Response to Walker's Rebuttal
Phil Karn
(Work in progress - last updated 16 Oct 2000)

Walker's rebuttal is yet another sloppy piece of writing that belies an almost total ignorance of the well-established mathematics of modulation and filtering. Not content to rest on his "unusual" interpretation of Claude Shannon's famous 1948 theorem, Walker apparently also rejects the work of Harry Nyquist (1928) and Jean Baptiste Joseph Fourier (1807) as well -- the very foundations of modern digital communications.

What limited knowledge Walker has of modulation seems to be almost entirely empirical. He appears to have little theoretical understanding of PM and FM beyond the usual Bessel series expansion of a carrier frequency modulated by a simple sine wave. [1]. But digital data does not consist of simple sine waves, and the analyses appropriate for simple sinusoids are inadequate and inappropriate for digital data.

In this response, I will address some of Walker's claims. Like my original analysis, this is a work in progress. The fact that I haven't addressed any specific claim does not mean that I necessarily agree with it. I might still be working on it, or I might still be trying to dissect Walker's thoroughly confused writing style.

Walker's "Facts"

After Walker begins by stating that my analysis is "absolutely, completely, totally incorrect", he asserts four interesting "facts". Let's look at them:

Fact: VMSK modulation is a form of modified BPSK modulation

Not only is this actually true, but I took pains to show exactly this in my critique! It is precisely because VMSK is an (inefficient) form of BPSK that VMSK needs at least as much spectrum as BPSK to achieve the same data rate: 1 bit/sec/Hz. [6]. So by conceding that VMSK is a form of BPSK, Walker has conceded my main point: VMSK cannot possibly achieve his claimed spectral efficiencies.

Fact: No one other than Karn attempts to analyze BPSK as Phase Modulation.

This one is so bizarre that I read it several times wondering if it could be a typo. I don't know of a textbook that discusses BPSK without analyzing it as phase modulation, because the "P" in BPSK stands for phase.

The textbooks also show that the special case of BPSK with a phase shift of +/-90 degrees has no carrier component, and it can also be seen as suppressed-carrier double-sideband AM. Both views are equally valid. BPSK with shifts other than +/-90 degrees can be generated either by direct phase modulation by the appropriate amount, or by combining a suppressed-carrier BPSK signal with a carrier in quadrature and at the appropriate amplitude.

Fact: All modulation information is in the side bands.

Another astonishing admission! The wide sidebands produced by VMSK modulation (what Walker calls "grass") is indeed where his information lies. That has been my assertion from the very beginning.

Walker was probably referring to the sidebands produced by the modulation of his main RF carrier by the baseband VMSK signal. This is also correct, though it does not diminish the fact that the wide sidebands ("grass") present in the baseband VMSK signal are all translated to RF along with his super-strong clock. As in the baseband signal, these wide sidebands are what carry all the user data at RF.

Fact: One sideband is transmitted.

This may be true for his system, but it is irrelevant. That Walker uses single-sideband techniques to shift the baseband VMSK signal to RF also does not diminish the fact that at least one full sideband ("grass"), out to at least the Nyquist limit of one half the data rate, must be carried to permit demodulation. The minimum RF bandwidth cannot be less than the minimum baseband bandwidth. [7]

Power Efficiency, again

Walker claims that my analysis of VMSK power efficiency is wrong:
On page 4 he shows the pulse width changes as carrying all of the energy. This is false. These pulse width changes only create the grass. The energy is in the total period of each positive and negative swing, as shown in the clock at the top, not in the changes. These larger pulse widths contain 16 times as much energy as the grass pulses.
As I took pains to point out, I disregard much of the energy in his pulses because it goes into a useless clock component that is the same for every bit, and does not help the receiver tell whether a one or a zero has been transmitted.

If, as Walker says, "the pulse width changes only create the grass", then the obvious solution to the "grass problem" is to eliminate the pulse width changes entirely and just send the exact same square pulse regardless of whether the data is a zero or a one. The silliness of this approach should be apparent even to Walker.

