Answer to Walker
Phil Karn
18 June 2001

This is in answer to Walker's essay The VMSK Delusion, which is in turn a rambling attempt to rebut my debunking of VMSK. Mr. Walker's text from the 13 June 2001 version of his document is quoted here in full as block-indented text along with my comments. His file is in PDF format, so I have tried to maintain style, mispellings, etc exactly as shown in the original in this conversion to HTML.

Why Am I Doing This?

Phil Karn from Qualcomm is conducting a one man crusade, or more correctly, a war, to prove that VMSK is a snare and a delusion.
I've been interested in technical education for a long time, especially in my own professional field of digital communications and computer networking. I wish the general public had a better basic understanding of the principles of digital technology in particular and of engineering, science, and the scientific method in general.

I also feel a strong moral obligation to object whenever I see others spout pseudo-scientific nonsense, as is clearly the case with Mr. Walker and VMSK. It also presents an educational opportunity, as pseudo-science can often serve as a springboard for teaching real science to the layman. After all, we learn from our mistakes (except for, it would seem, Mr. Walker).

Also, with human nature as it is, many people who would be quickly bored to tears by a straightforward lecture on information theory might well take interest in a conflict like that between Mr. Walker and myself and --horrors!-- end up learning something. Sneaky, eh?

Debunking VMSK has given me the opportunity to refresh my knowledge of basic signal theory, a subject I apply almost every day at work but haven't studied formally since I graduated from college over twenty years ago. So while Mr. Walker's characterization of my effort as a "war" is a little strong, I am using VMSK as educational "target practice", as it were.

Just Who Are All Those Scientists?

Unfortunately he is a few years too late and there are too many reputable scientists, who have worked with VMSK, who do not share his views. There are too many knowledgeable people and interested companies who do not feel they are being deluded. [changed to - PRK]

Mr. Walker has never named those "many reputable scientists", knowledgeable people and interested companies, and he has repeatedly refused to do so when asked. That is his right, but then we have the right to wonder whether they actually exist.

One might think we would be angry at Karn's efforts. Not so. We have begged Karn to continue this site. He has been a tremendous help to us technologically and financially. We could not hire a more industrious, capable and dedicated Devil's advocate. As a result of Karn's comments, we have rewritten, and are now rewriting, most of the files on this web site in an effort to clarify them. But best of all, Karn has caused us to look for the cause of the "grass" and find a way to eliminate it. This has resulted in new simplified hardware and a proprietary modulator, which we are adding to the patent pending list. In fact, he pushed us to come with the 4 th generation method - which has no grass. Karn wins our employee of the year award.
Walker's amusing sarcasm aside, he has hardly eliminated the "grass"; he has only reduced it in a way that necessarily impairs the operation of his scheme. Nevertheless, Hal, I'd be happy to collect any cash associated with your "employee of the year" award.

"Pay No Attention to That Man Behind the Computer!"

Phil has just added a simulation program that misses the mark. We have repeatedly emphasized that there is no known narrow band filter that functions at baseband. The full Nyquist BW must be used. This is seen in the "Nonsense" file and some others. If one uses a narrow band filter at baseband, only a sine wave at bit rate comes out. This is also why we cannot use conventional filters at RF and why simulation programs such as LabView, Matlab, SysView and EEsoft fail.

Of course there is no known narrow band filter that will allow VMSK to function at baseband. Showing that was the entire purpose of my simulation! What Walker still can't understand is that the exact same analysis applies at both baseband and RF. There is no reason why they should be different.

When a baseband signal is applied to a balanced modulator, the baseband frequency spectrum is shifted by the carrier frequency. The positive frequency components in the baseband signal become the upper sideband and the negative frequency components become the lower sideband. Because the negative baseband components are equal to the positive components, the lower modulation sideband is a mirror image of the upper sideband.

At baseband, the negative and positive frequencies overlap. But at RF, they are separated into two sidebands. Therefore the occupied RF bandwidth is fully twice that of the baseband signal. Single sideband techniques can remove one of the redundant sidebands, but they cannot reduce the occupied RF bandwidth to less than that of the original baseband signal.

When he successfully comes up with an RF simulation that shows the single frequency AND THE DETECTED OUTPUT shown in the "Grass is Gone" paper, the world will accept the gospel according to Karn.

What a strange statement. Of course I will never be able to simulate a pure single-frequency system that produces detected output because I've been saying all along that this is impossible!

On the other hand, I could simulate a working VMSK system where the spectral density of the data-carrying grass is low enough to fall below the noise floor of a real spectrum analyzer, making it appear that the narrowband clock is the only relevant signal component. For this I need the complete phase-frequency response curves of Walker's "magic filters" out to at least ten or twenty times the data rate, depending on his "code".

