Yet Another Proof: The Data is in the Grass

Here is yet another clear and simple proof that the data in VMSK is carried entirely in the wideband "grass".

We will again analyze the so-called (7,8,9) VMSK waveform. Its eye pattern looks like this:

Note that the traces for the 0- and 1-bits are laid on top of each other, as is customary in eye patterns. If we divide each bit interval into 16 uniform time periods, the waveform always has a value of +1 for the first 7 periods and -1 for the last 7 periods. Only during the middle two periods does the VMSK waveform depend on the data bit: -1 for a 0-bit and +1 for a 1-bit.

Using the principle of superposition, we can represent this VMSK time-domain waveform as the sum of two separate but time-synchronized component waveforms. This first component waveform represents the beginning and ending intervals that are always the same for every bit:

The second component waveform represents the middle interval that depends on each data bit.

The data-dependent component of VMSK is nothing more than a return-to-zero (RZ) pulse that occurs in the middle of each bit interval.

Now let's display the spectrum for each component waveform. First, the clock:

This clock spectrum consists entirely of narrow spectral lines, including the line at 1x the data rate that Walker claims to be his entire signal. Note that the clock spectrum isn't quite that of a square wave; in particular, it has even-order harmonics not present in a square wave. That's because our clock isn't a pure square wave: it's got that little "kink" in the middle where it goes to zero for 2/16 of the bit interval. The grass is gone, because this is the spectrum of a signal that contains no information at all. We removed that part and put it in our second component waveform.

And now, the spectrum of the data pulse:

Here we have nothing but grass! The spectral lines are gone, because they belong entirely to the clock.

Now let's look at the spectrum for the original, UN-separated VMSK signal:

Guess what! This is just the sum of the spectral plots for the individual (clock and data) components we just showed. That's because of the linearity property of Fourier transforms: the Fourier transform of a sum of signals is equal to the sum of their individual Fourier transforms. This is why it's OK to separate the components and analyze them separately.

The astute reader may notice a change in the amplitude of the grass between these last two spectral plots. That's due to an auto-scaling (i.e., AGC) mechanism in my spectral analysis routine. With the powerful clock removed from the data/grass-only plot, the gain was automatically increased. This is of no consequence for the analysis, once it is understood that the displayed amplitudes are relative, not absolute.

Why the grass is so wide

It's interesting to compare these VMSK spectra to that of unfiltered conventional NRZ, where the waveform simply goes to either +1 or -1 for the entire bit interval, depending on the bit being sent. Since NRZ is commonly used with BPSK, this is also the RF spectrum of BPSK -- just shift the frequencies shown on the horizontal axis by the RF carrier frequency.

Here are several things to note:

  • The lack of any narrow spectral lines. Unlike RZ/VMSK, NRZ/BPSK does not waste any transmitter power on a carrier or clock.
  • The first nulls for NRZ occur at +/-1x the data rate, while the first nulls for (7,8,9) VMSK occur at +/-8x the data rate. This follows directly from VMSK's use of narrow RZ data pulses; because they are only 1/8 of a bit long, their spectra are 8x as wide. This is a fundamental property of time and frequency: narrow in time implies wide in frequency, and narrow in frequency implies wide in time.
  • The spectrum in the vicinity of DC (0Hz) is much flatter for RZ/VMSK than for NRZ/BPSK. That's because both spectra follow a sin(x)/x or sinc shape, but VMSK's sinc function is expanded in frequency by a factor of 8 because of its shorter pulse width. If the RZ pulses in VMSK were further narrowed, they would spread and flatten even further. They would also weaken because less transmitter power would be devoted to them (and more to the clock). As the RZ/VMSK pulse widths approach zero, their spectra would become infinitely wide and infinitely weak. This is relevant to Walker's latest "breakthrough", where in fact he is narrowing his data pulses in an attempt to lower the grass.

    Many extra side lobes extend well past the first nulls for both NRZ and VMSK, although VMSK's secondary side lobes are not seen here because they start right at the edges of the plot. However, the Nyquist bandwidths for NRZ and RZ (VMSK) are exactly the same. With the proper filtering the bandwidth of either one can be reduced to the same lower limit: 1 bit/sec/Hz. Narrower filtering would necessarily introduce intersymbol interference.

    19 June 2001, Phil Karn