Nevertheless, there is a third way to analyze VMSK power efficiency that does use the full energy of the signal. And it just happens to give the same results as the two analyses presented earlier.

The best possible receiver for any form of digital modulation is the matched filter. One way to implement a matched filter is with a parallel bank of correlators. Each correlator is assigned one of the possible signalling pulses for each bit; for binary modulation (e.g., BPSK, BFSK and VMSK), there are two possible pulses and hence two correlators. Each correlator compares the received signal with their assigned local replicas. At the end of each bit, the correlator that produced the largest output is taken to be the one using the replica of the bit most likely to have been sent.

Let's first examine suppressed-carrier BPSK. To keep the arithmetic simple, we'll assume the BPSK waveform has a bit time of 1 time unit and an amplitude of +/- 1 unit. This gives a power of 1 power unit and an energy per bit of 1 energy unit.

In BPSK, the signal for a one bit is a function we'll call A(t), which is simply +1 for the bit interval from 0 to 1. [8]

For a zero, B(t) is -1 over the same interval.

The received signal is a function we'll call S(t). For any given bit interval, S(t) is the sum of noise plus either A(t) or B(t), depending on whether a one or zero bit is being sent.

At the receiver, the "one" correlator computes the integral over the bit time from 0 to 1 of S(t) * A(t) dt, while the "zero" correlator computes the integral of S(t) * B(t) dt.

If the received signal S(t) = A(t), i.e., a one-bit is transmitted and there is no noise, then the "one" correlator will produce a value of 1 and the "zero" correlator will produce a value of -1. Conversely, if a zero-bit is transmitted, then the "one" correlator produces -1 and the "zero" correlator +1.

The difference between the "right" and "wrong" correlator outputs is therefore +1 - (-1) or 2. [2] Noise can make this difference larger or smaller. If the noise makes it larger, there's no problem. But if the noise is large enough to make the difference go negative, then the wrong decision is made and a bit error occurs. So for best noise performance, we want to choose signalling pulses that maximize the difference between "correct" and "wrong" correlator outputs for a given transmitted bit energy.

Now let's look at VMSK. Again we'll use what Walker calls a (7,8,9) code. For a "one" bit, the signal remains at +1 for 9/16ths of the bit period and -1 for 7/16ths of a bit. For a "zero" bit, the signal remains at +1 for 7/16ths bit followed by 9/16ths bit at -1.

Note that these pulses have the same total energy as the BPSK pulses (1 energy unit) and as with BPSK, the "right" VMSK correlator produces an output of +1.

The value produced by the "wrong" VMSK correlator can be computed in parts. Regardless of which bit was transmitted, the received and replica VMSK waveforms always match during the first and last 7/16ths of the bit, so their product in these two intervals is always +1. So the definite integral over each of those two sub-periods is 7/16 energy units. Over the middle 2/16ths of the bit interval where they differ, their product is -1, making that part of the integral -2/16. The integral over the whole bit period is therefore 7/16 - 2/16 + 7/16 = 12/16 = 3/4 energy units.

The difference between the "right" and "wrong" correlators is therefore only 1-3/4 = 1/4 energy unit, vs 2 energy units for BPSK. This is a ratio of 2 / (1/4) = 8:1, or 9 dB. I.e., the power efficiency of VMSK is 9 dB worse than BPSK, exactly the same result we computed earlier in two different ways.


In response to my critique, Walker sent me a frequency sweep response for his filter. It showed pretty much what I expected to see: a sharp peak around his clock frequency, with attenuation increasing to about -30 dB on the low side at several hundred KHz from the clock. Although his sweep was insufficiently wide to capture the filter response out to the Nyquist limit for the data rate in use, it looked like the response had pretty much leveled out at about -30dB and wouldn't decrease much further. [5]

This easily explains why his system continues to pass data: the wideband "grass" that he refuses to accept as such an important part of his signal is only attenuated by about 30 dB. In a lab test or demo, one could easily have much more than 30 dB of link margin, meaning there would be enough wideband "grass" left to permit detection even after being filtered.