Walker has repeatedly ignored my requests for this information. What could he be hiding?

I'd also need an accurate model of any stray RF coupling that may exist between modulator and receiver on the lab bench that might bypass his filters. I note that such stray coupling is far more likely at RF than at baseband, as it can take many paths: free space between unshielded equipment, along common power supply leads, etc. At Qualcomm we often have to go to great lengths to preclude stray RF coupling in equipment tests, e.g., through the use of copper "screen rooms".

If stray RF coupling is bypassing Walker's filters, this could help explain Walker's delusion that VMSK can somehow tolerate tighter filtering at RF than at baseband.

We say it without tongue in cheek or double meaning. This work by Karn is darn good. The coding is excellent, even praiseworthy. His FFT is excellent. Unfortunately, he tried what about 6 or 7 others have tried unsuccessfully. VMSK cannot be analyzed at baseband unless you use the full Nyquist BW. See the "Engineering Summary".
Yet Walker no longer recommends VMSK for general use -- only at baseband. How odd!

Pointed Questions, Direct Answers

Here are a few pointed questions to Karn: VMSK is coded BPSK with end to end pulse width modulation transmitted single sideband.

1) Does BPSK work? If you say NO, then you had better argue that with some people at Qualcomm who use it for Chipping in CDMA.

Of course BPSK works; I never said it didn't. However, BPSK requires a minimum RF bandwidth equal to the data rate. Real systems generally use somewhat more. Qualcomm IS-95 CDMA chips at 1.2288 million chips per second, and the occupied RF bandwidth is about 1.25 MHz. A carefully designed digital filter (described in the IS-95 spec) is required to confine the CDMA signal to that bandwidth.

So if VMSK is coded BPSK, then it too requires an RF bandwidth equal to the Nyquist rate.

2) Can you filter off the upper harmonics and keep only the inner fundamentals? Ditto-ask Qualcomm and all the text book writers. ( See Feher and Rappaport).
If you mean the harmonics of your square wave clock, yes. You can remove these without harm because they are well outside the Nyquist limit. This can be shown quite easily with my simulation. But you must still pass (at some amplitude) all of the components within the Nyquist limit, or the resulting intersymbol interference will destroy your modulation.
3) Does pulse width and/or pulse position modulation work? If you say NO, then argue that Federal Radio, who made the gear and AT&T who used it extensively in the 50's. Also tell the text and reference book authors to please remove it from the old books.
Sure, PWM and PPM work just fine -- as long as you don't try to filter them below the Nyquist limit.
4) When using PPM, can you throw away the upper harmonics and keep the fundamentals only? If you say NO, then argue that with Federal Radio and AT&T, plus the texts. Notably Schwartz.
Sure, once again you can throw away anything beyond the Nyquist limit. But not within the limit.
5) Is the VMSK pulse width and rise time any different from that of a BPSK signal transmitting a 101010101 pattern? We don't think so. It is a sinx/x pulse with a pulse width = 2X the bit period and a rise time equal to the bit period.
As I have repeatedly explained, the modulation spectra produced by a short repeating bit pattern like 101010... consists of discrete spectral lines. But a modem that only lets the user send such a sequence is rather useless, wouldn't you say? The only meaningful tests are with random data. And random data always produces a broad, diffuse, noise-like spectrum. Just like the "grass" that has proven to be such an embarrassment to you.
6) Does Single Sideband work? A great many people ( including Hams who have been using it ) seem to think so.
Sure, SSB works fine. But it still requires an RF bandwidth equal to or greater than the baseband audio bandwidth.
7) Is the definition of Shannon's Channel Capacity - that it is the "sampling rate times the energy per sample correct"? If not, argue it with Schwartz and Proakis, who say it is in their texts. We accept Schwartz and Proakis. We sample at the data rate and change the filter BW to keep the proper value of C/N.

As worded, this question doesn't make sense. The "sampling rate times the energy per sample" gives the total received power. Exactly how does this relate to Shannon?

Actually, we don't even need to invoke Shannon to debunk VMSK. We can go back to Nyquist's work 20 years before Shannon. Nyquist showed how the bandwidth a signal must occupy depends directly on how rapidly changes are made to that signal (i.e., how fast it is modulated). This is basic, solid and utterly incontrovertible mathematics.

"But You Haven't Actually Tested My Perpetual Motion Machine!"

