This sounds a lot like Harold Walker's clearly unworkable VMSK, so the obvious question arises: is this the same thing?

First of all, xG appears to be an independent effort by Joseph A.
Bobier. Harold Walker says he knows Bobier and has worked
cooperatively with him in the past, but Bobier's work on xMax appears
to be independent. Walker is not mentioned on the xG website, and
Bobier has not responded to my emails asking for comment. It has been
very hard to tell just what xMax is or even in what way it is
supposedly better than existing technologies because xG has been so
tight-lipped and vague. But some recently granted patents and their
website can provide some insights. And while I'm still analyzing them,
my initial impression is that it doesn't look good at all. Considering
that xG Technology's current market capitalization is in the range of
US $1.5
*billion*, this is more than a little distressing to me.

The Technology Page on xG's website says the following:

xG Flash Signal uses breakthrough single cycle modulation to deliver longer range and lower power RF communications. Single cycle modulation is implemented when individual sinusoidal cycles of RF energy are modulated to represent one or more bits of data.One clear conclusion can be drawn from this paragraph: whoever wrote it doesn't have a clue about digital radio communications. For six decades we've known that there is an absolute lower limit on the RF energy required to send a bit of data over a noisy radio channel. Every competent radio engineer is familiar with this limit, and the state of the art is already very close to it.This proprietary modulation method differs from conventional approaches where tens to hundreds of thousands of RF cycles are required to convey a single bit of information. With each additional cycle representing more RF power, getting the job done with fewer cycles translates into more power efficient RF transmissions.

While many opportunities for innovation remain in the digital wireless field,
a breakthrough in required RF power due to a
new modulation scheme is *not* one of them.

The foundation for modern digital radio communications was laid almost
single-handedly in 1948 by the
late Claude
Shannon. It is hard to overstate the importance of his paper
A Mathematical Theory of Communication.
Among several other remarkable and fundamental insights, Shannon
proved that it is possible to communicate without error (or with an
arbitrarily low error rate) on a noisy channel as long as just one requirement
is met: the special ratio
E_{b}/N_{0}
must be greater than ln(2), the natural logarithm of
2. Numerically, that is about 0.693, or
-1.6
decibels (dB)
in electrical engineering terms. This number is now known naturally enough
as the
Shannon bound. This is a theoretical limit; any real receiver
will always do worse.

What is E_{b}/N_{0}?
Basically, it's closely related to the
signal-to-noise_ratio, or SNR.
It seems obvious that you can't receive a signal unless
it's stronger than the noise, and this is almost true.
Spelled out, E_{b}/N_{0} is the ratio, at the receiver, of the
signal energy per bit to the noise power spectral density. What does *that* mean?

The received energy per bit is, simply enough, the signal power at
the receiver in
watts
divided by the data rate in bits/sec. Power is energy per
unit time, so the energy per bit times the number of bits per second
gives the signal power. An important note: here
a *bit* refers to a user data bit, not the encoded bits from an
error correction code or a symbol from a modulator. Bits are what the end
users send and receive.

The noise power spectral density N_{0} takes a little more
explanation. Shannon assumes that the channel is impaired
by additive white Gaussian
noise (AWGN).
Not every channel is AWGN, but it's a good assumption because
AWGN is common in nature; it is essentially the "waterfall" noise you hear
or the "snow" that you see
when you tune to a radio or TV channel without a signal. "White" noise
consists of all frequencies in equal amounts just as
white light consists of all frequencies (colors) of light.
So the intensity of AWGN can be specified with a single number: the noise
power, in watts, per
hertz
of
bandwidth.
That's N_{0}. (Note that we're using the word "bandwidth" in its
original communications sense as "a range of frequencies" and *not*
in its newer and rather inaccurate computer networking sense as "a data rate". When we mean
data rate, we'll say *data rate*, not "bandwidth".)

An interesting thing about E_{b}/N_{0} is that
it's a dimensionless ratio. Energy is measured in watt-seconds, also
known as
joules,
while N_{0} is measured in watts per
hertz. But a hertz is a reciprocal second, so "watts per hertz" is
the same as watt-seconds or joules. This makes the ratio
E_{b}/N_{0} a dimensionless quantity, as it would have
to be to be represented in decibels.

