# the wideband SNR problem and function-following carriers

#### gojirasan

Joined Dec 17, 2011
22
I am trying to establish an ultra long distance link budget for wideband pulse communication and I'm continuously frustrated by the following noise equation:

[received power from noise] = [Boltzmann constant] x [receiver system temperature] x [receiver bandwidth] or Pn = Kb x Tsys x Br.

According to this equation the bandwidth or range of frequencies that the receiver is listening on is directly proportional to the noise level in the receiver. If you want more bandwidth you have to increase signal strength to the same degree or your signal to noise ratio plummets possibly making your signal undetectable, lost in a sea of noise.

The relationship between receiver bandwidth and noise does make some intuitive sense to me. In a narrow band receiver you are able to filter out a much larger number of frequencies that your receiver doesn't have to listen to at all. If you think of each frequency as a 'channel' the noise level is multiplied by every channel that you have to listen to. The noise multiplies, but the signal doesn't. Or something like that. I have no idea if this way of looking at it is correct. It is just how I am thinking about the problem.

So if the problem is that the receiver has to listen to a much broader window of frequencies, is there some way to solve or mitigate this problem at least for certain applications? Or is the direct bandwidth-noise relationship a fundamental law of nature which just has to be lived with?

My potential application would involve sending a series of raised cosine shaped 5 μs pulses with a carrier frequency of 9.3 Ghz. I believe that the short length of the pulses results in unintentional amplitude modulation and therefore frequency modulation of the carrier wave. A 5 μs pulse corresponds to a frequency of 200 kHz. I guess this is equivalent to modulating or mixing a 200 kHz sinusoidal wave with a 9.3 Ghz one resulting in 2 sideband frequencies of 9,300,200,000 Hz max and 9,299,800,000 Hz min for a total bandwidth of 400 Khz.

This results in a factor of 200,000 times more noise in the receiver compared to an ultra narrow band 1 Hz modulated carrier wave that only varies between 9,300,000,001 Hz and 9,299,999,999 Hz (1 Hz sidelobes) and results in an effective range that is reduced by a factor of 200,000. If a signal with a bandwidth of 200 kHz were strong enough to be received at a distance of 1 km, then a 1 Hz signal with the same EIRP could travel 200,000 km or about halfway to the moon. I'd call that difference non-trivial.

According to the above equation it seems like this is unavoidable. Just another of nature's TAINSTAFL scenarios. If you want a higher bitrate you'll need more bandwidth. If you want more bandwidth you'll get an equal quantity of noise in your receiver to go with it. Or in my case, if you want to transmit short pulses you'll get a proportional amount of carrier frequency variation and receiver noise to go with it. The shorter the pulse the more noise in the receiver and the higher the EIRP has to be to compensate.

But I have been thinking about spread spectrum frequency hopping systems. With SSFH both the transmitter and receiver constantly hop to different (presumably narrow band) frequencies. At any single point in time, during the dwell time on each channel, it seems like the scenario is not so different from traditional narrowband communication. At that point in time the receiver should only need to listen to a very narrow band of frequencies in order to receive the full signal.

What if you had a SSFH system, where instead of the transmitter and receiver pair hopping to discrete channels, they smoothly changed their oscillation frequency according to some function. For instance, a raised cosine function. The transmitter and receiver are designed to sync and change their frequency according to that function. Assuming an ideal system with no frequency drift, at any one time the receiver only has to pay attention to one and only one frequency.

Since the wideband SNR problem is a direct result of the receiver not being able to filter out as many frequencies it would seem like this kind of system could significantly reduce the bandwidth and thus significantly increase the range possible for a given transmitter output power and receiver sensitivity. Of course, there isn't any actual information being transmitted in this scenario. Amplitude modulation, phase modulation, and frequency modulation of the carrier would all result in an unpredictable dynamic change in this hypothetical function-following constantly changing carrier frequency.

Whether it could still be made to work (maybe by somehow measuring the changes in the carrier frequency which deviate from the ideal function) for those modulation schemes I'm not sure, but even if you could make it work you'd be back to side-lobes in the frequency domain again and bandwidth that is proportional to the symbol rate you want to send at. So for CW applications this really doesn't seem to get you anywhere (except for a level of interception/jamming security comparable to SSFH). But for pulsed applications the connection between pulse length and bandwidth would effectively be severed. I'm also thinking that pulse position modulation might allow for an effectively zero bandwidth (resulting in a nearly infinite range from an infinitesimally small EIRP? ) communication channel and shouldn't interfere with the sync function itself as something like pulse duration modulation would. So is the idea plausible or what?

#### w2aew

Joined Jan 3, 2012
219
In short - total noise power in a receiver is directly proportional to the BW in the receiver. Noise exists at all frequencies in the circuitry, and the receiver is going to integrate all of the noise in it's BW. There really isn't a way around these physics. Faster modulation or datarates require more BW, thus require more transmitted power.