We keep Karn's commentaries on line to point out the pitfalls. ( and for a snicker or two ) . He should take a look at the hardware, or at least the schematics, before making his rather obviously incorrect statements public.
I don't need to look at Walker's hardware to know that it cannot meet its claims any more than I need to examine a perpetual motion machine to know that it too cannot work as claimed.
Despite his disclaimer, Qualcomm does have a financial interest in discrediting VMSK. VMSK R&D is supported by the GSM Consortium, which does not like Qualcomm's CDMA.

This one is just too outrageously funny to ignore. Actually, my manager and colleagues think I'm just wasting my (personal) time debunking VMSK, because it is so obviously flawed that no competent professional could ever be fooled for very long. While Qualcomm (and I) may consider GSM to be inferior to CDMA, I have to assume that the GSM Consortium is comprised of largely competent individuals. So I find it rather hard to accept Walker's claim that they are funding him.

VMSK, which should be considered a form of 'Coded BPSK', was first demonstrated in 1996 to a group of scientists working on satellite communications. Since that time it has been demonstrated to over 200 scientists working for the Cellular Telephone industry and to a great many military contractors. It is being submitted as an industry standard to the ITU. There have been some skeptics who advanced Karn's theories, but for the most part they have been convinced otherwise by making their own measurements.
Once again, who are these "over 200 scientists"? Can you provide any names or references at all?
Karn has not seen the equipment in operation and has apparently not experimented with hardware. He bases his conclusions on mathematics with incorrect, or partially correct, input data. He has not been given all the facts.

The only "input data" I've ever needed to analyze VMSK is a description of the basic approach. And I've had that in hand since I first read about the scheme last August in EDN. That was all I needed to immediately make two conclusions: that what was being claimed violated basic mathematical principles that have been firmly established for 75 years; and some magazine writers and editors really should have known better than to publish this garbage.

Computers are regularly used to simulate complex digital communication systems before hardware is built. In fact, most modern communication systems are actually implemented as programs executing on digital computers, e.g., DSP chips. But the fallacy underlying VMSK is so obvious and fundamental that it can be dismissed out of hand by a simple application of basic signal theory; no simulation was necessary. However, my simulations do match my simple mathematical analyses and, I might add, are fully consistent with Walker's own spectrum analyzer plots. This includes the pesky "grass" that he can see with his own eyes, and which he hand-waves away as "data-dependent noise".

If Mr. Walker feels I have not been given all the facts, he only has himself to blame. He has repeatedly ignored my requests for more details, such as detailed filter response curves. But this information would be useful only to establish exactly how Walker deludes himself into thinking he has somehow beaten Shannon and Nyquist. Do his modems pass data because his custom filters have such broad skirts? Or is it because of stray RF coupling? Perhaps it's both. But those are mere details, akin to autopsy results. One does not need a full autopsy report to prove that someone is, in fact, dead.

"News Flash! RS-232 Can't Work!"

He completely ignores the fact that the detector uses ordinary CMOS gates that cannot respond to his theories- and since the 4 th generation has no "grass"- where does he go from here?
The definition of the continuous Fourier Transform (see The Real Facts of Life) shows that the spectral amplitude of a signal at any given frequency depends on the entire signal in time, and conversely the amplitude of the signal at any instant in time depends on the entire frequency spectrum. Far from being too weak to affect anything, all those little grass components (along with the clock) add up in phase to make the total signal voltage whatever it should be at any given instant. So ordinary CMOS gates can and do respond to the spectral "grass" in a VMSK signal, just as they respond to a BPSK signal (or even a baseband NRZ signal, e.g., RS-232) that consists entirely of grass whenever random data is sent.

Walker's so-called "4th generation" scheme has hardly eliminated the grass. He has merely reduced its total power and spread it even more thinly across the spectrum so as to make it invisible on his spectrum analyzer. But it's still there, and it can be easily seen in computer simulations that aren't affected by the inherent limitations of a spectrum analyzer. Ironically, in his Quixotic quest to build an "ultra-narrowband" modulation scheme, Walker has unwittingly created a system with some of the properties of spread spectrum!

Where There's Smoke -- There's Smoke

The 'biphase single sideband with suppressed carrier' concept has been under development for 15 years, with VMSK/2 as the latest development stage. It is patented world wide, with additional patents being filed every few months. Now there is a fourth generation under test.

15 years seems like a long time for such a development effort. One might reasonably ask why none of these innovations have ever appeared as actual products. At least Mr. Walker's patent attorney seems to be making a good living off his efforts.

