Mowing the Grass
3 Mar 2001

Several files on Walker's website try to rebut the simple fact that the "grass" in a VMSK signal actually contains the modulation.

Examples include:

  • The Grass is Bad
  • What's this Grass Nonsense

    These lengthy, rambling documents make the same unsubstantiated claims over and over: the "grass" in a VMSK signal is noncoherent noise, it does not contain the modulation information, and that it is too weak to trigger a real logic circuit like a CMOS gate.

    The spectrum of a random NRZ bit stream

    They are wrong on all three counts. The "grass" in a VMSK signal closely resembles what one sees at the output of a BPSK modulator fed a random NRZ data source. The output of the source is either +1 or -1, depending on the bit being sent, and remains constant until it is time to send the next bit. The output is never 0, hence the term Non-Return-to-Zero (NRZ).

    A random NRZ signal looks noise-like on a spectrum analyzer only because the random data is itself essentially "noise". But looks are deceiving: in actuality the "noise" is completely coherent. Logic gates -- which operate in the time domain without performing a spectral analysis -- are quite able respond to it. (Walker and his colleagues frequently confuse the time-domain and frequency-domain representations of signals.)

    More specifically, the random data NRZ/BPSK spectrum looks like noise on which is superimposed a sin(x)/x shape. Maximum spectral density occurs at the (suppressed) carrier frequency, with nulls occurring at the carrier frequency plus and minus non-zero integer multiples of the bit rate.

    This is nothing more than the spectral shape of the baseband NRZ signal shifted up to the carrier frequency, with a negative "mirror" sideband added. I.e., the baseband spectrum looks just like the upper sideband of the modulated BPSK signal shifted down to a carrier frequency of 0 Hz.

    If Walker were correct in his claim that CMOS gates cannot respond to such "noise-like" signals, then no data transmission system -- even those operating at baseband, such as RS-232 -- could possibly work. That they do work just fine shows just how confused Walker and his colleagues are.

    Other baseband formats: Manchester and RZ

    The preceding analysis assumes the use of NRZ at baseband. The spectral shape (both baseband and BPSK modulated) changes if other formats are used. For example, a Manchester (biphase) encoded signal has no energy at DC and a peak at the data rate (where NRZ has a null). If such a signal modulates a BPSK transmitter, one gets a RF spectrum with two distinct spectral peaks: one at Fc+r and the other at Fc-r, where Fc is the carrier frequency and r is the bit rate. If the BPSK modulator is perfectly balanced, there is no energy at all at Fc.

    Manchester can be seen as the result of binary phase-shift-keying a square wave "carrier" at the bit rate with an NRZ data stream. That's why the Manchester baseband format has a spectral peak at the data rate; it corresponds to the peak seen at the carrier frequency in a BPSK signal. When a Manchester signal is placed on a BPSK carrier, the Manchester clock becomes a "sub-carrier" that modulates the main carrier. There's no power at the main carrier frequency because that's the first BPSK null below the Manchester sub-carrier.

    Another baseband format is RZ (return-to-zero). It resembles NRZ except that the signal "returns to zero" (from either +1 or -1) before the end of each bit interval. A random bit stream in RZ format has a sin(x)/x spectrum resembling NRZ except that the spectral nulls no longer occur at every non-zero integer multiple of the bit rate. Their location depends on the "duty cycle" of the pulse, as does the peak spectral density and the total signal energy. If the RZ signal returns to zero halfway through the bit, then the spectral nulls occur at non-zero even multiples of the data rate, i.e., at 2x, 4x, 6x, etc, and the total energy in the RZ signal will be one half that of the NRZ signal. The spectral density will be one quarter that of the NRZ signal because one half as much total power is being spread over twice the bandwidth.

    The narrower the RZ pulse (i.e., the sooner it returns to zero after the beginning of the bit), the wider its spectrum becomes and the less total energy it contains. E.g., if the pulse lasts only 1/3 of the bit interval, the spectral nulls will occur at 3x, 6x, 9x, etc and the maximum spectral density will decrease to 1/9 that of the NRZ signal.

    In the limit as the pulse duty cycle approaches 0%, its spectrum becomes essentially flat with frequency. At this point you are signalling with impulses. These have a Fourier transform that is constant across all frequencies.

    RZ and VMSK

    The preceding discussion of RZ is directly relevant to VMSK, because a baseband VMSK signal is nothing more than RZ coding added to a strong clock signal. See my first critique of VMSK to see why this is true.

    Walker takes strong issue with this characterization of VMSK, but it is just a straightforward use of superposition -- a fundamental principle widely taught in engineering mathematics courses. And it shows precisely what happens as the time shift in the center of the VMSK bit is decreased: the equivalent RZ pulses get narrower. This simultaneously reduces the fraction of the total transmitter power devoted to the data-bearing RZ pulses and spreads what's left over a wider bandwidth. That's why the VMSK signal consists of a powerful zero-bandwidth clock signal summed with low-level "grass" that is even wider than an ordinary BPSK signal at the same data rate.

    If the grass were totally removed (not partially removed with one of Walker's "magic" filters) only the clock signal would be left. There would be no shift in the mid-bit timing from which to recover the data, because it is the summation of the wideband "grass" that, when added to the clock component, results in the movement of the intra-bit transition that allows the data to be extracted.

    Killing the clock

    Walker says that if the clock component is removed from a VMSK signal, the demodulator no longer functions. He claims this disproves my claim about the useless of the VMSK clock component. From that he jumps to the false conclusion that the narrow clock component represents the entire VMSK signal.

    I confess to a poor choice of words. When I said that the VMSK clock was "useless", I meant that it carries no information. This is still true. However, the demodulator can use it as a clock reference, so removing the clock (e.g., with a notch filter) would indeed cause it to stop working.

    It's entirely possible to build a VMSK demodulator that ignores the transmitted clock and instead regenerates it locally from the data sidebands ("grass"). Such techniques are widely used in PSK modems because sending a discrete carrier or clock component takes transmitter power away from the data-bearing sidebands. But because Walker has gone to the other extreme -- putting almost all of his transmitter power into the clock -- a VMSK demodulator that ignored the clock would perform even more poorly than Walker's designs. But it would work if the SNR is high enough.