There are techniques that can be used to detect/demodulate signals that appear buried in the noise, but these techniques typically employ some form of correlation or other long-term processing that can pull recognizable patterns and signals out of the din. GPS receivers operate this way - but as you can imagine, the GPS data is fairly slow.

#### gojirasan

Joined Dec 17, 2011
22
I think frequency hopping receivers change frequency channels precisely in sync with the transmitter. At any given point in time it only needs to listen to whatever relatively narrow band of frequencies that the transmitter is sending on.

You could design a transmitter that hops around between 1 kHz and 1 Ghz but if each channel only uses 100 Hz of bandwidth then I don't see why the receiver would have to listen with anything wider than a 100 Hz bandwidth. The receiver shouldn't have to listen to the entire 1 kHz to 1 Ghz range because it already knows in advance what the next narrowband channel is going to be. It doesn't have to scan for it or guess.

That's the principle I would like to use for pulsed communication. If the receiver knows in advance the width of the pulses then it should also be able to predict exactly how fast the carrier wave frequency will change during the pulse and match that rate of frequency change exactly so that it doesn't need to listen to the entire range of possible frequencies just to get the carrier wave itself.

#### Tesla23

Joined May 10, 2009
537
If you have a pulsed communications system where you send a '1' as a pulse and a '0' as absence of a pulse, then this is OOK modulation.

You need to look up Eb/No (http://en.wikipedia.org/wiki/Eb/N0) and you can find that for OOK to get a BER of 10^-3 you need an Eb/No of 11dB. (you can redo the calculation for other levels).

Assuming that the noise is dominated by receiver input noise, then the sensitivity of an ideal receiver is given by:

P(sens) = -174dBm/Hz + F(dB) + 10*log10(datarate) + EbNo

so if your 5us pulses allow a datarate of 200kbps, and if the receiver has a 3dB noise figure, then an ideal receiver has a sensitivity of

-174 + 3 + 53 + 11 = -107dBm for a BER of 10^-3.

To achieve this sensitivity you need to investigate matched filters, depending on how much effort you put into the receiver design you may be a few dB off the ideal figure, or you may get very close.

If your carrier changes frequency, then the carrier in the ideal receiver will track this.

This analysis is independent of pulse shape (that does affect the tx spectrum and the design of the receiver obviously).

#### Tesla23

Joined May 10, 2009
537
What if you had a SSFH system, where instead of the transmitter and receiver pair hopping to discrete channels, they smoothly changed their oscillation frequency according to some function. For instance, a raised cosine function. The transmitter and receiver are designed to sync and change their frequency according to that function. Assuming an ideal system with no frequency drift, at any one time the receiver only has to pay attention to one and only one frequency.

Since the wideband SNR problem is a direct result of the receiver not being able to filter out as many frequencies it would seem like this kind of system could significantly reduce the bandwidth and thus significantly increase the range possible for a given transmitter output power and receiver sensitivity. Of course, there isn't any actual information being transmitted in this scenario. Amplitude modulation, phase modulation, and frequency modulation of the carrier would all result in an unpredictable dynamic change in this hypothetical function-following constantly changing carrier frequency.

Whether it could still be made to work (maybe by somehow measuring the changes in the carrier frequency which deviate from the ideal function) for those modulation schemes I'm not sure, but even if you could make it work you'd be back to side-lobes in the frequency domain again and bandwidth that is proportional to the symbol rate you want to send at. So for CW applications this really doesn't seem to get you anywhere (except for a level of interception/jamming security comparable to SSFH). But for pulsed applications the connection between pulse length and bandwidth would effectively be severed. I'm also thinking that pulse position modulation might allow for an effectively zero bandwidth (resulting in a nearly infinite range from an infinitesimally small EIRP? ) communication channel and shouldn't interfere with the sync function itself as something like pulse duration modulation would. So is the idea plausible or what?
To be honest, this is just wishful thinking. Put some maths behind it - it's not that hard to work out the spectrum and BER for the sort of systems you propose, and then the key issues will become apparent.

#### gojirasan

Joined Dec 17, 2011
22
If you have a pulsed communications system where you send a '1' as a pulse and a '0' as absence of a pulse, then this is OOK modulation.
Actually I would probably be using pulse position modulation.

Tesla23 said:
You need to look up Eb/No (http://en.wikipedia.org/wiki/Eb/N0) and you can find that for OOK to get a BER of 10^-3 you need an Eb/No of 11dB. (you can redo the calculation for other levels).