The most important sources of radio noise are the thermal vibrations of the receiver's own atoms and objects seen by its antenna. The thermal noise power density emitted by an object is

N_{0}= kT, wherek = Boltzmann constant, 1.38*10

^{-23}J/K

T = absolute temperature in kelvins

So at a temperature of 300 kelvins (26.85 C or 80.33 F),

NThe noise power in a bandwidth B is_{0}= kT

= 1.38*10^{-23}J/K * 300 K

= 4.14*10^{-21}W/Hz (-203.83 dBJ)

N = NSo the total noise power from an object at 300 K in a bandwidth of 1 MHz is_{0}B

N = NThat's an incredibly small power, but it is nonetheless a critical limiting factor in most radio systems._{0}B

= 4.14*10^{-21}* 1*10^{6}

= 4.14*10^{-15}= 4.14 femtowatts = -143.83 dBW

E_{b}/N_{0} is closely related to
signal-to-noise_ratio, or S/N.
The relationship is

EE_{b}/N_{0}= (S/N) * (B/R), whereS = signal power, watts

N = noise power, watts

B = signal bandwidth, hertz

R = bit rate, bits/sec

Some older textbooks refer to E_{b}/N_{0}
as the "SNR per bit". This term fell out of use and
E_{b}/N_{0} became the preferred figure of merit for
digital communications for a simple reason: it's independent of
signal bandwidth.
Although bandwidth seems like a simple concept, it can be remarkably
tricky to measure or even define. It has several different
definitions, and while they give similar results on many signals, some
unusual signals can have very different bandwidths depending on which
definition is chosen. (The "ultra narrow band" delusion is entirely
about concocting pathological signals that are "narrow band" only by
an incorrect and irrelevant definition of the term).

A signal can also be deliberately widened in bandwidth to make it more
resistant to interference. This is
spread spectrum,
a technique with
military origins that's the basis of
CDMA digital cellular, some
forms of
WiFi
and many cordless phones. Spreading reduces the required
S/N by the amount that the signal is widened, making it even
harder to compare different kinds of modems.
So it's much easier all around to instead use
E_{b}/N_{0} as a figure of merit
that doesn't depend at all on a definition
of bandwidth or whether the signal is spread.

The Shannon bound of -1.6 dB is for infinite
bandwidth, the most favorable case. (It's a little counterintuitive
that using more bandwidth should let you use less power since
the total received noise power increases with bandwidth. But it's true.)
If your bandwidth is limited, then
the minimum E_{b}/N_{0} is greater. If you are strictly
limited to one hertz of bandwidth per bit per second of data rate, then the
E_{b}/N_{0} must be at least 0 dB. As your bandwidth
becomes even more constrained, the minimum E_{b}/N_{0}
rises exponentially. That's why telephone modems require good connections,
and it is why so-called "ultra narrow band" schemes like VMSK are
completely unworkable.

The most important things to understand about the Shannon bound is
that it comes from a mathematical *proof*, and that it
is *universal*. Shannon proved, and countless others have
verified his proof, that you cannot communicate
without error when E_{b}/N_{0} is worse than
the Shannon bound for the particular data rate and channel bandwidth.
This applies to *every*
modulation and coding scheme, whether or not it has yet been
invented. No matter how clever you are, and no matter how much computer
power you have, you cannot exceed the Shannon limit and communicate
without error. Period.

xG's statement implies you can also reduce the number of RF cycles needed to transmit a bit by increasing the transmitted energy in each RF cycle. This is true, but it completely defeats the claimed benefit of reducing the energy consumed per data bit.

So xG's claims simply don't make any sense.

According to xG, 35 milliwatts were used to transmit a 3.67 Mb/s signal over a distance of 18 miles on the 902 MHz "ISM" band. Other reports said they used 50 mW from a 204-meter tall transmission tower with 6.5 dBi of gain on the transmitting antenna and a 8 dBi directional receive antenna.

Florida is a very flat state, so we can easily compute the range to the horizon from the top of this tower as

range = R cosPlugging in R = 6378 km and h = 0.204 km,^{-1}(R/(R+h)), whereR = radius of the earth

h = tower height, same units

= 6378 cos^{-1}(6378/(6378+0.204))

= 6378 cos^{-1}(.99996801607)

= 51 km

The operating distance in the demo was 18 miles or 28.8 km, well within line of sight range of this tower. The atmosphere is transparent at 902 MHz, so the signal loss with distance is purely geometric: the receiving antenna simply captures a smaller fraction of the transmitted energy.

This is the inverse square law, and radio engineers routinely incorporate it and other well-understood physical properties of radio wave propagation into link budgets so they can predict the performance of communication systems even before they're built. So we'll do a link budget with xG's numbers.

The first component in the link budget is the inverse square path loss, computed as

path_loss(dB) = 20*logwhere D is the distance and λ is the wavelength in the same units. 18 miles is 28,968 meters and the wavelength of 915 MHz (the center of the 902-928 MHz band) is 0.3279 meters. Plugging in these numbers gives us:_{10}(4*π*D/λ)

20*log_{10}(4*π*28968/0.3279) = 120.9 dB

This is the "loss" between two isotropic antennas that radiate equally in all directions. Such antennas don't actually exist, but they're a useful reference. Real antennas achieve gains by focusing the power in a desired direction at the expense of others. Antenna gains are usually cited in dBi, decibels with respect to an ideal isotropic antenna. According to this article, the transmitter used an omnidirectional antenna with a gain of 6.5 dBi and a directional receive antenna with 8 dBi gain. Both figures seem reasonable, so we'll add them to the link budget.