At this time there are numerous University and commercial labs conducting tests. Over the air testing is underway in New Jersey now and repeat tests soon will be underway in Atlanta Ga. These tests are being sponsored by the Cellular operators he claims have been deluded while making their own tests.
Once again, where are the details? When Qualcomm conducted field tests of what became IS-95 CDMA in the early 1990s, they were open to all interested parties and the results were openly published. Where are yours?
Pegasus is holding 'hands on public demonstrations' for interested parties. The last was in Boston for MITRE, Lincoln Labs and MIT professors. It was attended by Army, Navy and Air Force research personnel as well. The next will be in Washington DC for the benefit of the National Association of Broadcasters, the FCC, various government agencies and contractors.
I was not present at the Boston meeting (I live in San Diego). But Steve Kelley was at the meeting, and here is what he had to say.
Development kits are being sent to licensees and prospective licensees world wide so that they can develop products. For others, the older schematics are available here.
One doesn't have to look very hard on the web to find companies selling "development kits" for perpetual motion machines. Where are the independent reports on the performance of these kits?
If VMSK did not work as claimed, all of the above would be for naught. Rest assured that it does work as claimed.
Sorry, this is simply not sufficient. Extraordinary claims require extraordinary evidence, not unsupported assertions. A modulation scheme that achieves 90 bps/Hz (or far more, as you've sometimes claimed) is an extraordinary claim.

No Honor Among Thieves

To reply to some of Karn's comments:

A former Internet Service Provider named AlphaCom has acquired a license to "make, use and sell" products using VPSK modulation", an earlier patented 'biphase' SSB concept. They were a multilevel marketing firm ( presently inactive ), with nearly 10,000 dealers world wide. The many sites found on the Internet touting VMSK are owned by these independent dealers, who are awaiting products to sell from AlphaCom. AlphaCom does not own the VMSK patents, but they have funded some R&D work toward developing products for their sales force. They have no products at this time. This premature advertising - with many false claims- is an embarrassment to Pegasus.

Pegasus Data Systems is not affiliated with AlphaCom Communications.

A lawsuit has been filed in Federal Court, Newark, to force AlphaCom to cease making false and misleading claims regarding ownership and performance.

I am not in a position to verify these facts, but given that multilevel marketing companies have long been used to market questionable products, VMSK seemed like a nice match. Too bad it didn't work out.

If You Can't Dazzle 'em With Brilliance...

The technical facts are that VMSK has a baseband and SSB - RF transmitted spectrum 1 Hz wide. As generated at the modulator, the bandwidth efficiency ( using the full Nyquist bandwidth ) is 1 bit/sec./Hz, but less than 1% of that Nyquist bandwidth is actually transmitted. All of the useful modulation energy, which is transmitted single sideband with suppressed carrier, is in that narrow band spectrum.
Sorry, but this is simply moronic nonsense. By definition, the Nyquist bandwidth of a digital signal is the absolute minimum bandwidth that the channel must have to pass the signal without introducing intersymbol interference. This is always one hertz of bandwidth per symbol per second of signalling rate, and this applies to every modulation method. If you have a good signal-to-noise ratio you can encode multiple bits per channel symbol, but that's not what you're doing.
How do you define the bandwidth efficiency of a modulation method with a spectrum 1 Hz wide? FCC style, it is equal to the bit rate at the transmitter, which could be in the millions of b/s/Hz. The more conventional method is to use the relationship [bit rate]/[filter noise bandwidth]. With the filters available, this value is anywhere from 65 to 250 bits/sec./Hz. Is there a definition problem here? See the papers on " VMSK Spectrum" "Interference".
Yes, there is indeed a definition problem here. The problem is that the FCC definition of bandwidth is arbitrary, and it is completely different from the concept of absolute bandwidth used by Nyquist and Shannon. You've found an amusing loophole in the FCC's definition, but it is merely a parlor trick. There's no breakthrough here. Indeed, competent communication engineers know that to fully use an assigned channel it is necessary (but not sufficient) for your signal to closely resemble uniform white noise that is band-limited to the channel.
WE can't help it if the mathematics and the spectrum analyzer say it is 1 Hz wide. That's just the way it turns out. The skeptics claim you must be able to see the modulation in terms of bandwidth spread. The paper "What's This Grass Nonsense" explains why it isn't visible and shows the single frequency with phase changes, as photographed, from the oscilloscope screen. The modulation is visible when looking at the single frequency on an oscilloscope.
And I can't help it if you're simply wrong. The mathematics (when correctly done) say nothing of the sort. They clearly show it is much wider, as do the simulations. The only reason you think your spectrum analyzer shows it is 1 Hz wide is because you don't understand its inherent limitations and the true nature of your signal: a strong spectral line combined with weak, broadband information.
Accusations that Shannon's Limit is being violated are false, since if that were true, it wouldn't work. It is true that if Shannon's Limit is incorrectly interpreted, as it applies to VMSK, one needs a few 'Billion Megawatts' of power to transmit across the room. Obviously the incorrectly calculated 250-300 dB signal to noise ratio needed for VMSK with a high bandwidth efficiency does not hold, or the transmitters in use would not be working.
You are correct; Shannon's limit is not being violated. The fact that it is alleged to work with less than several billion megawatts of power merely proves that its true bandwidth is far greater than you claim.

