then, its sidelobes remain about -30 db even out to 95 kHz. A log-log plot of the spectral response to a rectangular pulse shows that the side-lobe power decreased as 1/offset_freq. Needless to say, the longer rectangular pulse creates a much greater disturbance to other users of the radiowave spectrum than does the triangular pulse. This difference is entirely due to the slower rise and fall times of the triangular pulse.
I also have a comment concerning our licenses. Based on our anticipated modes of operation, we (not the licenser) identified the bandwidths that we needed to carry out our operations. We have identified the 50kHz to 60kHz that are required and no one has objected to that allocation. However, as I have mentioned earlier, we are operating with experimental licenses, and therefore we do not have the right to disturb others. If someone complains, we will be asked to change our mode of operation and we have no recourse but to do so. Therefore, we have nothing to gain and a lot to lose by operating our radars in an indiscriminate manner.
Having had a lot of positive contacts with FCC monitors over the years, I can tell you that they recognize SuperDARN transmissions and know that our typical bandwidths range from 30-50 kHz. They measure our signals, so if we suddenly start to put out exceptionally broad transmissions, it could be to our detriment. Bill has an advantage in Alaska where the population density is fairly low, but that is not the case in the lower 48, Europe, or Japan.
Now for a few comments on Bill's document.
1) In the calculated spectrum Bill shows at the bottom of his page 2, we see a full span of the central lobe and the first sidelobe toward higher and lower frequencies. The spectra also shows a few pimples which do not contain much power, but conveniently allowed Bill to set his reference level about 3 dB higher than he would have had he only referenced his level to the broader main peak. If he raised his entire spectrum by 3 dB then he would find that the first side lobes were also at the -20dB level so that his bandwidth was actually 2*1.2/tau, which is 60 kHz. This comment does not apply to the spectrum he shows on page 3.
2) The 300 us pulse spectra shown on page 4 looks very similar to the rectangular pulse spectra I presented. I would anticipate that if he had expanded the frequency range to 200 kHz, he would have found a large number of additional sidelobes above -30 dB that would significantly broaden the potential impact of his transmissions. This problem could be mitigated if a slower rise and fall time had been used.
3) In the Barker coded sequence the average power transmitted during the 1-chip sequences was significantly lower than the average power per chip of the 2 and 5 chip sequences. I would expect that the backscattered returns would be accordingly weaker and that this could impact his ability to maintain low side-lobes from the code. Is this true?
Well, I hope this gives you something to think about.
Appendix A: NTIA References
All references here refer to the Manual of Regulations and Procedures for Federal Radio Frequency Management (Redbook), Revision May 2011 of the January 2008 Edition
In Annex J on measuring bandwidth:
J.3 RADAR SYSTEMS
For radars the necessary bandwidth shall be determined at a point that is 20 dB below the peak envelope value of the spectrum by one of the following with the order of preference shown:
1. Computation in accordance with the radar formulas from Table A in this Annex.
2. Results of actual measurement.
3. Use of the best available technical information from other sources.
The B(-20) calculation is referenced on J-10 in Annex J:
If t/tr < 12.6, then: Bn=B(-20dB)=1.79/sqrt(tr*t), otherwise: Bn=B(-20dB)=6.36/t
The B(-40) calculation is referenced in the NTIA manual on page 5-31 in Section 18.104.22.168 noting the calculation is:
B(-40)=6.2/sqrt(tr*t) or 64/t whichever is less; where tr is the rise time (10% to 90%) and t is the time of 50% to 50%.
The B(-60) calculation could not be found within the NTIA manual.
The FCC 47 CFR Part 2 section on bandwidth (2.202) does not have any information on pulse radars.