The transmitted power is said in one place to be 35 milliwatts; in another, 50 mW. We'll use the smaller figure to be conservative, so that's -14.56 dBW, or decibels referenced to 1 watt. Therefore the received power would be:

P(rx) = P(tx) - path_loss + tx_gain + rx_gainThe data rate is stated to be 3.67 Mb/s, or 65.65 dB-Hz (decibels referenced to one Hz). So the received energy per bit, E

= -14.56 dBW - 120.9 dB + 6.5 dBi + 8 dBi

= -120.96 dBW

E_{b}= P(rx) - 10*log_{10}(bit_rate)

= -120.96 dBW - 10*log_{10}(3.67*10^{6})

= -120.96 dBW - 65.65 dB-Hz

= -186.61 dBJ, decibels referenced to one joule

Now we need to compute N_{0}, the received power spectral
density. The receiving equipment is not specified, but we can make an
educated guess about its performance. White Gaussian noise is
radiated
by every object not
at absolute zero temperature by the random thermal vibrations of its
molecules. If the object is hot enough, it visibly glows;
this is how the sun and incandescent light bulbs produce their heat and light.
At room temperature the radiated power is very small, but
highly significant in communications systems. It can be calculated as
N_{0} = k*T, where k is the
Boltzmann constant, 1.38*10^{-23} J/K or
-228.6 dBJ/K, and T is the absolute temperature in
kelvins. With
reasonably state-of-the-art receiver technology it is not hard to make
an amplifier at 915 MHz that produces considerably less noise than that
radiated by the earth and picked up by the antenna.
So we can assume the
system noise temperature
is merely the temperature of the earth,
about 300 kelvins (24.8 dBK, decibels referenced to one kelvin). This
makes:

NSo we finally have:_{0}= 10log_{10}(1.38*10^{-23}) + 10log_{10}(T)

= -228.6 dBJ/K + 24.8 dB-K = -203.8 dBJ

EThis is obviously well above the Shannon bound of -1.6 dB. If the transmitter power was 50 mW rather than 35 mW, the E_{b}/N_{0}= E_{b}(dBJ) - N_{0}(dBJ)

= -186.61 dBJ - (-203.8 dBJ) = +17.2 dB

As calculated above, the received signal power at the xMax receiver was -120.91 dBW or -90.91 dBm.
The Ruckus Wireless
802.11g transceiver specifies a receiver sensitivity of only -96 dBm at 6 Mb/s.
That's 5 dB less power for almost twice the data rate (6 Mb/s vs 3.7 Mb/s), or 7.1 dB less power for WiFi
compared to xMax at the same speed.
Note that this comparison does not require *any* estimates,
while the E_{b}/N_{0} calculation above required that we estimate the noise temperature of
the receiving system.
This is because receiver sensitivity figures combine the effects of system noise and demodulator/decoder
E_{b}/N_{0}.

It is possible that the xMax demo operated with link margin, meaning that it could have worked on a weaker signal than the one used. But one can reasonably presume that if this were the case, xG would have lowered the signal level and eliminated the margin to make its demo seem more impressive. In any event, xG Technology is the one claiming to have a new and vastly more power-efficient modulation method so it is entirely up to them to prove it.

The xMax demo may impress those who haven't done the calculations and are unaware of how little power it actually takes to transmit digital data over a benign line-of-sight path. But the same demonstrated performance could have been easily achieved with just about any conventional digital modulation scheme, including an off-the-shelf WiFi transceiver, and some of these schemes already come very close to the theoretical limits.

Broadcast stations use such powerful transmitters because they cannot assume line of sight paths; large margins are required to reliably overcome obstacles, fading, reflections and interference throughout their entire service areas.

This is not to say there can't be any more big developments in digital radio communications; far from it! But they will not come as fundamental modulation breakthroughs from xG Technologies or anyone else. Opportunities still abound in frequency reuse, the basis of modern cellular phones; cooperative ad-hoc networking; improved spectrum sharing; better adaptivity to changing or hostile radio link conditions; improved interference mitigation; multiple-in, multiple-out (MIMO) antenna arrays; improvements in RF hardware that will let us make better use of the underutilized higher frequency bands, and in other ways. But not from breakthroughs in modulation, because there won't be any more.

By way of analogy, dialup modems reached their theoretical limits years ago, but Internet access speeds continue to rise thanks to cable modems, fiber deployments, radio networking, and DSL. It is foolish to try to beat fundamental physical limits when you have workable alternatives!

See also:

Phil Karn, revised 1 March 2015