"If the World Disagrees With Me, They Must All Be Wrong"

The papers available here for down loading show some inconsistencies in the conventional interpretation of Shannon's Limit - namely, where it does not agree with the SNR limit used for calculating the Bit Error Rate ( Pe). References giving the correct interpretation are given. We prefer the SNR limit, which conforms to reality, both for VMSK and the well known M'ary FSK modulation method (OFSK).
The Shannon capacity formula is just that -- a capacity formula. It merely sets the upper theoretical limit on how many bits per second can possibly be sent without any errors at all on a channel with a given absolute bandwidth and signal-to-noise ratio. Although it can tell you that a particular scheme cannot possibly work (i.e., if it claims performance in excess of channel capacity) it cannot predict the actual performance of a scheme operating within the limit.

In particular, the Shannon capacity formula is the wrong tool to compute the bit error rate of a modulation scheme without forward error correction (FEC) coding. That's done by analyzing the probability of error based on the signal constellation and the noise probability distribution. That's how those classic BER curves for BPSK, QPSK, etc were drawn for the textbooks.

A great many engineers misinterpret this equation. The correct interpretation straight from the text books and Shannon's original paper is given in these papers. ( with references ). These inconsistencies and misinterpretations are illustrated with examples using the well know QAM modulation method. We do not make any claims that the formulas as applied in our papers represent the true state of affairs, merely that these inconsistencies offer food for theoretical thought and that they reconcile the various text book formulas with one another.
Except for students in an introductory communications theory course, the only engineer I've ever known to misinterpret Shannon's capacity channel equation is Harold Walker. At least the students usually figure it out pretty quickly.
The formulas given by Karn are correct. It is the incomplete data he feeds into them that is not correct. There is a computer adage that applies, " Garbage in = Garbage out".
I remain open to additional data, if you wish to provide it. Again, it may prove useful in figuring out exactly where you went wrong, but it certainly won't change the basic fact that VMSK cannot possibly meet its bandwidth claims.
VMSK is "Coded BPSK".
I suggest another term. The term "coded" normally means something like convolutional coding and Viterbi decoding. That certainly performs better than VMSK.
For BPSK, Shannon's Limit is 0 dB.
This is incorrect for reasons already stated. With forward error correction, BPSK can theoretically achieve an Eb/No down to -1.6 dB if infinite bandwidth is available. 0 dB is actually the theoretical limit for any combination of modulation and coding that achieves 1 bit/sec/Hz. Here a "bit" is a real user data bit, not counting those added for error control.
There is a net C/N improvement in practice of 2-3 dB over un-encoded BPSK. VMSK suffers from an effect similar to the FM knee, which limits the lower C/N values.
Sorry, but the power wasted on the clock ensures that VMSK will always under-perform BPSK on a power basis. The precise penalty relative to BPSK for what you call (7,8,9) VMSK is 9 dB.
The filters are described in the papers that can be down loaded here, with very little information being held back.
I have repeatedly asked for more detailed information on these filters, which you have declined to provide.
The theory is that if the filter represents a pure resistance at a single frequency, or, an infinite impedance at a single frequency, it has a very low group delay at that single frequency and can accept the very brief modulation phase reversals.
This is nonsense; at best, you've confused filter implementation with the underlying signal theory. It is a basic principle of filter theory that "low group delay" necessarily implies a wide bandwidth, so you are more correct than you realize when you say that VMSK requires low group delay filters. This is exactly because a signal that is narrow in time must be wide in frequency, and that a signal that is narrow in frequency must be wide in time. So it is utter nonsense to talk about a narrow (in frequency) filter with a low group delay at only one frequency. Such filters simply don't exist. And if you say they do, you don't know what you're talking about. If your filter can accept "brief modulation phase reversals", then by definition it must have wide bandwidth.

Note that I'm not talking about noise bandwidth here. Your filters appear to have a narrow region (several kHz) of low attenuation and a wide region (many MHz) of higher fairly constant attenuation (around 30 to 40 dB). The noise bandwidths of these filters may indeed be small, but that's simply because of how "noise bandwidth" is defined.

The important part of your signal is the grass. Because it is so wide, relatively little of it falls in the narrow center part of the filter passband. So the gain in that region doesn't affect the grass very much (although it does affect the amplitude of the narrow clock, which you use as a timing reference when demodulating the grass.) Most of the grass is spread over the much wider 30-40 dB attenuation region, and all it takes to overcome that filter loss is a stage or two of IF gain.

