Walker's so-called breakthrough

In March 2001, Harold Walker triumphantly announced that the "grass" in the VMSK signal spectrum that had proved so vexing "is now gone", thanks to some "simple" changes to the modulator.

He has apparently made two changes to his modulator. First, he further narrowed the difference between a "0" and a "1". Second, he apparently changed the shape of each pulse to maintain an average amplitude of zero, i.e., to eliminate the DC component.

Walker's first "fix"

The fallacy of the first "fix" is immediately apparent. A VMSK signal can be seen as the sum of an unmodulated square-wave clock signal and narrow data-dependent pulses that change the timing of the zero crossing in the middle of the bit:

Because the data-dependent pulses are so narrow, they occupy a much broader spectrum than an ordinary NRZ BPSK signal where the pulses occupy the entire bit interval. More specifically, the spectral amplitude of NRZ BPSK is sinc(f*T), where f is the frequency in Hz relative to the carrier and T is the bit duration in seconds. But in VMSK, the pulse duration is not that of the entire bit but only some fraction of it. E.g., for the so-called (7,8,9) VMSK code, the data pulses are only 1/16 of a bit duration so their spectrum becomes sinc(f*T*(1/16)).

In other words, narrowing the pulses doesn't cause the "grass" to disappear; it simply spreads it ever more thinly over a greater bandwidth. Also, as the pulses shorten, their energies decrease. This further reduces their spectral amplitude, making the "grass" even harder to see on a spectrum analyzer and also more difficult for a receiver to detect in the presence of noise.

Walker's second "fix"

Walker's second "fix" was a little harder to analyze mainly because his description of it was so confusing. It now appears that his "fix" involves adding DC offsets to the data-dependent pulses so that their average values over a bit interval is zero.

What does this do? Here are Fourier transforms of two VMSK signals. The first is the "classic" VMSK signal with pulse amplitudes of +/-1:

Note the strong spectral line at f=1. Walker believes this is the only significant portion of his signal, but in fact it is merely the strong unmodulated clock component that consumes most of his transmitter power. Note also the broad noise-like "grass" extending to nulls at f=+/- 16; this is where the data is actually carried. It appears noise-like simply because the data is random. The nulls occur where they do because the data pulses are only 1/16 of a bit wide.

Now let's look at a VMSK signal that has been modified so that each pulse has a zero DC component. That is, the narrow half of the bit has an amplitude of +/-1.0645813 and the wide half has an amplitude of +/-0.9393364; these figures are for the (7,8,9) code where the data pulses are 1/16 of a bit wide. They were also chosen to keep the average signal power equal to unity, to simplify comparisons to the unmodified VMSK signal.

Note the expected null at zero frequency. However, the rest of the spectrum is only slightly affected. It still extends out to first nulls at f = +/-16, and there are now ripples with peaks exceeding those for the unmodified signal. In particular, the spectral energy surrounding the clock at f=1 is still quite strong.

Here's another way to look at what's going on. The "DC" corrections to the data pulses constitute a conventional NRZ BPSK signal, i.e., with bits lasting the full duration of each bit interval. This correcting signal, with a spectrum of sinc(f*T), cancels the conventional BPSK component in the VMSK signal in the region surrounding zero frequency. But because the VMSK data pulses are only 1/16 of a bit interval wide, they have a spectrum of sinc(f*T*1/16). When f is nearly zero, sinc(f*T) is nearly equal to sinc(f*T*1/16), so there is good cancellation. But at larger values of f, sinc(f*T) falls nearly to zero while sinc(f*T*1/16) is still quite large. Hence there is poor cancellation at higher frequencies, and even an enhancement at certain frequencies (e.g., the region between f=1 and f=2.)

Once again, Walker has tried to fool nature and has failed. Yet, like the Black Knight in Monty Python and the Holy Grail, he just doesn't seem to know when to quit.

Phil Karn, 23 May 2001