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.

Here are several things to note:

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*