Assuming that the noise is dominated by receiver input noise, then the sensitivity of an ideal receiver is given by:

P(sens) = -174dBm/Hz + F(dB) + 10*log10(datarate) + EbNo

so if your 5us pulses allow a datarate of 200kbps, and if the receiver has a 3dB noise figure, then an ideal receiver has a sensitivity of

-174 + 3 + 53 + 11 = -107dBm for a BER of 10^-3.
What if I am just sending an unmodulated pulsed carrier wave containing no information at all? Would that just mean that the datarate drops out of the equation and everything else remains the same? When I say unmodulated I mean not even pulse position modulation or polarization modulation. Every pulse would have exactly the same width/duration and exactly the same interval between pulses. Just a perfectly regular contentless beacon.

Can you explicitly state the variables of that sensitivity equation or post a link? I haven't seen that before. I've only seen the Pn = Kb x Tsys x Br equation. I notice that your equation doesn't seem to have an explicit bandwidth term. Does the equation assume that you are choosing the maximum datarate that your bandwidth will allow? I think it is the gap between maximum datarate and actual datarate that I would be trying to take advantage of with my hypothetical method.

Tesla23 said:
To achieve this sensitivity you need to investigate matched filters, depending on how much effort you put into the receiver design you may be a few dB off the ideal figure, or you may get very close.
I am looking into matched filters which seem interesting, but a lot of this stuff is above my level. I'm just trying to look into the feasibility of the idea. This is part of signal theory or signal processing, right? Can you recommend any books on the subject?

Tesla23 said:
If your carrier changes frequency, then the carrier in the ideal receiver will track this.
Every modulated carrier or, if pulsed, even unmodulated carrier changes frequency, no? If the carrier in the receiver can track the carrier in the transmitter then I don't see why a wide pulsed bandwidth in the transmitter need result in a wide bandwidth in the receiver. At least if you are not making use of the full datarate available to you. If you are then you are making use of every Hz of bandwidth you've got and there is no way around that.

Rereading my initial post it looks like I may not have made it clear, but the whole point of this idea was to avoid being 'penalized' for short pulse lengths. A 5 μs pulse would result in a 200 kHz bandwidth which might allow something like a 200 kbps datarate. But what if you only wanted to send 1 bps? You end up with a huge range penalty that you wouldn't have had with a non-pulsed continuous wave carrier. With CW your bandwidth is just limited by your data rate. With pulsed communication you are limited by both your data rate and your pulse length. I would like to know if it might be possible to only be limited by the data rate even with pulsed communication.

Let's say you are sending only a single 5 μs pulse per day for a data rate of 0.116 μbps and a bandwidth of 200 kHz. If you had sent a 500 ms pulse with a bandwidth of 2 Hz instead of 200 Khz the receiver would have identified the pulsed just fine, but, because the 5 μs pulse is so wideband it is lost in a sea of noise at the receiver.

I believe it may be the change of frequencies caused by the amplitude modulation of the rising/falling edges of the pulses resulting in a kind of contentless frequency modulation which causes this range penalty that is unrelated to data rate. I'm wondering why the receiver cannot just follow along with those carrier frequency changes.

#### gojirasan

Joined Dec 17, 2011
22
To be honest, this is just wishful thinking. Put some maths behind it - it's not that hard to work out the spectrum and BER for the sort of systems you propose, and then the key issues will become apparent.
Well I'm going to get cracking on signal theory/analysis. I suspect I will need to review my calculus, but I'll start looking into some books soon. Although you haven't actually come right out and said it, it sounds like you are implying that my scheme wouldn't work. The additional range penalty due solely to short pulse lengths really sucks. Do you think this wideband SNR penalty is something we will eventually overcome say 100 or 1000 years from now or is it just an unchangeable law of nature?

#### gojirasan

Joined Dec 17, 2011
22
How about a light analogy? Something like a pulsed laser. A continuous wave laser would be monochromatic. A pulsed laser would be wideband. Eventually the pulses could get so short that the laser would be emitting frequencies throughout the entire visible range and you would have a white light laser. You want to send pulsed light communication signals to a space station orbiting Saturn, but the only way to make the pulses powerful enough is to make them very, very short, resulting in white light.

The power of the white laser is distributed over a wide range of frequencies. If the space station receiver watches for only a single frequency of light, filtering out all others, it will only be receiving a small fraction of the total light output. By doing so it will filter out most of the noise in the receiver but it will also filter out a lot of the signal itself.

So my idea is that the white laser isn't really white. It just appears that way if viewed over the relatively large time scale of the entire pulse. What is really happening is that the OOK modulation or amplitude modulation is changing the frequency of the light along with the amplitude. At any one point in time the laser is still monochromatic. It's just that the frequency at which it is transmitting is continuously changing throughout the length of the pulse. So it appears white. If the photodetector at the space station were tuned to change exactly in step with the intra-pulse frequency changes in the laser then the receiver could be more sensitive because it only has to watch for one particular frequency at any given point in time. When the laser was blue, the receiver would only be looking for blue. When it was yellow it would only be looking for yellow. And so on.

