Phil Karn

(Work in progress - last updated 16 Oct 2000)

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.

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.

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.

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.

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.

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.

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.

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.