Again you've fallen prey to the exact same fallacy as your use of the FCC's arbitrary definition of bandwidth rather than the notion of absolute bandwidth required by Shannon and Nyquist.

Some newer filter types are not discussed pending patent action. Karn has been given the response of a single stage filter.
I beg to differ. I have never gotten the filter frequency responses out to multiples of the data rate, as I've requested, and I've never gotten any information at all about phase responses.
Filters are cascaded to further reduce the shoulder response. The frequency plot is available in these papers as well. ( Photos).
If you cascade more filters, it would only take more IF gain to overcome the additional losses. Eventually, stray signal paths such as direct radiation and power supply coupling would undoubtedly become important -- if they aren't already. Either way, it's almost as if your filters weren't there at all.
FACT: The measured C/N for 10-6 BER using a 7,8,9 code and a bandwidth efficiency of 100 b/s/Hz [calculated from (bit rate)/(Filter Noise BW)] is approximately that of theoretical BPSK. Using accepted formulas, this results in an Eb/n that is below Shannon's Limit. WE do not claim this is right or wrong, we merely point out what the calculated values show. Those who attend our demonstrations make this measurement for themselves using HP test equipment.
Asserting this as fact won't make it so, no matter how many times you try. The real fact is that using the filter noise bandwidth as the signal bandwidth is incorrect, and that's why you get the wrong result.
Karn attempts to prove the modulation is in the 'grass', which is more than 40 dB down. This has been refuted many times over.
Refuted? How? I've proven it many times with both analysis and simulation and you have yet to refute it even once.
It is now a moot point, since a recent change in the modulator removes it.
It has hardly been removed. It has merely been spread even more thinly.
According to Karn, the strong narrow band 'clock' component does nothing, but all the modulation is in the grass. The papers "What's This Grass Nonsense" and "Intereference" shows very clearly in the scope photo, that this strong narrow band 'clock' component is the signal.
The narrowband clock is certainly a large part of the signal, since it consumes most of the power. But there would be no detectable variation in the zero crossings at all if it weren't for all those little "grass" components adding up in phase at the right moments to push it one way or the other.
The grass is a byproduct resulting from the modulation pattern, not the cause of the modulation. Karn confuses `Cause' with `Effect'. We eliminated the cause and no longer see the effect.
And exactly why do you suppose the grass is a "byproduct resulting from the modulation pattern"? Just what is the mechanism here? If it's noise, then why does it show up in a computer simulation where there is no noise at all? Fourier transform pairs are one-to-one; you cannot change a signal in the time domain without also changing its spectrum, and vice versa.
A thorough analysis of the modulation shows that the grass is a non-coherent very broadband noise that results from the integration of the Aav levels of the Fourier transform. No modulation is detectable from the grass. A new proprietary modulator has eliminated the integration, and along with it, the "grass".
Care to present this "thorough analysis"? And what are the "Aav levels" if not the data you're trying to send?
He apparently did not look at the schematics. That strong 'clock' ( it's his term, actually it is the modulation ) is used to trigger CMOS gates for detection. In any modulation method, the energy is in the sidebands. This 'clock' is one sideband that just happens to be a single frequency undergoing phase changes relative to a reference clock. The paper on Nonsense" contains a scope photo of the cycles of the single sideband undergoing this phase change.
Once again you repeat your fundamental misconception: you cannot change any parameter of a signal without also changing its spectrum; once again, Fourier transform pairs are one-to-one. You can't modulate a clock -- even by a tiny bit -- according to a random pattern (e.g., user data) without also changing its spectrum. The faster you change the signal parameter, the more bandwidth you will need to recover those changes at the other end. It's really quite simple, as Nyquist showed 75 years ago.
These gates need a strong signal to overcome chip and circuit noise. They only want to know when a zero crossing occurred. They don't give a hoot about the low level grass ( about 55 dB ) and can accept a relatively high level of Gaussian white noise. See the paper on "Interference". The newest modulators have this grass at -80 to -90 dB in mean power terms below the peak of the single frequency.
Indeed, the gates need a strong signal. Yet they are time-domain devices; they know nothing about Fourier transforms and spectrum analyzer displays. The "low level grass" is low level only when displayed in the frequency domain, but at those moments where the zero crossings occur, all those little components add up in the time domain to make exactly the change you should see. Again, if this weren't so, then how does an RS-232 line receiver function when carrying random data? Stick a spectrum analyzer on the line and you'll see nothing but "low level grass"!
One other simple way to show that grass has no effect is to transmit a repeating single character (byte) where there is no broadband grass, but there are some discrete energy bins. After filtering, the very weak energy bins, when added, contribute less than 0.1% to the high level phase reversing signal phasor, which is triggering the gates.
0.1% is about -30dB, which is consistent with typical VMSK parameters. So yes, the grass is weak. That's why VMSK performs so poorly in the presence of noise. But though it may be weak, it's what's carrying the data.
Using random data instead of the single character introduces the non-coherent grass, but there is no change in the appearance of the detected signal at the detector output. The detected output level does not drop when the grass is removed because the phase coherent detector is phase locked to the main energy lobe.
And why should the output level drop? The different data patterns have different spectral characteristics, that's all. The total grass energy is the same, it's simply distributed differently. If it weren't, it wouldn't be possible for the demodulator to tell the different data patterns apart.
The grass is a signal in quadrature to the main lobe when transmitting a repeated byte, and is therefor rejected by the detector. With random data, it spreads like broadband noise, so as to have no distinct frequency related to the data or the coherence reference.
No. The grass is nothing more than the Fourier transform of the data, stretched and shaped by the narrow VMSK pulse shape. Its spectrum appears noise-like precisely because it is random. It is fully coherent with the clock. This is exactly what's expected from a correct Fourier analysis of the VMSK waveform.