#### Tesla23

Joined May 10, 2009
537
Actually I would probably be using pulse position modulation.

What if I am just sending an unmodulated pulsed carrier wave containing no information at all? Would that just mean that the datarate drops out of the equation and everything else remains the same? When I say unmodulated I mean not even pulse position modulation or polarization modulation. Every pulse would have exactly the same width/duration and exactly the same interval between pulses. Just a perfectly regular contentless beacon.
I've no idea what you are asking. A waveform only contains information if you can't predict the next piece from what you have already received. You can listen to a continuous carrier, or a repetitious modulation stream, and you get no additional information. The transmitter modulator adds information to the waveform by changing the waveform in a way that cannot be predicted by the receiver. The accuracy with which the receiver can reconstruct what the transmitter did is a key system parameter that you typically try to optimise.

Can you explicitly state the variables of that sensitivity equation or post a link? I haven't seen that before. I've only seen the Pn = Kb x Tsys x Br equation. I notice that your equation doesn't seem to have an explicit bandwidth term. Does the equation assume that you are choosing the maximum datarate that your bandwidth will allow? I think it is the gap between maximum datarate and actual datarate that I would be trying to take advantage of with my hypothetical method.
The bandwidth comes in through the actual pulse shape you choose, this will determine the transmit spectrum and the receiver response.

I don't understand what gap you are talking about, as Shannon showed there is no limit to the data rate you can send through a fixed bandwidth, as you can trade off S/N for datarate.

I am looking into matched filters which seem interesting, but a lot of this stuff is above my level. I'm just trying to look into the feasibility of the idea. This is part of signal theory or signal processing, right? Can you recommend any books on the subject?
What books work for you depends on your background, I like books by Proakis, this is one of my standard references http://www.amazon.com/Communication-Systems-Engineering-John-Proakis/dp/0130617938/. There are lots of books, often the best one is the one that does a case study on a system very similar to the one you are working on.

#### gojirasan

Joined Dec 17, 2011
22
Since I started this thread I have found that my understanding of amplitude modulation was wrong. I had believed that amplitude modulation caused frequency modulation. In fact what happens is that the sideband frequencies are fixed and that RF energy is distributed over various sideband frequencies depending on the baseband modulating signal.

My understanding now is that a perfectly sinusoidal baseband signal amplitude modulating the carrier results in 3 monochromatic or nearly monochromatic waves. Thus even if the bandwidth of such a signal is very wide, most of the space between the lower sideband frequency, the carrier and the upper sideband frequency would be just empty space. Nearly all of the RF energy would be concentrated at those 3 frequencies. This is what motivates the following.

The actual case I am interested in is a 9.3 Ghz pulsed transmitter with 5 μs pulse durations. The interval between pulses would average 4 seconds. The modulation would be PPM which would change the interval between pulses between a range of something like 3.5 and 4.5 seconds. So the data rate would be around 0.25 bps.

The instantaneous bandwidth of a 5 μs pulse is 200 kHz. However, if you assume that pulsed transmission is equivalent to OOK modulation of a CW carrier then most of that 200 kHz might be empty of RF energy.

Unfortunately if you actually had a continuous sine wave baseband signal with peak to peak times of only 5 μs you would have an average bandwidth of 200 kHz and not just an instantaneous bandwidth of 200 kHz. I don't think the transmitter would be capable of that anyway. It needs some rest time between pulses. The closest I could actually get to a continuous sine wave would be a raised cosine shaped pulse. That may ruin things, but for now I am interested in the theoretical possibility.

There are two possibilities I can think of to take advantage of the empty space between sideband frequencies and the carrier. The first would be to use three narrow passband filters on the receiver to only listen for the 2 sideband frequencies and the carrier frequency, filtering out all other frequencies. In that scenario, if there wasn't too much RF energy between the 3 channels you should be able to detect the full strength of the signal.

The second possibility would be to just listen for the carrier frequency (filtering out both sidebands) or to just listen to one of the sideband frequencies (filtering out one sideband and the carrier). A slight variation of the second method would be to have 3 separate narrow band receivers. The first could listen for the lower sideband frequency. The second could listen for the carrier. The third could listen for the upper sideband frequency. This second option would presumably result in a much lower power signal at the receiver since 2 out of the 3 bands are just being ignored. However short pulses allow for a transmitter that is about 6 times more powerful. So that may make up for the lower received signal strength. One problem with this method is maybe the pulse position modulation would be lost somehow.

So, neglecting the issue of whether my actual baseband signal would be sufficiently sinusoidal to keep all of the signal in 3 narrow bands, would this scheme work? I could probably shape a pulse to approximate a cosine wave pretty well, but the cosine wave would be very discontinuous, appearing for 5 μs and then not appearing again for another 4 seconds or so. Who knows what the actual mixed output signal would look like on a spectrum analyzer.