Later on, Walker says

His analysis on page 10 is false. The sinx/x pulses do not completely overlap as he shows. They would, if the filters had a high group delay.

This is another curious passage where Walker concedes the argument without realizing it. Any real filter must be "causal". That is, it cannot possibly respond to an input signal before it arrives. [3] And according to the reciprocity principle of Fourier transforms, the narrower a filter response is in frequency, the wider its impulse response must be in time. So by saying that the sinx/x pulses would overlap only if the filters had a high group delay, Walker is conceding that to keep the pulses from overlapping it is necessary to use filters with wide passbands. And that is again precisely what I have been saying from the beginning.

By the way, my presentation of the sinx/x pulses can be found in any communications theory textbook, as it is the standard way to present the Nyquist sampling theorem. This theorem, published in 1928, was the foundation for Shannon's work 20 years later.


Walker objects to my position that he is deluding himself and others, claiming that "200 scientists have performed due diligence studies of VMSK" and "arrived at the ... idea that VMSK works". He also claims that unnamed Baby Bells are participating in field tests. But Walker has repeatedly refused to divulge any of those names or the details of any tests, claiming that "they wouldn't want to be bothered" or that "they have minds of their own" and "don't need to see" my analyses.

This attitude should immediately trigger red flags with anyone even remotely familiar with the basic precepts of science: peer review and open debate. Indeed, such an attitude is a time-honored hallmark of pseudo-science and snake oil.

Walker says he has 15 patents on various modulation concepts. I learned long ago that the existence of a patent does not in any way prove that the invention works as claimed. Contrary to popular belief, the US patent office still grants patents on perpetual motion machines. [4]

I do not need to buy one of Walker's VMSK development kits any more than I need to buy a development kit for a perpetual motion machine. Neither can possibly work as advertised.


1. This apparently explains his frequent use of terms like J0, J1 and J2, as these are commonly used to denote Bessel functions.

2. That a BPSK correlator looking for a "one" responds to an incoming "zero" (and vice versa) with equal amplitude but opposite polarity follows from the fact that BPSK is an "antipodal" signalling scheme. This allows a single correlator to be used in an actual demodulator, with the subtraction of the two virtual correlators happening automatically. The correlator output can then be compared to zero in a slicer to determine whether a one or a zero was sent.

Binary FSK is an orthogonal scheme that requires two separate correlators, one for each "tone". For non-coherent detection, each correlator usually consists of a bandpass filter followed by an energy detector. At the end of the bit, the filter producing the most energy indicates the bit that was sent.

3. Perhaps Walker's next patent will be for a time machine.

4. My recent favorite was for a windmill mounted on an electric vehicle that could charge the vehicle's battery as the car was driven. And there's the patent on the hyper-light-speed antenna that transmits signals faster than light -- an obvious match for VMSK.

5. I will include a copy of the filter response when Walker gives me permission to do so.

6. The spectral efficiency of both BPSK and VMSK can actually be doubled to 2 bits/sec/Hz (but no further) by using either single-sideband techniques or QPSK. See the next footnote.

7. There's a much easier way to avoid doubling the bandwidth of his baseband signal in the RF modulation process than by using single sideband. In QPSK, two independent BPSK signals can be sent on the same RF carrier but in phase quadrature, making them orthogonal (independent). With QPSK you must pass both sidebands, as they are no longer mirror images of each other. But you can carry twice as much data as BPSK in the same amount of spectrum: 2 bits/sec/Hz.

8. Here we're actually analyzing the baseband form of BPSK, i.e., BPSK on a zero frequency (DC) carrier. The analysis is the same regardless of the carrier frequency, and BPSK receivers generally downconvert the signal to zero frequency right before it is detected.