You are also wrong in claiming that your detector rejects the grass because it is in quadrature to the clock. All of the VMSK spectral components, regardless of phase, add vectorially to produce the received waveform at any moment in time.

Interference 40-50 dB above the grass can be added before the main phasor is affected at RF. Another way is to use VMSK encoding with both sidebands and a conventional filter. There is a loss in C/N, but there is no more broadband grass than is seen with conventional un-encoded BPSK. Using a narrow band SSB filter gains back the loss with a few dB extra.
With VMSK's undoubtedly wide true bandwidth, I would expect it to tolerate a certain amount of co-channel interference. This is a well-known property of spread spectrum modulation, and you've inadvertently designed a system with some of the properties of spread spectrum.
His explanation of the grass has been demonstrated in hardware to be incorrect. ( See the papers " The grass is gone" "What's This Grass Nonsense" and "The Grass is Bad").
Sorry, but this is simply wrong.
The grass can be created separately without the main SSB lobe, but it is non coherent noise that cannot be detected.
Actually, it can be detected -- just not with your hardware. Your hardware relies on the clock for bit timing, so if you remove the clock the data stops. That does not prove the data isn't in the grass.
The grass is coherent to the suppressed carrier, not to the sideband.
Just a second -- earlier you said the grass was noncoherent noise. Now you say it's coherent to the suppressed carrier. So is it coherent or not? If you're going to be wrong, you might as well be consistently wrong.

Actually, it's fully coherent to the clock. I don't know what you mean by "sideband". The ordinary usage of those terms would describe the clock as your carrier, and the grass as your sidebands.

The detected signal is in the phase reversing main SSB lobe, which is triggering the gates. This is a high energy phasor, which is compared to a reference phasor in the detector gate. The detected output corresponds to the phase difference between the phasors and is dependent upon the timing of a phase reversing change, not the level of a minor background noise. The grass is noise distortion that must be removed to make the FCC happy.
Once again, you cannot impress a high speed data stream on a signal without broadening its bandwidth. Nyquist proved otherwise 75 years ago. Sorry.
Removing the main lobe ( which Karn claims does nothing ) with a notch filter destroys the modulation. We have posted the detected output plots of the signal with grass alone and with the grass removed. With grass alone, the signal cannot be detected, since it is not coherent to anything. That should prove the point, since there is nothing to trigger the detector gate. These scope plots and the analysis are posted here under "The Grass is Bad". The latest modulators have no grass, so the point is moot.
Nope. See above. I bet that pulling the AC power cord also kills your demodulator. Does that prove that the data is coming in over the AC power line?
Before showing the errors of his earlier statements, let's concentrate on the errors of his latest posting.

It has been shown above that his primary error is in assuming the main lobe does nothing and the grass does everything. This has been refuted above and amply so in the papers on "Nonsense", "Interference" and "Grass". New papers in preparation, using a new modulator, show no grass at all.

What he refers to as the 'Clock' is actually the desired signal, which is a single frequency changing in phase with the coded time differences. This triggers the gates. See the papers on "Interference" and "Nonsense."

WE can't help it if the bandwidth efficiency after SSB filtering is extremely high. That's just the way the ball bounces and the way the accepted formulas read. Yes, as created, the BW efficiency is 1 b/s/Hz with broadband filtering. But after narrow band SSB filtering, it is closer to 100/b/s/Hz. See the "Illustrated" paper. (VMSK Simplified) and the "VMSK Spectrum"

No need to answer these again.
Isn't it terrible that one stage of our low group delay filters has approximately the same degree of shoulder rejection as a normal 2 pole crystal filter?
And how much is that? Actually, I don't care about your individual components. I want to know their combined characteristics.
Now to go over some of his earlier claims.

His statement that removing the grass contributes to ISI is totally incorrect. The reverse is true. See " The Grass is Bad" for plots of the detector output. The grass is caused by Fourier amplitude offsets from the previous bit patterns integrating over time, hence must interfere with the present bit.

Sorry, but you're simply wrong. Any textbook discussion of the Nyquist theorem will demonstrate why.
His pictorial of the sinx/x functions was totally incorrect. See " Wirecom.pdf ". Oscilloscope photos show the description in this paper is correct. The sinx/x pulses do not overlay, but appear in a timed phase reversing sequence just as they do for normal BPSK.
My pictorial showed what must happen if the bandwidth were truly as small as you claimed. The fact that they do not in fact overlap proves merely that the effective channel bandwidth is far larger than you claim. Mathematicians call this "proof by contradiction".
He assumed incorrectly that the pulses are narrower than 1 bit width. They are one bit width +- a small deviation, not the pulses of 1/16 bit width that he used in his analysis, or that we used in our hardware simulation of the broadband grass. ( "The Grass is Bad" ).
No, I modeled your pulses exactly as you describe them. My decomposition of your pulses into a static "clock" and narrow "data" pulses is a direct application of a fundamental principle known as "superposition". This is a handy tool that can be applied to any linear system; the concept is taught early in introductory engineering courses.

In any event, I have modeled the pulses both in this manner and as whole pulses with the given shapes, and I get the same answers each time.

This 100% +- is approximately the same pulse width as in normal BPSK transmitting a 1010101 pattern. There is no difficulty whatever in separating the main lobes of these sinx/x pulses by using a phase detector.
The devil is in the details, and here the devil is in the two characters "+-". Yes, your signal does closely resemble BPSK sending 101010. That's why it's dominated by that unmodulated spectral line. But it's the tiny "+-" part that carries all the data, and that's also the part that generates all of the grass. Narrow in time, wide in frequency -- remember?
As for being 'creamed' by an adjacent channel, we do very well thank you in an AMPS Cellular environment, or when placing a VMSK signal in quadrature between cable TV analog channels. See " Interference".
Test results, please? And what exactly do you mean by "in quadrature between cable TV analog channels"? If VMSK is as narrow as you claim, its phase relationships to the adjacent TV channels shouldn't matter as long as the spectral power density of the TV signal is low within the claimed VMSK bandwidth.

But if the phase of the TV carrier with respect to the VMSK carrier does matter, that's conclusive proof that the TV signal at frequencies far removed from your claimed VMSK bandwidth are, in fact, interfering with it. In other words, VMSK is far wider than you've been claiming.

No one should attempt to analyze VMSK as phase modulation. It is modified BPSK, which is usually analyzed as AM so that there are no BESSEL products to contend with. The phasor timing does create some signal on the quadrature ( Q) axis ( grass), but it is the timing of the 180 degree phase alternations ( I axis ) that is being detected, just as it is in conventional BPSK. If one insists on using PM, then use the +-90 degree changes used with BPSK, or FM with a modulation index where the Sine of the modulation angle is 1.0. It is incorrect to assume the modulation index for a 7,8,9 code is pi/16 radians. The VMSK encoder timing does not permit that. The correct phase shift index is pi/2+- , where is 1/2 the small change in pulse width. See the paper "Errata".

VMSK should be analyzed as a form of PPM modulation, not PM.

It is often possible to analyze a single digital communication system in several different but equally valid ways. All will necessarily give the same answers. For example, BPSK can be seen as either phase modulation with a +/-90 degree phase shift or as double-sideband amplitude modulation with suppressed carrier; both views are equally correct. And that's exactly how I've analyzed VMSK in three different yet equally valid ways: as BPSK plus a strong carrier; as binary orthogonal PPM plus a strong carrier; and as their original non-orthogonal constituent pulses with "matched filter" detection. And I got exactly the same answers each time. Please be more specific in your objections to my approaches, and explain why all three are wrong.
The paper "Illustrated Explanation of VMSK" is must reading for the beginner. The illustrations clear up a number of questions. The paper on "Interference" should clear up the "grass" argument, but for the 'die hards', look at the scope plots in " The Grass is Bad" and "What's This Grass Nonsense". The latter is undergoing drastic revision.
The only thing made clear by these illustrations is the depth of your own utter confusion and inability to learn.